Package 'PSinference'

Canonical Empirical Distribution

Description

This function calculates the empirical distribution of the pivotal random variable that can be used to perform inferential procedures for the regression of one subset of variables on the other based on the released Single Synthetic data generated under Plug-in Sampling, assuming that the original dataset is normally distributed.

Usage

canodist(part, nsample, pvariates, iterations)

Arguments

part	Number of variables in the first subset.
nsample	Sample size.
pvariates	Number of variables.
iterations	Number of iterations for simulating values from the distribution and finding the quantiles. Default is 10000.

Details

We define

$T_4^\star|\boldsymbol{\Delta} = \frac{(|\boldsymbol{S}^{\star}_{12} (\boldsymbol{S}^{\star}_{22})^{-1}-\boldsymbol{\Delta}) \boldsymbol{S}^{\star}_{22}(\boldsymbol{S}^{\star}_{12}) (\boldsymbol{S}^{\star}_{22})^{-1}-\boldsymbol{\Delta})^\top|} {|\boldsymbol{S}^{\star}_{11.2}|}$

where $\boldsymbol{S}^\star = \sum_{i=1}^n (v_i - \bar{v})(v_i - \bar{v})^{\top}$, $v_i$ is the $i$th observation of the synthetic dataset, considering $\boldsymbol{S}^\star$ partitioned as

$\boldsymbol{S}^{\star}=\left[\begin{array}{lll} \boldsymbol{S}^{\star}_{11}& \boldsymbol{S}^{\star}_{12}\\ \boldsymbol{S}^{\star}_{21} & \boldsymbol{S}^{\star}_{22} \end{array}\right].$

For $\Delta = \boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}$, where $\boldsymbol{\Sigma}$ is partitioned the same way as $\boldsymbol{S}^{\star}$ its distribution is stochastic equivalent to

$\frac{|\boldsymbol{\Omega}_{12}\boldsymbol{\Omega}_{22}^{-1} \boldsymbol{\Omega}_{21}|}{|\boldsymbol{\Omega}_{11}-\boldsymbol{\Omega}_{12} \boldsymbol{\Omega}_{22}^{-1}\boldsymbol{\Omega}_{21}|}$

where $\boldsymbol{\Omega} \sim \mathcal{W}_p(n-1, \frac{\boldsymbol{W}}{n-1})$, $\boldsymbol{W} \sim \mathcal{W}_p(n-1, \mathbf{I}_p)$ and $\boldsymbol{\Omega}$ partitioned in the same way as $\boldsymbol{S}^{\star}$. To test $\mathcal{H}_0: \boldsymbol{\Delta} =\boldsymbol{\Delta}_0$, compute the value of $T_{4}^\star$, $\widetilde{T_{4}^\star}$, with the observed values and reject the null hypothesis if $\widetilde{T_{4}^\star}>t^\star_{4,1-\alpha}$ for $\alpha$-significance level, where $t^\star_{4,\gamma}$ is the $\gamma$th percentile of $T_4^\star$.

Value

a vector of length iterations that recorded the empirical distribution's values.

References

Klein, M., Moura, R. and Sinha, B. (2021). Multivariate Normal Inference based on Singly Imputed Synthetic Data under Plug-in Sampling. Sankhya B 83, 273–287.

Examples

# generate original data
library(MASS)
n_sample = 100
p = 4
mu <- c(1,2,3,4)
Sigma = matrix(c(1,   0.5, 0.1, 0.7,
                 0.5,   2, 0.4, 0.9,
                 0.1, 0.4,   3, 0.2,
                 0.7, 0.9, 0.2,   4), nr = 4, nc = 4, byrow = TRUE)

df = mvrnorm(n_sample, mu = mu, Sigma = Sigma)
# generate synthetic data
df_s = simSynthData(df)
#Decompose Sigma and Sstar
part = 2
Sigma_12 = partition(Sigma,nrows = part, ncol = part)[[2]]
Sigma_22 = partition(Sigma,nrows = part, ncol = part)[[4]]
Delta0 = Sigma_12 %*% solve(Sigma_22)

Sstar = cov(df_s)
Sstar_11 = partition(Sstar,nrows = part, ncol = part)[[1]]
Sstar_12 = partition(Sstar,nrows = part, ncol = part)[[2]]
Sstar_21 = partition(Sstar,nrows = part, ncol = part)[[3]]
Sstar_22 = partition(Sstar,nrows = part, ncol = part)[[4]]


DeltaEst = Sstar_12 %*% solve(Sstar_22)
Sstar11_2 = Sstar_11 - Sstar_12 %*% solve(Sstar_22) %*% Sstar_21


T4_obs = det((DeltaEst-Delta0)%*%Sstar_22%*%t(DeltaEst-Delta0))/det(Sstar11_2)

T4 <- canodist(part = part, nsample = n_sample, pvariates = p, iterations = 10000)
q95 <- quantile(T4, 0.95)

T4_obs > q95 #False means that we don't have statistical evidences to reject Delta0
print(T4_obs)
print(q95)
# When the observed value is smaller than the 95% quantile,
# we don't have statistical evidences to reject the Sphericity property.
#
# Note that the value is very close to zero

---

Generalized Variance Empirical Distribution

Description

This function calculates the empirical distribution of the pivotal random variable that can be used to perform inferential procedures for the Generalized Variance of the released Single Synthetic dataset generated under Plug-in Sampling, assuming that the original distribution is normally distributed.

Usage

GVdist(nsample, pvariates, iterations = 10000)

Arguments

nsample	Sample size.
pvariates	Number of variables.
iterations	Number of iterations for simulating values from the distribution and finding the quantiles. Default is 10000.

Details

We define

$T_1^\star = (n-1)\frac{|\boldsymbol{S}^*|}{|\boldsymbol{\Sigma}|},$

where $\boldsymbol{S}^\star = \sum_{i=1}^n (v_i - \bar{v})(v_i - \bar{v})^{\top}$, $\boldsymbol{\Sigma}$ is the population covariance matrix and $v_i$ is the $i$th observation of the synthetic dataset. Its distribution is stochastic equivalent to

$\prod_{i=1}^n \chi_{n-i}^2 \prod_{i=1}^p \chi_{n-i}^2$

where $\chi_{n-i}^2$ are all independent chi-square random variables. The $(1-\alpha)$ level confidence interval for $|\boldsymbol{\Sigma}|$ is given by

$\left(\frac{(n-1)^p|\tilde{\boldsymbol{S}}^\star|}{t^\star_{1,1-\alpha/2}}, \frac{(n-1)^p|\tilde{\boldsymbol{S}}^\star|}{t^\star_{1,\alpha/2}} \right)$

where $\tilde{\boldsymbol{S}}^\star$ is the observed value of $\boldsymbol{S}^\star$ and $t^\star_{1,\gamma}$ is the $\gamma$th percentile of $T_1$.

Value

a vector of length iterations that recorded the empirical distribution's values.

References

Klein, M., Moura, R. and Sinha, B. (2021). Multivariate Normal Inference based on Singly Imputed Synthetic Data under Plug-in Sampling. Sankhya B 83, 273–287.

Examples

# Original data creation
library(MASS)
mu <- c(1,2,3,4)
Sigma <- matrix(c(1, 0.5, 0.5, 0.5,
                  0.5, 1, 0.5, 0.5,
                  0.5, 0.5, 1, 0.5,
                  0.5, 0.5, 0.5, 1), nrow = 4, ncol = 4, byrow = TRUE)
seed = 1
n_sample = 100
# Create original simulated dataset
df = mvrnorm(n_sample, mu = mu, Sigma = Sigma)

# Synthetic data created

df_s = simSynthData(df)


# Gather the 0.025 and 0.975 quantiles and construct confident interval for sigma^2
# Check that sigma^2 is inside in both cases
p = dim(df_s)[2]

T <- GVdist(100, p, 10000)
q975 <- quantile(T, 0.975)
q025 <- quantile(T, 0.025)

left <- (n_sample-1)^p * det(cov(df_s)*(n_sample-1))/q975
right <- (n_sample-1)^p * det(cov(df_s)*(n_sample-1))/q025

cat(left,right,'\n')
print(det(Sigma))
# The synthetic value is inside the confidence interval of GV

---

Independence Empirical Distribution

Description

This function calculates the empirical distribution of the pivotal random variable that can be used to perform inferential procedures and test the independence of two subsets of variables based on the released Single Synthetic data generated under Plug-in Sampling, assuming that the original dataset is normally distributed.

Usage

Inddist(part, nsample, pvariates, iterations)

Arguments

part	Number of variables in the first subset.
nsample	Sample size.
pvariates	Number of variables.
iterations	Number of iterations for simulating values from the distribution and finding the quantiles. Default is 10000.

Details

We define

$T_3^\star = \frac{|\boldsymbol{S}^{\star}|} {|\boldsymbol{S}^{\star}_{11}||\boldsymbol{S}^{\star}_{22}|}$

where $\boldsymbol{S}^\star = \sum_{i=1}^n (v_i - \bar{v})(v_i - \bar{v})^{\top}$, $v_i$ is the $i$th observation of the synthetic dataset, considering $\boldsymbol{S}^\star$ partitioned as

$\boldsymbol{S}^{\star}=\left[\begin{array}{lll} \boldsymbol{S}^{\star}_{11}& \boldsymbol{S}^{\star}_{12}\\ \boldsymbol{S}^{\star}_{21} & \boldsymbol{S}^{\star}_{22} \end{array}\right].$

Under the assumption that $\boldsymbol{\Sigma}_{12} = \boldsymbol{0}$, its distribution is stochastic equivalent to

$\frac{|\boldsymbol{\Omega}|}{|\boldsymbol{\Omega}_{11}||\boldsymbol{\Omega}_{22}|}$

where $\boldsymbol{\Omega} \sim \mathcal{W}_p(n-1, \frac{\boldsymbol{W}}{n-1})$, $\boldsymbol{W} \sim \mathcal{W}_p(n-1, \mathbf{I}_p)$ and $\boldsymbol{\Omega}$ partitioned in the same way as $\boldsymbol{S}^{\star}$. To test $\mathcal{H}_0: \boldsymbol{\Sigma}_{12} = \boldsymbol{0}$, compute the value of $T_{3}^\star$, $\widetilde{T_{3}^\star}$, with the observed values and reject the null hypothesis if $\widetilde{T_{3}^\star}<t^\star_{3,\alpha}$ for $\alpha$-significance level, where $t^\star_{3,\gamma}$ is the $\gamma$th percentile of $T_3^\star$.

Value

a vector of length iterations that recorded the empirical distribution's values.

References

Klein, M., Moura, R. and Sinha, B. (2021). Multivariate Normal Inference based on Singly Imputed Synthetic Data under Plug-in Sampling. Sankhya B 83, 273–287.

Examples

#generate original data with two independent subsets of variables
library(MASS)
n_sample = 100
p = 4
mu <- c(1,2,3,4)
Sigma = matrix(c(1,   0.5,   0,     0,
                 0.5,   2,   0,     0,
                 0,     0,   3,   0.2,
                 0,     0,   0.2,   4), nr = 4, nc = 4, byrow = TRUE)
df = mvrnorm(n_sample, mu = mu, Sigma = Sigma)
# generate synthetic data
df_s = simSynthData(df)

#Decompose Sstar in 4 parts
part = 2

Sstar = cov(df_s)
Sstar_11 = partition(Sstar,nrows = part, ncol = part)[[1]]
Sstar_12 = partition(Sstar,nrows = part, ncol = part)[[2]]
Sstar_21 = partition(Sstar,nrows = part, ncol = part)[[3]]
Sstar_22 = partition(Sstar,nrows = part, ncol = part)[[4]]

#Compute observed T3_star
T3_obs = det(Sstar)/(det(Sstar_11)*det(Sstar_22))

alpha = 0.05

# colect the quantile from the distribution assuming independence between the two subsets
T3 <- Inddist(part = part, nsample = n_sample, pvariates = p, iterations = 10000)
q5 <- quantile(T3, alpha)

T3_obs < q5 #False means that we don't have statistical evidences to reject independence
print(T3_obs)
print(q5)
# Note that the value of the observed T3_obs is close to one as expected

---

Split a matrix into blocks

Description

This function split a matrix into a list of blocks (either by rows and columns).

Usage

partition(Matrix, nrows, ncols)

Arguments

Matrix	a matrix to split .
nrows	positive integer indicating the number of rows blocks.
ncols	positive integer indicating the number of columns blocks.

Value

a list of partitioned submatrices

Examples

Mat = matrix(c(1,0.5,0,0,
                  0.5,2,0,0,
                  0,0,3,0.2,
                  0, 0, 0.2,4), nrow = 4, ncol = 4, byrow = TRUE)
partition(Matrix = Mat, nrows = 2, ncols = 2)

---

Plug-in Sampling Single Synthetic Dataset Generation

Description

This function is used to generate a single synthetic version of the original data via Plug-in Sampling.

Usage

simSynthData(X, n_imp = dim(X)[1])

Arguments

X	matrix or dataframe
n_imp	sample size

Details

Assume that $\mathbf{X}=\left(\mathbf{x}_1, \dots, \mathbf{x}_n\right)$ is the original data, assumed to be normally distributed, we compute $\bar{\mathbf{x}}$ as the sample mean and $\hat{\boldsymbol{\Sigma}}=\mathbf{S}/(n-1)$ as the sample covariance matrix, where $\mathbf{S}$ is the sample Wishart matrix. We generate $\mathbf{V}=\left(\mathbf{v}_1, \dots, \mathbf{v}_n\right)$, by drawing

$\mathbf{v}_i\stackrel{i.i.d.}{\sim}N_p(\bar{\mathbf{x}},\hat{\boldsymbol{\Sigma}}).$

Value

a matrix of generated dataset

References

Klein, M., Moura, R. and Sinha, B. (2021). Multivariate Normal Inference based on Singly Imputed Synthetic Data under Plug-in Sampling. Sankhya B 83, 273–287.

Examples

library(MASS)
n_sample = 1000
mu=c(0,0,0,0)
Sigma=diag(1,4,4)
# Create original simulated dataset
df_o = mvrnorm(n_sample, mu, Sigma)
# Create singly imputed synthetic dataset
df_s = simSynthData(df_o)
#Estimators synthetic
mean_s <- colMeans(df_s)
S_s <- (t(df_s)- mean_s) %*% t(t(df_s)- mean_s)
# careful about this computation
# mean_o is a column vector and if you are thinking as n X p matrices and
# row vectors you should be aware of this.
print(mean_s)
print(S_s/(dim(df_s)[1]-1))

---

Spherical Empirical Distribution

Description

This function calculates the empirical distribution of the pivotal random variable that can be used to perform the Sphericity test of the population covariance matrix $\boldsymbol{\Sigma}$ that is $\boldsymbol{\Sigma} = \sigma^2 \mathbf{I}_p$, based on the released Single Synthetic data generated under Plug-in Sampling, assuming that the original dataset is normally distributed.

Usage

Sphdist(nsample, pvariates, iterations)

Arguments

nsample	Sample size.
pvariates	Number of variables.
iterations	Number of iterations for simulating values from the distribution and finding the quantiles. Default is 10000.

Details

We define

$T_2^\star = \frac{|\boldsymbol{S}^{\star}|^{\frac{1}{p}}}{tr(\boldsymbol{S}^{\star})/p}$

where $\boldsymbol{S}^\star = \sum_{i=1}^n (v_i - \bar{v})(v_i - \bar{v})^{\top}$, $v_i$ is the $i$th observation of the synthetic dataset. For $\boldsymbol{\Sigma} = \sigma^2 \mathbf{I}_p$, its distribution is stochastic equivalent to

$\frac{|\boldsymbol{\Omega}_{1}\boldsymbol{\Omega}_{2}|^{\frac{1}{p}}}{tr(\boldsymbol{\Omega}_{1}\boldsymbol{\Omega}_{2})/p}$

where $\boldsymbol{\Omega}_1$ and $\boldsymbol{\Omega}_2$ are Wishart random variables, $\boldsymbol{\Omega}_1 \sim \mathcal{W}_p(n-1, \frac{\mathbf{I}_p}{n-1})$ is independent of $\boldsymbol{\Omega}_2 \sim \mathcal{W}_p(n-1, \mathbf{I}_p)$. To test $\mathcal{H}_0: \boldsymbol{\Sigma} = \sigma^2 \mathbf{I}_p$, compute the observed value of $T_{2}^\star$, $\widetilde{T_{2}^\star}$, with the observed values and reject the null hypothesis if $\widetilde{T_{2}^\star}>t^\star_{2,\alpha}$ for $\alpha$-significance level, where $t^\star_{2,\gamma}$ is the $\gamma$th percentile of $T_2^\star$.

Value

a vector of length iterations that recorded the empirical distribution's values.

References

Klein, M., Moura, R. and Sinha, B. (2021). Multivariate Normal Inference based on Singly Imputed Synthetic Data under Plug-in Sampling. Sankhya B 83, 273–287.

Examples

# Original data created
library(MASS)
mu <- c(1,2,3,4)
Sigma <- matrix(c(1, 0, 0, 0,
                  0, 1, 0, 0,
                  0, 0, 1, 0,
                  0, 0, 0, 1), nrow = 4, ncol = 4, byrow = TRUE)
seed = 1
n_sample = 100
# Create original simulated dataset
df = mvrnorm(n_sample, mu = mu, Sigma = Sigma)

# Synthetic data created

df_s = simSynthData(df)


# Gather the 0.95 quantile

p = dim(df_s)[2]

T_sph <- Sphdist(nsample = n_sample, pvariates = p, iterations = 10000)
q95 <- quantile(T_sph, 0.95)

# Compute the observed value of T from the synthetic dataset
S_star = cov(df_s*(n_sample-1))

T_obs = (det(S_star)^(1/p))/(sum(diag(S_star))/p)

print(q95)
print(T_obs)

#Since the observed value is bigger than the 95% quantile,
#we don't have statistical evidences to reject the Sphericity property.
#
#Note that the value is very close to one

---