| Title: | Efficient Computation and Testing of the Bergsma-Dassios Sign Covariance |
|---|---|
| Description: | Computes the t* statistic corresponding to the tau* population coefficient introduced by Bergsma and Dassios (2014) <DOI:10.3150/13-BEJ514> and does so in O(n^2) time following the algorithm of Heller and Heller (2016) <DOI:10.48550/arXiv.1605.08732> building off of the work of Weihs, Drton, and Leung (2016) <DOI:10.1007/s00180-015-0639-x>. Also allows for independence testing using the asymptotic distribution of t* as described by Nandy, Weihs, and Drton (2016) <DOI:10.1214/16-EJS1166>. |
| Authors: | Luca Weihs [aut], Emin Martinian [ctb] (Created the red-black
tree library included in package.), Julian D. Karch [cre]
|
| Maintainer: | Julian D. Karch <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 1.1.7 |
| Built: | 2024-12-12 18:04:36 UTC |
| Source: | CRAN |
Computes the t* statistic corresponding to the tau star population coefficient introduced by Bergsma and Dassios (2014) <DOI:10.3150/13-BEJ514> and does so in O(n^2*log(n)) time following the algorithm of Weihs, Drton, and Leung (2016) <DOI:10.1007/s00180-015-0639-x>. Also allows for independence testing using the asymptotic distribution of t* as described by Nandy, Weihs, and Drton (2016) <http://arxiv.org/abs/1602.04387>. To directly compute the t* statistic see the function tStar. If otherwise interested in performing tests of independence then see the function tauStarTest.
Maintainer: Julian D. Karch [email protected] (ORCID)
Authors:
Luca Weihs [email protected]
Other contributors:
Emin Martinian (Created the red-black tree library included in package.) [contributor]
Bergsma, Wicher; Dassios, Angelos. A consistent test of independence based
on a sign covariance related to Kendall's tau. Bernoulli 20 (2014), no.
2, 1006–1028.
Luca Weihs, Mathias Drton, and Dennis Leung. Efficient Computation of the
Bergsma-Dassios Sign Covariance. Computational Statistics, x:x-x,
2016. to appear.
Preetam Nandy, Luca Weihs, and Mathias Drton. Large-Sample Theory for the
Bergsma-Dassios Sign Covariance. arXiv preprint arXiv:1602.04387. 2016.
library(TauStar) # Compute t* for a concordant quadruple tStar(c(1, 2, 3, 4), c(1, 2, 3, 4)) # == 2/3 # Compute t* for a discordant quadruple tStar(c(1, 2, 3, 4), c(1, -1, 1, -1)) # == -1/3 # Compute t* on random normal iid normal data set.seed(23421) tStar(rnorm(4000), rnorm(4000)) # near 0 # Compute t* as a v-statistic set.seed(923) tStar(rnorm(100), rnorm(100), vStatistic = TRUE) # Compute an approximation of tau* via resampling set.seed(9492) tStar(rnorm(10000), rnorm(10000), resample = TRUE, sampleSize = 30, numResamples = 5000 ) # Perform a test of independence using continuous data set.seed(123) x <- rnorm(100) y <- rnorm(100) testResults <- tauStarTest(x, y) print(testResults$pVal) # big p-value # Now make x and y correlated so we expect a small p-value y <- y + x testResults <- tauStarTest(x, y) print(testResults$pVal) # small p-valuelibrary(TauStar) # Compute t* for a concordant quadruple tStar(c(1, 2, 3, 4), c(1, 2, 3, 4)) # == 2/3 # Compute t* for a discordant quadruple tStar(c(1, 2, 3, 4), c(1, -1, 1, -1)) # == -1/3 # Compute t* on random normal iid normal data set.seed(23421) tStar(rnorm(4000), rnorm(4000)) # near 0 # Compute t* as a v-statistic set.seed(923) tStar(rnorm(100), rnorm(100), vStatistic = TRUE) # Compute an approximation of tau* via resampling set.seed(9492) tStar(rnorm(10000), rnorm(10000), resample = TRUE, sampleSize = 30, numResamples = 5000 ) # Perform a test of independence using continuous data set.seed(123) x <- rnorm(100) y <- rnorm(100) testResults <- tauStarTest(x, y) print(testResults$pVal) # big p-value # Now make x and y correlated so we expect a small p-value y <- y + x testResults <- tauStarTest(x, y) print(testResults$pVal) # small p-value
Computes the pth quantile of a cumulative distribution function using a simple binary serach algorithm. This can be extremely slow but has the benefit of being trivial to implement.
binaryQuantileSearch(pDistFunc, p, lastLeft, lastRight, error = 10^-4)binaryQuantileSearch(pDistFunc, p, lastLeft, lastRight, error = 10^-4)
pDistFunc |
a cumulative distribution function on the real numbers, it should take a single argument x and return the cumualtive distribution function evaluated at x. |
p |
the quantile |
lastLeft |
binary search works by continuously decreasing the search space from the left and right. lastLeft should be a lower bound for the quantile p. |
lastRight |
similar to lastRight but should be an upper bound. |
error |
the error tolerated from the binary search |
the quantile (within error).
Computes the eigenvalues needed to determine the asymptotic distributions in the mixed/discrete cases. See Nandy, Weihs, and Drton (2016) <http://arxiv.org/abs/1602.04387> for more details.
eigenForDiscreteProbs(p)eigenForDiscreteProbs(p)
p |
a vector of probabilities that sum to 1. |
the eigenvalues associated to the matrix generated by p
Attempts to determine if the input data is from a discrete distribution. Will return true if the data type is of type integer or there are non-unique values.
isDiscrete(x)isDiscrete(x)
x |
a vector which should be determined if discrete or not. |
the best judgement of whether or not the data was discrete
Checks if the input vector has a single entry that is between 0 and 1
isProb(prob)isProb(prob)
prob |
the probability to check |
TRUE if conditions are met, FALSE if otherwise
Checks if the input vector has entries that sum to 1 and are non-negative
isProbVector(probs)isProbVector(probs)
probs |
the probability vector to check |
TRUE if conditions are met, FALSE if otherwise
Determines if input vector is a valid vector of real valued observations
isValidDataVector(x)isValidDataVector(x)
x |
the vector to be tested |
TRUE or FALSE
Density, distribution function, quantile function and random generation for the asymptotic null distribution of t* in the discrete case. That is, in the case that t* is generated from a sample of jointly discrete independent random variables X and Y.
pDisHoeffInd(x, probs1, probs2, lower.tail = TRUE, error = 10^-5) dDisHoeffInd(x, probs1, probs2, error = 10^-3) rDisHoeffInd(n, probs1, probs2) qDisHoeffInd(p, probs1, probs2, error = 10^-4)pDisHoeffInd(x, probs1, probs2, lower.tail = TRUE, error = 10^-5) dDisHoeffInd(x, probs1, probs2, error = 10^-3) rDisHoeffInd(n, probs1, probs2) qDisHoeffInd(p, probs1, probs2, error = 10^-4)
x |
the value (or vector of values) at which to evaluate the function. |
probs1 |
a vector of probabilities corresponding to the (ordered)
support of X. That is if your first random variable has support
|
probs2 |
just as probs1 but for the second random variable Y. |
lower.tail |
a logical value, if TRUE (default), probabilities are
|
error |
a tolerated error in the result. This should be considered as a guide rather than an exact upper bound to the amount of error. |
n |
the number of observations to return. |
p |
the probability (or vector of probabilities) for which to get the quantile. |
dDisHoeffInd gives the density, pDisHoeffInd gives the distribution function, qDisHoeffInd gives the quantile function, and rDisHoeffInd generates random samples.
Density, distribution function, quantile function and random generation for the asymptotic null distribution of t* in the continuous case. That is, in the case that t* is generated from a sample of jointly continuous independent random variables.
pHoeffInd(x, lower.tail = TRUE, error = 10^-5) rHoeffInd(n) dHoeffInd(x, error = 1/2 * 10^-3) qHoeffInd(p, error = 10^-4)pHoeffInd(x, lower.tail = TRUE, error = 10^-5) rHoeffInd(n) dHoeffInd(x, error = 1/2 * 10^-3) qHoeffInd(p, error = 10^-4)
x |
the value (or vector of values) at which to evaluate the function. |
lower.tail |
a logical value, if TRUE (default), probabilities are
|
error |
a tolerated error in the result. This should be considered as a guide rather than an exact upper bound to the amount of error. |
n |
the number of observations to return. |
p |
the probability (or vector of probabilities) for which to get the quantile. |
dHoeffInd gives the density, pHoeffInd gives the distribution function, qHoeffInd gives the quantile function, and rHoeffInd generates random samples.
Density, distribution function, quantile function and random generation for the asymptotic null distribution of t* in the mixed case. That is, in the case that t* is generated a sample from an independent bivariate distribution where one coordinate is marginally discrete and the other marginally continuous.
pMixHoeffInd(x, probs, lower.tail = TRUE, error = 10^-6) dMixHoeffInd(x, probs, error = 10^-3) rMixHoeffInd(n, probs, error = 10^-8) qMixHoeffInd(p, probs, error = 10^-4)pMixHoeffInd(x, probs, lower.tail = TRUE, error = 10^-6) dMixHoeffInd(x, probs, error = 10^-3) rMixHoeffInd(n, probs, error = 10^-8) qMixHoeffInd(p, probs, error = 10^-4)
x |
the value (or vector of values) at which to evaluate the function. |
probs |
a vector of probabilities corresponding to the (ordered)
support the marginally discrete random variable. That is, if the
marginally discrete distribution has support |
lower.tail |
a logical value, if TRUE (default), probabilities are
|
error |
a tolerated error in the result. This should be considered as a guide rather than an exact upper bound to the amount of error. |
n |
the number of observations to return. |
p |
the probability (or vector of probabilities) for which to get the quantile. |
dMixHoeffInd gives the density, pMixHoeffInd gives the distribution function, qMixHoeffInd gives the quantile function, and rMixHoeffInd generates random samples.
A simple print function for tstest (Tau* test) objects.
## S3 method for class 'tstest' print(x, ...)## S3 method for class 'tstest' print(x, ...)
x |
the tstest object to be printed |
... |
ignored. |
No return value, prints to console.
Performs a (consistent) test of independence between two input vectors using the asymptotic (or permutation based) distribution of the test statistic t*. The asymptotic results hold in the case that x is generated from either a discrete or continous distribution and similarly for y (in particular it is allowed for one to be continuous while the other is discrete). The asymptotic distributions were computed in Nandy, Weihs, and Drton (2016) <http://arxiv.org/abs/1602.04387>.
tauStarTest(x, y, mode = "auto", resamples = 1000)tauStarTest(x, y, mode = "auto", resamples = 1000)
x |
a vector of sampled values. |
y |
a vector of sampled values corresponding to x, y must be the same length as x. |
mode |
should be one of five possible values: "auto", "continuous", "discrete", "mixed", or "permutation". If "auto" is selected then the function will attempt to automatically determine whether x,y are discrete or continuous and then perform the appropriate asymptotic test. In cases "continuous", "discrete", and "mixed" we perform the associated asymptotic test making the given assumption. Finally if "permutation" is selected then the function runs a Monte-Carlo permutation test for some given number of resamplings. |
resamples |
the number of resamplings to do if mode = "permutation". Otherwise this value is ignored. |
a list with class "tstest" recording the outcome of the test.
Preetam Nandy, Luca Weihs, and Mathias Drton. Large-Sample Theory for the Bergsma-Dassios Sign Covariance. arXiv preprint arXiv:1602.04387. 2016.
set.seed(123) x <- rnorm(100) y <- rnorm(100) testResults <- tauStarTest(x, y) print(testResults$pVal) # big p-value y <- y + x # make x and y correlated testResults <- tauStarTest(x, y) print(testResults$pVal) # small p-valueset.seed(123) x <- rnorm(100) y <- rnorm(100) testResults <- tauStarTest(x, y) print(testResults$pVal) # big p-value y <- y + x # make x and y correlated testResults <- tauStarTest(x, y) print(testResults$pVal) # small p-value
Computes the t* U-statistic for input data pairs (x_1,y_1), (x_2,y_2), ..., (x_n,y_n) using the algorithm developed by Heller and Heller (2016) <arXiv:1605.08732> building off of the work of Weihs, Drton, and Leung (2015) <DOI:10.1007/s00180-015-0639-x>.
tStar( x, y, vStatistic = FALSE, resample = FALSE, numResamples = 500, sampleSize = min(length(x), 1000), method = "fastest", slow = FALSE )tStar( x, y, vStatistic = FALSE, resample = FALSE, numResamples = 500, sampleSize = min(length(x), 1000), method = "fastest", slow = FALSE )
x |
A numeric vector of x values (length >= 4). |
y |
A numeric vector of y values, should be of the same length as x. |
vStatistic |
If TRUE then will compute the V-statistic version of t*, otherwise will compute the U-Statistic version of t*. Default is to compute the U-statistic. |
resample |
If TRUE then will compute an approximation of t* using a subsettting approach: samples of size sampleSize are taken from the data numResample times, t* is computed on each subsample, and all subsample t* values are then averaged. Note that this only works when vStatistic == FALSE, in general you probably don't want to compute the V-statistic via resampling as the size of the bias depends on the sampleSize irrespective numResamples. Default is resample == FALSE so that t* is computed on all of the data, this may be slow for very large sample sizes. Resampling can only be used when the method argument is using its default. |
numResamples |
See resample variable description for details, this value is ignored if resample == FALSE (ignored by default). |
sampleSize |
See resample variable description for details, this value is ignored if resample == FALSE (ignored by default). |
method |
which method to use to compute the statistic. Default is "fastest" which uses the fastest available method (currently "heller"). The options are "heller" described in Heller and Heller (2016), "weihs", using the algorithm from Weihs et al. (2015), and "naive" using a naive algorithm. |
slow |
a deprecated option kept for backwards compatability. If TRUE then will override the method parameter and compute the t* statistic using a naive O(n^4) algorithm. |
The numeric value of the t* statistic.
Bergsma, Wicher; Dassios, Angelos. A consistent test of independence based
on a sign covariance related to Kendall's tau. Bernoulli 20 (2014),
no. 2, 1006–1028.
Heller, Yair and Heller, Ruth. "Computing the Bergsma Dassios
sign-covariance." arXiv preprint arXiv:1605.08732 (2016).
Weihs, Luca, Mathias Drton, and Dennis Leung. "Efficient Computation of the
Bergsma-Dassios Sign Covariance." arXiv preprint arXiv:1504.00964 (2015).
library(TauStar) # Compute t* for a concordant quadruple tStar(c(1, 2, 3, 4), c(1, 2, 3, 4)) # == 2/3 # Compute t* for a discordant quadruple tStar(c(1, 2, 3, 4), c(1, -1, 1, -1)) # == -1/3 # Compute t* on random normal iid normal data set.seed(23421) tStar(rnorm(4000), rnorm(4000)) # near 0 # Compute t* as a v-statistic set.seed(923) tStar(rnorm(100), rnorm(100), vStatistic = TRUE) # Compute an approximation of tau* via resampling set.seed(9492) tStar(rnorm(10000), rnorm(10000), resample = TRUE, sampleSize = 30, numResamples = 5000 )library(TauStar) # Compute t* for a concordant quadruple tStar(c(1, 2, 3, 4), c(1, 2, 3, 4)) # == 2/3 # Compute t* for a discordant quadruple tStar(c(1, 2, 3, 4), c(1, -1, 1, -1)) # == -1/3 # Compute t* on random normal iid normal data set.seed(23421) tStar(rnorm(4000), rnorm(4000)) # near 0 # Compute t* as a v-statistic set.seed(923) tStar(rnorm(100), rnorm(100), vStatistic = TRUE) # Compute an approximation of tau* via resampling set.seed(9492) tStar(rnorm(10000), rnorm(10000), resample = TRUE, sampleSize = 30, numResamples = 5000 )