A Test for First-Order Markovianness

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Description

Performs a test for first-order Markovianness of a data series by inferring the sequence of i.i.d. U(0,1) random noise that might have generated it.

Usage

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markov.test(x, type = c("lb", "ks"), method = "holm", lag = 20, ...)

Arguments

x

the data series as a vector.

type

the procedures to use to test whether or not the disturbance series is independently and identically distributed on the unit interval. See ‘Details’.

method

the correction method to be used for adjusting the p-values. It is identical to the method argument of the p.adjust function, which is called to adjust the p-values.

lag

the number of lags to use when applying the Ljung-Box (portmanteau) test (lb.test).

...

parameters to pass on to functions that can be subsequently called.

Details

This function tests a symbolic sequence for first-order Markovianness (also known as the Markov property). It does this by reverse-engineering the sequence to obtain a sample of the kind of output from a pseudo-random number generator that would have produced the observed sequence if it had been generated by simulating a Markov chain .The sample output is then tested to see if it is an independent and identically distributed siequence of uniform numbers in the range 0-1. this involves the application of at least two tests, one for independence and another for uniformity over the unit interval. One concludes that the sequence is Markovian if the sample output passes the tests (that is, all null hypotheses are accepted) and non-Markovian otherwise.

The test is set up as follows:

H0: the sequence is first-order Markov
H1: the sequence is not first-order Markov

To simplify the use of the test, correction for multiple testing is carried out, which yields a single adjusted p- value. If this p-value is less than the significance level established for the test procedure, the null hypothesis of Markovianness is rejected. Otherwise, the null hypothesis should be accepted.

To correctly apply the test, use the type argument to specify at least one test of independence and one test of uniformity from the options displayed in the following table.

Category Function Test
Uniformity ks.unif.test Kolmogorov-Smirnov test for uniform$(0,1)$ data
chisq.unif.test Pearson's chi-squared test for discrete uniform data,
Independence lb.test Ljung-Box $Q$ test for uncorrelated data
diffsign.test signed difference test of independence
turningpoint.test turning point test of independence
rank.test rank test of independence

If type is not specified, lb.test and ks.unif.test are used by default.

As this procedure performs multiple tests in order to assess if the sequence has a Markovian dependence structure, it is necessary to adjust the p-values for multiple testing. By default, the Holm-Bonferroni method (holm) is used to correct for multiple testing, but this can be overridden via the method argument. The adjusted p-values are displayed when the result of the test is printed.

The smallest adjusted p-value constitutes the overall p-value for the test. If this p-value is less than the significance level fixed for the test procedure, the null hypothesis of first-order Markovianness is rejected. Otherwise, the null hypothesis should be accepted.

Value

A list with class "multiplehtest" containing the following components:

method

the character string “Composite test for a first-order (finite state) Markov chain”.

statistics

the values of the test statistic for all the tests.

parameters

parameters for all the tests. Exactly one parameter is recorded for each test, for example, df for lb.test. Any additional parameters are not saved, for example, the a and b parameters of chisq.unif.test.

p.values

p-values of all the tests.

methods

a vector of character strings indicating what type of tests were performed.

adjusted.p.values

the adjusted p-values.

data.name

a character string giving the name of the data.

adjust.method

indicates which correction method was used to adjust the p-values for multiple testing.

estimate

the transition matrix estimated to fit a first-order Markov chain to the data and used to generate the infered random disturbance.

Note

Sometimes, a warning message advising that ties should not be present for the Kolmogorov-Smirnov test can arise when analysing long sequences. If you do receive this warning, it means that the results of the Kolmogorov-Smirnov test (ks.unif.test) should not be trusted. In this case, Pearson's chi-squared test (chisq.unif.test) should be used instead of the Kolmogorov-Smirnov test.

Author(s)

Andrew Hart and Servet Mart<ed>nez

References

Hart, A.G. and Mart<ed>nez, S. (2011) Statistical testing of Chargaff's second parity rule in bacterial genome sequences. Stoch. Models 27(2), 1–46.

Hart, A.G. and Mart<ed>nez, S. (2014) Markovianness and Conditional Independence in Annotated Bacterial DNA. Stat. Appl. Genet. Mol. Biol. 13(6), 693-716. arXiv:1311.4411 [q-bio.QM].

See Also

diid.test, ks.unif.test, chisq.unif.test, diffsign.test, turningpoint.test, rank.test, lb.test

Examples

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#Generate an IID uniform DNA sequence
seq <- simulateMarkovChain(5000, matrix(0.25, 4, 4), states=c("a","c","g","t"))
markov.test(seq)

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