Package 'MRAM'

Estimate the Multivariate Regression Association Measure

Description

Compute $T_n$ and its standard error estimates using the nearest neighbor method and the $m$-out-of-$n$ bootstrap.

Usage

mram(
  y_data,
  x_data,
  z_data = NULL,
  bootstrap = FALSE,
  B = 1000,
  g_vec = seq(0.4, 0.9, by = 0.05)
)

Arguments

y_data	A $n \times d$ matrix of responses, where $n$ is the sample size.
x_data	A $n \times p$ matrix of predictors.
z_data	A $n \times q$ matrix of conditional predictors. The default value is NULL.
bootstrap	Perform the $m$-out-of-$n$ bootstrap if TRUE. The default value is FALSE.
B	Number of bootstrap replications. The default value is 1000.
g_vec	A vector of candidate values for $\gamma$ between 0 and 1, used to generate a collection of rules for the $m$-out-of-$n$ bootstrap. The default value is seq(0.4,0.9,by = 0.05).

Details

Let $\{({\bf X}_i,{\bf Y}_i,{\bf Z}_i)\}_{i = 1}^n$ be independent and identically distributed data from the population $({\bf X},{\bf Y},{\bf Z})$. The estimate $T_n({\bf X},{\bf Y})$ for the unconditional measure (z_data = NULL) is given as

$T_n({\bf X},{\bf Y}) = \binom{n}{2}^{-1} \sum_{i < j} \langle S({{\bf Y}_i - {\bf Y}_j}), S({{\bf Y}_{N(i)} - {\bf Y}_{N(j)}}) \rangle,$

where $\langle \cdot, \cdot \rangle$ is the dot product, $S(\cdot)$ is the spatial sign function, and $N(i)$ is the index $j$ such that ${\bf X}_j$ is the nearest neighbor of ${\bf X}_i$ according to the Euclidean distance. The estimate $T_n({\bf X},{\bf Y} \mid {\bf Z})$ for the conditional measure is given as

$T_n({\bf X},{\bf Y} \mid {\bf Z} ) = \frac{T_n(({\bf X},{\bf Z}),{\bf Y} ) - T_n({\bf Z},{\bf Y} )}{1 - T_n({\bf Z},{\bf Y} )}.$

See the paper Shih and Chen (2026) for more details.

For the $m$-out-of-$n$ bootstrap, the rule (resample size) is set to be $m = \lfloor n^\gamma \rfloor$, where $\lfloor x \rfloor$ denotes the largest integer that is smaller than or equal to $x$ and $0 < \gamma < 1$ takes values from the vector g_vec. It is recommended to use T_se_cluster, the standard error estimate obtained based on the cluster rule. See Dette and Kroll (2024) for more details.

The mram function is used in vs_mram function for variable selection.

Value

T_est	The estimate of the multivariate regression association measure. The value returned by T_est is between $-1$ and $1$. However, it is between $0$ and $1$ asymptotically. A small value indicates that x_data has low predictability for y_data condition on z_data in the sense of the considered measure. On the other hand, a large value indicates that x_data has high predictability for y_data condition on z_data. If z_data = NULL, the returned value indicates the unconditional predictability.
T_se_cluster	The standard error estimate based on the cluster rule.
m_vec	The vector of $m$ generated by g_vec.
T_se_vec	The vector of standard error estimates obtained from the $m$-out-of-$n$ bootstrap, where $m$ is equal to m_vec.
J_cluster	The index of the best m_vec chosen by the cluster rule.

References

Dette and Kroll (2024) A Simple Bootstrap for Chatterjee’s Rank Correlation, Biometrika, asae045.

Shih and Chen (2026) Measuring multivariate regression association via spatial sign, Computational Statistics & Data Analysis, 215, 108288.

See Also

vs_mram

Examples

library(MRAM)

n = 100

set.seed(1)
x_data = matrix(rnorm(n*2),n,2)
y_data = matrix(0,n,2)
y_data[,1] = x_data[,1]*x_data[,2]+x_data[,1]+rnorm(n)
y_data[,2] = x_data[,1]*x_data[,2]-x_data[,1]+rnorm(n)

mram(y_data,x_data[,1],x_data[,2])
mram(y_data,x_data[,2],x_data[,1])
mram(y_data,x_data[,1])
mram(y_data,x_data[,2])

## Not run: 

# perform the m-out-of-n bootstrap
mram(y_data,x_data[,1],x_data[,2],bootstrap = TRUE)
mram(y_data,x_data[,2],x_data[,1],bootstrap = TRUE)
mram(y_data,x_data[,1],bootstrap = TRUE)
mram(y_data,x_data[,2],bootstrap = TRUE)

## End(Not run)

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Variable Selection via the Multivariate Regression Association Measure

Description

Select a subset of $\bf X$ which can be used to predict $\bf Y$ based on $T_n$.

Usage

vs_mram(y_data, x_data)

Arguments

y_data	A $n \times d$ matrix of responses, where $n$ is the sample size.
x_data	A $n \times p$ matrix of predictors.

Details

vs_mram performs forward stepwise variable selection based on the multivariate regression association measure proposed in Shih and Chen (2026). At each step, it selects the predictor with the highest conditional predictability for the response given the previously selected predictors. The algorithm is modified from the FOCI algorithm from Azadkia and Chatterjee (2021).

Value

The vector containing the indices of the selected predictors in the order they were chosen.

References

Azadkia and Chatterjee (2021) A simple measure of conditional dependence, Annals of Statistics, 46(6): 3070-3102.

Shih and Chen (2026) Measuring multivariate regression association via spatial sign, Computational Statistics & Data Analysis, 215, 108288.

See Also

mram

Examples

library(MRAM)

n = 200
p = 10

set.seed(1)
x_data = matrix(rnorm(p*n),n,p)
colnames(x_data) = paste0(rep("X",p),seq(1,p))

y_data = x_data[,1]*x_data[,2]+x_data[,1]-x_data[,3]+rnorm(n)
colnames(x_data)[vs_mram(y_data,x_data)] # selected variables

## Not run: 

n = 500
p = 10

set.seed(1)
x_data = matrix(rnorm(p*n),n,p)
colnames(x_data) = paste0(rep("X",p),seq(1,p))

# Linear
y_data = matrix(0,n,2)
y_data[,1] = x_data[,1]*x_data[,2]+x_data[,1]-x_data[,3]+rnorm(n)
y_data[,2] = x_data[,2]*x_data[,4]+x_data[,2]-x_data[,5]+rnorm(n)
colnames(x_data)[vs_mram(y_data,x_data)] # selected variables

# Nonlinear
y_data = matrix(0,n,2)
y_data[,1] = x_data[,1]*x_data[,2]+sin(x_data[,1]*x_data[,3])+0.3*rnorm(n)
y_data[,2] = cos(x_data[,2]*x_data[,4])+x_data[,3]-x_data[,4]+0.3*rnorm(n)
colnames(x_data)[vs_mram(y_data,x_data)] # selected variables

# Non-additive error
y_data = matrix(0,n,2)
y_data[,1] = abs(x_data[,1]+runif(n))^(sin(x_data[,2])-cos(x_data[,3]))
y_data[,2] = abs(x_data[,2]-runif(n))^(sin(x_data[,3])-cos(x_data[,4]))
colnames(x_data)[vs_mram(y_data,x_data)] # selected variables

## End(Not run)

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