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Bayes and his Theorem
My earlier post on Bayesian probability seems to have generated quite a lot of readers, so this lunchtime I thought I’d add a little bit of background. The previous discussion started from the result
$P(B|AC) = K^{-1}P(B|C)P(A|BC) = K^{-1} P(AB|C)$
where
$K=P(A|C).$
Although this is called Bayes’ theorem, the general form of it as stated here was actually first written down, not by Bayes but by Laplace. What Bayes’ did was derive the special case of this formula for “inverting” the binomial distribution. This distribution gives the probability of x successes in n independent “trials” each having the same probability of success, p; each “trial” has only two possible outcomes (“success” or “failure”). Trials like this are usually called Bernoulli trials, after Daniel Bernoulli. If we ask the question “what is the probability of exactly x successes from the possible n?”, the answer is given by the binomial distribution:
$P_n(x|n,p)= C(n,x) p^x (1-p)^{n-x}$
where
$C(n,x)= n!/x!(n-x)!$
is the number of distinct combinations of x objects that can be drawn from a pool of n.
You can probably see immediately how this arises. The probability of x consecutive successes is p multiplied by itself x times, or px. The probability of (n-x) successive failures is similarly (1-p)n-x. The last two terms basically therefore tell us the probability that we have exactly x successes (since there must be n-x failures). The combinatorial factor in front takes account of the fact that the ordering of successes and failures doesn’t matter.
The binomial distribution applies, for example, to repeated tosses of a coin, in which case p is taken to be 0.5 for a fair coin. A biased coin might have a different value of p, but as long as the tosses are independent the formula still applies. The binomial distribution also applies to problems involving drawing balls from urns: it works exactly if the balls are replaced in the urn after each draw, but it also applies approximately without replacement, as long as the number of draws is much smaller than the number of balls in the urn. I leave it as an exercise to calculate the expectation value of the binomial distribution, but the result is not surprising: E(X)=np. If you toss a fair coin ten times the expectation value for the number of heads is 10 times 0.5, which is five. No surprise there. After another bit of maths, the variance of the distribution can also be found. It is np(1-p).
So this gives us the probability of x given a fixed value of p. Bayes was interested in the inverse of this result, the probability of p given x. In other words, Bayes was interested in the answer to the question “If I perform n independent trials and get x successes, what is the probability distribution of p?”. This is a classic example of inverse reasoning. He got the correct answer, eventually, but by very convoluted reasoning. In my opinion it is quite difficult to justify the name Bayes’ theorem based on what he actually did, although Laplace did specifically acknowledge this contribution when he derived the general result later, which is no doubt why the theorem is always named in Bayes’ honour.
This is not the only example in science where the wrong person’s name is attached to a result or discovery. In fact, it is almost a law of Nature that any theorem that has a name has the wrong name. I propose that this observation should henceforth be known as Coles’ Law.
So who was the mysterious mathematician behind this result? Thomas Bayes was born in 1702, son of Joshua Bayes, who was a Fellow of the Royal Society (FRS) and one of the very first nonconformist ministers to be ordained in England. Thomas was himself ordained and for a while worked with his father in the Presbyterian Meeting House in Leather Lane, near Holborn in London. In 1720 he was a minister in Tunbridge Wells, in Kent. He retired from the church in 1752 and died in 1761. Thomas Bayes didn’t publish a single paper on mathematics in his own name during his lifetime but despite this was elected a Fellow of the Royal Society (FRS) in 1742. Presumably he had Friends of the Right Sort. He did however write a paper on fluxions in 1736, which was published anonymously. This was probably the grounds on which he was elected an FRS.
The paper containing the theorem that now bears his name was published posthumously in the Philosophical Transactions of the Royal Society of London in 1764.
P.S. I understand that the authenticity of the picture is open to question. Whoever it actually is, he looks to me a bit like Laurence Olivier…
11 Responses to “Bayes and his Theorem”
1. Bryn Jones Says:
The Royal Society is providing free access to electronic versions of its journals until the end of this month. Readers of this blog might like to look at Thomas Bayes’s two posthumous publications in the Philosophical Transactions.
The first is a short paper about series. The other is the paper about statistics communicated by Richard Price. (The statistics paper may be accessible on a long-term basis because it is one of the Royal Society’s Trailblazing papers the society provides access to as part of its 350th anniversary celebrations.)
Incidentally, both Thomas Bayes and Richard Price were buried in the Bunhill Fields Cemetery in London and their tombs can be seen there today.
2. Steve Warren Says:
You may be remembered in history as the discoverer of coleslaw, but you weren’t the first.
• Anton Garrett Says:
For years I thought it was “cold slaw” because it was served cold. A good job I never asked for warm slaw.
3. telescoper Says:
My surname, in Spanish, means “Cabbages”. So it was probably one of my ancestors who invented the chopped variety.
4. Anton Garrett Says:
Thomas Bayes is now known to have gone to Edinburgh University, where his name appears in the records. He was barred from English universities because his nonconformist family did not have him baptised in the Church of England. (Charles Darwin’s nonconformist family covered their bets by having baby Charles baptised in the CoE, although perhaps they believed it didn’t count as a baptism since Charles had no say in it. Tist is why he was able to go to Christ’s College, Cambridge.)
5. “Cole” is an old English word for cabbage, which survives in “cole slaw”. The German word is “Kohl”. (Somehow, I don’t see PM or President Cabbage being a realistic possibility. 🙂 )
Note that Old King Cole is unrelated (etymologically). Of course, this discussion could cause Peter to post a clip of
Nat “King” Cole
(guess what his real surname is).
To remind people to pay attention to spelling when they hear words, we’ll close with the Quote of the Day:
It’s important to pay close attention in school. For years I thought that
bears masturbated all winter.
—Damon R. Milhem
6. Of course, this discussion could cause Peter to post a clip of
Nat King Cole
(giess what his real surname is).
7. Of course, this discussion could cause Peter to post a clip of
Nat King Cole
(giess what his real surname is).
The first typo was my fault; the extra linebreaks in the second attempt
(tested again here) appear to be a new “feature”.
8. telescoper Says:
The noun “cole” can be found in English dictionaries as a generic name for plants of the cabbage family. It is related to the German kohl and scottish kail or kale. These are all derived from the latin word colis (or caulis) meaning a stem, which is also the root of the word cauliflower.
The surname “Cole” and the variant “Coles” are fairly common in England and Wales, but are not related to the latin word for cabbage. Both are diminutives of the name “Nicholas”.
9. […] I posted a little piece about Bayesian probability. That one and the others that followed it (here and here) proved to be surprisingly popular so I’ve been planning to add a few more posts […]
10. It already has a popular name: Stigler’s law of eponymy. |
Physical Quantity Analogous to Inductance
1. May 12, 2013
tapan_ydv
Hi,
I understand that some physical quantities in electromagnetism are analogous to physical quantities in heat transfer. For instance, electric field is analogous to temperature gradient.
I want to know which physical quantity in heat transfer is analogous to Inductance ("L") ?
Regards,
2. May 12, 2013
tiny-tim
welcome to pf!
hi tapan_ydv! welcome to pf!
i don't know about a heat transfer analogy,
but a hydraulics analogy is a paddle-wheel
A heavy paddle wheel placed in the current. The mass of the wheel and the size of the blades restrict the water's ability to rapidly change its rate of flow (current) through the wheel due to the effects of inertia, but, given time, a constant flowing stream will pass mostly unimpeded through the wheel, as it turns at the same speed as the water flow …
(from http://en.wikipedia.org/wiki/Hydraulic_analogy#Component_equivalents )
3. May 12, 2013
technician
In mechanics.....inertia
4. May 12, 2013
tiny-tim
how?
5. May 12, 2013
technician
Reluctance to change...as in a paddle wheel.
Last edited: May 12, 2013 |
# MicroEJ Test Suite Engine¶
## Introduction¶
The MicroEJ Test Suite Engine is a generic tool made for validating any development project using automatic testing.
This section details advanced configuration for users who wish to integrate custom test suites in their build flow.
The MicroEJ Test Suite Engine allows the user to test any kind of projects within the configuration of a generic Ant file.
The MicroEJ Test Suite Engine is already pre-configured for running test suites on a MicroEJ Platform (either on Simulator or on Device).
## Using the MicroEJ Test Suite Ant Tasks¶
Multiple Ant tasks are available in the testsuite-engine.jar provided in the Build Kit:
• testsuite allows the user to run a given test suite and to retrieve an XML report document in a JUnit format.
• javaTestsuite is a subtask of the testsuite task, used to run a specialized test suite for Java (will only run Java classes).
• htmlReport is a task which will generate an HTML report from a list of JUnit report files.
### The testsuite Task¶
The following attributes are mandatory:
testsuite task mandatory attributes
Attribute Name Description
outputDir The output folder of the test suite. The final report will be generated at [outputDir]/[label]/[reportName].xml, see the testsuiteReportFileProperty and testsuiteReportDirProperty attributes.
harnessScript The harness script must be an Ant script and it is the script which will be called for each test by the test suite engine. It is called with a basedir located at output location of the current test.
The test suite engine provides the following properties to the harness script giving all the informations to start the test:
harnessScript properties
Attribute Name Description
testsuite.test.name The output name of the current test in the report. Default value is the relative path of the test. It can be manually set by the user. More details on the output name are available in the section Specific Custom Properties.
testsuite.test.path The current test absolute path in the filesystem.
testsuite.test.properties The absolute path to the custom properties of the current test (see the property customPropertiesExtension)
testsuite.common.properties The absolute path to the common properties of all the tests (see the property commonProperties)
testsuite.report.dir The absolute path to the directory of the final report.
The following attributes are optional:
testsuite task optional attributes
Attribute Name Description Default value
timeOut The time in seconds before any test is considerated as unknown. Set it to 0 to disable the time-out. 60
verboseLevel The required level to output messages from the test suite. Can be one of those values: error, warning, info, verbose, debug. info
reportName The final report name (without extension). testsuite-report
customPropertiesExtension The extension of the custom properties for each test. For instance, if it is set to .options, a test named xxx/Test1.class will be associated with xxx/Test1.options. If a file exists for a test, the property testsuite.test.properties is set with its absolute path and given to the harnessScript. If the test path references a directory, then the custom properties path is the concatenation of the test path and the customPropertiesExtension value. .properties
commonProperties The properties to apply to every test of the test suite. Those options might be overridden by the custom properties of each test. If this option is set and the file exists, the property testsuite.common.properties is set to the absolute path of the harnessScript file. no common properties
label The build label. timestamp of when the test suite was invoked.
productName The name of the current tested product. TestSuite
jvm The location of your Java VM to start the test suite (the harnessScript is called as is: [jvm] [...] -buildfile [harnessScript]). java.home location if the property is set, java otherwise.
jvmargs The arguments to pass to the Java VM started for each test. None.
testsuiteReportFileProperty The name of the Ant property in which the path of the final report is stored. Path is [outputDir]/[label]/[reportName].xml testsuite.report.file
testsuiteReportDirProperty The name of the Ant property in which is store the path of the directory of the final report. Path is [outputDir]/[label]. testsuite.report.dir
testsuiteResultProperty The name of the Ant property in which you want to have the result of the test suite (true or false), depending if every tests successfully passed the test suite or not. Ignored tests do not affect this result. None
Finally, you have to give as nested element the path containing the tests.
testsuite task nested elements
Element Name Description
testPath Containing all the file of the tests which will be launched by the test suite.
testIgnoredPath (optional) Any test in the intersection between testIgnoredPath and testPath will be executed by the test suite, but will not appear in the JUnit final report. It will still generate a JUnit report for each test, which will allow the HTML report to let them appears as “ignored” if it is generated. Mostly used for known bugs which are not considered as failure but still relevant enough to appears on the HTML report.
Example of test suite task invocation
<!-- Launch the testusite engine -->
<testsuite:testsuite
timeOut="${microej.kf.testsuite.timeout}" outputDir="${target.test.xml}/testkf"
harnessScript="${com.is2t.easyant.plugins#microej-kf-testsuite.microej-kf-testsuite-harness-jpf-emb.xml.file}" commonProperties="${microej.kf.launch.propertyfile}"
testsuiteResultProperty="testkf.result"
testsuiteReportDirProperty="testkf.testsuite.report.dir"
productName="${module.name} testkf" jvmArgs="${microej.kf.testsuite.jvmArgs}"
lockPort="${microej.kf.testsuite.lockPort}" verboseLevel="${testkf.verbose.level}"
>
<testPath refid="target.testkf.path"/>
</testsuite:testsuite>
### The javaTestsuite Task¶
This task extends the testsuite task, specializing the test suite to only start real Java class. This task retrieves the classname of the tests from the classfile and provides new properties to the harness script:
javaTestsuite task properties
Property Name Description
testsuite.test.class The classname of the current test. The value of the property testsuite.test.name is also set to the classname of the current test.
testsuite.test.classpath The classpath of the current test.
<!-- Launch test suite -->
<testsuite:javaTestsuite
verboseLevel="${microej.testsuite.verboseLevel}" timeOut="${microej.testsuite.timeout}"
outputDir="${target.test.xml}/@{prefix}" harnessScript="${harness.file}"
commonProperties="${microej.launch.propertyfile}" testsuiteResultProperty="@{prefix}.result" testsuiteReportDirProperty="@{prefix}.testsuite.report.dir" productName="${module.name} @{prefix}"
jvmArgs="${microej.testsuite.jvmArgs}" lockPort="${microej.testsuite.lockPort}"
retryCount="${microej.testsuite.retry.count}" retryIf="${microej.testsuite.retry.if}"
retryUnless="${microej.testsuite.retry.unless}" > <testPath refid="target.@{prefix}.path"/> <testIgnoredPath refid="tests.@{prefix}.ignored.path" /> </testsuite:javaTestsuite> ### The htmlReport Task¶ This task allow the user to transform a given path containing a sample of JUnit reports to an HTML detailed report. Here is the attributes to fill: • A nested fileset element containing all the JUnit reports of each test. Take care to exclude the final JUnit report generated by the test suite. • A nested element report: • format: The format of the generated HTML report. Must be noframes or frames. When noframes format is choosen, a standalone HTML file is generated. • todir: The output folder of your HTML report. • The report tag accepts the nested tag param with name and expression attributes. These tags can pass XSL parameters to the stylesheet. The built-in stylesheets support the following parameters: • PRODUCT: the product name that is displayed in the title of the HTML report. • TITLE: the comment that is displayed in the title of the HTML report. Note It is advised to set the format to noframes if your test suite is not a Java test suite. If the format is set to frames, with a non-Java MicroEJ Test Suite, the name of the links will not be relevant because of the non-existency of packages. Example of htmlReport task invocation <!-- Generate HTML report --> <testsuite:htmlReport> <fileset dir="${@{prefix}.testsuite.report.dir}">
<include name="**/*.xml"/> <!-- include unary reports -->
<exclude name="**/bin/**/*.xml"/> <!-- exclude test bin files -->
<exclude name="*.xml"/> <!-- exclude global report -->
</fileset>
<report format="noframes" todir="\${target.test.html}/@{prefix}"/>
</testsuite:htmlReport>
## Using the Trace Analyzer¶
This section will shortly explains how to use the Trace Analyzer. The MicroEJ Test Suite comes with an archive containing the Trace Analyzer which can be used to analyze the output trace of an application. It can be used from different forms;
• The FileTraceAnalyzer will analyze a file and research for the given tags, failing if the success tag is not found.
• The SerialTraceAnalyzer will analyze the data from a serial connection.
Here is the common options to all TraceAnalyzer tasks:
• successTag: the regular expression which is synonym of success when found (by default .*PASSED.*).
• failureTag: the regular expression which is synonym of failure when found (by default .*FAILED.*).
• verboseLevel: int value between 0 and 9 to define the verbose level.
• waitingTimeAfterSuccess: waiting time (in s) after success before closing the stream (by default 5).
• noActivityTimeout: timeout (in s) with no activity on the stream before closing the stream. Set it to 0 to disable timeout (default value is 0).
• stopEOFReached: boolean value. Set to true to stop analyzing when input stream EOF is reached. If false, continue until timeout is reached (by default false).
• onlyPrintableCharacters: boolean value. Set to true to only dump ASCII printable characters (by default false).
Here is the specific options of the FileTraceAnalyzer task:
• traceFile: path to the file to analyze.
Here is the specific options of the SerialTraceAnalyzer task:
• port: the comm port to open.
• baudrate: serial baudrate (by default 9600).
• databits: databits (5|6|7|8) (by default 8).
• stopBits: stopbits (0|1|3 for (1_5)) (by default 1).
• parity: none | odd | event (by default none).
## Appendix¶
The goal of this section is to explain some tips and tricks that might be useful in your usage of the test suite engine.
### Specific Custom Properties¶
Some custom properties are specifics and retrieved from the test suite engine in the custom properties file of a test.
• The testsuite.test.name property is the output name of the current test. Here are the steps to compute the output name of a test:
• If the custom properties are enabled and a property named testsuite.test.name is find on the corresponding file, then the output name of the current test will be set to it.
• Otherwise, if the running MicroEJ Test Suite is a Java test suite, the output name is set to the class name of the test.
• Otherwise, from the path containing all the tests, a common prefix will be retrieved. The output name will be set to the relative path of the current test from this common prefix. If the common prefix equals the name of the test, then the output name will be set to the name of the test.
• Finally, if multiples tests have the same output name, then the current name will be followed by _XXX, an underscore and an integer.
• The testsuite.test.timeout property allow the user to redefine the time out for each test. If it is negative or not an integer, then global timeout defined for the MicroEJ Test Suite is used. |
T-Test In R On Microarray Data
3
1
Entering edit mode
10.4 years ago
Diana ▴ 900
Hello everyone,
I'm trying to do a simple t-test on my microarray sample in R. My sample looks like this:
gene_id gene sample_1 value_1 sample_2 value_2
XLOC_000001 LOC425783 Renal 20.8152 Heart 14.0945
XLOC_000002 GOLGB1 Renal 10.488 Heart 8.89434
So the t-test is between sample 1 and sample 2 and my code looks like this:
ttestfun = function(x) t.test(x[4], x[6])$p.value p.value = apply(expression_data, 1, ttestfun) It gives me the following error: Error in t.test.default(x[6], x[8]) : not enough 'x' observations In addition: Warning message: In mean.default(x) : argument is not numeric or logical: returning NA What am I doing wrong? Please help. Many thanks. r microarray • 15k views ADD COMMENT 8 Entering edit mode Nag your supervisor to provide some more arrays and allow you to run the experiment again. The arguments to convince him or her are possibly that: • a nonreplicated experiment does not meet the standards of research in the field (does it in any field?) • the data will therefore not be publishable • the money and time invested in the first screen will therefore be wasted ADD REPLY 3 Entering edit mode +1 because I can't give +2 or more. ADD REPLY 9 Entering edit mode 10.4 years ago I think there's some misconceptions operating here from the original questioner. First and foremost, a t-test is not just a way of calculating p-values, it is a statistical test to determine whether two populations have varying means. The p-value that results from the test is a useful indicator for whether or not to support your null hypothesis (that the two populations have the same mean), but is not the purpose of the test. In order to carry out a t-test between two populations, you need to know two things about those populations: 1) the mean of the observations and 2) the variance about that mean. The single value you have for each population could be a proxy for the mean (although it is a particularly bad one - see below), but there is no way that you can know the variance from only one observation. This is why replicates are required for microarray analysis, not a nice optional extra. The reason a single observation on a single microarray is a bad proxy for the population mean is because you have no way of knowing whether the individual tested is typical for the population concerned. Assuming the expression of a given gene is normally distributed among your population (and this is an assumption that you have to make in order for the t-test to be a valid test anyway), your single individual could come from anywhere on the bell curve. Yes, it is most likely that the observation is somewhere near the mean (by definition, ~68% within 1 standard deviation, see the graph), but there is a significant chance that it could have come from either extreme. Finally, I've read what you suggest about the hypergeometric test in relation to RNA-Seq data recently, but again the use of this test is based on a flawed assumption (that the variance of a gene between the 2 populations is equivalent to the population variance). Picking a random statistical test out of the bag, just because it is able to give you a p-value in your particular circumstance is almost universally bad practise. You need to be able to justify it in light of the assumptions you are making in order to apply the test. BTW, your data does not look like it is in log2 scale (if it is, there's an ~32-fold difference between the renal and heart observations for the first gene above) - how have you got the data into R & normalised it? ADD COMMENT 0 Entering edit mode +1 excellent explaination for beginners ADD REPLY 3 Entering edit mode 10.4 years ago It looks like you are trying to do a t-test with one value per group. That is a statistical impossibility (hence, the "not enough 'x' observations" error). Your only real option is to calculate a fold-change between the two samples by calculating a ratio. expression_data$ratio = expression_data[,3]-expression_data[,5] # assumes log scaled data
You can choose 2-fold changed genes by:
expression_data_filtered = expression_data[abs(expression_data$ratio)>2,] After you obtain replicates, you will want to use limma for gene expression analysis. Unmoderated t-tests are probably not the best way to go. ADD COMMENT 0 Entering edit mode Thank you so much Ben and Sean. Actually I'm trying to answer which of the genes are differentially expressed between these two samples and these are the only values I have. I don't have replicate experiments. Basically I want to associate some kind of significance to the differential expression and I thought calculating p-values would do that and hence the t-test. So there's no way I can calculate p-value for each gene with this data? ADD REPLY 3 Entering edit mode Hi, Diana. Unfortunately there is no way a statistical test can be performed without replication. The only option you have to compute p-values is to repeat the experiment. ADD REPLY 0 Entering edit mode Your interpretation is correct--no p-values with the data that you have in hand. ADD REPLY 0 Entering edit mode I don't know if this is a stupid question again, but someone whose working on such data suggested to me that a hypergeometric test can be done with only these values in hand. I wanted to confirm before I embarked on a useless journey. What do you all think? ADD REPLY 0 Entering edit mode How would you apply that test? ADD REPLY 0 Entering edit mode The hypergeometric distribution is used for the analysis of overlaps of gene sets, e.g. given 2 gene sets selected by some arbitrary choice, what is the probability that 100 or more out of the 1000 genes in each set are common to both both. That doesn't fit because you cannot make sensible gene sets yet. ADD REPLY 0 Entering edit mode Another point. The way you are approaching your problem is detrimental to the solution. Instead of responding by picking some random methods which you seemingly don't understand, you should: - respond to our proposal to replicate the experiment (what did your boss say about replication?) - try to understand how tests work ADD REPLY 0 Entering edit mode Thanks. No replicates for now. Maybe in near future. ADD REPLY 2 Entering edit mode 10.4 years ago Ben ★ 2.0k You are applying the t-test to the 4th and 6th value in each row; firstly R doesn't use zero-indexing so you don't seem to have a 6th column and secondly you are comparing two single values each time. For an (unpaired) t-test comparing expression_data$value_1 and expression_data$value_2 try: t.test(expression_data[,3], expression_data[,5])$p.value
edit: of course it's probably more useful to keep the whole returned list than just the p-value
0
Entering edit mode
Thanks a lot. I want to put all pairwise p-values in one object. When I try to use a loop, it gives me the same error again.
for(i in 1:38620)
{
u = t.test(expression_data[i,3], expression_data[i,5])
}
Error in t.test.default(RNA[i, 3], RNA[i, 5]) : not enough 'x' observations
What's wrong with my loop?
3
Entering edit mode
Again, you're trying to perform a t-test on two values... I think you need to look at what a t-test is and think about what you're trying to find from this data. You likely just want to add paired=T to the code I gave you above. See ?t.test in R too.
0
Entering edit mode
I need to do a t-test for each gene and I will be using two values for comparison. My question is: how can I do the pairwise t-test for each of the two values quickly...I was thinking a loop but its giving me an error. I don't want to do a t-test for each gene individually because I have a lot of genes
0
Entering edit mode
As Ben and I point out, you cannot perform a t-test between groups with only 1 member in them. As an aside, using a for-loop like this in R is usually not the best way to go. See the "apply" function for a better approach (can be orders-of-magnitude faster than a for loop). |
# Thread: how to compare multiple files?
1. Visitor
Join Date
Jun 2013
Posts
4
## how to compare multiple files?
hi,
I dont have much exp with BC 3 and scripting, i would like to build script to compare four files for example:
scenario1:
file 1: \\server1\folder1\text.txt
file 2: \\server2\folder1\text.txt
scenario2:
file 3: \\server3\folder1\text.txt
file 4: \\server4\folder1\text.txt
ofc, I would like to have these two comparisons done in the same time.
2. Team Scooter
Join Date
Oct 2007
Location
Posts
11,375
Hello,
Would you like to generate a report comparing file1 to file2, then generate a 2nd report comparing file3 to file4?
This can be done in scripting using the command line:
bcompare.exe "@c:\bcscript.txt"
Then the script file example could be:
[CODE]
text-report layout:side-by-side output-to:"c:bcreport1.html" output-options:html-color "\\server1\folder1\text.txt" "\\server2\folder1\text.txt"
text-report layout:side-by-side output-to:"c:bcreport2.html" output-options:html-color "\\server3\folder1\text.txt" "\\server4\folder1\text.txt"
Scripting actions follow the general actions you can perform in the graphical interface. Could you provide more details on the steps you are following in the interface and the reports you are generating from there? We can then help with the script to follow similar steps.
3. Visitor
Join Date
Jun 2013
Posts
4
would it be possible to have output in one file instead of multiple files? for example:
bcreport.html
also, where exactly output file bcreport.html will be saved?
4. Visitor
Join Date
Jun 2013
Posts
4
also, would it be possible to note only file differences (if any)?
5. Team Scooter
Join Date
Oct 2007
Location
Posts
11,375
It is not possible to have a single HTML report file for multiple text comparisons unless you open a folder compare, select the multiple files you want to compare, then generate the report. If you pass in pairs of files on the command line, we do not support appended reports together.
Code:
log verbose "c:\bclog.txt"
criteria rules-based
expand all
select diff.files
text-report layout:side-by-side options:display-mismatches output-to:"c:\bcreport.html" output-options:html-color
For a plain text report, you could append them together using a batch file:
Code:
bcompare.exe "@c:\script.txt" "c:\file1" "c:\file2"
type tempReport.txt >> mainreport.txt
bcompare.exe "@c:\script.txt" "c:\file3" "c:\file4"
type tempReport.txt >> mainreport.txt
Where script.txt is
Code:
text-report layout:side-by-side options:display-mismatches output-to:"c:\tempReport.txt" "%1" "%2"
6. Team Scooter
Join Date
Oct 2007
Location
Posts
11,375
To show only differences, add the "options:display-mismatches" parameter to the text-report command. Detailed documentation can be found in the Help file -> Scripting Reference, or in the Help file -> Using Beyond Compare -> Automating with Script chapter.
7. Visitor
Join Date
Jun 2013
Posts
4
thank you, this was very useful! |
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# Sampling distribution
Article Id: WHEBN0000520670
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Title: Sampling distribution Author: World Heritage Encyclopedia Language: English Subject: Collection: Statistical Theory Publisher: World Heritage Encyclopedia Publication Date:
### Sampling distribution
In statistics a sampling distribution or finite-sample distribution is the probability distribution of a given statistic based on a random sample. Sampling distributions are important in statistics because they provide a major simplification en route to statistical inference. More specifically, they allow analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution of all the individual sample values.
## Contents
• Introduction 1
• Standard error 2
• Examples 3
• Statistical inference 4
• References 5
## Introduction
The sampling distribution of a statistic is the distribution of that statistic, considered as a random variable, when derived from a random sample of size n. It may be considered as the distribution of the statistic for all possible samples from the same population of a given size. The sampling distribution depends on the underlying distribution of the population, the statistic being considered, the sampling procedure employed, and the sample size used. There is often considerable interest in whether the sampling distribution can be approximated by an asymptotic distribution, which corresponds to the limiting case either as the number of random samples of finite size, taken from an infinite population and used to produce the distribution, tends to infinity, or when just one equally-infinite-size "sample" is taken of that same population.
For example, consider a normal population with mean μ and variance σ². Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean \scriptstyle \bar x for each sample – this statistic is called the sample mean. Each sample has its own average value, and the distribution of these averages is called the "sampling distribution of the sample mean". This distribution is normal \scriptstyle \mathcal{N}(\mu,\, \sigma^2/n) (n is the sample size) since the underlying population is normal, although sampling distributions may also often be close to normal even when the population distribution is not (see central limit theorem). An alternative to the sample mean is the sample median. When calculated from the same population, it has a different sampling distribution to that of the mean and is generally not normal (but it may be close for large sample sizes).
The mean of a sample from a population having a normal distribution is an example of a simple statistic taken from one of the simplest statistical populations. For other statistics and other populations the formulas are more complicated, and often they don't exist in closed-form. In such cases the sampling distributions may be approximated through Monte-Carlo simulations[1][p. 2], bootstrap methods, or asymptotic distribution theory.
## Standard error
The standard deviation of the sampling distribution of a statistic is referred to as the standard error of that quantity. For the case where the statistic is the sample mean, and samples are uncorrelated, the standard error is:
\sigma_{\bar x} = \frac{\sigma}{\sqrt{n}}
where \sigma is the standard deviation of the population distribution of that quantity and n is the sample size (number of items in the sample).
An important implication of this formula is that the sample size must be quadrupled (multiplied by 4) to achieve half (1/2) the measurement error. When designing statistical studies where cost is a factor, this may have a role in understanding cost–benefit tradeoffs.
## Examples
Population Statistic Sampling distribution
Normal: \mathcal{N}(\mu, \sigma^2) Sample mean \bar X from samples of size n \bar X \sim \mathcal{N}\Big(\mu,\, \frac{\sigma^2}{n} \Big)
Bernoulli: \operatorname{Bernoulli}(p) Sample proportion of "successful trials" \bar X n \bar X \sim \operatorname{Binomial}(n, p)
Two independent normal populations:
\mathcal{N}(\mu_1, \sigma_1^2) and \mathcal{N}(\mu_2, \sigma_2^2)
Difference between sample means, \bar X_1 - \bar X_2 \bar X_1 - \bar X_2 \sim \mathcal{N}\! \left(\mu_1 - \mu_2,\, \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} \right)
Any absolutely continuous distribution F with density ƒ Median X_{(k)} from a sample of size n = 2k − 1, where sample is ordered X_{(1)} to X_{(n)} f_{X_{(k)}}(x) = \frac{(2k-1)!}{(k-1)!^2}f(x)\Big(F(x)(1-F(x))\Big)^{k-1}
Any distribution with distribution function F Maximum M=\max\ X_k from a random sample of size n F_M(x) = P(M\le x) = \prod P(X_k\le x)= \left(F(x)\right)^n
## Statistical inference
In the theory of statistical inference, the idea of a sufficient statistic provides the basis of choosing a statistic (as a function of the sample data points) in such a way that no information is lost by replacing the full probabilistic description of the sample with the sampling distribution of the selected statistic.
In frequentist inference, for example in the development of a statistical hypothesis test or a confidence interval, the availability of the sampling distribution of a statistic (or an approximation to this in the form of an asymptotic distribution) can allow the ready formulation of such procedures, whereas the development of procedures starting from the joint distribution of the sample would be less straightforward.
In Bayesian inference, when the sampling distribution of a statistic is available, one can consider replacing the final outcome of such procedures, specifically the conditional distributions of any unknown quantities given the sample data, by the conditional distributions of any unknown quantities given selected sample statistics. Such a procedure would involve the sampling distribution of the statistics. The results would be identical provided the statistics chosen are jointly sufficient statistics.
## References
1. ^
• Merberg, A. and S.J. Miller (2008). "The Sample Distribution of the Median". Course Notes for Math 162: Mathematical Statistics, on the web at http://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/162/Handouts/MedianThm04.pdf pgs 1–9. |
By Kimserey Lam with
# Conemu A Better Command Prompt For Windows
Jul 22nd, 2017 - written by Kimserey with .
When developing multiple Web api under multiple Visual Studio solutions, it can become very tedious to maintain, run and debug. Opening multiple instances of Visual Studio is very costly in term of memory and running all at once also clutter the screen which rapidly becomes irritating. With the advent of dotnet CLI tools, it has been clear that the next step would be to move out of the common “right click/build, F5” of Visual Studio and toward “dotnet run” on a command prompt. Last month I was looking for a Windows alternative of the bash terminal which can be found on Mac and I found ConEmu. ConEmu provides access to all typical shells via an enhanced UI. Today we will see how we can use ConEmu to ease our development process by leveraging only 2 of its features; the tasks and environment setup.
1. dotnet CLI
2. Setup environment
4. Apply to multiple services
## 1. dotnet CLI
We can start first by getting ConEmu from the repository releases https://github.com/Maximus5/ConEmu/releases. From now we can start straight using ConEmu as a command prompt. Multi tabs are supported by default, win + w hotkey opens a new tab.
Next what we can do is navigate to our Web API project and run dotnet run. This will run the Web API service in the command prompt, here in ConEmu.
It is also possible to restore packages with dotnet restore and build a project without running with dotnet build.
When the project is ran, it is ran in production mode. This is the default behaviour since usually the production setup is the most restrictive one. In order to have the environment set to development we can set it by setting it in the current command prompt context:
1
set ASPNETCORE_ENVIRONMENT=Development
We would need to run this on every new command prompt window. If we want to persist it, we can set it as a global Windows variable but this will affect the whole operating system. Lucky us ConEmu provides a way to run repeated commands on start of prompt which we will see now.
## 2. Setup environment
At each prompt start, ConEmu allows us to run a set of commands. Those can be used to set environment variables or to set aliases which will exist only in ConEmu context.
In order to access the environment setup, go to settings > startup > environment and the following window will show:
From here we can see that we can set variables, here I’ve set ASPNETCORE_ENVIRONMENT and also the base path of all my projects. And I also set an alias ns which helps me to quickly serve an Angular app with Angular CLI ng serve.
ConEmuBaseDir is the base directory containing ConEmu files. As we can see, %ConEmuBaseDir%\Scripts is also set to the path. This \Scripts folder is provided by ConEmu and already set to path for us to place scripts in which are then easy access for our tasks.
Now that we know how to setup environment variables, we will no longer need to manually set the ASPNETCORE_ENVIRONMENT variable as it will be done automatically. What we still need to do is to navigate to our service and dotnet run the project manually. Lucky us, again, ConEmu has a way to automate that by creating a script and setting it to a hotkey with ConEmu tasks which we will see next.
Let’s say we have a Web API located in C:\Projects\MyApi\MyApi.Web. In order to run it, we could do the following:
1
2
3
title My Api
cd C:\Projects\MyApi\MyApi.Web
dotnet run
This would set the title of the prompt to My Api then navigate to the service folder and run the project under development environment (since it was set in 2.). What we can do now is put those 3 lines in MyApi.cmd file which we will place under ConEmu \Scripts folder.
1
\ConEmu\ConEmu\Scripts\MyApi.cmd
Since the \Scripts folder is added to PATH in each prompt, we should be able to launch it straight from anywhere.
1
> MyApi.cmd
This is already pretty neat as it cut down a lot of time for quick launching but we can go a step further by defining a task.
We start by opening the task settings settings > startup > tasks.
From there we can set a task which will start a new prompt and run the MyApi.cmd script. We do that by clicking on +, naming the service Services::My Api and adding the command cmd.exe /k MyApi.cmd.
The naming convention allows grouping of tasks for easy access through the UI, [Group]::[Task] which is accessable from + on the main UI page.
A Hotkey can also be set with a combination of keys for even quicker access.
## 4. Apply to multiple services
All we have to do left is to create a script and task per service that we have. We can then create a global task which we can call Services::Multi containing all services:
1
2
3
4
5
cmd.exe /k MyApi.cmd
cmd.exe /k MyApi2.cmd
cmd.exe /k MyApi3.cmd
This task when ran will open 3 tabs and launch one script per tab which will result in a start of all services in one click.
# Conclusion
Today we saw how to configure ConEmu to environment and task to allow us to start multiple services running ASP NET Core Web API in a single click. The ease of use and the support of multi tab make ConEmu a major contributor to reducing the amount of time wasted in development cycle. Hope you enjoyed reading this post as much as I enjoyed writing it. If you have any questions leave it here or hit me on Twitter @Kimserey_Lam. See you next time!
Designed, built and maintained by Kimserey Lam. |
# Why does the DSolve not solve the PDE giving the 'Arbitrary functions'?
Posted 1 month ago
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Hello, I have two PDEs (strainDisp11 & strainDisp22) in 2 variables x1 and x2. strainDisp11 is a PDE with the partial differential term in x1 whereas, strainDisp22 is a PDE with the partial differential term in x2 I am trying to solve these two PDEs separately using DSolve (Last two command lines in the attached file), however, the solution is not generated along with the required arbitrary functions C1[1] which should be f1[x2] and C1[1] which should be f2[x1] in the respective solutions of the PDEs. Attached is Notebook for your reference. Appreciate your help.
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Posted 1 month ago
A Tip: Don't use Subscript , because causes problems.
Posted 1 month ago
Thanks! Very much appreciated.
Posted 11 days ago
Hello, I have two PDEs in 2 variables 'r' and 'theta'. I am trying to solve these two PDEs separately using DSolve (The last two command lines in the attached file). The solution is generated as expected for the 1st PDE (Integration with respect to variable 'r'), however, the solution is not generated for the 2nd PDE (Integration with respect to 'theta'). I cannot understand why Mathematica does not solve all the terms and has replaced 'theta' by K[1] in the unsolved integral with limits? Attached is Notebook for your reference. Appreciate your help.
Posted 11 days ago
Maybe: solDispRR = DSolve[strainDispRR == 0, uR, {r, \[Theta]}] // Flatten; solDisp\[Theta]\[Theta] = DSolve[strainDisp\[Theta]\[Theta] == 0, u\[Theta], {r, \[Theta]}] // Flatten; uRFunctionTemp = uR[r, \[Theta]] /. solDispRR[[1]] u\[Theta]FunctionTemp = (u\[Theta][r, \[Theta]] /. solDisp\[Theta]\[Theta][[1]] /. solDispRR[[1]]) // Activate // ExpandAll Looks like MMA can't integrate, a workaround: u\[Theta]FunctionTemp = (Integrate[#, {K[1], 1, \[Theta]}] & /@ (u\[Theta]FunctionTemp[[1, 1]])) + u\[Theta]FunctionTemp[[2]] (*Integrate[-C[1][K[1]], {K[1], 1, \[Theta]}] + (2*P*\[Nu]^2*Log[r]*(Sin[1] - Sin[\[Theta]]))/(Pi*\[DoubleStruckCapitalE]) + (2*P*\[Nu]*(-Sin[1] + Sin[\[Theta]]))/(Pi*\[DoubleStruckCapitalE]) + (2*P*\[Nu]^2*(-Sin[1] + Sin[\[Theta]]))/(Pi*\[DoubleStruckCapitalE]) + (2*P*Log[r]*(-Sin[1] + Sin[\[Theta]]))/(Pi*\[DoubleStruckCapitalE]) + C[1][r]*) In this line: Integrate[-C[1][K[1]], {K[1], 1, \[Theta]}] what answer do you expect? |
# Math Help - partial derivative notation question
1. ## partial derivative notation question
what does the notation at the bottom mean? the second derivative wrt z over the partial of y times the partial of x. Is that right? and what does that mean procedurally?
2. ## Re: partial derivative notation question
It means to first take the partial derivative of z with respect to y, then take the partial derivative of this result with respect to x.
For a function like this which is continuous and the respective partials exist, the order of differentiation does not matter, i.e.:
$\frac{\delta^2 z}{\delta x\delta y}=\frac{\delta^2 z}{\delta y\delta x}$ |
# Compute the Frequency Response of a Multistage Decimator
Figure 1a shows the block diagram of a decimation-by-8 filter, consisting of a low-pass finite impulse response (FIR) filter followed by downsampling by 8 [1]. A more efficient version is shown in Figure 1b, which uses three cascaded decimate-by-two filters. This implementation has the advantages that only FIR 1 is sampled at the highest sample rate, and the total number of filter taps is lower.
The frequency response of the single-stage decimator before downsampling is just the response of the FIR filter from f = 0 to fs/2. After downsampling, remaining signal components above fs/16 create aliases at frequencies below fs/16. It’s not quite so clear how to find the frequency response of the multistage filter: after all, the output of FIR 3 has unique spectrum extending only to fs/8, and we need to find the response from 0 to fs/2. Let’s look at an example to see how to calculate the frequency response. Although the example uses decimation-by-2 stages, our approach applies to any integer decimation factor.
Figure 1. Decimation by 8. (a) Single-stage decimator. (b) Three-stage decimator.
For this example, let the input sample rate of the decimator in Figure 1b equal 1600 Hz. The three FIR filters then have sample rates of 1600, 800, and 400 Hz. Each is a half-band filter [2 - 4] with passband of at least 0 to 75 Hz. Here is Matlab code that defines the three sets of filter coefficients (See Appendix):
b1= [-1 0 9 16 9 0 -1]/32; % fs = 1600 Hz
b2= [23 0 -124 0 613 1023 613 0 -124 0 23]/2048; % fs/2 = 800 Hz
b3= [-11 0 34 0 -81 0 173 0 -376 0 1285 2050 1285 0 -376 0 173 0 ...
-81 0 34 0 -11]/4096; % fs/4 = 400 Hz
The frequency responses of these filters are plotted in Figure 2. Each response is plotted over f = 0 to half its sampling rate:
FIR 1: 0 to 800 Hz
FIR 2: 0 to 400 Hz
FIR 3: 0 to 200 Hz
Figure 2. Frequency Responses of halfband decimation filters.
Now, to find the overall response at fs = 1600 Hz, we need to know the time or frequency response of FIR 2 and FIR 3 at this sample rate. Converting the time response is just a matter of sampling at fs instead of at fs /2 or fs /4 – i.e., upsampling. For example, the following Matlab code upsamples the FIR 2 coefficients by 2, from fs/2 to fs:
b2_up= zeros(1,21);
b2_up(1:2:21)= b2;
Figure 3 shows the coefficients b2 and b2_up. The code has inserted samples of value zero halfway between each of the original samples of b2 to create b2_up. b2_up now has a sample rate of fs. But although we have a new representation of the coefficients, upsampling has no effect on the math: b2_up and b2 have the same coefficient values and the same time interval between the coefficients.
For FIR 3, we need to upsample by 4 as follows:
b3_up= zeros(1,89);
b3_up(1:4:89)= b3;
Figure 4 shows the coefficients b3 and b3_up. Again, the upsampled version is mathematically identical to the original version. Now we have three sets of coefficients, all sampled at fs = 1600 Hz. A block diagram of the cascade of these coefficients is shown in Figure 5.
Figure 3. Top: Halfband filter coefficients b2. Bottom: Coefficients upsampled by 2.
Figure 4. Top: Halfband filter coefficients b3. Bottom: Coefficients upsampled by 4.
Figure 5. Conceptual diagram showing cascade of FIR 1 and upsampled versions of FIR 2 and FIR 3, used for computing frequency response of decimator of Figure 1b.
Using the DFT, we can compute and plot the frequency response of each filter stage, as shown in Figure 6. Upsampling b2 and b3 has allowed us to compute the DFT at the input sampling frequency fs for those sections. The sampling theorem [5] tells us that the frequency response of b2, which has a sample rate of 800 Hz, has an image between 400 and 800 Hz. Since b2_up has a sample rate of 1600 Hz, this image appears in its DFT (middle plot). Similarly, the DFT of b3_up has images from 200 to 400; 400 to 600; and 600 to 800 Hz (bottom plot).
Each decimation filter response in Figure 6 has stopband centered at one-half of its original sample frequency, shown as a red horizontal line (see Appendix). This attenuates spectrum in that band prior to downsampling by 2.
Figure 6. Frequency responses of decimator stages, fs = 1600 Hz.
Top: FIR 1 (b1 ) Middle: FIR 2 (b2_up) Bottom: FIR 3 (b3_up)
Now let’s find the overall frequency response. To do this, we could a) find the product of the three frequency responses in Figure 6, or b) compute the impulse response of the cascade of b1, b2_up, and b3_up, then use it to find H(z). Taking the latter approach, the overall impulse response is:
b123 = b1 ⊛ (b2up ⊛ b3up)
where ⊛ indicates convolution. The Matlab code is:
b23= conv(b2_up,b3_up);
b123= conv(b23,b1); % overall impulse response at fs= 1600 Hz
The impulse response is plotted in Figure 7. It is worth comparing the length of this response to that of the decimator stages. The impulse response has 115 samples; that is, it would take a 115-tap FIR filter to implement the decimator as a single stage FIR sampled at 1600 Hz. Of the 115 taps, 16 are zero. By contrast, the length of the three decimator stages are 7, 11, and 23 taps, of which a total of 16 taps are zero. So the multistage approach saves taps, and furthermore, only the first stage operates at 1600 Hz. Thus, the multistage decimator uses significantly fewer resources than a single stage decimator.
Calculating the frequency response from b_123:
fs= 1600; % Hz decimator input sample rate
[h,f]= freqz(b123,1,256,fs);
H= 20*log10(abs(h)); % overall freq response magnitude
The frequency response magnitude is plotted in Figure 8, with the stopband specified in the Appendix shown in red.
Here is a summary of the steps to compute the decimator frequency response:
1. Upsample the coefficients of all of the decimator stages (except the first stage) so that their sample rate equals the input sample rate.
2. Convolve all the coefficients from step 1 to obtain the overall impulse response at the input sample rate.
3. Take the DFT of the overall impulse response to obtain the frequency response.
Our discussion of upsampling may bring to mind the use of that process in interpolators. As in our example, upsampling in an interpolator creates images of the signal spectrum at multiples of the original sample frequency. The interpolation filter then attenuates those images [6].
We don’t want to forget aliasing, so we’ll take a look at that next.
Figure 7. Overall impulse response of three-stage decimator at fs = 1600 Hz (length = 115).
Figure 8. Overall frequency response of Decimator at fs= 1600 Hz.
## Taking Aliasing into Account
The output sample rate of the decimator in Figure 1b is fs out = 1600/8 = 200 Hz. If we apply sinusoids to its input, they will be filtered by the response of Figure 8, but then any components above fs out /2 (100 Hz) will produce aliases in the band of 0 to fs out /2. Let’s apply equal level sinusoids at 75, 290, and 708 Hz, as shown in Figure 9. The response in the bottom of Figure 9 shows the expected attenuation at 290 Hz is about 52 dB and at 708 Hz is about 53 dB (red dots). For reference, the component at 75 Hz has 0 dB attenuation. After decimation, the components at 290 and 708 Hz alias as follows:
f1 = 290 – fs out = 290 – 200 = 90 Hz
f= 4*fs out – 708 = 800 – 708 = 92 Hz
So, after decimation, we expect a component at 90 MHz that is about 52 dB below the component at 75 Hz, and a component at 92 Hz that is about 53 dB down. This is in fact what we get when we go through the filtering and downsampling operations: see Figure 10.
Note that the sines at 290 and 708 MHz are not within the stopbands as defined in the Appendix for FIR 1 and FIR 2. For that reason, the aliased components are greater than the specified stopband of -57 dB. This is not necessarily a problem, however, because they fall outside the passband of 75 Hz. They can be further attenuated by a subsequent channel filter.
Figure 9. Top: Multiple sinusoidal input to decimator at 75, 290, and 708 Hz.
Bottom: Decimator overall frequency response. Note fs out = fs/8.
Figure 10. Decimator output spectrum for input of Figure 9. fs out = fs/8 = 200 Hz.
## Appendix: Decimation Filter Synthesis
The halfband decimators were designed by the window method [3] using Matlab function fir1. We obtain halfband coefficients by setting the cutoff frequency to one-quarter of the sample rate. The order of each filter was chosen to meet the passband and stopband requirements shown in the table. Frequency responses are plotted in Figure 2 of the main text. We could have made the stopband attenuation of FIR 3 equal to that of the other filters, at the expense of more taps.
Common parameters:
Passband: > -0.1 dB at 75 Hz
Window function: Chebyshev, -47 dB
Section Sample rate Stopband edge Stopband atten Order FIR 1 fs = 1600 Hz fs/2 – 75 = 725 Hz 57 dB 6 FIR 2 fs/2 = 800 Hz fs/4 – 75 = 325 Hz 57 dB 10 FIR 3 fs/4 = 400 Hz fs/8 - 75 = 125 Hz 43 dB 22
Note that the filters as synthesized by fir1 have zero-valued coefficients on each end, so the actual filter order is two less than that in the function call. Using N = 6 and 10 in fir1 (instead of 8 and 12) would eliminate these superfluous zero coefficients, but would result in somewhat different responses.
% dec_fil1.m 1/31/19 Neil Robertson
% synthesize halfband decimators using window method
% fc = (fs/4)/fnyq = (fs/4)/(fs/2) = 1/2
% resulting coeffs have zeros on the each end,so actual filter order is N-2.
%
> fc= 1/2; % -6 dB freq divided by nyquist freq
%
% b1: halfband decimator from fs= 1600 Hz to 800 Hz
N= 8;
win= chebwin(N+1,47); % chebyshev window function, -47 dB
b= fir1(N,fc,win); % filter synthesis by window method
b1= round(b*32)/32; % fixed-point coefficients
%
% b2: halfband decimator from fs= 800 Hz to 400 Hz
N= 12;
win= chebwin(N+1,47);
b= fir1(N,fc,win);
b2= round(b*2048)/2048;
%
% b3: halfband decimator from fs= 400 Hz to 200 Hz
N= 24;
win= chebwin(N+1,47);
b= fir1(N,fc,win);
b3= round(b*4096)/4096;
## References
1. Lyons, Richard G. , Understanding Digital Signal Processing, 2nd Ed., Prentice Hall, 2004, section 10.1.
2. Mitra, Sanjit K.,Digital Signal Processing, 2nd Ed., McGraw-Hill, 2001, p 701-702.
3. Robertson, Neil, “Simplest Calculation of Halfband Filter Coefficients”, DSP Related website, Nov, 2017 https://www.dsprelated.com/showarticle/1113.php
4. Lyons, Rick, “Optimizing the Half-band Filters in Multistage Decimation and Interpolation”, DSP Related website, Jan, 2016 https://www.dsprelated.com/showarticle/903.php
5. Oppenheim, Alan V. and Shafer, Ronald W., Discrete-Time Signal Processing, Prentice Hall, 1989, Section 3.2.
6. Lyons, Richard G. , Understanding Digital Signal Processing, 2nd Ed., Prentice Hall, 2004, section 10.2.
Neil Robertson February 2019
[ - ]
Comment by February 11, 2019
Hi Neil.
This is a great blog. Your Figure 10 shows a very important principle that we sometimes forget. That principle is: After decimation by 8, *ALL* of the spectral energy that exists in the freq range of 0 -to- 800 Hz in the filter's output in Figure 8 is folded down and shows up in the decimated-by-8 signal's spectrum that you show your Figure 10. Good job!
[ - ]
Comment by February 12, 2019
Thanks Rick, I appreciate the encouragement!
To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments. |
# Why are vacancy rate and unemployment rate negatively correlated?
Why is this the case?
Since Vacancy rate is defined as following,
let $A,Q,U$ denote number of vacancies in the economy, labor force, unemployed respectively.
$$\frac{A}{A+Q-U}$$
Here we can see that if unemployed increase vacancy rate would go up?
Why is there a negatively correlation then? Take the beveridge curve as an example :
https://en.wikipedia.org/wiki/Beveridge_curve
• I have no idea what you are asking here. Maybe rephrase the question. – Jamzy Nov 1 '16 at 22:05
• Are you asking 'why is unemployment lower when job vacancies are higher?'. Unemployed people are people are looking for work. When you increase the thing that they are looking for (work), there will be less of them. – Jamzy Nov 1 '16 at 22:08
Adopting your notation, the vacancy rate at any given time is defined as $A/Q$. There is no mechanical relationship between the unemployment rate $U/Q$ and vacancy rate (A/Q). |
Issue No. 08 - August (2008 vol. 19)
ISSN: 1045-9219
pp: 1099-1110
ABSTRACT
Peer-to-peer (P2P) networks often demand scalability, low communication latency among nodes, and low system-wide overhead. For scalability, a node maintains partial states of a P2P network and connects to a few nodes. For fast communication, a P2P network intends to reduce the communication latency between any two nodes as much as possible. With regard to a low system-wide overhead, a P2P network minimizes its traffic in maintaining its performance efficiency and functional correctness. In this paper, we present a novel tree-based P2P network with low communication delay and low system-wide overhead. The merits of our tree-based network include: $(i)$ a tree-shaped P2P network which guarantees that the degree of a node is constant in probability regardless of the system size. The network diameter in our tree-based network increases logarithmically with an increase of the system size. Specially, given a physical network with a power-law latency expansion property, we show that the diameter of our tree network is constant. $(ii)$ Our proposal has the provable performance guarantees. We evaluate our proposal by rigorous performance analysis, and validate by extensive simulations.
INDEX TERMS
Distributed networks, Distributed Systems, Multicast
CITATION
H. Hsiao and C. He, "A Tree-Based Peer-to-Peer Network with Quality Guarantees," in IEEE Transactions on Parallel & Distributed Systems, vol. 19, no. , pp. 1099-1110, 2007.
doi:10.1109/TPDS.2007.70798 |
## Category Archives: Pre-RMO
### Rules for Inequalities
If a, b and c are real numbers, then
1. $a < b \Longrightarrow a + c< b + c$
2. $a < b \Longrightarrow a - c < b - c$
3. $a < b \hspace{0.1in} and \hspace{0.1in}c > 0 \Longrightarrow ac < bc$
4. $a < b \hspace{0.1in} and \hspace{0.1in}c < 0 \Longrightarrow bc < ac$ special case: $a < b \Longrightarrow -b < -a$
5. $a > 0 \Longrightarrow \frac{1}{a} > 0$
6. If a and b are both positive or both negative, then $a < b \Longrightarrow \frac{1}{b} < \frac{1}{a}$.
Remarks:
Notice the rules for multiplying an inequality by a number: Multiplying by a positive number preserves the inequality; multiplying by a negative number reverses the inequality. Also, reciprocation reverses the inequality for numbers of the same sign.
Regards,
Nalin Pithwa.
### Set Theory, Relations, Functions Preliminaries: II
Relations:
Concept of Order:
Let us say that we create a “table” of two columns in which the first column is the name of the father, and the second column is name of the child. So, it can have entries like (Yogesh, Meera), (Yogesh, Gopal), (Kishor, Nalin), (Kishor, Yogesh), (Kishor, Darshna) etc. It is quite obvious that “first” is the “father”, then “second” is the child. We see that there is a “natural concept of order” in human “relations”. There is one more, slightly crazy, example of “importance of order” in real-life. It is presented below (and some times also appears in basic computer science text as rise and shine algorithm) —-
Rise and Shine algorithm:
When we get up from sleep in the morning, we brush our teeth, finish our morning ablutions; next, we remove our pyjamas and shirt and then (secondly) enter the shower; there is a natural order here; first we cannot enter the shower, and secondly we do not remove the pyjamas and shirt after entering the shower. 🙂
Ordered Pair: Definition and explanation:
A pair $(a,b)$ of numbers, such that the order, in which the numbers appear is important, is called an ordered pair. In general, ordered pairs (a,b) and (b,a) are different. In ordered pair (a,b), ‘a’ is called first component and ‘b’ is called second component.
Two ordered pairs (a,b) and (c,d) are equal, if and only if $a=c$ and $b=d$. Also, $(a,b)=(b,a)$ if and only if $a=b$.
Example 1: Find x and y when $(x+3,2)=(4,y-3)$.
Solution 1: Equating the first components and then equating the second components, we have:
$x+3=4$ and $2=y-3$
$x=1$ and $y=5$
Cartesian products of two sets:
Let A and B be two non-empty sets then the cartesian product of A and B is denoted by A x B (read it as “A cross B”),and is defined as the set of all ordered pairs (a,b) such that $a \in A$, $b \in B$.
Thus, $A \times B = \{ (a,b): a \in A, b \in B\}$
e.g., if $A = \{ 1,2\}$ and $B = \{ a,b,c\}$, tnen $A \times B = \{ (1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}$.
If $A = \phi$ or $B=\phi$, we define $A \times B = \phi$.
Number of elements of a cartesian product:
By the following basic counting principle: If a task A can be done in m ways, and a task B can be done in n ways, then the tasks A (first) and task B (later) can be done in mn ways.
So, the cardinality of A x B is given by: $n(A \times B)= n(A) \times n(B)$.
So, in general if a cartesian product of p finite sets, viz, $A_{1}, A_{2}, A_{3}, \ldots, A_{p}$ is given by $n(A_{1} \times A_{2} \times A_{3} \ldots A_{p}) = n(A_{1}) \times n(A_{2}) \times \ldots \times n(A_{p})$
Definitions of relations, arrow diagrams (or pictorial representation), domain, co-domain, and range of a relation:
Consider the following statements:
i) Sunil is a friend of Anil.
ii) 8 is greater than 4.
iii) 5 is a square root of 25.
Here, we can say that Sunil is related to Anil by the relation ‘is a friend of’; 8 and 4 are related by the relation ‘is greater than’; similarly, in the third statement, the relation is ‘is a square root of’.
The word relation implies an association of two objects according to some property which they possess. Now, let us some mathematical aspects of relation;
Definition:
A and B are two non-empty sets then any subset of $A \times B$ is called relation from A to B, and is denoted by capital letters P, Q and R. If R is a relation and $(x,y) \in R$ then it is denoted by $xRy$.
y is called image of x under R and x is called pre-image of y under R.
Let $A=\{ 1,2,3,4,5\}$ and $B=\{ 1,4,5\}$.
Let R be a relation such that $(x,y) \in R$ implies $x < y$. We list the elements of R.
Solution: Here $A = \{ 1,2,3,4,5\}$ and $B=\{ 1,4,5\}$ so that $R = \{ (1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)\}$ Note this is the relation R from A to B, that is, it is a subset of A x B.
Check: Is a relation $R^{'}$ from B to A defined by x<y, with $x \in B$ and $y \in A$ — is this relation $R^{'}$ *same* as R from A to B? Ans: Let us list all the elements of R^{‘} explicitly: $R^{'} = \{ (1,2),(1,3),(1,4),(1,5),(4,5)\}$. Well, we can surely compare the two sets R and $R^{'}$ — the elements “look” different certainly. Even if they “look” same in terms of numbers, the two sets $R$ and $R^{'}$ are fundamentally different because they have different domains and co-domains.
Definition : Domain of a relation R: The set of all the first components of the ordered pairs in a relation R is called the domain of relation R. That is, if $R \subseteq A \times B$, then domain (R) is $\{ a: (a,b) \in R\}$.
Definition: Range: The set of all second components of all ordered pairs in a relation R is called the range of the relation. That is, if $R \subseteq A \times B$, then range (R) = $\{ b: (a,b) \in R\}$.
Definition: Codomain: If R is a relation from A to B, then set B is called co-domain of the relation R. Note: Range is a subset of co-domain.
Type of Relations:
One-one relation: A relation R from a set A to B is said to be one-one if every element of A has at most one image in B and distinct elements in A have distinct images in B. For example, let $A = \{ 1,2,3,4\}$, and let $B=\{ 2,3,4,5,6,7\}$ and let $R_{1}= \{ (1,3),(2,4),(3,5)\}$ Then $R_{1}$ is a one-one relation. Here, domain of $R_{1}= \{ 1,2,3\}$ and range of $R_{1}$ is $\{ 3,4,5\}$.
Many-one relation: A relation R from A to B is called a many-one relation if two or more than two elements in the domain A are associated with a single (unique) element in co-domain B. For example, let $R_{2}=\{ (1,4),(3,7),(4,4)\}$. Then, $R_{2}$ is many-one relation from A to B. (please draw arrow diagram). Note also that domain of $R_{1}=\{ 1,3,4\}$ and range of $R_{1}=\{ 4,7\}$.
Into Relation: A relation R from A to B is said to be into relation if there exists at least one element in B, which has no pre-image in A. Let $A=\{ -2,-1,0,1,2,3\}$ and $B=\{ 0,1,2,3,4\}$. Consider the relation $R_{1}=\{ (-2,4),(-1,1),(0,0),(1,1),(2,4) \}$. So, clearly range is $\{ 0,1,4\}$ and $range \subseteq B$. Thus, $R_{3}$ is a relation from A INTO B.
Onto Relation: A relation R from A to B is said to be ONTO relation if every element of B is the image of some element of A. For example: let set $A= \{ -3,-2,-1,1,3,4\}$ and set $B= \{ 1,4,9\}$. Let $R_{4}=\{ (-3,9),(-2,4), (-1,1), (1,1),(3,9)\}$. So, clearly range of $R_{4}= \{ 1,4,9\}$. Range of $R_{4}$ is co-domain of B. Thus, $R_{4}$ is a relation from A ONTO B.
Binary Relation on a set A:
Let A be a non-empty set then every subset of $A \times A$ is a binary relation on set A.
Illustrative Examples:
E.g.1: Let $A = \{ 1,2,3\}$ and let $A \times A = \{ (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$. Now, if we have a set $R = \{ (1,2),(2,2),(3,1),(3,2)\}$ then we observe that $R \subseteq A \times A$, and hence, R is a binary relation on A.
E.g.2: Let N be the set of natural numbers and $R = \{ (a,b) : a, b \in N and 2a+b=10\}$. Since $R \subseteq N \times N$, R is a binary relation on N. Clearly, $R = \{ (1,8),(2,6),(3,4),(4,2)\}$. Also, for the sake of completeness, we state here the following: Domain of R is $\{ 1,2,3,4\}$ and Range of R is $\{ 2,4,6,8\}$, codomain of R is N.
Note: (i) Since the null set is considered to be a subset of any set X, so also here, $\phi \subset A \times A$, and hence, $\phi$ is a relation on any set A, and is called the empty or void relation on A. (ii) Since $A \times A \subset A \times A$, we say that $A \subset A$ is a relation on A called the universal relation on A.
Note: Let the cardinality of a (finite) set A be $n(A)=p$ and that of another set B be $n(B)=q$, then the cardinality of the cartesian product $n(A \times B)=pq$. So, the number of possible subsets of $A \times B$ is $2^{pq}$ which includes the empty set.
Types of relations:
Let A be a non-empty set. Then, a relation R on A is said to be: (i) Reflexive: if $(a,a) \in R$ for all $a \in A$, that is, aRa for all $a \in A$. (ii) Symmetric: If $(a,b) \in R \Longrightarrow (b,a) \in R$ for all $a,b \in R$ (iii) Transitive: If $(a,b) \in R$, and $(b,c) \in R$, then so also $(a,c) \in R$.
Equivalence Relation:
A (binary) relation on a set A is said to be an equivalence relation if it is reflexive, symmetric and transitive. An equivalence appears in many many areas of math. An equivalence measures “equality up to a property”. For example, in number theory, a congruence modulo is an equivalence relation; in Euclidean geometry, congruence and similarity are equivalence relations.
Also, we mention (without proof) that an equivalence relation on a set partitions the set in to mutually disjoint exhaustive subsets.
Illustrative examples continued:
E.g. Let R be an equivalence relation on $\mathbb{Q}$ defined by $R = \{ (a,b): a, b \in \mathbb{Q}, (a-b) \in \mathbb{Z}\}$. Prove that R is an equivalence relation.
Proof: Given that $R = \{ (a,b) : a, b \in \mathbb{Q}, (a-b) \in \mathbb{Z}\}$. (i) Let $a \in \mathbb{Q}$ then $a-a=0 \in \mathbb{Z}$, hence, $(a,a) \in R$, so relation R is reflexive. (ii) Now, note that $(a,b) \in R \Longrightarrow (a-b) \in \mathbb{Z}$, that is, $(a-b)$ is an integer $\Longrightarrow -(b-a) \in \mathbb{Z} \Longrightarrow (b-a) \in \mathbb{Z} \Longrightarrow (b,a) \in R$. That is, we have proved $(a,b) \in R \Longrightarrow (b,a) \in R$ and so relation R is symmetric also. (iii) Now, let $(a,b) \in R$, and $(b,c) \in R$, which in turn implies that $(a-b) \in \mathbb{Z}$ and $(b-c) \in \mathbb{Z}$ so it $\Longrightarrow (a-b)+(b-c)=a-c \in \mathbb{Z}$ (as integers are closed under addition) which in turn $\Longrightarrow (a,c) \in R$. Thus, $(a,b) \in R$ and $(b,c) \in R$ implies $(a,c) \in R$ also, Hence, given relation R is transitive also. Hence, R is also an equivalence relation on $\mathbb{Q}$.
Illustrative examples continued:
E.g.: If $(x+1,y-2) = (3,4)$, find the values of x and y.
Solution: By definition of an ordered pair, corresponding components are equal. Hence, we get the following two equations: $x+1=3$ and $y-2=4$ so the solution is $x=2,y=6$.
E.g.: If $A = (1,2)$, list the set $A \times A$.
Solution: $A \times A = \{ (1,1),(1,2),(2,1),(2,2)\}$
E.g.: If $A = \{1,3,5 \}$ and $B=\{ 2,3\}$, find $A \times B$, and $B \times A$, check if cartesian product is a commutative operation, that is, check if $A \times B = B \times A$.
Solution: $A \times B = \{ (1,2),(1,3),(3,2),(3,3),(5,2),(5,3)\}$ whereas $B \times A = \{ (2,1),(2,3),(2,5),(3,1),(3,3),(3,5)\}$ so since $A \times B \neq B \times A$ so cartesian product is not a commutative set operation.
E.g.: If two sets A and B are such that their cartesian product is $A \times B = \{ (3,2),(3,4),(5,2),(5,4)\}$, find the sets A and B.
Solution: Using the definition of cartesian product of two sets, we know that set A contains as elements all the first components and set B contains as elements all the second components. So, we get $A = \{ 3,5\}$ and $B = \{ 2,4\}$.
E.g.: A and B are two sets given in such a way that $A \times B$ contains 6 elements. If three elements of $A \times B$ are $(1,3),(2,5),(3,3)$, find its remaining elements.
Solution: We can first observe that $6 = 3 \times 2 = 2 \times 3$ so that A can contain 2 or 3 elements; B can contain 3 or 2 elements. Using definition of cartesian product of two sets, we get that $A= \{ 1,2,3\}$ and $\{ 3,5\}$ and so we have found the sets A and B completely.
E.g.: Express the set $\{ (x,y) : x^{2}+y^{2}=25, x, y \in \mathbb{W}\}$ as a set of ordered pairs.
Solution: We have $x^{2}+y^{2}=25$ and so
$x=0, y=5 \Longrightarrow x^{2}+y^{2}=0+25=25$
$x=3, y=4 \Longrightarrow x^{2}+y^{2}=9+16=25$
$x=4, y=3 \Longrightarrow x^{2}+y^{2}=16+9=25$
$x=5, y=0 \Longrightarrow x^{2}+y^{2}=25+0=25$
Hence, the given set is $\{ (0,5),(3,4),(4,3),(5,0)\}$
E.g.: Let $A = \{ 1,2,3\}$ and $B = \{ 2,4,6\}$. Show that $R = \{ (1,2),(1,4),(3,2),(3,4)\}$ is a relation from A to B. Find the domain, co-domain and range.
Solution: Here, $A \times B = \{ (1,2),(1,4),(1,6),(2,2),(2,4),(2,6),(3,2),(3,4),(3,6)\}$. Clearly, $R \subseteq A \times B$. So R is a relation from A to B. The domain of R is the set of first components of R (which belong to set A, by definition of cartesian product and ordered pair) and the codomain is set B. So, Domain (R) = $\{ 1,3\}$ and co-domain of R is set B itself; and Range of R is $\{ 2,4\}$.
E.g.: Let $A = \{ 1,2,3,4,5\}$ and $B = \{ 1,4,5\}$. Let R be a relation from A to B such that $(x,y) \in R$ if $x. List all the elements of R. Find the domain, codomain and range of R. (as homework quiz, draw its arrow diagram);
Solution: Let $A = \{ 1,2,3,4,5\}$ and $B = \{ 1,4,5\}$. So, we get R as $(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)$. $domain(R) = \{ 1,2,3,4\}$, $codomain(R) = B$, and $range(R) = \{ 4,5\}$.
E.g. Let $A = \{ 1,2,3,4,5,6\}$. Define a binary relation on A such that $R = \{ (x,y) : y=x+1\}$. Find the domain, codomain and range of R.
Solution: By definition, $R \subseteq A \times A$. Here, we get $R = \{ (1,2),(2,3),(3,4),(4,5),(5,6)\}$. So we get $domain (R) = \{ 1,2,3,4,5\}$, $codomain(R) =A$, $range(R) = \{ 2,3,4,5,6\}$
Tutorial problems:
1. If $(x-1,y+4)=(1,2)$, find the values of x and y.
2. If $(x + \frac{1}{3}, \frac{y}{2}-1)=(\frac{1}{2} , \frac{3}{2} )$
3. If $A=\{ a,b,c\}$ and $B = \{ x,y\}$. Find out the following: $A \times A$, $B \times B$, $A \times B$ and $B \times A$.
4. If $P = \{ 1,2,3\}$ and $Q = \{ 4\}$, find the sets $P \times P$, $Q \times Q$, $P \times Q$, and $Q \times P$.
5. Let $A=\{ 1,2,3,4\}$ and $\{ 4,5,6\}$ and $C = \{ 5,6\}$. Find $A \times (B \bigcap C)$, $A \times (B \bigcup C)$, $(A \times B) \bigcap (A \times C)$, $A \times (B \bigcup C)$, and $(A \times B) \bigcup (A \times C)$.
6. Express $\{ (x,y) : x^{2}+y^{2}=100 , x, y \in \mathbf{W}\}$ as a set of ordered pairs.
7. Write the domain and range of the following relations: (i) $\{ (a,b): a \in \mathbf{N}, a < 6, b=4\}$ (ii) $\{ (a,b): a,b \in \mathbf{N}, a+b=12\}$ (iii) $\{ (2,4),(2,5),(2,6),(2,7)\}$
8. Let $A=\{ 6,8\}$ and $B=\{ 1,3,5\}$. Let $R = \{ (a,b): a \in A, b \in B, a+b \hspace{0.1in} is \hspace{0.1in} an \hspace{0.1in} even \hspace{0.1in} number\}$. Show that R is an empty relation from A to B.
9. Write the following relations in the Roster form and hence, find the domain and range: (i) $R_{1}= \{ (a,a^{2}) : a \hspace{0.1in} is \hspace{0.1in} prime \hspace{0.1in} less \hspace{0.1in} than \hspace{0.1in} 15\}$ (ii) $R_{2} = \{ (a, \frac{1}{a}) : 0 < a \leq 5, a \in N\}$
10. Write the following relations as sets of ordered pairs: (i) $\{ (x,y) : y=3x, x \in \{1,2,3 \}, y \in \{ 3,6,9,12\}\}$ (ii) $\{ (x,y) : y>x+1, x=1,2, y=2,4,6\}$ (iii) $\{ (x,y) : x+y =3, x, y \in \{ 0,1,2,3\}\}$
More later,
Nalin Pithwa
### Set Theory, Relations, Functions Preliminaries: I
In these days of conflict between ancient and modern studies there must surely be something to be said of a study which did not begin with Pythagoras and will not end with Einstein. — G H Hardy (On Set Theory)
In every day life, we generally talk about group or collection of objects. Surely, you must have used the words such as team, bouquet, bunch, flock, family for collection of different objects.
It is very important to determine whether a given object belongs to a given collection or not. Consider the following conditions:
i) Successful persons in your city.
ii) Happy people in your town.
iii) Clever students in your class.
iv) Days in a week.
v) First five natural numbers.
Perhaps, you have already studied in earlier grade(s) —- can you state which of the above mentioned collections are sets? Why? Check whether your answers are as follows:
First three collections are not examples of sets but last two collections represent sets. This is because in first three collections, we are not sure of the objects. The terms ‘successful persons’, ‘happy people’, ‘clever students’ are all relative terms. Here, the objects are not well-defined. In the last two collections, we can determine the objects clearly (meaning, uniquely, or without ambiguity). Thus, we can say that the objects are well-defined.
So what can be the definition of a set ? Here it goes:
A collection of well-defined objects is called a set. (If we continue to “think deep” about this definition, we are led to the famous paradox, which Bertrand Russell had discovered: Let C be a collection of all sets such which are not elements of themselves. If C is allowed to be a set, a contradiction arises when one inquires whether or not C is an element of itself. Now plainly, there is something suspicious about the idea of a set being an element of itself, and we shall take this as evidence that the qualification “well-defined” needs to be taken seriously. Bertrand Russell re-stated this famous paradox in a very interesting way: In the town of Seville lives a barber who shaves everyone who does not shave himself. Does the barber shave himself?…)
The objects in a set are called elements or members of that set.
We denote sets by capital letters : A, B, C etc. The elements of a set are represented by small letters : a, b, c, d, e, f ….etc. If x is an element of a set A, we write $x \in A$. And, we read it as “x belongs to A.” If x is not an element of a set A, we write $x \not\in A$, and read as ‘x does not belong to A.’e.g., 1 is a “whole” number but not a “natural” number.
Hence, $0 \in W$, where W is the set of whole numbers and $0 \not\in N$, where N is a set of natural numbers.
There are two methods of representing a set:
a) Roster or Tabular Method or List Method (b) Set-Builder or Ruler Method
a) Roster or Tabular or List Method:
Let A be the set of all prime numbers less than 20. Can you enumerate all the elements of the set A? Are they as follows?
$A=\{ 2,3,5,7,11,15,17,19\}$
Can you describe the roster method? We can describe it as follows:
In the Roster method, we list all the elements of the set within braces $\{, \}$ and separate the elements by commas.
In the following examples, state the sets using Roster method:
i) B is the set of all days in a week
ii) C is the set of all consonants in English alphabets.
iii) D is the set of first ten natural numbers.
2) Set-Builder Method:
Let P be the set of first five multiples of 10. Using Roster Method, you must have written the set as follows:
$P = \{ 10, 20, 30, 40, 50\}$
Question: What is the common property possessed by all the elements of the set P?
Answer: All the elements are multiples of 10.
Question: How many such elements are in the set?
Answer: There are 5 elements in the set.
Thus, the set P can be described using this common property. In such a case, we say that set-builder method is used to describe the set. So, to summarize:
In the set-builder method, we describe the elements of the set by specifying the property which determines the elements of the set uniquely.
Thus, we can write : $P = \{ x: x =10n, n \in N, n \leq 5\}$
In the following examples, state the sets using set-builder method:
i) Y is the set of all months of a year
ii) M is the set of all natural numbers
iii) B is the set of perfect squares of natural numbers.
Also, if elements of a set are repeated, they are written once only; while listing the elements of a set, the order in which the elements are listed is immaterial. (but this situation changes when we consider sets from the view-point of permutations and combinations. Just be alert in set-theoretic questions.)
Subset: A set A is said to be a subset of a set B if each element of set A is an element of set B. Symbolically, $A \subseteq B$.
Superset: If $A \subset B$, then B is called the superset of set A. Symbolically: $B \supset A$
Proper Subset: A non empty set A is said to be a proper subset of the set B, if and only if all elements of set A are in set B, and at least one element of B is not in A. That is, if $A \subseteq B$, but $A \neq B$ then A is called a proper subset of B and we write $A \subset B$.
Note: the notations of subset and proper subset differ from author to author, text to text or mathematician to mathematician. These notations are not universal conventions in math.
Intervals:
1. Open Interval : given $a < b$, $a, b \in R$, we say $a is an open interval in $\Re^{1}$.
2. Closed Interval : given $a \leq x \leq b = [a,b]$
3. Half-open, half-closed: $a , or $a \leq x
4. The set of all real numbers greater than or equal to a : $x \geq a =[a, \infty)$
5. The set of all real numbers less than or equal to a is $(-\infty, a] = x \leq a$
Types of Sets:
1. Empty Set: A set containing no element is called the empty set or the null set and is denoted by the symbol $\phi$ or $\{ \}$ or void set. e.g., $A= \{ x: x \in N, 1
2. Singleton Set: A set containing only one element is called a singleton set. Example : (i) Let A be a set of all integers which are neither positive nor negative. Then, $A = \{ 0\}$ and example (ii) Let B be a set of capital of India. Then $B= \{ Delhi\}$
We will define the following sets later (after we giving a working definition of a function): finite set, countable set, infinite set, uncountable set.
3. Equal sets: Two sets are said to be equal if they contain the same elements, that is, if $A \subseteq B$ and $B \subseteq A$. For example: Let X be the set of letters in the word ‘ABBA’ and Y be the set of letters in the word ‘BABA’. Then, $X= \{ A,B\}$ and $Y= \{ B,A\}$. Thus, the sets $X=Y$ are equal sets and we denote it by $X=Y$.
How to prove that two sets are equal?
Let us say we are given the task to prove that $A=B$, where A and B are non-empty sets. The following are the steps of the proof : (i) TPT: $A \subset B$, that is, choose any arbitrary element $x \in A$ and show that also $x \in B$ holds true. (ii) TPT: $B \subset A$, that is, choose any arbitrary element $y \in B$, and show that also $y \in A$. (Note: after we learn types of functions, we will see that a fundamental way to prove two sets (finite) are equal is to show/find a bijection between the two sets).
PS: Note that two sets are equal if and only if they contain the same number of elements, and the same elements. (irrespective of order of elements; once again, the order condition is changed for permutation sets; just be alert what type of set theoretic question you are dealing with and if order is important in that set. At least, for our introduction here, order of elements of a set is not important).
PS: Digress: How to prove that in general, $x=y$? The standard way is similar to above approach: (i) TPT: $x < y$ (ii) TPT: $y < x$. Both (i) and (ii) together imply that $x=y$.
4. Equivalent sets: Two finite sets A and B are said to be equivalent if $n(A)=n(B)$. Equal sets are always equivalent but equivalent sets need not be equal. For example, let $A= \{ 1,2,3 \}$ and $B = \{ 4,5,6\}$. Then, $n(A) = n(B)$, so A and B are equivalent. Clearly, $A \neq B$. Thus, A and B are equivalent but not equal.
5. Universal Set: If in a particular discussion all sets under consideration are subsets of a set, say U, then U is called the universal set for that discussion. You know that the set of natural numbers the set of integers are subsets of set of real numbers R. Thus, for this discussion is a universal set. In general, universal set is denoted by or X.
6. Venn Diagram: The pictorial representation of a set is called Venn diagram. Generally, a closed geometrical figures are used to represent the set, like a circle, triangle or a rectangle which are known as Venn diagrams and are named after the English logician John Venn.
In Venn diagram the elements of the sets are shown in their respective figures.
Now, we have these “abstract toys or abstract building-blocks”, how can we get new such “abstract buildings” using these “abstract building blocks”. What I mean is that we know that if we are a set of numbers like 1,2,3, …, we know how to get “new numbers” out of these by “adding”, subtracting”, “multiplying” or “dividing” the given “building blocks like 1, 2…”. So, also what we want to do now is “operations on sets” so that we create new, more interesting or perhaps, more “useful” sets out of given sets. We define the following operations on sets:
1. Complement of a set: If A is a subset of the universal set U then the set of all elements in U which are not in A is called the complement of the set A and is denoted by $A^{'}$ or $A^{c}$ or $\overline{A}$ Some properties of complements: (i) ${A^{'}}^{'}=A$ (ii) $\phi^{'}=U$, where U is universal set (iii) $U^{'}= \phi$
2. Union of Sets: If A and B are two sets then union of set A and set B is the set of all elements which are in set A or set B or both set A and set B. (this is the INCLUSIVE OR in digital logic) and the symbol is : \$latex A \bigcup B
3. Intersection of sets: If A and B are two sets, then the intersection of set A and set B is the set of all elements which are both in A and B. The symbol is $A \bigcap B$.
4. Disjoint Sets: Let there be two sets A and B such that $A \bigcap B=\phi$. We say that the sets A and B are disjoint, meaning that they do not have any elements in common. It is possible that there are more than two sets $A_{1}, A_{2}, \ldots A_{n}$ such that when we take any two distinct sets $A_{i}$ and $A_{j}$ (so that $i \neq j$, then $A_{i}\bigcap A_{j}= \phi$. We call such sets pairwise mutually disjoint. Also, in case if such a collection of sets also has the property that $\bigcup_{i=1}^{i=n}A_{i}=U$, where U is the Universal Set in the given context, We then say that this collection of sets forms a partition of the Universal Set.
5. Difference of Sets: Let us say that given a universal set U and two other sets A and B, $B-A$ denotes the set of elements in B which are not in A; if you notice, this is almost same as $A^{'}=U-A$.
6. Symmetric Difference of Sets: Suppose again that we are two given sets A and B, and a Universal Set U, by symmetric difference of A and B, we mean $(A-B)\bigcup (B-A)$. The symbol is $A \triangle B.$ Try to visualize this (and describe it) using a Venn Diagram. You will like it very much. Remark : The designation “symmetric difference” for the set $A \triangle B$ is not too apt, since $A \triangle B$ has much in common with the sum $A \bigcup B$. In fact, in $A \bigcup B$ the statements “x belongs to A” and “x belongs to B” are joined by the conjunction “or” used in the “either …or …or both…” sense, while in $A \triangle B$ the same two statements are joined by “or” used in the ordinary “either…or….” sense (as in “to be or not to be”). In other words, x belongs to $A \bigcup B$ if and only if x belongs to either A or B or both, while x belongs to $A \triangle B$ if and only if x belongs to either A or B but not both. The set $A \triangle B$ can be regarded as a kind of a “modulo-two-sum” of the sets A and B, that is, a sum of the sets A and B in which elements are dropped if they are counted twice (once in A and once in B).
Let us now present some (easily provable/verifiable) properties of sets:
1. $A \bigcup B = B \bigcup A$ (union of sets is commutative)
2. $(A \bigcup B) \bigcup C = A \bigcup (B \bigcup C)$ (union of sets is associative)
3. $A \bigcup \phi=A$
4. $A \bigcup A = A$
5. $A \bigcup A^{'}=U$ where U is universal set
6. If $A \subseteq B$, then $A \bigcup B=B$
7. $U \bigcup A=U$
8. $A \subseteq (A \bigcup B)$ and also $B \subseteq (A \bigcup B)$
Similarly, some easily verifiable properties of set intersection are:
1. $A \bigcap B = B \bigcap A$ (set intersection is commutative)
2. $(A \bigcap B) \bigcap C = A \bigcap (B \bigcap C)$ (set intersection is associative)
3. $A \bigcap \phi = \phi \bigcap A= \phi$ (this matches intuition: there is nothing common in between a non empty set and an empty set :-))
4. $A \bigcap A =A$ (Idempotent law): this definition carries over to square matrices: if a square matrix is such that $A^{2}=A$, then A is called an Idempotent matrix.
5. $A \bigcap A^{'}=\phi$ (this matches intuition: there is nothing in common between a set and another set which does not contain any element of it (the former set))
6. If $A \subseteq B$, then $A \bigcap B =A$
7. $U \bigcap A=A$, where U is universal set
8. $(A \bigcap B) \subseteq A$ and $(A \bigcap B) \subseteq B$
9. i: $A \bigcap (B \bigcap )C = (A \bigcap B)\bigcup (A \bigcap C)$ (intersection distributes over union) ; (9ii) $A \bigcup (B \bigcap C)=(A \bigcup B) \bigcap (A \bigcup C)$ (union distributes over intersection). These are the two famous distributive laws.
The famous De Morgan’s Laws for two sets are as follows: (it can be easily verified by Venn Diagram):
For any two sets A and B, the following holds:
i) $(A \bigcup B)^{'}=A^{'}\bigcap B^{'}$. In words, it can be captured beautifully: the complement of union is intersection of complements.
ii) $(A \bigcap B)^{'}=A^{'} \bigcup B^{'}$. In words, it can be captured beautifully: the complement of intersection is union of complements.
Cardinality of a set: (Finite Set) : (Again, we will define the term ‘finite set’ rigorously later) The cardinality of a set is the number of distinct elements contained in a finite set A and we will denote it as $n(A)$.
Inclusion Exclusion Principle:
For two sets A and B, given a universal set U: $n(A \bigcup B) = n(A) + n(B) - n(A \bigcap B)$.
For three sets A, B and C, given a universal set U: $n(A \bigcup B \bigcup C)=n(A) + n(B) + n(C) -n(A \bigcap B) -n(B \bigcap C) -n(C \bigcup A) + n(A \bigcap B \bigcap C)$.
Homework Quiz: Verify the above using Venn Diagrams.
Power Set of a Set:
Let us consider a set A (given a Universal Set U). Then, the power set of A is the set consisting of all possible subsets of set A. (Note that an empty is also a subset of A and that set A is a subset of A itself). It can be easily seen (using basic definition of combinations) that if $n(A)=p$, then $n(power set A) = 2^{p}$. Symbol: $P(A)$.
Homework Tutorial I:
1. Describe the following sets in Roster form: (i) $\{ x: x \hspace{0.1in} is \hspace{0.1in} a \hspace{0.1in} letter \hspace{0.1in} of \hspace{0.1in} the \hspace{0.1in} word \hspace{0.1in} PULCHRITUDE\}$ (II) $\{ x: x \hspace{0.1in } is \hspace{0.1in} an \hspace{0.1in} integer \hspace{0.1in} with \hspace{0.1in} \frac{-1}{2} < x < \frac{1}{2} \}$ (iii) $\{x: x=2n, n \in N\}$
2. Describe the following sets in Set Builder form: (i) $\{ 0\}$ (ii) $\{ 0, \pm 1, \pm 2, \pm 3\}$ (iii) $\{ \}$
3. If $A= \{ x: 6x^{2}+x-15=0\}$ and $B= \{ x: 2x^{2}-5x-3=0\}$, and $x: 2x^{2}-x-3=0$, then find (i) $A \bigcup B \bigcup C$ (ii) $A \bigcap B \bigcap C$
4. If A, B, C are the sets of the letters in the words, ‘college’, ‘marriage’, and ‘luggage’ respectively, then verify that $\{ A-(B \bigcup C)\}= \{ (A-B) \bigcap (A-C)\}$
5. If $A= \{ 1,2,3,4\}$, $B= \{ 3,4,5, 6\}$, $C= \{ 4,5,6,7,8\}$ and universal set $X= \{ 1,2,3,4,5,6,7,8,9,10\}$, then verify the following:
5i) $A\bigcup (B \bigcap C) = (A\bigcup B) \bigcap (A \bigcup C)$
5ii) $A \bigcap (B \bigcup C)= (A \bigcap B) \bigcup (A \bigcap C)$
5iii) $A= (A \bigcap B)\bigcup (A \bigcap B^{'})$
5iv) $B=(A \bigcap B)\bigcup (A^{'} \bigcap B)$
5v) $n(A \bigcup B)= n(A)+n(B)-n(A \bigcap B)$
6. If A and B are subsets of the universal set is X, $n(X)=50$, $n(A)=35$, $n(B)=20$, $n(A^{'} \bigcap B^{'})=5$, find (i) $n(A \bigcup B)$ (ii) $n(A \bigcap B)$ (iii) $n(A^{'} \bigcap B)$ (iv) $n(A \bigcap B^{'})$
7. In a class of 200 students who appeared certain examinations, 35 students failed in MHTCET, 40 in AIEEE, and 40 in IITJEE entrance, 20 failed in MHTCET and AIEEE, 17 in AIEEE and IITJEE entrance, 15 in MHTCET and IITJEE entrance exam and 5 failed in all three examinations. Find how many students (a) did not flunk in any examination (b) failed in AIEEE or IITJEE entrance.
8. From amongst 2000 literate and illiterate individuals of a town, 70 percent read Marathi newspaper, 50 percent read English newspapers, and 32.5 percent read both Marathi and English newspapers. Find the number of individuals who read
8i) at least one of the newspapers
8ii) neither Marathi and English newspaper
8iii) only one of the newspapers
9) In a hostel, 25 students take tea, 20 students take coffee, 15 students take milk, 10 students take both tea and coffee, 8 students take both milk and coffee. None of them take the tea and milk both and everyone takes at least one beverage, find the number of students in the hostel.
10) There are 260 persons with a skin disorder. If 150 had been exposed to chemical A, 74 to chemical B, and 36 to both chemicals A and B, find the number of persons exposed to (a) Chemical A but not Chemical B (b) Chemical B but not Chemical A (c) Chemical A or Chemical B.
11) If $A = \{ 1,2,3\}$ write down the power set of A.
12) Write the following intervals in Set Builder Form: (a) $(-3,0)$ (b) $[6,12]$ (c) $(6,12]$ (d) $[-23,5)$
13) Using Venn Diagrams, represent (a) $(A \bigcup B)^{'}$ (b) $A^{'} \bigcup B^{'}$ (c) $A^{'} \bigcap B$ (d) $A \bigcap B^{'}$
Regards,
Nalin Pithwa.
### References for IITJEE Foundation Mathematics and Pre-RMO (Homi Bhabha Foundation/TIFR)
1. Algebra for Beginners (with Numerous Examples): Isaac Todhunter (classic text): Amazon India link: https://www.amazon.in/Algebra-Beginners-Isaac-Todhunter/dp/1357345259/ref=sr_1_2?s=books&ie=UTF8&qid=1547448200&sr=1-2&keywords=algebra+for+beginners+todhunter
2. Algebra for Beginners (including easy graphs): Metric Edition: Hall and Knight Amazon India link: https://www.amazon.in/s/ref=nb_sb_noss?url=search-alias%3Dstripbooks&field-keywords=algebra+for+beginners+hall+and+knight
3. Elementary Algebra for School: Metric Edition: https://www.amazon.in/Elementary-Algebra-School-H-Hall/dp/8185386854/ref=sr_1_5?s=books&ie=UTF8&qid=1547448497&sr=1-5&keywords=elementary+algebra+for+schools
4. Higher Algebra: Hall and Knight: Amazon India link: https://www.amazon.in/Higher-Algebra-Knight-ORIGINAL-MASPTERPIECE/dp/9385966677/ref=sr_1_6?s=books&ie=UTF8&qid=1547448392&sr=1-6&keywords=algebra+for+beginners+hall+and+knight
5. Plane Trigonometry: Part I: S L Loney: https://www.amazon.in/Plane-Trigonometry-Part-1-S-L-Loney/dp/938592348X/ref=sr_1_16?s=books&ie=UTF8&qid=1547448802&sr=1-16&keywords=plane+trigonometry+part+1+by+s.l.+loney
The above references are a must. Best time to start is from standard VII or standard VIII.
-Nalin Pithwa.
### Pre RMO Practice question: 2018: How long does it take for a news to go viral in a city? And, a cyclist vs horseman
Problem 1:
Some one arrives in a city with very interesting news and within 10 minutes tells it to two others. Each of these tells the news within 10 minutes to two others(who have not heard it yet), and so on. How long will it take before everyone in the city has heard the news if the city has three million inhabitants?
Problem 2:
A cyclist and a horseman have a race in a stadium. The course is five laps long. They spend the same time on the first lap. The cyclist travels each succeeding lap 1.1 times more slowly than he does the preceding one. On each lap the horseman spends d minutes more than he spent on the preceding lap. They each arrive at the finish line at the same time. Which of them spends the greater amount of time on the fifth lap and how much greater is this amount of time?
I hope you enjoy “mathematizing” every where you see…
Good luck for the Pre RMO in Aug 2018!
Nalin Pithwa.
### How to solve equations: Dr. Vicky Neale: useful for Pre-RMO or even RMO training
Dr. Neale simply beautifully nudges, gently encourages mathematics olympiad students to learn to think further on their own…
### A nice dose of practice problems for IITJEE Foundation math and PreRMO
It is said that “practice makes man perfect”.
Problem 1:
Six boxes are numbered 1 through 6. How many ways are there to put 20 identical balls into these boxes so that none of them is empty?
Problem 2:
How many ways are there to distribute n identical balls in m numbered boxes so that none of the boxes is empty?
Problem 3:
Six boxes are numbered 1 through 6. How many ways are there to distribute 20 identical balls between the boxes (this time some of the boxes can be empty)?
Finish this triad of problems now!
Nalin Pithwa.
### IITJEE Foundation Math and PRMO (preRMO) practice: another random collection of questions
Problem 1: Find the value of $\frac{x+2a}{2b--x} + \frac{x-2a}{2a+x} + \frac{4ab}{x^{2}-4b^{2}}$ when $x=\frac{ab}{a+b}$
Problem 2: Reduce the following fraction to its lowest terms:
$(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) \div (\frac{x+y+z}{x^{2}+y^{2}+z^{2}-xy-yz-zx} - \frac{1}{x+y+z})+1$
Problem 3: Simplify: $\sqrt[4]{97-56\sqrt{3}}$
Problem 4: If $a+b+c+d=2s$, prove that $4(ab+cd)^{2}-(a^{2}+b^{2}-c^{2}-d^{2})^{2}=16(s-a)(s-b)(s-c)(s-d)$
Problem 5: If a, b, c are in HP, show that $(\frac{3}{a} + \frac{3}{b} - \frac{2}{c})(\frac{3}{c} + \frac{3}{b} - \frac{2}{a})+\frac{9}{b^{2}}=\frac{25}{ac}$.
May u discover the joy of Math! 🙂 🙂 🙂
Nalin Pithwa.
### Pre-RMO (PRMO) Practice Problems
Pre-RMO days are back again. Here is a list of some of my random thoughts:
Problem 1:
There are five different teacups, three saucers, and four teaspoons in the “Tea Party” store. How many ways are there to buy two items with different names?
Problem 2:
We call a natural number “odd-looking” if all of its digits are odd. How many four-digit odd-looking numbers are there?
Problem 3:
We toss a coin three times. How many different sequences of heads and tails can we obtain?
Problem 4:
Each box in a 2 x 2 table can be coloured black or white. How many different colourings of the table are there?
Problem 5:
How many ways are there to fill in a Special Sport Lotto card? In this lotto, you must predict the results of 13 hockey games, indicating either a victory for one of two teams, or a draw.
Problem 6:
The Hermetian alphabet consists of only three letters: A, B and C. A word in this language is an arbitrary sequence of no more than four letters. How many words does the Hermetian language contain?
Problem 7:
A captain and a deputy captain must be elected in a soccer team with 11 players. How many ways are there to do this?
Problem 8:
How many ways are there to sew one three-coloured flag with three horizontal strips of equal height if we have pieces of fabric of six colours? We can distinguish the top of the flag from the bottom.
Problem 9:
How many ways are there to put one white and one black rook on a chessboard so that they do not attack each other?
Problem 10:
How many ways are there to put one white and one black king on a chessboard so that they do not attack each other?
I will post the answers in a couple of days.
Nalin Pithwa.
### Three in a row !!!
If my first were a 4,
And, my second were a 3,
What I am would be double,
The number you’d see.
For I’m only three digits,
Just three in a row,
So what must I be?
Don’t say you don’t know!
Cheers,
Nalin Pithwa. |
# Probability of getting 2 Aces, 2 Kings and 1 Queen in a five card poker hand (Part II)
So I reworked my formula in method 1 after getting help with my original question - Probability of getting 2 Aces, 2 Kings and 1 Queen in a five card poker hand. But I am still getting results that differ...although they are much much closer than before, but I must still be making a mistake somewhere in method 1. Anyone know what it is?
Method 1
$P(2A \cap 2K \cap 1Q) = P(Q|2A \cap 2K)P(2A|2K)P(2K)$
$$= \frac{1}{12}\frac{{4 \choose 2}{46 \choose 1}}{50 \choose 3}\frac{{4 \choose 2}{48 \choose 3}}{52 \choose 5}$$
$$= \frac{(6)(17296)(6)(46)}{(2598960)(19600)(12)}$$
$$= 4.685642 * 10^{-5}$$
Method 2
$$\frac{{4 \choose 2} {4 \choose 2}{4 \choose 1}}{52 \choose 5} = \frac{3}{54145}$$
$$5.540678 * 10^{-5}$$
-
Please make an effort to make the question self-contained and provide a link to your earlier question. – Sasha Oct 28 '12 at 19:56
I think we would rather ahve you edit your initial question by adding your new progress. This avoids having loss of answer and keeps track of progress – Jean-Sébastien Oct 28 '12 at 19:56
But there already answers to my original question so those answers would not make sense now that I am using a new formula for method 1. – sonicboom Oct 28 '12 at 20:03
Conditional probability arguments can be delicate. Given that there are exactly two Kings, what's the $46$ doing? That allows the possibility of more Kings. – André Nicolas Oct 28 '12 at 20:26
The $46$ is because have already taken two kings from the pack leaving us with 50. And now we have chosen 2 aces and we have to pick the other 1 card from the 50 remaining cards less the 4 aces? – sonicboom Oct 28 '12 at 20:42
show 1 more comment
$$\frac{1}{11}\frac{{4 \choose 2}{44 \choose 1}}{48 \choose 3}\frac{{4 \choose 2}{48 \choose 3}}{52 \choose 5}$$
If you wrote this as $$\frac{{4 \choose 2}{48 \choose 3}}{52 \choose 5}\frac{{4 \choose 2}{44 \choose 1}}{48 \choose 3}\frac{{4 \choose 1}{40 \choose 0}}{44 \choose 1}$$ it might be more obvious why they are the same. |
# [R] Sweave: controlling pointsize (pdf)
Lauri Nikkinen lauri.nikkinen at iki.fi
Fri Jun 27 13:37:49 CEST 2008
Yes, I think so too. I already tried with
options(SweaveHooks=list(fig=function() pdf(pointsize=10)))
but as you said it tries to open pdf device and Sweaving fails...
Best
Lauri
2008/6/27, Duncan Murdoch <murdoch at stats.uwo.ca>:
> On 27/06/2008 7:12 AM, Lauri Nikkinen wrote:
> > pdf.options() seems to be a new function (from 2.7.0), so I quess I'll
> > have to upgrade or write my own hook function for Sweave. Thanks.
> >
>
> I'd recommend upgrading. I think it would be difficult to do this with a
> hook function: you'd basically need to close the pdf file that Sweave
> opened, and reopen it with new args --- but I don't think it's easy for you
> to determine the filename that Sweave would have used. You probably need to
> look at sys.frame(-2)$chunkprefix or something equally ugly. > > Duncan Murdoch > > > > > > Best > > Lauri > > > > 2008/6/27, Duncan Murdoch <murdoch at stats.uwo.ca>: > > > > > On 27/06/2008 6:23 AM, Lauri Nikkinen wrote: > > > > > > > I'm working with Windows XP and R 2.6.0 > > > > > > > > > > > > > R.Version() > > > > > > > > > > > > > >$platform
> > > > [1] "i386-pc-mingw32"
> > > >
> > > > -Lauri
> > > >
> > > > 2008/6/27, Lauri Nikkinen <lauri.nikkinen at iki.fi>:
> > > >
> > > >
> > > > > Hello,
> > > > >
> > > > > Is there a way to control pointsize of pdf:s produced by Sweave? I
> > > > > would like to have the same pointsize from (not a working example)
> > > > >
> > > > >
> > > >
> > > You could use a pdf.options() call in an early chunk in the file, and it
> > > will apply to subsequent chunks.
> > >
> > > For some other cases you might want code to be executed before every
> figure;
> > > that could be put in a hook function (as described in ?Sweave, and in
> the
> > > Sweave manual).
> > >
> > > Duncan Murdoch
> > >
> > >
> > > >
> > > > > pdf(file="C:/temp/example.pdf", width=7, height=7, bg="white",
> > > > >
> > > >
> > > pointsize=10)
> > >
> > > >
> > > > > plot(1:10)
> > > > > etc..
> > > > > dev.off()
> > > > >
> > > > > as
> > > > >
> > > > > \documentclass[a4paper]{article}
> > > > > \usepackage[latin1]{inputenc}
> > > > > \usepackage[finnish]{babel}
> > > > > \usepackage[T1]{fontenc}
> > > > >
> > > > >
> > > >
> > >
> \usepackage{C:/progra\string~1/R/R-26\string~1.0/share/texmf/Sweave}
> > >
> > > >
> > > > > <<fig=TRUE, width=7, height=7>>=
> > > > > plot(1:10)
> > > > > etc..
> > > > > @
> > > > >
> > > > > \end{document}
> > > > >
> > > > > Regards
> > > > > Lauri
> > > > >
> > > > >
> > > > >
> > > > ______________________________________________
> > > > R-help at r-project.org mailing list
> > > > https://stat.ethz.ch/mailman/listinfo/r-help
> > > >
> > > http://www.R-project.org/posting-guide.html
> > >
> > > > and provide commented, minimal, self-contained, reproducible code.
> > > >
> > > >
> > >
> > >
> >
> > ______________________________________________
> > R-help at r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-help |
# Amplified risk of spatially compounding droughts during co-occurrences of modes of natural ocean variability
## Abstract
Spatially compounding droughts over multiple regions pose amplifying pressures on the global food system, the reinsurance industry, and the global economy. Using observations and climate model simulations, we analyze the influence of various natural Ocean variability modes on the likelihood, extent, and severity of compound droughts across ten regions that have similar precipitation seasonality and cover important breadbaskets and vulnerable populations. Although a majority of compound droughts are associated with El Niños, a positive Indian Ocean Dipole, and cold phases of the Atlantic Niño and Tropical North Atlantic (TNA) can substantially modulate their characteristics. Cold TNA conditions have the largest amplifying effect on El Niño-related compound droughts. While the probability of compound droughts is ~3 times higher during El Niño conditions relative to neutral conditions, it is ~7 times higher when cold TNA and El Niño conditions co-occur. The probability of widespread and severe compound droughts is also amplified by a factor of ~3 and ~2.5 during these co-occurring modes relative to El Niño conditions alone. Our analysis demonstrates that co-occurrences of these modes result in widespread precipitation deficits across the tropics by inducing anomalous subsidence, and reducing lower-level moisture convergence over the study regions. Our results emphasize the need for considering interactions within the larger climate system in characterizing compound drought risks rather than focusing on teleconnections from individual modes. Understanding the physical drivers and characteristics of compound droughts has important implications for predicting their occurrence and characterizing their impacts on interconnected societal systems.
## Introduction
Weather and climate extremes pose substantial risks to people, property, infrastructure, natural resources and ecosystems1,2,3. Although a majority of risk assessment studies have focused on single stressor hazards occurring in specific regions, the Intergovernmental Panel on Climate Change (IPCC) Special Report on Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation (SREX) highlights the importance of considering compound extremes resulting from the simultaneous or sequential occurrence of multiple climate hazards in the same region, for improved modeling and risk estimation of their impacts4. Since then, several studies have analyzed the risks and mechanisms of such compound events5,6,7,8,9. Another emerging category of compound events that involve the simultaneous occurrence of extremes across multiple regions, referred to as spatially compounding extremes, is gaining prominence due to the potential for their cascading impacts on the global food system, disaster management resources, international aid, reinsurance industries, and the global economy10,11,12.
Recent work has started to build an understanding of the physical mechanisms that connect the occurrence of extremes across different regions. Kornhuber et al.17 found that the co-occurring summer 2018 heatwaves across North America, Western Europe, and the Caspian Sea region were driven by a recurrent wave-7 circulation pattern in the Northern-hemisphere mid-latitude jet stream. More generally, the occurrence of Rossby wave numbers 5 and 7 are found to substantially increase the probability of spatially compounding heat extremes over multiple mid-latitude regions including Central North America, Eastern Europe, and Eastern Asia, reducing global average crop production by nearly 4%18. The occurrence of Rossby waves can also link extreme events in the mid-latitudes and subtropics. For instance, Lau and Kim19 identified the role of Rossby wave trains in linking two record-setting extreme events during summer 2010—the persistent Russian heat wave and catastrophic flooding in Pakistan—with land-atmosphere feedbacks amplifying the Russian heat wave and moisture transport from the Bay of Bengal sustaining and amplifying the rains over Pakistan. Such compound extremes simultaneously affected millions of people and triggered a global food price spike associated with an approximately 30% loss in grain production in Russia20, which is a leading contributor to the global wheat trade21.
While recent studies of compound extremes have focused on the Northern Hemisphere mid-latitudes, the processes influencing compound extremes across the lower latitudes have received relatively little attention. Singh et al.22 investigated the underlying mechanisms of one such event—compound severe droughts across South Asia, East Asia, Brazil, and North and South Africa during 1876–1878, which were linked to the famines that contributed to the Late Victorian Holocausts23. The severity, duration, and extent of this compound drought event was shaped by the co-occurrence of a record-breaking El Niño (1877–1878), a record strong Indian Ocean Dipole (IOD) (1877), and record warm conditions in the North Atlantic Ocean (1878)22. El Niño Southern Oscillation (ENSO) is one of the main modes of variability that can cause simultaneous droughts and consequently affect food production in multiple remote regions. For instance, the reduction in global maize production in 1983 resulting from simultaneous crop failures across multiple regions13 is linked to the strong 1982–1983 El Niño event24. ENSO teleconnections lead to correlated climate risks between agricultural regions in North and South America and across the Pacific in Northern China and Australia25. For example, maize and soybean growing conditions in the US and southeast South America are favorable during the El-Niño phase, while the conditions are unfavorable in northern China, Brazil, and Southern Mexico25. In addition to ENSO, modes of variability in the Indian and Atlantic Ocean such as the Indian Ocean Dipole (IOD), tropical Atlantic variability, and the North Atlantic Oscillation are found to substantially affect the production of globally-aggregated maize, soybean, and wheat24. The influence of the interaction between these modes of natural variability on spatially compounding droughts across various regions has not yet been investigated.
Here, we examine the influence of four modes of natural climate variability on compound droughts across ten regions (Fig. 1a) defined in the SREX2 - Amazon (AMZ), Central America (CAM), Central North America (CNA), East Africa (EAF), East Asia (EAS), East North America (ENA), South Asia (SAS), Southeast Asia (SEA), Tibetan Plateau (TIB), and West Africa (WAF). We select these regions for three main reasons: (1) these regions include areas that receive a majority of their annual precipitation during the summer season (June–September) and experience high monthly precipitation variability, (2) several of these regions are physically connected by the global summer monsoon system26,27, and (3) climate variability across these regions are affected by similar modes of sea surface temperature (SST) variability. Our analysis only focuses on areas within these regions that meet the criteria of predominantly summer season precipitation and high monthly variability, which are identified based on the Shannon Entropy Index. These regions include major population centers with high levels of poverty and food insecurity and a number of major grains producing regions of the world, making them important in the context of global food security.
The predominant influence of tropical Pacific SSTs (El-Niño or La-Nina condition) on precipitation variability over these regions is well-known28,29. In addition, previous studies have highlighted the significant influence of other modes of variability such as the IOD, the Atlantic Niño and the Tropical North Atlantic (TNA) alongside El-Niño on individual regions such as SAS30,31, WAF/EAF32,33, EAS34,35, SEA36, and AMZ37,38. We aim to understand how the co-occurrence of these modes of variability influence the characteristics of spatially compound droughts across the ten SREX regions. By advancing the knowledge of the physical drivers of compound droughts, the findings from this study have relevance for quantifying the cascading risk to critical, globally connected socio-economic sectors such as agriculture and thereby to regional and global food security and disaster risk management. By identifying SST conditions that have prediction skill on seasonal timescales39,40,41, our findings also highlight the potential for predictability of such events that can aid in predicting and managing their impacts42.
## Results and discussion
### Compound drought characteristics and their physical drivers
To identify summer season (June–September) compound droughts across the ten SREX regions (Fig. 1a), we utilize the Standardized Precipitation Index (SPI), which is a commonly-used measure of meteorological drought. Our analysis is limited to grid cells within each region that have high entropy values (Fig. 1a), signifying substantial summer season precipitation and high monthly precipitation variability. We define drought at a grid cell when SPI is below −1 standard deviation (< −1σ) and consider a region under drought when total number of grids with SPI < −1σ exceeds 80th percentile of the historical drought area for that region (see “Methods” section; Fig. 1c). Based on these definitions, we find 11 years since 1981 that have at least three regions simultaneously experiencing droughts (Fig. 1b), which we hereafter refer to as compound droughts.
El Niño exhibits the strongest influence on the occurrences of compound droughts in the observations. 8 of the 11 observed compound droughts in CHIRPS are associated with anomalously warm SSTs in the Niño3.4 region, with seven of them classified as El Niño events (≥0.5σ; Fig. 2). A majority of compound droughts occur during the developing phase of moderate to strong El Niño (SST anomaly >1σ) (Fig. S1) and only two compound droughts are associated with anomalously cold SST over Niño3.4 region. For instance, the strong El Niños of 1982, 1997, and 2015 resulted in widespread and severe compound droughts that simultaneously affected over five of the study regions. In each case, the total drought affected area across all ten regions exceeded the historical 90th percentile (referred to as widespread droughts) and average SPI across all regions remained in the lowest historical 10th percentile (referred to as severe droughts; Fig. 2). However, not all strong El Niño years led to compound droughts (e.g., 1987) and substantial SST anomalies across the Atlantic and Indian Ocean basins were also present during the 11 compound droughts (Fig. S2), indicating the possibility of a more complex interplay of multiple modes of ocean variability. Therefore, we seek to investigate the influence of individual and co-occurring natural modes of ocean variability on the characteristics of compound droughts. Specifically, we consider El Niño co-occurrences with IOD, Atlantic Niño, and TNA, since their influences on the interannual precipitation variability in our study regions are well established30,43. We note that 7 of 12 positive IOD (IOD+; DMI > 0.5σ), 5 of 11 negative Atlantic Niño (AtlNiño; SST anomaly < −0.5σ), and 7 of 14 negative TNA (TNA; SST anomaly < −0.5σ) co-occurred with compound droughts (Fig. 2). Overall, more than 60% (7 out of 11) of the observed compound droughts occurred during the years when two or more of these modes of ocean variability were active (Fig. 2).
The apparent dominance of El Niño as a major player during the episodes of compound droughts is not sensitive to the choice of threshold used to define drought. For instance, if classification of a region under drought is based on 90th percentile of the historical drought area for that region instead of the 80th percentile, the total number of compound droughts in the last four decades expectedly reduces (5 instead of 11) (Fig. S1), however, 80% of them are still during strong El Niño events. These findings are also insensitive to the choice of the observational dataset. For instance, use of precipitation from Climate Research Unit (CRU) and SSTs from Extended Reconstructed Sea Surface Temperature (ERSST) NOAA V544 over 1901–2018 yields nearly 70% (12 of the 17) of compound and widespread droughts during strong El Niño events (Fig. S3). Similar to CHIRPS, more than half (~60%) of the compound droughts are associated with the co-occurrence of two or more modes of ocean variability (Fig. S3). While we do find 8 of the 39 compound droughts in the 118-year record associated with opposite phases of two or more of these variability modes (Figs. 2 and S3), those conditions are comparatively rare45.
### Identifying relevant phases of natural variability modes
To establish the relationship between these modes of ocean variability and SPI in the study regions, we perform a multiple linear regression analysis (Fig. S4). Our analyses reveal a widespread and consistent negative influence of the Niño3.4 SST anomalies (Fig. S4a) and positive influence of the Atlantic Niño SST anomalies (Fig. S4b) on SPI in most regions, suggesting that Niño3.4+ and AtlNiño conditions are conducive to droughts in these regions. In contrast, we find that the TNA SST anomalies (Fig. S4c) and the IOD (Fig. S4d) have a varied influence across these regions. For instance, the IOD has a positive influence on SPI over parts of WAF, EAF and SAS but a negative influence over parts of CNA and SEA. This indicates that IOD+ conditions promote droughts over CNA and SEA. Similarly, the TNA SST anomalies exhibit a negative influence over parts of AMZ, but positive influence over parts of CAM, SEA, and WAF, which suggest that TNA conditions favor droughts over the latter regions.
We also calculate the fraction of the total drought events in each region during different phases of these modes of ocean variability (Fig. S5). Positive Niño3.4 SST anomalies (>0.5σ; Niño3.4+) are linked to a substantial fraction of historical drought events over several regions. Niño3.4+ (nine events historically) conditions are associated with ≥75% of droughts over CAM, SAS, and SEA, and ≥50% of droughts over EAF, WAF, and TIB. Similarly, TNA conditions are coincident with ≥75% of droughts over CAM and SAS, and ≥50% over SEA, TIB, and WAF. AtlNiño- events coincide with ≥50% of droughts over AMZ, EAS, and WAF while IOD+ is present during ≥75% of droughts over SEA and ≥50% of droughts over AMZ, CAM, CAN, SAS, TIB, and WAF (Fig. S5). In contrast, the opposite phases of IOD, TNA, and AtlNiño are associated with a small fraction of droughts over only one region. Collectively, these results suggest the predominant influence of Niño3.4+, IOD+, TNA, and AtlNiño on individual regions and compound droughts. These conditions are also more likely to co-occur. For instance, El Niño conditions are more likely to co-occur with IOD+ conditions as they tend to drive warmer SSTs over the western Indian ocean through the atmospheric bridge and cooler SSTs over the eastern Indian ocean via oceanic Indonesian throughflow45. Similarly, cold SSTs over the tropical north Atlantic Ocean can induce warm conditions over the Pacific Ocean by influencing the Walker circulations45, making cold TNA conditions and El Niños more likely. Therefore, we further explore how IOD+, TNA, and AtlNiño modes interact with El Niño to influence drought characteristics over individual regions and consequently, compound droughts.
### Amplifying effect of co-occurring modes with El Niño
The interplay of Niño3.4+ with other modes of ocean variability requires several instances of their co-occurrences for robustly distinguishing their individual and combined influence. Given the limited length of the observed record, we primarily study their interactions in a multicentury (1800 years) preindustrial climate simulation from the National Center for Atmospheric Research (NCAR) Community Earth System Model (CESM)46. CESM skillfully represents precipitation over the study regions and SST variability representing various oceanic modes relevant to this study47,48. We have included comparisons of CESM with observations, where feasible (Fig. 3). The 1800-year preindustrial simulation provides a substantially larger number of events to examine the relative and combined influence of natural modes of variability without any changes resulting from external climate forcing (Fig. 3). We compare regional drought characteristics during three types of conditions (see “Methods” section)—(1) El Niño co-occurring with other modes (either IOD+ or/and TNA or/and AtlNiño; referred to as co-occurring conditions), (2) El Niño occurring alone (referred to as Niño3.4+ conditions), and (3) neutral conditions, when none of them are active (Fig. 3). It should be noted that there are no neutral conditions in the 38-year observational record, and limited instances of the other two conditions does not allow their robust comparisons (e.g., there are 2 Niño3.4+ and 7 co-occurring conditions).
During Niño3.4+, a large fraction of all tropical regions—AMZ, CAM, EAF, WAF, SAS, and SEA—experience abnormally dry anomalies (Fig. S6), consistent with well-known observed ENSO teleconnections49,50,51. The co-occurrence of Niño3.4+ with other modes intensifies dry conditions over EAS, SEA, CAM, and AMZ, while the opposite impact is experienced over EAF (Fig. S6). The simulated composites show consistency with both observed datasets over most regions, with the exception of biases in the extent and intensity of precipitation deficits over parts of SAS, WAF and EAF between model and observations (Fig. S6). We also quantify the aggregate drought area and intensity across the individual regions (Fig. 3). In the CESM preindustrial simulation, two regions—CAM and SEA—experience significantly larger drought areas during both Niño3.4+ and co-occurring conditions relative to neutral conditions (indicated by gray arrows in Fig. 3a), while two regions—AMZ and SAS only show significantly larger droughts during co-occurring conditions but not during Niño3.4+ relative to neutral conditions (box plots, Fig. 3a). In addition, co-occurring conditions expand the drought area over AMZ and SEA and significantly reduce drought area over EAF relative to Niño3.4+ (indicated by green arrows in Fig. 3a), consistent with observations (solid circles, Fig. 3a), highlighting their role in shaping drought characteristics. Moreover, Niño3.4+ significantly increases drought intensity over EAF, SEA, and WAF relative to neutral conditions. Further, co-occurring conditions are associated with significantly higher drought intensity over AMZ, CAM, WAF, EAF, SAS, and SEA relative to Niño3.4+ (Fig. 3b), consistent with observations (solid circles, Fig. 3b). Overall, these findings highlight the complex interplay of Niño3.4+ and other modes of ocean variability that control the spatial footprint and severity of over studies regions (Figs. 3 and S6).
While Niño3.4+ exhibits the strongest influence on regional precipitation characteristics, (Fig. 3 and S6), the frequency, severity and spatial extent of compound droughts is substantially enhanced when Niño3.4+ co-occurs with other natural modes of ocean variability (Fig. S7). For instance, the probability of compound droughts in CESM increases from 0.09 during neutral conditions to ~0.27 during Niño3.4+ conditions and ~0.43 during co-occurring conditions (Fig. S7d). Likewise, the probability of widespread and severe droughts is nearly 70% higher during co-occurring conditions relative to Niño3.4+ conditions alone (Fig. S7e, f). These model-based findings are mostly consistent with observations (Fig. S7a–c), except that the simulated number of drought-affected regions during co-occurring conditions is not significantly higher even though the probability of simulated compound droughts is ~20% higher relative to Niño3.4+ conditions in observations (Fig. S7a).
### Influence of co-occurring modes on regional droughts
Next, we isolate the influence of each individual mode of variability and their co-occurrence with Niño3.4+ on precipitation characteristics (Fig. 4). AtlNiño is associated with anomalously dry conditions (relative to neutral) over WAF, central AMZ, northern TIB and EAS (Fig. 4a, b). Its co-occurrence with Niño3.4+ significantly influences precipitation anomalies in the Atlantic Rim regions, including stronger precipitation deficits over WAF and the AMZ and reversal of Niño3.4+ forced anomalies (wet to dry) over CNA (Fig. 4e, f). More intense and widespread drying over WAF and AMZ during Niño3.4+/AtlNiño occurs without substantial increase in SST anomalies over the Niño3.4 region, which indicates an additive influence of these modes on regional drought characteristics. Likewise, co-occurring TNA-/Niño3.4+ conditions also appear to have an additive influence though the composites do indicate significantly higher SST anomalies over the part of Niño3.4 region indicative of slightly stronger Niño3.4+ conditions (Fig. 4g). Individually, TNA are associated with dry conditions over WAF, EAF, CAM, southern SAS, and northern TIB relative to neutral conditions (Fig. 4a, c, e). Co-occurring TNA/Niño3.4+ conditions amplify the Niño3.4+-related drying over CAM, AMZ, EAF, northern TIB, central EAS, and SEA. In addition, there are more widespread precipitation deficits across WAF, EAF and SAS over areas that would experience wet anomalies during Niño3.4+ (Fig. 4, e, g).
In contrast to the relatively consistent drying influence of these modes across multiple regions, IOD+ exhibits a dipolar influence across the regions surrounding the Indian Ocean. IOD+ is associated with anomalous drying over western SEA, northern SAS, TIB, northeast EAS and parts of CNA and anomalous wet conditions over EAF52 and WAF49 (Fig. 4a, d). Therefore, Niño3.4+/IOD+ co-occurrence dampens the drying impacts of Niño3.4+across the latter regions, while it expands and intensifies precipitation deficits over SAS and SEA. These findings are consistent with Preethi et al.49 suggesting the co-occurrence of IOD+ conditions can dampen the influence of tropical drivers over Africa. One confounding factor in determining the modulating influence of the IOD+ on Niño3.4+-related drought effects is that intensity of Niño3.4+ is substantially higher during IOD+ (Fig. 4e, h), as studies suggest that strong Niño3.4+ events force IOD+ conditions53,54,55, which is perhaps partly responsible for the intensification of drought severity over SEA and parts of SAS during Niño3.4+/IOD+ co-occurrence.
Given the substantial effect of all four natural variability modes on regional precipitation (Figs. 3 and 4), we assess the individual and combined influence of each of the combinations on aggregate drought area and intensity across a subset of six SREX regions that are substantially affected by these variability modes (Fig. 5). Amongst the four modes, Niño3.4+ significantly increases drought area over the largest number of these regions—CAM, EAF, and SEA-relative to neutral conditions, followed by TNA- that increases drought area over CAM and EAF (indicated by gray arrows in Fig. 5a). The individual influence of other modes is limited to fewer regions—AtlNiño significantly increases drought area over WAF, whereas IOD+ significantly decreases drought area over EAF and WAF and increases it over SAS. However, their co-occurrence with Niño3.4+ has significant effects over multiple regions. Co-occurring Niño3.4+/TNA are associated with significantly higher drought area over all regions but AMZ relative to neutral conditions (gray arrows in Fig. 5a). In addition, the co-occurring Niño3.4+/TNA significantly (at 5% significance level) increase drought area over SEA and CAM while Niño3.4+/IOD+ co-occurrence significantly decreases drought area over EAF and increases drought area over SEA, relative to Niño3.4+ alone (indicated by green arrows in Fig. 5a). AMZ, which experiences no significant change in drought area under Niño3.4+ relative to neutral conditions, has a significantly higher drought area when Niño3.4+/AtlNiño or Niño3.4+/IOD+ co-occur. Similarly, WAF only shows significantly higher drought area during co-occurring Niño3.4+/TNA and Niño3.4+/AtlNiño but not during Niño3.4+ alone.
Unlike the influence on drought area, we find a more limited influence of the individual occurrences of these modes on drought intensity over most regions, except an increase in drought intensity over WAF during TNA and AtlNiño and over EAF, SEA, and WAF during Niño3.4+ relative to neutral conditions (indicated by gray arrows in Fig. 5b). However, co-occurring modes significantly increase drought intensity over all six regions relative to neutral conditions. For instance, despite no substantial difference in drought intensity over CAM and SAS between Niño3.4+ and neutral conditions, co-occurring Niño3.4+/TNA lead to significantly higher drought intensity over these regions and over SEA and WAF (indicated by gray arrows in Fig. 5a). In addition, co-occurring Niño3.4+/IOD+ are associated with significantly higher drought intensity over SEA and SAS and Niño3.4+/TNA are associated with significantly higher drought intensity over CAM and SAS, relative to Niño3.4+ alone (indicated by green arrows in Fig. 5b).
### Influence of co-occurring modes on compound droughts
The individual and co-occurring influences of these modes on regional drought characteristic also leads to the episodes of compound droughts across ten SREX regions when at least three regions simultaneously experience drought during the same season (Fig. 6). The probability of experiencing compound droughts increases approximately threefold during AtlNiño- (probability = 0.25), TNA (probability = 0.24) and Niño3.4+ (probability = 0.27) relative to neutral conditions (probability = 0.09) (gray arrows in Fig. 6a), which is further amplified during their co-occurrences. For instance, co-occurring Niño3.4+/IOD+ or Niño3.4+/AtlNiño increase the probability of compound droughts by a factor of ~5 while co-occurring TNA-/ Niño3.4+ increase it by a factor of ~7 relative to neutral conditions. Overall, the co-occurring Niño3.4+/TNA conditions are associated with the largest amplification of compound drought risk (~2.5 or ~150%) over their probability during Niño3.4+ conditions.
Similarly, the total compound drought area measured across all ten SREX regions shows a significant increase during TNA- and Niño3.4+ relative to neutral conditions (Fig. 6b). Niño3.4+ increases the probability of widespread droughts, events with drought area in the top 90th percentile (~21%), to 0.19 compared to ~0 during neutral conditions. Co-occurrence of other natural variability modes with Niño3.4+ also substantially increase compound drought area compared to neutral conditions. Most notably, co-occurring TNA/Niño3.4+ raises the probability of widespread droughts by a factor of ~3 relative to Niño3.4+. Likewise, co-occurrence of various ocean variability modes amplifies the probability of severe compound droughts events with the area-weighted average drought intensity across all regions in the lowest 10th percentile (~−1.52) (Fig. 6c). Co-occurring Niño3.4+/TNA are associated with a 2.5 times higher probability of severe droughts relative to Niño3.4+. The co-occurring Niño3.4+/IOD+ and Niño3.4+/AtlNiño- also increase the probability of severe droughts by a factor of ~2 and ~1.5, respectively relative to Niño3.4+. Overall, these analyses suggest that Niño3.4+ leads to the largest increase in the probability, extent and intensity of compound droughts relative to the neutral conditions, and the co-occurrence of IOD+, and/or TNA, and/or AtlNiño- with Niño3.4+ can significantly amplify these characteristics through their influence on drought intensity and extent over one or multiple SREX regions.
### Physical mechanisms associated with compound droughts
We investigate the underlying physical mechanisms that connect simultaneous precipitation anomalies over several terrestrial regions with SST anomalies in various oceanic basins by analyzing upper level (200 hPa) velocity potential (VP) and low-level (at 850 hPa) moisture flux convergence (MFC) anomalies corresponding to the individual and co-occurring modes (Fig. 7). The VP describes large-scale horizontal convergence and divergence centers of the atmospheric circulation and is particularly useful in identifying anomalies in the tropical circulations. It is well known that El Niño modulates tropical/sub-tropical precipitation via forcing anomalies in the Walker circulation56,57. Climatologically, the strongest upper-level divergence centers (also known as the ascending branches of the Walker circulation) during the boreal summer are located in the western Pacific and eastern Indian Oceans and their subsiding branches are located in the eastern Pacific, southwestern Indian, and Atlantic Oceans (Fig. S8). These upper-level divergence centers coincide with the strong monsoon-driven convection across Asia. During Niño3.4+, the ascending (subsiding) branches of the Walker circulation in the western Pacific and eastern Indian (eastern Pacific and south Atlantic) weaken, leading to anomalous upper level convergence (divergence) anomalies that are reflected in the positive (negative) VP anomalies (Fig. 7a). Such changes in the tropical circulations weaken boreal summer monsoons, reduce low-level moisture convergence and consequently, support drier conditions over those regions (Fig. 7a, e). The associated anomalies in the South Atlantic high also induce changes in the trade winds over the equatorial Atlantic which influence moisture supply over AMZ, CAM, and WAF (Fig. 7e).
The co-occurrence of AtlNiño with Niño3.4+ noticeably amplifies the positive VP anomalies over WAF during Niño3.4+ and reduces the anomalous ascent of the Walker circulation over CAM and AMZ, (Figs. 5a and 7f). These circulation changes along with cooler than normal SSTs in the region lead to reduced moisture convergence, expanding the precipitation deficits over these regions relative to during Niño3.4+ (Figs. 4e, f and 7e, f). Co-occurring Niño3.4+/TNA exhibit the strongest and most widespread positive VP anomalies over the studied regions that influence the large-scale monsoon circulations (Fig. 7a, c) and low-level moisture availability (Fig. 7e, g), which further intensify the strength of Niño3.4+-induced drying as reflected in Figs. 4g and 5b. Earlier studies also note that TNA influences precipitation over the African regions by altering the northward extent of the West African Monsoon49 and moisture transport from the Atlantic Ocean and Gulf of Guinea58, and over CAM through the modulation of low-level moisture convergence over the Caribbean region and the strength of the Atlantic northeasterly trades59. The main influence of IOD+ is seen over the African and Asian regions. IOD+ reduces (strengthens) the influence of Niño3.4+ on the upper-level circulation over Africa (EAS and SEA), which reduces (intensifies) the extent and intensity of dry anomalies (Fig. 5a, b). Our findings are consistent with previous studies that have found that IOD+ weakens the African Easterly Jet and strengthens the Tropical Easterly Jet, while Niño3.4+ generally drives the opposite response49. Similarly, the anomalously cool SSTs surrounding SEA during IOD+ contribute to reducing the low-level moisture convergence (Fig. 7e, h), and thereby amplify the regional drying associated with substantial weakening of Walker circulation60. Overall, we note that the simultaneous occurrence of other modes of ocean variability oftentimes intensifies and/or expands the large-scale circulation anomalies associated with Niño3.4+, resulting in more intense or widespread moisture deficits over several regions.
## Summary and conclusions
Spatially compound extremes impose amplifying pressures on the disaster risk management resources and the global food system. As the impacts of such extremes are increasingly being recognized, recent studies have started to investigate their probability of occurrence and associated mechanisms7,12,14,18,24. While previous studies have focused on the mechanisms of compound temperature extremes across the mid-latitudes18, we examine the drivers of compound droughts across ten SREX regions that predominantly experience summer precipitation with high variability, identified based on the Shannon Entropy index. We use the 38-year observational record and an 1800-year CESM preindustrial climate simulation to examine the characteristics of compound droughts and the influence of natural ocean variability modes. We identify 11 historical compound droughts in the observational records, of which seven are associated with strong El Niño conditions. In addition to the central role of El Niño in driving these events, our analysis based on observational and the preindustrial simulation demonstrates substantial influence of three other modes of ocean variability—IOD, TNA, and AtlNiño conditions—that amplify various characteristics of regional droughts and global occurrences of compound droughts.
El Niño leads to a significant increase in the drought area and intensity over the largest number of regions relative to the other modes of natural variability (Figs. 3 and 5), and in turn, increase the probability of compound droughts by a factor of ~3, compared to their probability during neutral conditions (Fig. 6). Additionally, El Niño heightens the probability of widespread and severe droughts to 0.19 and 0.17, respectively, relative to 0 during neutral conditions. Other modes of natural variability show a varying influence on drought extent and intensity over specific regions and therefore, by themselves have an overall smaller impact on the probability of compound droughts compared to the impact of El Niño. The TNA mode has the largest influence among the three other modes, with TNA significantly amplifies drought area across CAM and SEA, and drought intensity over CAM and SAS during its co-occurrence with El Niño, contributing to a 2.5-fold, 3-fold, and 2.5-fold increase in probability of compound, widespread and severe droughts, respectively (Fig. 6). In contrast, because IOD+ dampens the influence of El Niño on drought area in EAF but amplifies it in SEA, and its co-occurrence with Niño3.4+ leads to a relatively moderate 1.6-fold increase in the probability of compound, widespread, and severe droughts (Fig. 6).
Overall, our analyses reveal the importance of considering other modes of ocean variability in addition to El Niño for assessing the risk, extent, and severity of compound droughts. We highlight a few caveats and limitations of this study. First, because of the relatively small sample size of the precipitation record in several of the study regions, our analysis of the individual and combined influence of natural variability modes largely depends on the long preindustrial climate simulation. Second, although the CESM simulation largely captures the relationship between various modes of variability considered in this study, it demonstrates stronger than observed correlations between TNA and ENSO, and IOD and ENSO at different lead times. Third, while we utilize the CESM model, which is one of the most skillful climate models in representing El Niño conditions48, we do not investigate intermodel differences in the identified relationships that may arise due to varying representations of precipitation processes, natural variability characteristics and teleconnections. Four, we do not consider the potential lead-lag relationships between some of these modes of variability and their regional impacts on precipitation45,61,62,63. Efforts to comprehensively assess these relationships and interactions between modes on various timescales can support predictability efforts. In addition, our future work will also focus on investigating the physical processes underlying the interactions between these modes and the regional and global impacts of their co-occurrence.
Compound droughts have the potential to induce synchronous crop failures and simultaneously cause other impacts across various societal sectors in multiple regions, leading to cascading global consequences. In the backdrop of the global interconnectivity of our socio-economic and physical systems, our study highlights the importance of considering the occurrence of and interactions between multiple modes of natural variability that represent the large-scale state of climate in characterizing the compound drought risks and their impacts on global food security, rather than solely focusing on individual modes that drive region-specific droughts. Our study presents the first step towards understanding the factors that influence compound droughts and their characteristics, which can help understand how they might change in response to the projected increases in extreme El Niño conditions47 and positive IOD conditions64. Understanding the factors that shape the characteristics of compound droughts have important implications for enhancing society’s resilience to the multitude impacts of droughts including food insecurity and water scarcity. A better understanding of compound drought risks is relevant for informing agriculture insurance companies to design more optimal crop insurance schemes, which are presently based on the historical probabilities of extreme events in individual regions without considering their spatial relationships. By identifying how interactions among different modes of natural variability can influence compound droughts, our study highlights the potential for seasonal prediction of such events to aid in the management of their impacts. Several modes of SST variability have skillful predictions at varying lead times including up to 9-months for El Niño39, up to 6 months for the IOD40 and 4 months for tropical Atlantic Ocean SSTs41. Timely predictions of droughts and drought-induced shocks in agricultural production will help manage potential food insecurity in several vulnerable regions42. Additionally, predictions of such events have implications for international trade, where the agribusiness industry and grain producers can get enough time to minimize the economic losses due to anticipated disasters.
## Methods
### Data
We primarily use precipitation from the widely-used high-resolution (0.25° × 0.25°) Climate Hazards group Infrared Precipitation with Stations (CHIRPS version 2) dataset (1981 to present). The CHIRPS daily precipitation dataset has been used for the assessment of daily, monthly, seasonal, and annual precipitation characteristics in several regions of the world65,66,67,68. CHIRPS blends satellite-based precipitation estimates with in situ observations, and models of terrain-based precipitation modification to provide high resolution, spatially-complete, and continuous long-term data from 1981 to present, providing distinct advantages over rain-gauge-based products that include variations in station density or remotely sensed data that have a limited temporal extent69,70.
In order to establish the robustness of our findings, we also compare our analyses with data from the Climate Prediction Center (CPC; 0.5° × 0.5°) and Climatic Research Unit (CRU; 0.5° × 0.5°), by comparing the Standardized Precipitation Index (SPI) across all ten SREX regions from all three datasets (Fig. S9). The SPI from CHIRPS and CRU are strongly correlated ($$\rho$$ > 0.72) over all regions but CPC-based estimates exhibit comparatively lower correlations over some regions including EAF, WAF, SAS, and EAS. We find that CPC-based SPI does not capture documented droughts over AMZ71, SAS72 during the record breaking El Niño year 2015, and over SAS73, EAF74, and EAS75 in another well-known El Niño year 2009 (Fig. S9). Therefore, of these three datasets, we use CHIRPS for the remainder of our analysis. Further, while the Global Historical Climate Network has station-data availability over a longer period of time over some regions, we do not include it in this analysis due to the non-uniform density of stations across the study area, and temporal discontinuities in data availability. We obtain sea surface temperatures (SST) from the National Oceanic and Atmospheric Administration (NOAA) High Resolution (0.25° spatial resolution) Optimum Interpolation (OI) SST dataset version 2 (V2), which has temporal coverage from 1981 to present76. Although our observational analysis is based on precipitation from CHIRPS and SST from OI NOAA V2 due to their finer spatial resolution, we perform complementary analyses with the long-term observed precipitation from CRU77 (0.5° spatial resolution) and SSTs (2° spatial resolution) from Extended Reconstructed Sea Surface Temperature (ERSST) NOAA V544 during 1901–2018.
Given the limited length of the observed record, we further characterize the influence of various SST variability modes on precipitation variability in the ten SREX regions using an 1800-year preindustrial simulation from the CESM46. Since the simulation has constant preindustrial climate forcing, it isolates the influence of unforced natural climate variability from the confounding influence of changing external climate forcings46. We select the CESM model simulation because it is one of the most skillful modern climate models in reproducing El Niño behavior and its teleconnections47,48.
### Selection of regions
We examine compound droughts across ten SREX regions4,9,78, which are selected based on the similarity in their precipitation characteristics. Specifically, we consider regions that show high variability in summer precipitation and receive a majority of their precipitation during the summer season. To identify the subregions that meet these criteria, we compute the Shannon Entropy Index for summer season precipitation, which is a concept drawn from information theory to measure the variability of a random variable79. The Shannon Entropy index is defined as measure of variability and has been used in hydroclimatic studies to assess the spatial and temporal variability of precipitation time series80. The Shannon entropy H can be computed as80,
$$H = - {\sum} {p_{\rm{i}}\log _2p_{\rm{i}}},$$
(1)
where p is the probability of each ith observation of the variable time series. We restrict our analyses to regions that have high entropy values over more than 30% of the total domain. Only ten tropical and mid-latitude SREX regions meet this criterion. Within these regions, we only consider areas with entropy values exceeding 4.86, which is the median entropy value across the regions considered.
### Drought definitions
We define drought at each grid cell based on SPI calculated with accumulated summer season (June–September; JJAS) precipitation. Following the method developed by McKee et al.81, the probability of accumulated JJAS precipitation from all season is transformed to a standard normal distribution. The estimated JJAS SPI is similar to the JJAS precipitation anomaly, but the standardization makes it comparable across space and time. The SPI time series is linearly de-trended to eliminate long-term trends and capture interannual precipitation variability. We define a grid cell under drought if its SPI is less than –1 standard deviation (σ) of the long-term (1981–2018) mean SPI. We define a region under drought if the fractional area experiencing drought (SPI < −1σ) within that region exceeds the 80th percentile of the seasonal drought area distribution. We choose the 80th percentile threshold to define a region under drought because it captures several documented droughts across various regions and, compared to higher percentile thresholds, it is relatively less sensitive to the length of observational records. Additionally, higher percentiles (>80th percentile) also substantially limit the drought events sample size, limiting the statistical robustness of our findings. The drought extent is defined as the fraction of the area within a region with SPI < −1σ and the drought intensity is defined as the area weighted-average SPI value over all the grid cells experiencing drought. We define compound droughts as at least three of ten SREX regions simultaneously experiencing droughts. We define widespread drought as events in which the fraction of total area across all ten regions simultaneously affected by drought exceeds the 90th percentile of the long-term average drought area. We define severe drought as events in which average SPI across all drought affected areas falls below the 10th percentile of the long-term average SPI.
### Multiple linear regression (MLR)
We perform a MLR analysis to understand the individual influence of Niño3.4, TNA, IOD, and Atlantic Niño indices on SPI across all SREX regions. Using MLR, we compute the regression coefficients (slope) between SPI (dependent variable) and these SST-based indices (independent variable). To examine the multicollinearity in this multiple regression model, we estimate the variation inflation factor (VIF) corresponding to each independent variable82. We found relatively low VIFs for all four indices (TNA—1.05; Atlantic Niño—1.17; Niño3.4—1.46; IOD—1.27), which suggests a minimal concern of multicollinearity in our regression model.
### Natural variability modes
The Niño3.4 index is used to define ENSO as the average SST anomalies over 5°S–5°N, 170°–120°W83. The TNA index is estimated as the average SST anomalies over 5.5°–23.5°N, 15°–57.5°W84. The Atlantic Niño (AtlNiño) index is calculated from average SST anomalies over 5°S–5°N and 20°W–0°85, and IOD is identified by using the Dipole Mode Index (DMI), which is calculated as the SST difference between the western (50°–70°E, 10°S–10°N) and eastern (90°–110°E, 10°S– Equator) equatorial Indian Ocean22,86. The spatial extent of all regions used to calculate these indices are highlighted in Fig. 1. All indices are calculated for the summer. Niño3.4+ refers to El Niño conditions when JJAS positive SST anomaly over the Niño3.4 region is >0.5σ. TNA and AtlNiño refer to cold phases of these indices that are identified based on negative JJAS SST anomalies (< −0.5σ) over their corresponding regions. IOD+ refers to positive IOD when JJAS DMI is >0.5σ. Since, we aim to investigate the relationship between modes of ocean variability and compound droughts on interannual timescales, we remove the climate change signal by detrending the observed timeseries of all modes, SSTs and SPI, which makes the identified relationships more comparable between observations and preindustrial simulations.
To understand the influence of El-Niño and its interactions with other modes of natural variability on drought characteristics, we first categorize all available seasons in the observed record into Niño3.4+-only and co-occurring conditions. Niño3.4+-only conditions are defined as years when Niño3.4+ is active while all other modes are in their neutral phase (<±0.5). Co-occurring conditions are defined as years when Niño3.4+ co-occurs with AtlNiño, TNA, or IOD+ conditions. There are two Niño3.4+ and seven co-occurring conditions during the 38-year observed period. To get a larger distribution of compound droughts under various anomalous SST conditions, we examine these interactions in a 1800-year CESM preindustrial climate simulation.
In addition, we categorize years based on the individual occurrences of each variability mode, and their combined occurrences with Niño3.4+ to understand their individual and combined influence on drought characteristics relative to neutral conditions. Neutral conditions are defined as years without any substantial phase of either of the four modes of ocean variability. Niño3.4+/AtlNiño, Niño3.4+/IOD+, and Niño3.4+/TNA refer to years when Niño3.4+ co-occur with AtlNiño, IOD+, and TNA, respectively, while the other modes are in their neutral conditions.
We evaluate the lead correlations between each mode and JJAS SPI over study regions during 1901–2018 to assess the validity of using contemporaneous (JJAS) SSTs in each basin. Given that El Niño events typically peak in winter87, we examine correlations between the 4-month moving average of the Niño3.4 index starting from November of the previous year to September of the current year (Fig. S10). Although some regions show significant correlations at several month lag times, they constitute a relatively small fraction of the all regions considered (~12%) (Fig. S10a). The area with significant correlations between JJAS(0) (“0” refers to the months of the current year) SPI and ENSO increases substantially with reduced lead time of the ENSO index. Specifically, ~40% of the studied area shows the strongest correlation with instantaneous impact of summer ENSO conditions49,50,88,89,90 (Fig. S10a). In addition, JJAS(0) SPI shows the strongest correlation with contemporaneous ENSO (Fig. S10b). Similarly, we assess the correlations of JJAS(0) SPI with other modes of variability and find that the strongest and most widespread correlations across all regions altogether are with contemporaneous IOD and Atlantic Niño. The TNA index has its strongest correlations at a short lead time though the correlations are not substantially different than during the JJAS season (Fig. S10a, b).
We also note that there are some contemporaneous and lagged correlations between ENSO and other modes of variability61,62,63(Fig. S11). Consistent with previous studies, we find an insignificant contemporaneous correlation between co-occurring AtlNiño and ENSO62,63 but weak lead correlations up to 6 months in advance61. Further, we find insignificant correlations between TNA and ENSO on most timescales in observations. Correlations between IOD and ENSO are the strongest in JJAS (Fig. S11a). The simulations generally capture these relationships but indicate stronger than observed correlations between TNA and ENSO, and IOD and ENSO at nearly all lead times (Fig. S11b). These lagged correlations between modes61,62,63 highlight the potential for their predictability and their associated regional precipitation anomalies91 and warrant further investigation. However, our analyses are constrained to the influence of contemporaneous states of all modes on regional precipitation, given the overall strongest and most widespread influence of most modes on regional precipitation in these regions. Our choice of using contemporaneous SSTs follows numerous studies that have identified the importance of contemporaneous Pacific, Atlantic, and Indian Ocean SST conditions on monsoons, which govern precipitation over a majority of these regions49,50,92,93.
### Statistical significance
We use the permutation test to assess the statistical significance of the differences in mean of drought characteristics during the occurrence of various combinations of natural ocean variability modes94. Permutation tests are becoming increasingly common to estimate the significance level of certain statistical analyses95. The non-parametric permutation test does not make any assumptions pertaining to sample size and distribution of the data, and is therefore suitable for a variety of situations, including for comparing distributions of different sizes, as is the case here. Here, we use the difference in the means of the two distributions as the test statistic. We first quantify the test statistic from the two original distributions and then randomly permute the samples from the two distributions, and re-estimate the test statistic from the resampled distributions. We repeat this procedure 10,000 times to obtain an empirical distribution of the test statistic, which represents the possible outcomes if the distributions were totally random. If the original test statistic is higher (lower) than the 95th (5th) percentile of the statistic from these permuted samples, we consider the mean of the distributions to be significantly different at the 5% level.
## Data availability
All datasets used in the manuscript are publicly available and their sources are provided in the “Methods” section.
## Code availability
The scripts developed to analyze these datasets can be made available on request from the corresponding author.
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## Acknowledgements
We would like to thank the National Oceanic and Atmospheric Administration (NOAA), National Center for Atmospheric Research (NCAR), Climatic Research Unit (CRU) University of East Anglia, and Climate Hazards Center UC Santa Barbara for archiving and enabling public access to their data. We thank Washington State University for the startup funding that has supported J.S. and D.S. W.B.A. acknowledges funding from Earth Institute Postdoctoral Fellowship. M.A. was supported by the National Climate‐Computing Research Center, which is located within the National Center for Computational Sciences at the ORNL and supported under a Strategic Partnership Project, 2316‐T849‐08, between DOE and NOAA. This manuscript has been co-authored by employees of Oak Ridge National Laboratory, managed by UT Battelle, LLC, under contract DE-AC05-00OR22725 with the U.S. Department of Energy (DOE). The publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
## Author information
Authors
### Contributions
All authors contributed to the design of the study. J.S. collected the data and performed the analyses. All authors were involved in discussions of the results. J.S. and D.S. wrote the manuscript with feedback from all authors.
### Corresponding author
Correspondence to Jitendra Singh.
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### Competing interests
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Singh, J., Ashfaq, M., Skinner, C.B. et al. Amplified risk of spatially compounding droughts during co-occurrences of modes of natural ocean variability. npj Clim Atmos Sci 4, 7 (2021). https://doi.org/10.1038/s41612-021-00161-2 |
# Relation between independent increments and Markov property
Independent increments and Markov property.do not imply each other. I was wondering
• if being one makes a process closer to being the other?
• if there are cases where one implies the other?
Thanks and regards!
-
To see this, assume that $(X_n)_{n\ge0}$ has independent increments, that is, $X_0=0$ and $X_n=Y_1+\cdots+Y_n$ for every $n\ge1$, where $(Y_n)_{n\ge1}$ is a sequence of independent random variables. The filtration of $(X_n)_{n\ge0}$ is $(\mathcal{F}^X_n)_{n\ge0}$ with $\mathcal{F}^X_n=\sigma(X_k;0\le k\le n)$. Note that $$\mathcal{F}^X_n=\sigma(Y_k;1\le k\le n),$$ hence $X_{n+1}=X_n+Y_{n+1}$ where $X_n$ is $\mathcal{F}^X_n$ measurable and $Y_{n+1}$ is independent on $\mathcal{F}^X_n$. This shows that the conditional distribution of $X_{n+1}$ conditionally on $\mathcal{F}^X_n$ is $$\mathbb{P}(X_{n+1}\in\mathrm{d}y|\mathcal{F}^X_n)=Q_n(X_n,\mathrm{d}y), \quad \mbox{where}\quad Q_n(x,\mathrm{d}y)=\mathbb{P}(x+Y_{n+1}\in\mathrm{d}y).$$ Hence $(X_n)_{n\ge0}$ is a Markov chain with transition kernels $(Q_n)_{n\ge0}$.
@Didier: Thanks! But I think it doesn't because of the following. First $P(X(t_3) | X(t_2), X(t_1)) = P(X(t_3)-X(t_2)|X(t_2), X(t_2)-X(t_1))$. Next $P(X(t_3)-X(t_2)|X(t_2), X(t_2)-X(t_1)) = P(X(t_3)-X(t_2)|X(t_2))$, if and only if $X(t_3)-X(t_2)$ and $X(t_2)-X(t_1))$ are conditionally independent given $X(t_2)$, which can not be implied by $X(t_3)-X(t_2)$ and $X(t_2)-X(t_1))$ are independent. Any mistake? – Tim Apr 29 '11 at 20:54
What is $P(W|U,V)$ for three random variables $U$, $V$, $W$? – Did Apr 29 '11 at 22:43
Why should "independent increments" require that $Y_j$ are independent of $X_0$? $X_0$ is not an increment. – Robert Israel Apr 29 '11 at 23:08
@Didier: Thanks! 1) I still have no clue how to explain and correct (2) in my last comment. Would you point me where in what texts/materials? 2) Generally when saying increments of a stochastic process, is $X_0$ an increment? Does the definition of an independent-increment process require $X_0=0$? – Tim May 3 '11 at 12:29
Invoking "smartness" here is a way to avoid following the explicit suggestions I made, which would lead you to understand the problem. It is also a cheap shot at my advice, considering the time and work I spent on your questions. // Since once again you are stopped by matters of definitions I suggest to come back to definitions: consider random variables $\xi$ and $\eta$ and a sigma-algebra $G$ such that $\xi$ is independent on $H=\sigma(\eta)\vee G$. Why is $E(u(\xi+\eta)\mid H)=E(u(\xi+\eta)\mid\eta)$ for every bounded $u$? Why is this related to your question? .../... – Did Nov 6 '11 at 8:48 |
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# Small Open Web Math Dataset
A 10k-sample subset of OpenWebMath, focused on high-quality mathematical text.
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