kb.test: Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests
In QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances
kb.test R Documentation
Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests
Description
This function performs the kernel-based quadratic distance goodness-of-fit
tests. It includes tests for multivariate normality, two-sample tests and
k-sample tests.
Usage
kb.test(
x,
y = NULL,
h = NULL,
method = "subsampling",
B = 150,
b = NULL,
Quantile = 0.95,
mu_hat = NULL,
Sigma_hat = NULL,
centeringType = "Nonparam",
K_threshold = 10,
alternative = "skewness"
)
## S4 method for signature 'ANY'
kb.test(
x,
y = NULL,
h = NULL,
method = "subsampling",
B = 150,
b = 0.9,
Quantile = 0.95,
mu_hat = NULL,
Sigma_hat = NULL,
centeringType = "Nonparam",
K_threshold = 10,
alternative = "skewness"
)
## S4 method for signature 'kb.test'
show(object)
Arguments
x
Numeric matrix or vector of data values.
y
Numeric matrix or vector of data values. Depending on the input
y, the corresponding test is performed.
if y = NULL, the function performs the tests for normality on
x
if y is a data matrix, with same dimensions of x, the
function performs the two-sample test between x and y.
if y is a numeric or factor vector, indicating the group
memberships for each observation, the function performs the k-sample
test.
h
Bandwidth for the kernel function. If a value is not provided, the
algorithm for the selection of an optimal h is performed
automatically. See the function select_h for more
details.
method
The method used for critical value estimation ("subsampling",
"bootstrap", or "permutation")(default: "subsampling").
B
The number of iterations to use for critical value estimation
(default: 150).
b
The size of the subsamples used in the subsampling algorithm
(default: 0.8).
Quantile
The quantile to use for critical value estimation, 0.95 is
the default value.
mu_hat
Mean vector for the reference distribution.
Sigma_hat
Covariance matrix of the reference distribution.
centeringType
String indicating the method used for centering the
normal kernel ('Param' or 'Nonparam').
K_threshold
maximum number of groups allowed. Default is 10. It is a
control parameter. Change in case of more than 10 samples.
alternative
Family of alternative chosen for selecting h, between
"location", "scale" and "skewness" (only if h
is not provided).
object
Object of class kb.test
Details
The function kb.test performs the kernel-based quadratic
distance tests using the Gaussian kernel with bandwidth parameter h.
Depending on the shape of the input y the function performs the tests
of multivariate normality, the non-parametric two-sample tests or the
k-sample tests.
The quadratic distance between two probability distributions F and
G is
defined as
d_{K}(F,G)=\iint K(x,y)d(F-G)(x)d(F-G)(y),
where G is a distribution whose goodness of fit we wish to assess and
K denotes the Normal kernel defined as
K_{{h}}(\mathbf{s}, \mathbf{t}) = (2 \pi)^{-d/2}
\left(\det{\mathbf{\Sigma}_h}\right)^{-\frac{1}{2}}
\exp\left\{-\frac{1}{2}(\mathbf{s} - \mathbf{t})^\top
\mathbf{\Sigma}_h^{-1}(\mathbf{s} - \mathbf{t})\right\},
for every \mathbf{s}, \mathbf{t} \in \mathbb{R}^d \times
\mathbb{R}^d, with covariance matrix \mathbf{\Sigma}_h=h^2 I and
tuning parameter h.
-
Test for Normality:
Let x_1, x_2, ..., x_n be a random sample with empirical
distribution function \hat F. We test the null hypothesis of
normality, i.e. H_0:F=G=\mathcal{N}_d(\mu, \Sigma).
We consider the U-statistic estimate of the sample KBQD
U_{n}=\frac{1}{n(n-1)}\sum_{i=2}^{n}\sum_{j=1}^{i-1}
K_{cen}(\mathbf{x}_{i}, \mathbf{x}_{j}),
then the first test statistics is
T_{n}=\frac{U_{n}}{\sqrt{Var(U_{n})}},
with Var(U_n) computed exactly following Lindsay et al.(2014),
and the V-statistic estimate
V_{n} = \frac{1}{n}\sum_{i=1}^{n}
\sum_{j=1}^{n}K_{cen}(\mathbf{x}_{i}, \mathbf{x}_{j}),
where K_{cen} denotes the Normal kernel K_h with parametric
centering with respect to the considered normal distribution
G = \mathcal{N}_d(\mu, \Sigma).
The asymptotic distribution of the V-statistic is an infinite combination
of weighted independent chi-squared random variables with one degree of
freedom. The cutoff value is obtained using the Satterthwaite
approximation c \cdot \chi_{DOF}^2, where c and DOF
are computed exactly following the formulas in Lindsay et al.(2014).
For the U-statistic the cutoff is determined empirically:
Generate data from the considered normal distribution ;
Compute the test statistics for B Monte Carlo(MC) replications;
Compute the 95th quantile of the empirical distribution of the test
statistic.
-
k-sample test:
Consider k random samples of i.i.d. observations
\mathbf{x}^{(i)}_1,
\mathbf{x}^{(i)}_{2},\ldots, \mathbf{x}^{(i)}_{n_i} \sim F_i,
i = 1, \ldots, k.
We test if the samples are generated from the same unknown distribution,
that is H_0: F_1 = F_2 = \ldots = F_k versus
H_1: F_i \not = F_j, for some 1 \le i \not = j \le k.
We construct a matrix distance \hat{\mathbf{D}}, with
off-diagonal elements
\hat{D}_{ij} = \frac{1}{n_i n_j} \sum_{\ell=1}^{n_i}
\sum_{r=1}^{n_j}K_{\bar{F}}(\mathbf{x}^{(i)}_\ell,\mathbf{x}^{(j)}_r),
\qquad \mbox{ for }i \not= j
and in the diagonal
\hat{D}_{ii} = \frac{1}{n_i (n_i -1)} \sum_{\ell=1}^{n_i}
\sum_{r\not= \ell}^{n_i} K_{\bar{F}}(\mathbf{x}^{(i)}_\ell,
\mathbf{x}^{(i)}_r), \qquad \mbox{ for }i = j,
where K_{\bar{F}} denotes the Normal kernel K_h
centered non-parametrically with respect to
\bar{F} = \frac{n_1 \hat{F}_1 + \ldots + n_k \hat{F}_k}{n},
\quad \mbox{ with } n=\sum_{i=1}^k n_i.
We compute the trace statistic
\mathrm{trace}(\hat{\mathbf{D}}_n) = \sum_{i=1}^{k}\hat{D}_{ii}
and D_n, derived considering all the possible pairwise comparisons
in the k-sample null hypothesis, given as
D_n = (k-1) \mathrm{trace}(\hat{\mathbf{D}}_n)
- 2 \sum_{i=1}^{k}\sum_{j> i}^{k}\hat{D}_{ij}.
We compute the empirical critical value by employing numerical techniques
such as the bootstrap, permutation and subsampling algorithms:
Generate k-tuples, of total size n_B, from the pooled sample
following one of the sampling methods;
Compute the k-sample test statistic;
Repeat B times;
Select the 95^{th} quantile of the obtained values.
-
Two-sample test:
Let x_1, x_2, ..., x_{n_1} \sim F and
y_1, y_2, ..., y_{n_2} \sim G be
random samples from the distributions F and G, respectively.
We test the null hypothesis that the two samples are generated from
the same unknown distribution, that is H_0: F=G vs
H_1:F\not=G. The test statistics coincide with the k-sample
test statistics when k=2.
Kernel centering
The arguments mu_hat and Sigma_hat indicate the normal model
considered for the normality test, that is H_0: F = N(mu_hat,
Sigma_hat).
For the two-sample and k-sample tests, mu_hat and
Sigma_hat can
be used for the parametric centering of the kernel, in the case we want to
specify the reference distribution, with centeringType = "Param".
This is the default method when the test for normality is performed.
The normal kernel centered with respect to
G \sim N_d(\mathbf{\mu}, \mathbf{V}) can be computed as
K_{cen(G)}(\mathbf{s}, \mathbf{t}) =
K_{\mathbf{\Sigma_h}}(\mathbf{s}, \mathbf{t}) -
K_{\mathbf{\Sigma_h} + \mathbf{V}}(\mathbf{\mu}, \mathbf{t})
- K_{\mathbf{\Sigma_h} + \mathbf{V}}(\mathbf{s}, \mathbf{\mu}) +
K_{\mathbf{\Sigma_h} + 2\mathbf{V}}(\mathbf{\mu}, \mathbf{\mu}).
We consider the non-parametric centering of the kernel with respect to
\bar{F}=(n_1 F_1 + \ldots n_k F_k)/n where n=\sum_{i=1}^k n_i,
with centeringType = "Nonparam", for the two- and k-sample
tests.
Let \mathbf{z}_1,\ldots, \mathbf{z}_n denote the pooled sample. For any
s,t \in \{\mathbf{z}_1,\ldots, \mathbf{z}_n\}, it is given by
K_{cen(\bar{F})}(\mathbf{s},\mathbf{t}) = K(\mathbf{s},\mathbf{t}) -
\frac{1}{n}\sum_{i=1}^{n} K(\mathbf{s},\mathbf{z}_i) -
\frac{1}{n}\sum_{i=1}^{n} K(\mathbf{z}_i,\mathbf{t}) +
\frac{1}{n(n-1)}\sum_{i=1}^{n} \sum_{j \not=i}^{n}
K(\mathbf{z}_i,\mathbf{z}_j).
Value
An S4 object of class kb.test containing the results of the
kernel-based quadratic distance tests, based on the normal kernel. The object
contains the following slots:
-
method: Description of the kernel-based quadratic
distance test performed.
-
x Data list of samples X (and Y).
-
Un The value of the U-statistic.
-
H0_Un A logical value indicating whether or not the null
hypothesis is rejected according to Un.
-
CV_Un The critical value computed for the test Un.
-
Vn The value of the V-statistic (if available).
-
H0_Vn A logical value indicating whether or not the null
hypothesis is rejected according to Vn (if available).
-
CV_Vn The critical value computed for the test Vn
(if available).
-
h List with the value of bandwidth parameter used for the
normal kernel function. If select_h is used, the matrix of computed
power values and the corresponding power plot are also provided.
-
B Number of bootstrap/permutation/subsampling replications.
-
var_Un exact variance of the kernel-based U-statistic.
-
cv_method The method used to estimate the critical value
(one of "subsampling", "permutation" or "bootstrap").
Note
For the two- and k-sample tests, the slots Vn, H0_Vn and
CV_Vn are empty, while the computed statistics are both reported in
slots Un, H0_Un and CV_Un.
A U-statistic is a type of statistic that is used to estimate a population
parameter. It is based on the idea of averaging over all possible distinct
combinations of a fixed size from a sample.
A V-statistic considers all possible tuples of a certain size, not just
distinct combinations and can be used in contexts where unbiasedness is not
required.
References
Markatou, M. and Saraceno, G. (2024). “A Unified Framework for
Multivariate Two- and k-Sample Kernel-based Quadratic Distance
Goodness-of-Fit Tests.”
https://doi.org/10.48550/arXiv.2407.16374
Lindsay, B.G., Markatou, M. and Ray, S. (2014) "Kernels, Degrees of Freedom,
and Power Properties of Quadratic Distance Goodness-of-Fit Tests", Journal
of the American Statistical Association, 109:505, 395-410,
DOI: 10.1080/01621459.2013.836972
See Also
kb.test for the class definition.
Examples
# create a kb.test object
x <- matrix(rnorm(100),ncol=2)
y <- matrix(rnorm(100),ncol=2)
# Normality test
my_test <- kb.test(x, h=0.5)
my_test
# Two-sample test
my_test <- kb.test(x,y,h=0.5, method="subsampling",b=0.9,
centeringType = "Nonparam")
my_test
# k-sample test
z <- matrix(rnorm(100,2),ncol=2)
dat <- rbind(x,y,z)
group <- rep(c(1,2,3),each=50)
my_test <- kb.test(x=dat,y=group,h=0.5, method="subsampling",b=0.9)
my_test
QuadratiK documentation built on Oct. 29, 2024, 5:08 p.m.
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| kb.test | R Documentation |
Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests
Description
This function performs the kernel-based quadratic distance goodness-of-fit
tests. It includes tests for multivariate normality, two-sample tests and
k-sample tests.
Usage
kb.test(
x,
y = NULL,
h = NULL,
method = "subsampling",
B = 150,
b = NULL,
Quantile = 0.95,
mu_hat = NULL,
Sigma_hat = NULL,
centeringType = "Nonparam",
K_threshold = 10,
alternative = "skewness"
)
## S4 method for signature 'ANY'
kb.test(
x,
y = NULL,
h = NULL,
method = "subsampling",
B = 150,
b = 0.9,
Quantile = 0.95,
mu_hat = NULL,
Sigma_hat = NULL,
centeringType = "Nonparam",
K_threshold = 10,
alternative = "skewness"
)
## S4 method for signature 'kb.test'
show(object)
Arguments
x |
Numeric matrix or vector of data values. |
y |
Numeric matrix or vector of data values. Depending on the input
|
h |
Bandwidth for the kernel function. If a value is not provided, the
algorithm for the selection of an optimal h is performed
automatically. See the function |
method |
The method used for critical value estimation ("subsampling", "bootstrap", or "permutation")(default: "subsampling"). |
B |
The number of iterations to use for critical value estimation (default: 150). |
b |
The size of the subsamples used in the subsampling algorithm (default: 0.8). |
Quantile |
The quantile to use for critical value estimation, 0.95 is the default value. |
mu_hat |
Mean vector for the reference distribution. |
Sigma_hat |
Covariance matrix of the reference distribution. |
centeringType |
String indicating the method used for centering the normal kernel ('Param' or 'Nonparam'). |
K_threshold |
maximum number of groups allowed. Default is 10. It is a control parameter. Change in case of more than 10 samples. |
alternative |
Family of alternative chosen for selecting h, between
"location", "scale" and "skewness" (only if |
object |
Object of class |
Details
The function kb.test performs the kernel-based quadratic
distance tests using the Gaussian kernel with bandwidth parameter h.
Depending on the shape of the input y the function performs the tests
of multivariate normality, the non-parametric two-sample tests or the
k-sample tests.
The quadratic distance between two probability distributions F and
G is
defined as
d_{K}(F,G)=\iint K(x,y)d(F-G)(x)d(F-G)(y),
where G is a distribution whose goodness of fit we wish to assess and
K denotes the Normal kernel defined as
K_{{h}}(\mathbf{s}, \mathbf{t}) = (2 \pi)^{-d/2}
\left(\det{\mathbf{\Sigma}_h}\right)^{-\frac{1}{2}}
\exp\left\{-\frac{1}{2}(\mathbf{s} - \mathbf{t})^\top
\mathbf{\Sigma}_h^{-1}(\mathbf{s} - \mathbf{t})\right\},
for every \mathbf{s}, \mathbf{t} \in \mathbb{R}^d \times
\mathbb{R}^d, with covariance matrix \mathbf{\Sigma}_h=h^2 I and
tuning parameter h.
-
Test for Normality:
Letx_1, x_2, ..., x_nbe a random sample with empirical distribution function\hat F. We test the null hypothesis of normality, i.e.H_0:F=G=\mathcal{N}_d(\mu, \Sigma).We consider the U-statistic estimate of the sample KBQD
U_{n}=\frac{1}{n(n-1)}\sum_{i=2}^{n}\sum_{j=1}^{i-1} K_{cen}(\mathbf{x}_{i}, \mathbf{x}_{j}),then the first test statistics is
T_{n}=\frac{U_{n}}{\sqrt{Var(U_{n})}},with
Var(U_n)computed exactly following Lindsay et al.(2014), and the V-statistic estimateV_{n} = \frac{1}{n}\sum_{i=1}^{n} \sum_{j=1}^{n}K_{cen}(\mathbf{x}_{i}, \mathbf{x}_{j}),where
K_{cen}denotes the Normal kernelK_hwith parametric centering with respect to the considered normal distributionG = \mathcal{N}_d(\mu, \Sigma).The asymptotic distribution of the V-statistic is an infinite combination of weighted independent chi-squared random variables with one degree of freedom. The cutoff value is obtained using the Satterthwaite approximation
c \cdot \chi_{DOF}^2, wherecandDOFare computed exactly following the formulas in Lindsay et al.(2014).For the
U-statistic the cutoff is determined empirically:Generate data from the considered normal distribution ;
Compute the test statistics for
BMonte Carlo(MC) replications;Compute the 95th quantile of the empirical distribution of the test statistic.
-
k-sample test:
Considerkrandom samples of i.i.d. observations\mathbf{x}^{(i)}_1, \mathbf{x}^{(i)}_{2},\ldots, \mathbf{x}^{(i)}_{n_i} \sim F_i,i = 1, \ldots, k. We test if the samples are generated from the same unknown distribution, that isH_0: F_1 = F_2 = \ldots = F_kversusH_1: F_i \not = F_j, for some1 \le i \not = j \le k.
We construct a matrix distance\hat{\mathbf{D}}, with off-diagonal elements\hat{D}_{ij} = \frac{1}{n_i n_j} \sum_{\ell=1}^{n_i} \sum_{r=1}^{n_j}K_{\bar{F}}(\mathbf{x}^{(i)}_\ell,\mathbf{x}^{(j)}_r), \qquad \mbox{ for }i \not= jand in the diagonal
\hat{D}_{ii} = \frac{1}{n_i (n_i -1)} \sum_{\ell=1}^{n_i} \sum_{r\not= \ell}^{n_i} K_{\bar{F}}(\mathbf{x}^{(i)}_\ell, \mathbf{x}^{(i)}_r), \qquad \mbox{ for }i = j,where
K_{\bar{F}}denotes the Normal kernelK_hcentered non-parametrically with respect to\bar{F} = \frac{n_1 \hat{F}_1 + \ldots + n_k \hat{F}_k}{n}, \quad \mbox{ with } n=\sum_{i=1}^k n_i.We compute the trace statistic
\mathrm{trace}(\hat{\mathbf{D}}_n) = \sum_{i=1}^{k}\hat{D}_{ii}and
D_n, derived considering all the possible pairwise comparisons in the k-sample null hypothesis, given asD_n = (k-1) \mathrm{trace}(\hat{\mathbf{D}}_n) - 2 \sum_{i=1}^{k}\sum_{j> i}^{k}\hat{D}_{ij}.We compute the empirical critical value by employing numerical techniques such as the bootstrap, permutation and subsampling algorithms:
Generate k-tuples, of total size
n_B, from the pooled sample following one of the sampling methods;Compute the k-sample test statistic;
Repeat
Btimes;Select the
95^{th}quantile of the obtained values.
-
Two-sample test:
Letx_1, x_2, ..., x_{n_1} \sim Fandy_1, y_2, ..., y_{n_2} \sim Gbe random samples from the distributionsFandG, respectively. We test the null hypothesis that the two samples are generated from the same unknown distribution, that isH_0: F=GvsH_1:F\not=G. The test statistics coincide with thek-sample test statistics whenk=2.
Kernel centering
The arguments mu_hat and Sigma_hat indicate the normal model
considered for the normality test, that is H_0: F = N(mu_hat,
Sigma_hat).
For the two-sample and k-sample tests, mu_hat and
Sigma_hat can
be used for the parametric centering of the kernel, in the case we want to
specify the reference distribution, with centeringType = "Param".
This is the default method when the test for normality is performed.
The normal kernel centered with respect to
G \sim N_d(\mathbf{\mu}, \mathbf{V}) can be computed as
K_{cen(G)}(\mathbf{s}, \mathbf{t}) =
K_{\mathbf{\Sigma_h}}(\mathbf{s}, \mathbf{t}) -
K_{\mathbf{\Sigma_h} + \mathbf{V}}(\mathbf{\mu}, \mathbf{t})
- K_{\mathbf{\Sigma_h} + \mathbf{V}}(\mathbf{s}, \mathbf{\mu}) +
K_{\mathbf{\Sigma_h} + 2\mathbf{V}}(\mathbf{\mu}, \mathbf{\mu}).
We consider the non-parametric centering of the kernel with respect to
\bar{F}=(n_1 F_1 + \ldots n_k F_k)/n where n=\sum_{i=1}^k n_i,
with centeringType = "Nonparam", for the two- and k-sample
tests.
Let \mathbf{z}_1,\ldots, \mathbf{z}_n denote the pooled sample. For any
s,t \in \{\mathbf{z}_1,\ldots, \mathbf{z}_n\}, it is given by
K_{cen(\bar{F})}(\mathbf{s},\mathbf{t}) = K(\mathbf{s},\mathbf{t}) -
\frac{1}{n}\sum_{i=1}^{n} K(\mathbf{s},\mathbf{z}_i) -
\frac{1}{n}\sum_{i=1}^{n} K(\mathbf{z}_i,\mathbf{t}) +
\frac{1}{n(n-1)}\sum_{i=1}^{n} \sum_{j \not=i}^{n}
K(\mathbf{z}_i,\mathbf{z}_j).
Value
An S4 object of class kb.test containing the results of the
kernel-based quadratic distance tests, based on the normal kernel. The object
contains the following slots:
-
method: Description of the kernel-based quadratic distance test performed. -
xData list of samples X (and Y). -
UnThe value of the U-statistic. -
H0_UnA logical value indicating whether or not the null hypothesis is rejected according to Un. -
CV_UnThe critical value computed for the test Un. -
VnThe value of the V-statistic (if available). -
H0_VnA logical value indicating whether or not the null hypothesis is rejected according to Vn (if available). -
CV_VnThe critical value computed for the test Vn (if available). -
hList with the value of bandwidth parameter used for the normal kernel function. Ifselect_his used, the matrix of computed power values and the corresponding power plot are also provided. -
BNumber of bootstrap/permutation/subsampling replications. -
var_Unexact variance of the kernel-based U-statistic. -
cv_methodThe method used to estimate the critical value (one of "subsampling", "permutation" or "bootstrap").
Note
For the two- and k-sample tests, the slots Vn, H0_Vn and
CV_Vn are empty, while the computed statistics are both reported in
slots Un, H0_Un and CV_Un.
A U-statistic is a type of statistic that is used to estimate a population parameter. It is based on the idea of averaging over all possible distinct combinations of a fixed size from a sample. A V-statistic considers all possible tuples of a certain size, not just distinct combinations and can be used in contexts where unbiasedness is not required.
References
Markatou, M. and Saraceno, G. (2024). “A Unified Framework for
Multivariate Two- and k-Sample Kernel-based Quadratic Distance
Goodness-of-Fit Tests.”
https://doi.org/10.48550/arXiv.2407.16374
Lindsay, B.G., Markatou, M. and Ray, S. (2014) "Kernels, Degrees of Freedom, and Power Properties of Quadratic Distance Goodness-of-Fit Tests", Journal of the American Statistical Association, 109:505, 395-410, DOI: 10.1080/01621459.2013.836972
See Also
kb.test for the class definition.
Examples
# create a kb.test object
x <- matrix(rnorm(100),ncol=2)
y <- matrix(rnorm(100),ncol=2)
# Normality test
my_test <- kb.test(x, h=0.5)
my_test
# Two-sample test
my_test <- kb.test(x,y,h=0.5, method="subsampling",b=0.9,
centeringType = "Nonparam")
my_test
# k-sample test
z <- matrix(rnorm(100,2),ncol=2)
dat <- rbind(x,y,z)
group <- rep(c(1,2,3),each=50)
my_test <- kb.test(x=dat,y=group,h=0.5, method="subsampling",b=0.9)
my_test
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