Package 'KPC'

2017 Korea presidential election data

Description

A dataset containing 9 variables, consists of the voting results earned by the top five candidates from 250 electoral districts in Korea.

Usage

ElecData

Format

A data frame with 1250 rows and 9 variables:

PrecinctCode

250 precinct codes designated by the election committee (4 digits)

CityCode

250 city codes of administrative standard code management system (5 digits)

CandidateName

Symbols 1-5, corresponding to Moon Jae-in, Hong Jun-pyo, Ahn Cheol-soo, Yoo Seung-min, Shim Sang-jung

AveAge

Average age of voters in 17 years: statistics on resident registration population of the Ministry of Government Administration and Home Affairs

AveYearEdu

Average number of years of education for voters

AveHousePrice

Average price per square meter in 17 years

AveInsurance

The average insurance premium for each city, county, district

VoteRate

Vote rate by candidate

NumVote

Number of votes by candidate

Source

https://github.com/OhmyNews/2017-Election

---

Kernel Feature Ordering by Conditional Independence

Description

Variable selection with KPC using directed K-NN graph or minimum spanning tree (MST)

Usage

KFOCI(
  Y,
  X,
  k = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)),
  Knn = min(ceiling(NROW(Y)/20), 20),
  num_features = NULL,
  stop = TRUE,
  numCores = parallel::detectCores(),
  verbose = FALSE
)

Arguments

Y	a matrix of responses (n by dy)
X	a matrix of predictors (n by dx)
k	a function $k(y, y')$ of class kernel. It can be the kernel implemented in kernlab e.g., Gaussian kernel: rbfdot(sigma = 1), linear kernel: vanilladot().
Knn	a positive integer indicating the number of nearest neighbor; or "MST". The suggested choice of Knn is 0.05n for samples up to a few hundred observations. For large n, the suggested Knn is sublinear in n. That is, it may grow slower than any linear function of n. The computing time is approximately linear in Knn. A smaller Knn takes less time.
num_features	the number of variables to be selected, cannot be larger than dx. The default value is NULL and in that case it will be set equal to dx. If stop == TRUE (see below), then num_features is the maximal number of variables to be selected.
stop	If stop == TRUE, then the automatic stopping criterion (stops at the first instance of negative Tn, as mentioned in the paper) will be implemented and continued till num_features many variables are selected. If stop == FALSE then exactly num_features many variables are selected.
numCores	number of cores that are going to be used for parallelizing the process.
verbose	whether to print each selected variables during the forward stepwise algorithm

Details

A stepwise forward selection of variables using KPC. At each step it selects the $X_j$ that maximizes $\hat{\rho^2}(Y,X_j |$selected $X_i)$. It is suggested to normalize the predictors before applying KFOCI. Euclidean distance is used for computing the K-NN graph and the MST.

Value

The algorithm returns a vector of the indices from 1,...,dx of the selected variables in the same order that they were selected. The variables at the front are expected to be more informative in predicting Y.

See Also

KPCgraph, KPCRKHS, KPCRKHS_VS

Examples

n = 200
p = 10
X = matrix(rnorm(n * p), ncol = p)
Y = X[, 1] * X[, 2] + sin(X[, 1] * X[, 3])
KFOCI(Y, X, kernlab::rbfdot(1), Knn=1, numCores=1)
# 1 2 3

---

A near linear time analogue of KMAc

Description

Calculate $\hat{\eta}_n^{\mbox{lin}}$ (the unconditional version of graph-based KPC) using directed K-NN graph or minimum spanning tree (MST). The computational complexity is O(nlog(n))

Usage

Klin(
  Y,
  X,
  k = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)),
  Knn = 1
)

Arguments

Y	a matrix of response (n by dy)
X	a matrix of predictors (n by dx)
k	a function $k(y, y')$ of class kernel. It can be the kernel implemented in kernlab e.g. rbfdot(sigma = 1), vanilladot()
Knn	the number of K-nearest neighbor to use; or "MST". A small Knn (e.g., Knn=1) is recommended.

Details

$\hat{\eta}_n$ is an estimate of the population kernel measure of association, based on data $\{(X_i,Y_i)\}_{i=1}^n$ from $\mu$. For K-NN graph, $\hat{\eta}_n$ can be computed in near linear time (in $n$). In particular,

$\hat{\eta}_n^{\mbox{lin}}:=\frac{n^{-1}\sum_{i=1}^n d_i^{-1}\sum_{j:(i,j)\in\mathcal{E}(G_n)} k(Y_i,Y_j)-(n-1)^{-1}\sum_{i=1}^{n-1} k(Y_i,Y_{i+1})}{n^{-1}\sum_{i=1}^n k(Y_i,Y_i)-(n-1)^{-1}\sum_{i=1}^{n-1} k(Y_i,Y_{i+1})}$

, where all symbols have their usual meanings as in the definition of $\hat{\eta}_n$. Euclidean distance is used for computing the K-NN graph and the MST.

Value

The algorithm returns a real number ‘Klin’: an empirical kernel measure of association which can be computed in near linear time when K-NN graphs are used.

References

Deb, N., P. Ghosal, and B. Sen (2020), “Measuring association on topological spaces using kernels and geometric graphs” <arXiv:2010.01768>.

See Also

KPCgraph, KMAc

Examples

library(kernlab)
Klin(Y = rnorm(100), X = rnorm(100), k = rbfdot(1), Knn = 1)

---

KMAc (the unconditional version of graph-based KPC) with geometric graphs.

Description

Calculate $\hat{\eta}_n$ (the unconditional version of graph-based KPC) using directed K-NN graph or minimum spanning tree (MST).

Usage

KMAc(
  Y,
  X,
  k = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)),
  Knn = 1
)

Arguments

Y	a matrix of response (n by dy)
X	a matrix of predictors (n by dx)
k	a function $k(y, y')$ of class kernel. It can be the kernel implemented in kernlab e.g., Gaussian kernel: rbfdot(sigma = 1), linear kernel: vanilladot()
Knn	the number of K-nearest neighbor to use; or "MST". A small Knn (e.g., Knn=1) is recommended for an accurate estimate of the population KMAc.

Details

$\hat{\eta}_n$ is an estimate of the population kernel measure of association, based on data $\{(X_i,Y_i)\}_{i=1}^n$ from $\mu$. For K-NN graph, ties will be broken at random. MST is found using package emstreeR. In particular,

$\hat{\eta}_n:=\frac{n^{-1}\sum_{i=1}^n d_i^{-1}\sum_{j:(i,j)\in\mathcal{E}(G_n)} k(Y_i,Y_j)-(n(n-1))^{-1}\sum_{i\neq j}k(Y_i,Y_j)}{n^{-1}\sum_{i=1}^n k(Y_i,Y_i)-(n(n-1))^{-1}\sum_{i\neq j}k(Y_i,Y_j)},$

where $G_n$ denotes a MST or K-NN graph on $X_1,\ldots , X_n$, $\mathcal{E}(G_n)$ denotes the set of edges of $G_n$ and $(i,j)\in\mathcal{E}(G_n)$ implies that there is an edge from $X_i$ to $X_j$ in $G_n$. Euclidean distance is used for computing the K-NN graph and the MST.

Value

The algorithm returns a real number ‘KMAc’, the empirical kernel measure of association

References

Deb, N., P. Ghosal, and B. Sen (2020), “Measuring association on topological spaces using kernels and geometric graphs” <arXiv:2010.01768>.

See Also

KPCgraph, Klin

Examples

library(kernlab)
KMAc(Y = rnorm(100), X = rnorm(100), k = rbfdot(1), Knn = 1)

---

Kernel partial correlation with geometric graphs

Description

Calculate the kernel partial correlation (KPC) coefficient with directed K-nearest neighbor (K-NN) graph or minimum spanning tree (MST).

Usage

KPCgraph(
  Y,
  X,
  Z,
  k = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)),
  Knn = 1,
  trans_inv = FALSE
)

Arguments

Y	a matrix (n by dy)
X	a matrix (n by dx) or NULL if $X$ is empty
Z	a matrix (n by dz)
k	a function $k(y, y')$ of class kernel. It can be the kernel implemented in kernlab e.g., Gaussian kernel: rbfdot(sigma = 1), linear kernel: vanilladot().
Knn	a positive integer indicating the number of nearest neighbor to use; or "MST". A small Knn (e.g., Knn=1) is recommended for an accurate estimate of the population KPC.
trans_inv	TRUE or FALSE. Is $k(y, y)$ free of $y$?

Details

The kernel partial correlation squared (KPC) measures the conditional dependence between $Y$ and $Z$ given $X$, based on an i.i.d. sample of $(Y, Z, X)$. It converges to the population quantity (depending on the kernel) which is between 0 and 1. A small value indicates low conditional dependence between $Y$ and $Z$ given $X$, and a large value indicates stronger conditional dependence. If X == NULL, it returns the KMAc(Y,Z,k,Knn), which measures the unconditional dependence between $Y$ and $Z$. Euclidean distance is used for computing the K-NN graph and the MST. MST in practice often achieves similar performance as the 2-NN graph. A small K is recommended for the K-NN graph for an accurate estimate of the population KPC, while if KPC is used as a test statistic for conditional independence, a larger K can be beneficial.

Value

The algorithm returns a real number which is the estimated KPC.

See Also

KPCRKHS, KMAc, Klin

Examples

library(kernlab)
n = 2000
x = rnorm(n)
z = rnorm(n)
y = x + z + rnorm(n,1,1)
KPCgraph(y,x,z,vanilladot(),Knn=1,trans_inv=FALSE)

n = 1000
x = runif(n)
z = runif(n)
y = (x + z) %% 1
KPCgraph(y,x,z,rbfdot(5),Knn="MST",trans_inv=TRUE)

discrete_ker = function(y1,y2) {
    if (y1 == y2) return(1)
    return(0)
}
class(discrete_ker) <- "kernel"
set.seed(1)
n = 2000
x = rnorm(n)
z = rnorm(n)
y = rep(0,n)
for (i in 1:n) y[i] = sample(c(1,0),1,prob = c(exp(-z[i]^2/2),1-exp(-z[i]^2/2)))
KPCgraph(y,x,z,discrete_ker,1)
##0.330413

---

Kernel partial correlation with RKHS method

Description

Compute estimate of Kernel partial correlation (KPC) coefficient using conditional mean embeddings in the reproducing kernel Hilbert spaces (RKHS).

Usage

KPCRKHS(
  Y,
  X = NULL,
  Z,
  ky = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)),
  kx = kernlab::rbfdot(1/(2 * stats::median(stats::dist(X))^2)),
  kxz = kernlab::rbfdot(1/(2 * stats::median(stats::dist(cbind(X, Z)))^2)),
  eps = 0.001,
  appro = FALSE,
  tol = 1e-05
)

Arguments

Y	a matrix (n by dy)
X	a matrix (n by dx) or NULL if $X$ is empty
Z	a matrix (n by dz)
ky	a function $k(y, y')$ of class kernel. It can be the kernel implemented in kernlab e.g., Gaussian kernel: rbfdot(sigma = 1), linear kernel: vanilladot().
kx	the kernel function for $X$
kxz	the kernel function for $(X, Z)$ or for $Z$ if $X$ is empty
eps	a small positive regularization parameter for inverting the empirical cross-covariance operator
appro	whether to use incomplete Cholesky decomposition for approximation
tol	tolerance used for incomplete Cholesky decomposition (implemented by the function inchol in the package kernlab)

Details

The kernel partial correlation (KPC) coefficient measures the conditional dependence between $Y$ and $Z$ given $X$, based on an i.i.d. sample of $(Y, Z, X)$. It converges to the population quantity (depending on the kernel) which is between 0 and 1. A small value indicates low conditional dependence between $Y$ and $Z$ given $X$, and a large value indicates stronger conditional dependence. If X = NULL, it measures the unconditional dependence between $Y$ and $Z$.

Value

The algorithm returns a real number which is the estimated KPC.

See Also

KPCgraph

Examples

n = 500
set.seed(1)
x = rnorm(n)
z = rnorm(n)
y = x + z + rnorm(n,1,1)
library(kernlab)
k = vanilladot()
KPCRKHS(y, x, z, k, k, k, 1e-3/n^(0.4), appro = FALSE)
# 0.4854383 (Population quantity = 0.5)
KPCRKHS(y, x, z, k, k, k, 1e-3/n^(0.4), appro = TRUE, tol = 1e-5)
# 0.4854383 (Population quantity = 0.5)

---

Variable selection with RKHS estimator

Description

The algorithm performs a forward stepwise variable selection using RKHS estimators.

Usage

KPCRKHS_VS(
  Y,
  X,
  num_features,
  ky = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)),
  kS = NULL,
  eps = 0.001,
  appro = FALSE,
  tol = 1e-05,
  numCores = parallel::detectCores(),
  verbose = FALSE
)

Arguments

Y	a matrix of responses (n by dy)
X	a matrix of predictors (n by dx)
num_features	the number of variables to be selected, cannot be larger than dx.
ky	a function $k(y, y')$ of class kernel. It can be the kernel implemented in kernlab e.g., Gaussian kernel: rbfdot(sigma = 1), linear kernel: vanilladot()
kS	a function that takes X and a subset of indices S as inputs, and then outputs the kernel for X_S. The first argument of kS is X, and the second argument is a vector of positive integer. If kS == NULL, Gaussian kernel with empitical bandwidth kernlab::rbfdot(1/(2*stats::median(stats::dist(X[,S]))^2)) will be used.
eps	a positive number; the regularization parameter for the RKHS estimator
appro	whether to use incomplete Cholesky decomposition for approximation
tol	tolerance used for incomplete Cholesky decomposition (inchol in package kernlab)
numCores	number of cores that are going to be used for parallelizing the process.
verbose	whether to print each selected variables during the forward stepwise algorithm

Details

A stepwise forward selection of variables using KPC. At each step it selects the $X_j$ that maximizes $\tilde{\rho^2}(Y,X_j |$selected $X_i)$. It is suggested to normalize the features before applying the algorithm.

Value

The algorithm returns a vector of the indices from 1,...,dx of the selected variables in the same order that they were selected. The variables at the front are expected to be more informative in predicting Y.

See Also

KPCgraph, KPCRKHS, KFOCI

Examples

n = 200
p = 10
X = matrix(rnorm(n * p), ncol = p)
Y = X[, 1] * X[, 2] + sin(X[, 1] * X[, 3])
library(kernlab)
kS = function(X,S) return(rbfdot(1/length(S)))
KPCRKHS_VS(Y, X, num_features = 3, rbfdot(1), kS, eps = 1e-3, appro = FALSE, numCores = 1)
kS = function(X,S) return(rbfdot(1/(2*stats::median(stats::dist(X[,S]))^2)))
KPCRKHS_VS(Y, X, num_features = 3, rbfdot(1), kS, eps = 1e-3, appro = FALSE, numCores = 1)

---

Medical data from 35 patients

Description

A dataset containing three variables (creatinine clearance C; digoxin clearance D; urine flow U) from 35 patients.

Usage

med

Format

A data frame with 35 rows and 3 variables:

C

creatinine clearance, in ml/min/1.73m^2

D

digoxin clearance, in ml/min/1.73m^2

U

urine flow, in ml/min

Source

Edwards, D. (2012). Introduction to graphical modelling, Section 3.1.4, Springer Science & Business Media.

---

Tn with geometric graphs

Description

Calculate $T_n$ using directed K-NN graph or minimum spanning tree (MST).

Usage

TnKnn(Y, X, k, Knn = 1)

Arguments

Y	a matrix of response (n by dy)
X	a matrix of predictors (n by dx)
k	a function $k(y, y')$ of class kernel. It can be the kernel implemented in kernlab e.g. Gaussian kernel: rbfdot(sigma = 1), linear kernel: vanilladot().
Knn	the number of K-nearest neighbor to use; or "MST".

Details

$T_n$ is an estimate of $E[E[k(Y_1,Y_1')|X]]$, with $Y_1$, $Y_1'$ drawn iid from $Y|X$, given $X$. For K-NN graph, ties will be broken at random. Algorithm finding the MST is implemented the package emstreeR.

Value

The algorithm returns a real number which is the value of Tn.

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