Package 'KSPM' reference manual

Title:	Kernel Semi-Parametric Models
Description:	To fit the kernel semi-parametric model and its extensions. It allows multiple kernels and unlimited interactions in the same model. Coefficients are estimated by maximizing a penalized log-likelihood; penalization terms and hyperparameters are estimated by minimizing leave-one-out error. It includes predictions with confidence/prediction intervals, statistical tests for the significance of each kernel, a procedure for variable selection and graphical tools for diagnostics and interpretation of covariate effects. Currently it is implemented for continuous dependent variables. The package is based on the paper of Liu et al. (2007), <doi:10.1111/j.1541-0420.2007.00799.x>.
Authors:	Catherine Schramm [aut, cre], Aurelie Labbe [ctb], Celia M. T. Greenwood [ctb]
Maintainer:	Catherine Schramm <[email protected]>
License:	GPL-3
Version:	0.2.1
Built:	2024-11-24 03:54:28 UTC
Source:	https://github.com/cran/KSPM

Case names of fitted models

Description

Simple utility returning names of cases involved in a kernel semi parametric model.

Usage

## S3 method for class 'kspm'
case.names(object, ...)
## S3 method for class 'kspm'
case.names(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`...`	additional optional argument (currently unused).

Value

a character vector.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Extract Model Coefficients

Description

Returns linear and kernel coefficients for a model of class "kspm".

Usage

## S3 method for class 'kspm'
coef(object, ...)
## S3 method for class 'kspm'
coef(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`...`	additional optional argument (currently unused).

Value

Two matrices of coefficients.

`linear`	A vector of coefficients for linear part. One row is one variable.
`kernel`	A matrix of coefficients for linear part. One row is one subject, one column is one kernel part.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
coef(fit)

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
coef(fit)

Confidence interavls for linear part of model parameters

Description

Computes confidence intervals for one or more parameters in the linear part of a fitted model of class "kspm".

Usage

## S3 method for class 'kspm'
confint(object, parm = NULL, level = 0.95, ...)
## S3 method for class 'kspm'
confint(object, parm = NULL, level = 0.95, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`parm`	a vector of names specifying which parameters are to be given confidence intervals. If missing, all parameters are considered.
`level`	the confidence level required. By default 0.95.
`...`	additional optional argument (currently unused).

Details

For objects of class "kspm", the confidence interval is based on student distribution and effective degree of freedom of the model.

Value

A matrix with column giving lower and upper confidence limits for each parameter. These are labelled as $\frac{1-level}{2}$ and $1 - \frac{1-level}{2}$ in percentage.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
confint(fit)

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
confint(fit)

Cook's distance for a Kernel Semi Parametric Model Fit

Description

Computes the Cook's distance method for an object of class "kspm".

Usage

## S3 method for class 'kspm'
cooks.distance(model, ...)
## S3 method for class 'kspm'
cooks.distance(model, ...)

Arguments

`model`	an model of class "kspm", usually, a result of a call to `kspm`.
`...`	furter arguments passed to or from other methods (currently unused).

Details

Cook's distance values ( $C_i$ ) are computed as follows: $C_i = \frac{e_i^2 h_{ii}}{\hat{\sigma}^2 tr(H) (1-h_{ii})^2}$ where e_i is the residual of subject i, h_ii is the i th diagonal element of Hat matrix H corresponding to the leverage associated with subject i and tr(H) is the trace of the Hat matrix H.

Value

A vector containing Cook's distance values.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Conventional and Social media features of 187 movies.

Description

A dataset containing the ratings and other attributes of 187 movies.

Usage

csm
csm

Format

A data frame with 187 rows and 13 variables:

Year: year at which movies were projected on the screens
Ratings: ratings
Genre: genre of the movie
Gross: gross income in USD
Budget: budget in USD
Screens: number of screens in USA
Sequel: sequel
Sentiment: sentiment score
Views: number of views of movie trailer on Youtube
Likes: number of likes of movie trailer on Youtube
Dislikes: number of dislikes of movie trailer on Youtube
Comments: number of comments of movie trailer on Youtube
Aggregate.Followers: aggregate actor followers on Twitter

Source

https://archive.ics.uci.edu/ml/index.php

References

AHMED, Mehreen, JAHANGIR, Maham, AFZAL, Hammad, et al. Using Crowd-source based features from social media and Conventional features to predict the movies popularity. In : Smart City/SocialCom/SustainCom (SmartCity), 2015 IEEE International Conference on. IEEE, 2015. p. 273-278.

Computing kernel function derivatives

Description

derivatives is a function for "kspm" object computing pointwise partial derivatives of $h(Z)$ accroding to each $Z$ variable.

Usage

derivatives(object)
derivatives(object)

Arguments

object

an object of class "kspm", usually, a result of a call to kspm.

Details

derivatives are not computed for interactions. If a variable is included in several kernels, the user may obtain the corresponding pointwise derivatives by summing the pointwise derivatives associated with each kernel.

Value

an object of class 'derivatives'

`derivmat`	a list of $n \times d$ matrix (one for each kernel) where $n$ is the number of subjects and $d$ the number of variables included in the kernel
`rawmat`	a $n \times q$ matrix with all variables included in the kernel part of the model $q$ the number of variables included in the whole kernel part
`scalemat`	scaled version of rawmat
`modelmat`	matrix of correspondance between variable and kernels

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Kim, Choongrak, Byeong U. Park, and Woochul Kim. "Influence diagnostics in semiparametric regression models." Statistics and probability letters 60.1 (2002): 49:58.

Model deviance

Description

Returns the deviance of a fitted model object of class "kspm".

Usage

## S3 method for class 'kspm'
deviance(object, ...)
## S3 method for class 'kspm'
deviance(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`, for which the deviance is desired.
`...`	additional optional argument (currently unused).

Details

This function extracts deviance of a model fitted using kspm function. The returned deviance is the residual sum of square (RSS).

Value

The value of the deviance extracted from the object object.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Examples

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
deviance(fit)

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
deviance(fit)

Energy consumption measuring hourly during 22 days

Description

A dataset containing the energy consumption and other attributes during 22 days.

Usage

energy
energy

Format

A data frame with 504 rows and 7 variables:

power: energy consumption
date: date
Temperature: temperature
P: pression
HR: humidity rate
hour: hour (categorical)
hour.num: hour (numerical)

Source

https://iles-ponant-edf-sei.opendatasoft.com, https://www.infoclimat.fr

Extract AIC from a Kernel Semi Parametric Model

Description

Computes the Akaike Information Criterion (AIC) for a kspm fit.

Usage

## S3 method for class 'kspm'
extractAIC(fit, scale = NULL, k = 2,
  correction = FALSE, ...)
## S3 method for class 'kspm'
extractAIC(fit, scale = NULL, k = 2,
  correction = FALSE, ...)

Arguments

`fit`	fitted model, usually the result of kspm.
`scale`	option not available for kspm fit.
`k`	numeric specifying the 'weight' of the effective degrees of freedom (edf) part in the AIC formula. See details.
`correction`	boolean indicating if the corrected AIC should be computed instead of standard AIC, may be `TRUE` only for `k=2`. See details.
`...`	additional optional argument (currently unused).

Details

The criterion used is $AIC = n log(RSS) + k (n-edf)$ where $RSS$ is the residual sum of squares and $edf$ is the effective degree of freedom of the model. k = 2 corresponds to the traditional AIC, using k = log(n) provides Bayesian Information Criterion (BIC) instead. For k=2, the corrected Akaike's Information Criterion (AICc) is obtained by $AICc = AIC + \frac{2 (n-edf) (n-edf+1)}{(edf-1)}$ .

Value

extractAIC.kspm returns a numeric value corresponding to AIC. Of note, the AIC obtained here differs from a constant to the AIC obtained with extractAIC applied to a lm object. If one wants to compare a kspm model with a lm model, it is preferrable to compute again the lm model using kspm function by specifying kernel = NULL and apply extractAIC method on this model.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Examples

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
extractAIC(fit)

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
extractAIC(fit)

Extract Model Fitted values

Description

Returns fitted values for a model of class "kspm".

Usage

## S3 method for class 'kspm'
fitted(object, ...)
## S3 method for class 'kspm'
fitted(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`...`	additional optional argument (currently unused).

Value

The vector of fitted values.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Examples

x <- 1:15
z <- runif(15, 1, 6)
y <- 3*x + z^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(z,
kernel.function = "polynomial", d = 2, rho = 1, gamma = 0))
fitted(fit)

x <- 1:15
z <- runif(15, 1, 6)
y <- 3*x + z^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(z,
kernel.function = "polynomial", d = 2, rho = 1, gamma = 0))
fitted(fit)

Summarizing Kernel Semi parametric Model Fits with flexible parameters for Davies' approximation method

Description

for flexibility in summary method for an object of class "summary.kspm"

Usage

flexible.summary(object, method = "davies", acc = 1e-06, lim = 10000)
flexible.summary(object, method = "davies", acc = 1e-06, lim = 10000)

Arguments

`object`	an object of class "summary.kspm", usually, a result of a call to `summary.kspm`.
`method`	method to approximate the chi square distribution in p-value computation, default is 'davies', another possibility is 'imhof'.
`acc`, `lim`	see davies and imhof functions in CompQuadForm package.

Details

the description of the model, including coefficients for the linear part and if asked for, test(s) of variance components associated with kernel part.

Value

Computes and returns the followimg summary statistics of the fitted kernel semi parametric model given in object

`residuals`	residuals
`coefficients`	a $p \times 4$ matrix with columns for the estimated coefficient, its standard error, t statistic and corresponding (two sided) p value for the linear part of the model.
`sigma`	the square root of the estimated variance of the random error $\sigma^2 = \frac{RSS}{edf}$ where $RSS$ is the residual sum of squares and $edf$ is the effective degree of freedom.
`edf`	effective degrees of freedom
`r.squared`	$R^2$ , the fraction of variance explained by the model, $1 - \frac{\sum e_i^2}{\sum(y_i - y^{\ast})^2}$ where $y^{\ast}$ is the mean of $y_i$ if there is an intercept and zero otherwise.
`adj.r.squared`	the above $R^2$ statistics, adjusted, penalizing for higher $p$ .
`score.test`	a $q \times 3$ matrix with colums for the estimated lambda, tau and p value for the q kernels for which a test should be performed.
`global.p.value`	p value from the score test for the global model.
`sample.size`	sample size (all: global sample size, inc: complete data sample size).

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Schweiger, Regev, et al. "RL SKAT: an exact and efficient score test for heritability and set tests." Genetics (2017): genetics 300395.

Li, Shaoyu, and Yuehua Cui. "Gene centric gene gene interaction: A model based kernel machine method." The Annals of Applied Statistics 6.3 (2012): 1134:1161.

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
summary.fit <- summary(fit)
flexible.summary(summary.fit, acc = 0.000001, lim = 1000)

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
summary.fit <- summary(fit)
flexible.summary(summary.fit, acc = 0.000001, lim = 1000)

compute Kernel Semi Parametric model parameters

Description

internal function to compute model parameters

Usage

get.parameters(X = NULL, Y = NULL, kernelList = NULL,
  free.parameters = NULL, n = NULL, not.missing = NULL,
  compute.kernel = NULL)
get.parameters(X = NULL, Y = NULL, kernelList = NULL,
  free.parameters = NULL, n = NULL, not.missing = NULL,
  compute.kernel = NULL)

Arguments

`X`	X matrix
`Y`	response matrix
`kernelList`	list of kernels
`free.parameters`	free parameters
`n`	number of samples
`not.missing`	number of non missing samples
`compute.kernel`	boolean indicating if kernel should be computed

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Extract Model Hyper-parameter

Description

Returns hyper-parameters for a model of class "kspm".

Usage

hypercoef(object, ...)
hypercoef(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`...`	additional optional argument (currently unused).

Value

A list of parameter.

`lambda`	A vector of penalisation arameters.
`kernel`	A vector of tunning parameters.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
hypercoef(fit)


x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
hypercoef(fit)

Giving information about Kernel Semi parametric Model Fits

Description

gives information about Kernel Semi parametric Model Fits

Usage

info.kspm(object, print = TRUE)
info.kspm(object, print = TRUE)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`print`	logical, if `TRUE`, table of information are printed.

Value

info.kspm returns a table of information whose each row corresponds to a kernel included in the model and columns are:

`type`	type of object used to define the kernel
`dim`	dimension of data used in the model
`type.predict`	type of object the user should provide in predict.kspm function
`dim.predict`	dimension of object the user should provide in predict.kspm function

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Create a Kernel Object

Description

Create a kernel object, to use as variable in a model formula.

Usage

Kernel(x, kernel.function, scale = TRUE, rho = NULL, gamma = NULL,
  d = NULL)
Kernel(x, kernel.function, scale = TRUE, rho = NULL, gamma = NULL,
  d = NULL)

Arguments

`x`	a formula, a vector or a matrix of variables grouped in the same kernel. It could also be a symetric matrix representing the Gram matrix, associated to a kernel function, already computed by the user.
`kernel.function`	type of kernel. Possible values are `"gaussian"`, `"linear"`, `"polynomial"`, `"sigmoid"`, `"inverse.quadratic"` or `"equality"`. See details below. If `x` is a Gram matrix, associated to a kernel function, already computed by the user, `kernel.function` should be equal to `"gram.matrix"`.
`scale`	boolean indicating if variables should be scaled before computing the kernel.
`rho`, `gamma`, `d`	kernel function hyperparameters. See details below.

Details

To use inside kspm() function. Given two $p-$ dimensional vectors $x$ and $y$ ,

the Gaussian kernel is defined as $k(x,y) = exp\left(-\frac{\parallel x-y \parallel^2}{\rho}\right)$ where $\parallel x-y \parallel$ is the Euclidean distance between $x$ and $y$ and $\rho > 0$ is the bandwidth of the kernel,
the linear kernel is defined as $k(x,y) = x^Ty$ ,
the polynomial kernel is defined as $k(x,y) = (\rho x^Ty + \gamma)^d$ with $\rho > 0$ , $d$ is the polynomial order. Of note, a linear kernel is a polynomial kernel with $\rho = d = 1$ and $\gamma = 0$ ,
the sigmoid kernel is defined as $k(x,y) = tanh(\rho x^Ty + \gamma)$ which is similar to the sigmoid function in logistic regression,
the inverse quadratic function defined as $k(x,y) = \frac{1}{\sqrt{\parallel x-y \parallel^2 + \gamma}}$ with $\gamma > 0$ ,
the equality kernel defined as $k(x,y) = \left\lbrace \begin{array}{ll} 1 & if x = y \\ 0 & otherwise \end{array}\right.$ .

Of note, Gaussian, inverse quadratic and equality kernels are measures of similarity resulting to a matrix containing 1 along the diagonal.

Value

A Kernel object including all parameters needed in computation of the model

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Kernel Functions

Description

These functions transform a $n \times p$ matrix into a $n \times n$ kernel matrix.

Usage

kernel.gaussian(x, rho = ncol(x))

kernel.linear(x)

kernel.polynomial(x, rho = 1, gamma = 0, d = 1)

kernel.sigmoid(x, rho = 1, gamma = 1)

kernel.inverse.quadratic(x, gamma = 1)

kernel.equality(x)
kernel.gaussian(x, rho = ncol(x))

kernel.linear(x)

kernel.polynomial(x, rho = 1, gamma = 0, d = 1)

kernel.sigmoid(x, rho = 1, gamma = 1)

kernel.inverse.quadratic(x, gamma = 1)

kernel.equality(x)

Arguments

`x`	a $n \times p$ matrix
`gamma`, `rho`, `d`	kernel hyperparameters (see details)

Details

Given two $p-$ dimensional vectors $x$ and $y$ ,

the Gaussian kernel is defined as $k(x,y) = exp\left(-\frac{\parallel x-y \parallel^2}{\rho}\right)$ where $\parallel x-y \parallel$ is the Euclidean distance between $x$ and $y$ and $\rho > 0$ is the bandwidth of the kernel,
the linear kernel is defined as $k(x,y) = x^Ty$ ,
the polynomial kernel is defined as $k(x,y) = (\rho x^Ty + \gamma)^d$ with $\rho > 0$ , $d$ is the polynomial order. Of note, a linear kernel is a polynomial kernel with $\rho = d = 1$ and $\gamma = 0$ ,
the sigmoid kernel is defined as $k(x,y) = tanh(\rho x^Ty + \gamma)$ which is similar to the sigmoid function in logistic regression,
the inverse quadratic function defined as $k(x,y) = \frac{1}{\sqrt{\parallel x-y \parallel^2 + \gamma}}$ with $\gamma > 0$ ,
the equality kernel defined as $k(x,y) = \left\lbrace \begin{array}{ll} 1 & if x = y \\ 0 & otherwise \end{array}\right.$ .

Of note, Gaussian, inverse quadratic and equality kernels are measures of similarity resulting to a matrix containing 1 along the diagonal.

Value

A $n \times n$ matrix.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

List of kernel parts included in the kernel semi parametric model

Description

internal method for listing all kernel parts included in the model

Usage

kernel.list(formula, data, names)
kernel.list(formula, data, names)

Arguments

`formula`	kernel part formula provided in the `kspm` function.
`data`	data provided in the `kspm` function.
`names`	row names of samples as they are evaluated in `kspm` function.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Kernel matrix

Description

These functions transform a $n \times p$ matrix into a $n \times n$ kernel matrix.

Usage

kernel.matrix(Z, whichkernel, rho = NULL, gamma = NULL, d = NULL)
kernel.matrix(Z, whichkernel, rho = NULL, gamma = NULL, d = NULL)

Arguments

`Z`	a $n \times p$ matrix
`whichkernel`	kernel function
`gamma`, `rho`, `d`	kernel hyperparameters (see details)

Details

Given a $n \times p$ matrix, this function returns a $n \times n$ matrix where each cell represents the similarity between two samples defined by two $p-$ dimensional vectors $x$ and $y$ ,

the Gaussian kernel is defined as $k(x,y) = exp\left(-\frac{\parallel x-y \parallel^2}{\rho}\right)$ where $\parallel x-y \parallel$ is the Euclidean distance between $x$ and $y$ and $\rho > 0$ is the bandwidth of the kernel,
the linear kernel is defined as $k(x,y) = x^Ty$ ,
the polynomial kernel is defined as $k(x,y) = (\rho x^Ty + \gamma)^d$ with $\rho > 0$ , $d$ is the polynomial order. Of note, a linear kernel is a polynomial kernel with $\rho = d = 1$ and $\gamma = 0$ ,
the sigmoid kernel is defined as $k(x,y) = tanh(\rho x^Ty + \gamma)$ which is similar to the sigmoid function in logistic regression,
the inverse quadratic function defined as $k(x,y) = \frac{1}{\sqrt{\parallel x-y \parallel^2 + \gamma}}$ with $\gamma > 0$ ,
the equality kernel defined as $k(x,y) = \left\lbrace \begin{array}{ll} 1 & if x = y \\ 0 & otherwise \end{array}\right.$ .

Value

A $n \times n$ matrix.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

some internal methods in computation of kernel semi parametric model

Description

internal methods

Usage

comb(x, ...)

check.integer(N)

asOneSidedFormula(object)

splitFormula(form, sep = "/")

computes.Kernel(x, ind, nameKernel, not.missing = NULL)

computes.Kernel.interaction(x, ind, nameKernel, not.missing = NULL)

computes.KernelALL(kernelList, not.missing = NULL)

renames.Kernel(object, names)

objects.Kernel(formula)
comb(x, ...)

check.integer(N)

asOneSidedFormula(object)

splitFormula(form, sep = "/")

computes.Kernel(x, ind, nameKernel, not.missing = NULL)

computes.Kernel.interaction(x, ind, nameKernel, not.missing = NULL)

computes.KernelALL(kernelList, not.missing = NULL)

renames.Kernel(object, names)

objects.Kernel(formula)

Arguments

`x`	list of objects
`...`	other arguments
`N`	numeric value
`object`	formula provided in the kernel part of `kspm` function
`form`	formula
`sep`	separator
`ind`	index value
`nameKernel`	name of kernel
`not.missing`	non missing values
`kernelList`	list of kernels
`names`	name of kernel
`formula`	formula

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Fitting Kernel Semi Parametric model

Description

kspm is used to fit kernel semi parametric models.

Usage

kspm(response, linear = NULL, kernel = NULL, data = NULL,
  level = 1, control = kspmControl())
kspm(response, linear = NULL, kernel = NULL, data = NULL,
  level = 1, control = kspmControl())

Arguments

`response`	a character with the name of the response variable or a vector containing the outcome or a matrix with outcome in the first column.
`linear`	an optional object of class "formula": a symbolic description of the linear part of the model to be fitted or a vector or a matrix containing covariates included in the linear part of the model. Default is intercept only. The details of model specification are given under ‘Details’.
`kernel`	an object of class "formula": a symbolic description of the kernel part of the model to be fitted. If missing a linear model is fitted using lm function. The details of model specification are given under ‘Details’.
`data`	an optional data frame containing the variables in the model. If NULL (default), data are taken from the workspace.
`level`	printed information about the model (0: no information, 1: information about kernels included in the model (default))
`control`	see kspmControl.

Details

The kernel semi parametric model refers to the following equation $Y_i = X_i\beta + h(Z_i) + e_i$ with $i=1..n$ where $n$ is the sample size, $Y$ is the univariate response, $X\beta$ is the linear part, $h(Z)$ is the kernel part and $e$ are the residuals. The linear part is defined using the linear argument by specifying the covariates $X$ . It could be either a formula, a vector of length $n$ if only one variable is included in the linear part or a $n \times p$ design matrix containing the values of the $p$ covariates included in the linear part (columns), for each individuals (rows). By default, an intercept is included. To remove the intercept term, use formula specification and add the term -1, as usual. Kernel part is defined using the kernel argument. It should be a formula of Kernel object(s). For a multiple kernel semi parametric model, Kernel objects are separated by the usual signs "+", "*" and ":" to specify addition and interaction between kernels. Specification formats of each Kernel object may be different. See Kernel for more information about their specification.

Value

kspm returns an object of class kspm.

An object of class kspm is a list containing the following components:

`linear.coefficients`	matrix of coefficients associated with linear part, the number of coefficients is the number of terms included in linear part
`kernel.coefficients`	matrix of coefficients associated with kernel part, the number of rows is the sample size included in the analysis and the number of columns is the number of kernels included in the model
`lambda`	penalization parameter(s)
`fitted.values`	the fitted mean values
`residuals`	the residuals, that is response minus the fitted values
`sigma`	standard deviation of residuals
`Y`	vector of responses
`X`	design matrix for linear part
`K`	kernel matrices computed by the model
`n.total`	total sample size
`n`	sample size of the model (model is performed on complete data only)
`edf`	effective degree of freedom
`linear.formula`	formula corresponding to the linear part of the model
`kernel.info`	information about kernels included in the model such as matrices of covariates (`Z`), kernel function (`type`), values of hyperparameters (`rho`, `gamma`, `d`). A boolean indicates if covariates were scaled (`kernel.scale`) and if `TRUE`, `kernel.mean`, `kernel.sd` and `Z.scale` give information about scaling. `kernel.formula` indicates the formula of the kernel and `free.parameters` indicates the hyperparameters that were estimated by the model.
`Hat`	The hat matrix $H$ such that $\hat{Y} = HY$
`L`	A matrix corresponding to $I - \sum\limits_{\ell = 1}^L K_{\ell} G_{\ell}^{-1} M_{\ell}$ according to our notations
`XLX_inv`	A matrix corresponding to $(XLX)^{-1}$
`GinvM`	A list of matrix, each corresponding to a kernel and equaling $G_{\ell}^{-1}M_{\ell}$ according to our notations
`control`	List of control parameters

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Kim, Choongrak, Byeong U. Park, and Woochul Kim. "Influence diagnostics in semiparametric regression models." Statistics and probability letters 60.1 (2002): 49:58.

Oualkacha, Karim, et al. "Adjusted sequence kernel association test for rare variants controlling for cryptic and family relatedness." Genetic epidemiology 37.4 (2013): 366:376.

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
summary(fit)

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
summary(fit)

Control various aspects of the optimisation problem

Description

Allow the user to set some characteristics of the optimisation algorithm

Usage

kspmControl(interval.upper = NA, interval.lower = NA, trace = FALSE,
  optimize.tol = .Machine$double.eps^0.25, NP = NA, itermax = 500,
  CR = 0.5, F = 0.8, initialpop = NULL, storepopfrom = itermax + 1,
  storepopfreq = 1, p = 0.2, c = 0,
  reltol = sqrt(.Machine$double.eps), steptol = itermax,
  parallel = FALSE)
kspmControl(interval.upper = NA, interval.lower = NA, trace = FALSE,
  optimize.tol = .Machine$double.eps^0.25, NP = NA, itermax = 500,
  CR = 0.5, F = 0.8, initialpop = NULL, storepopfrom = itermax + 1,
  storepopfreq = 1, p = 0.2, c = 0,
  reltol = sqrt(.Machine$double.eps), steptol = itermax,
  parallel = FALSE)

Arguments

`interval.upper`	integer or vetor of initial maximum value(s) allowed for parameter(s)
`interval.lower`	integer or vetor of initial maximum value(s) allowed for parameter(s)
`trace`	boolean. If TRUE parameters value at each iteration are displayed.
`optimize.tol`	if optimize function is used. See optimize
`NP`	if DEoptim function is used. See DEoptim.control
`itermax`	if DEoptim function is used. See DEoptim.control
`CR`	if DEoptim function is used. See DEoptim.control
`F`	if DEoptim function is used. See DEoptim.control
`initialpop`	if DEoptim function is used. See DEoptim.control
`storepopfrom`	if DEoptim function is used. See DEoptim.control
`storepopfreq`	if DEoptim function is used. See DEoptim.control
`p`	if DEoptim function is used. See DEoptim.control
`c`	if DEoptim function is used. See DEoptim.control
`reltol`	if DEoptim function is used. See DEoptim.control
`steptol`	if DEoptim function is used. See DEoptim.control
`parallel`	if DEoptim function is used. See DEoptim.control

Details

When only one hyperparameter should be estimated, the optimisation problem calls the optimize function from stats basic package. Otherwise, it calls the DEoptim function from the package DEoptim. In both case, the parameters are choosen among the initial interval defined by interval.lower and interval.upper.

Value

search.parameters is an iterative algorithm estimating model parameters and returns the following components:

`lambda`	tuning parameters for penalization.
`beta`	vector of coefficients associated with linear part of the model, the size being the number of variable in linear part (including an intercept term).
`alpha`	vector of coefficients associated with kernel part of the model, the size being the sample size.
`Ginv`	a matrix used in several calculations. $Ginv = (\lambda I + K)^{-1}$ .

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Log Likelihood of a kspm Object

Description

Returns the Log Likelihood value of the kernel semi parametric model represented by obect evaluated at the estimated coefficients.

Usage

## S3 method for class 'kspm'
logLik(object, ...)
## S3 method for class 'kspm'
logLik(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to kspm.
`...`	additional optional argument (currently unused).

Details

The function returns the Log Likelihood computed as follow: $logLik = -\frac{1}{2} RSS$ where $RSS$ is the residual sum of squares.

Value

logLik of kspm fit

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Examples

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
logLik(fit)

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
logLik(fit)

Computation of the leave one out error (LOOE) in kernel semi parametric model

Description

internal function to optimize model for estimating hyperparameters based on LOOE

Usage

lossFunction.looe(param. = NULL, Y. = NULL, X. = NULL,
  kernelList. = NULL, n. = NULL, not.missing. = NULL,
  compute.kernel. = NULL, print.lambda. = FALSE)
lossFunction.looe(param. = NULL, Y. = NULL, X. = NULL,
  kernelList. = NULL, n. = NULL, not.missing. = NULL,
  compute.kernel. = NULL, print.lambda. = FALSE)

Arguments

`param.`	initial parameter values.
`Y.`	response matrix.
`X.`	X matrix (linear part).
`kernelList.`	list of kernels (kernel part).
`n.`	nb of samples.
`not.missing.`	nb of non missing samples.
`compute.kernel.`	boolean. If TRUE, the kernel matrix is computed at each iteration. Should be TRUE when hyperparameters of kernel functions should be estimated by the model.
`print.lambda.`	boolean. If TRUE, values of tunning parameters (lambda) are printed at each iteration.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Extract the number of observations from a Kernel Semi parametric Model Fit

Description

Extract the number of observations use to estimate the model coefficients. This is principally intented to be used in computing BIC (see extractAIC.kspm).

Usage

## S3 method for class 'kspm'
nobs(object, ...)
## S3 method for class 'kspm'
nobs(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`...`	additional optional argument (currently unused).

Value

A single number (integer).

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Examples

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
nobs(fit)

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
nobs(fit)

Plot derivatives of a kspm object

Description

Plot of derivatives for kernel part of a kspm model.

Usage

## S3 method for class 'derivatives'
plot(x, subset = NULL, xlab = NULL,
  ylab = NULL, ...)
## S3 method for class 'derivatives'
plot(x, subset = NULL, xlab = NULL,
  ylab = NULL, ...)

Arguments

`x`	an object of class "derivatives", usually, a result of a call to `derivatives`.
`subset`	if a subset of the plots is required, specify the names of the variable for which plot of derivatives is required.
`xlab`	x label
`ylab`	y label
`...`	further arguments passed to or from other methods.

Details

X axis represents the raw data used as input in kernel part of the model. Y axis represents the pointwise derivative values i.e. the derivatives of fitted value according to the variable of interest.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Kim, Choongrak, Byeong U. Park, and Woochul Kim. "Influence diagnostics in semiparametric regression models." Statistics and probability letters 60.1 (2002): 49:58.

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
plot(derivatives(fit))

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
plot(derivatives(fit))

Plot Diagnostics for a kspm Object

Description

Five plots (selectable by which) are currently available: a plot of residuals against fitted values, a scale Location plot of $\sqrt{\mid residuals \mid}$ against fitted values, a Normal Q Q plot for residuals, a plot of Cook's distances versus row labels and a plot of residuals against leverages. By default, the first three and 5 are provided.

Usage

## S3 method for class 'kspm'
plot(x, which = c(1:3, 5), cook.levels = c(0.5, 1),
  id.n = 3, labels.id = names(x$residuals), cex.id = 0.75,
  col.id = "blue", ...)
## S3 method for class 'kspm'
plot(x, which = c(1:3, 5), cook.levels = c(0.5, 1),
  id.n = 3, labels.id = names(x$residuals), cex.id = 0.75,
  col.id = "blue", ...)

Arguments

`x`	an object of class "kspm", usually, a result of a call to `kspm`.
`which`	if a subset of the plots is required, specify a subset of the numbers 1:5.
`cook.levels`	levels of Cook's distance at which to draw contours.
`id.n`	number of points to be labelled in each plot, starting with the most extreme.
`labels.id`	vector of labels, from which the labels for extreme points will be chosen. NULL uses names associated to response specified in `kspm`.
`cex.id`	size of point labels.
`col.id`	color of point labels.
`...`	further arguments passed to or from other methods.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Kim, Choongrak, Byeong U. Park, and Woochul Kim. "Influence diagnostics in semiparametric regression models." Statistics and probability letters 60.1 (2002): 49:58.

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
plot(fit)

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
plot(fit)

Predicting Kernel Semi parametric Model Fits

Description

predict method for class "kspm".

Usage

## S3 method for class 'kspm'
predict(object, newdata.linear = NULL,
  newdata.kernel = NULL, interval = "none", level = 0.95, ...)
## S3 method for class 'kspm'
predict(object, newdata.linear = NULL,
  newdata.kernel = NULL, interval = "none", level = 0.95, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`newdata.linear`	should be a data frame or design matrix of variables used in the linear part
`newdata.kernel`	a list containing data frame or design matrix of variables used in each kernel part depending on the specification format of each kernel. When a kernel has been specified using `kernel.function = "gram.matrix"` in `Kernel` function, the user should also provide the Gram matrix associated to the new data points in `newdata.kernel`. The function info.kspm may help to correctly specify it.
`interval`	type of interval calculation. If `"none"` (default), no interval is computed, if `"confidence"`, the confidence interval is computed, if `"prediction"`, the prediction interval is computed.
`level`	confidence level. Default is `level = 0.95` meaning 95% confidence/prediction interval.
`...`	further arguments passed to or from other methods.

Details

predict.kspm produces predicted values. If a new dataset is not specified, it will return the fitted values from the original data (complete data used in the model specification). If predict.kspm is applied to a new dataset, all variables used in the original model should be provided in newdata.linear and newdata.kernel arguments but only complete data may be provided. Setting interval specifies computation of confidence or prediction intervals at the specified level.

Value

predict.kspm returns a vector of predictions or a matrix containing the following components if interval is set:

`fit`	predictions.
`lwr`	lower bound of confidence/prediction intervals.
`upr`	upper bound of confidence/prediction intervals.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
predict(fit, interval = "confidence")

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
predict(fit, interval = "confidence")

Print results from a Kernel Semi parametric Model Fit

Description

print method for class "kspm".

Usage

## S3 method for class 'kspm'
print(x, ...)

## S3 method for class 'summary.kspm'
print(x, ...)
## S3 method for class 'kspm'
print(x, ...)

## S3 method for class 'summary.kspm'
print(x, ...)

Arguments

`x`	an object used to select a method. Usually, a result of a call to `kspm` or a result from `summary.kspm`.
`...`	additional optional argument (currently unused).

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Extract residuals from a Kernel Semi Parametric Model

Description

Returns the vector of residuals for a model fit of class "kspm".

Usage

## S3 method for class 'kspm'
residuals(object, ...)
## S3 method for class 'kspm'
residuals(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`...`	additional optional argument (currently unused).

Value

A vector of residuals. The vector length is the number of observations used in model coefficients estimation (see nobs.kspm).

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Examples

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
residuals(fit)

x <- 1:15
y <- 3*x + rnorm(15, 0, 2)
fit <- kspm(y, kernel = ~ Kernel(x, kernel.function = "linear"))
residuals(fit)

Standardized residuals for Kernel Semi parametric Model Fits

Description

computes standardized residuals for an object of class "kspm".

Usage

## S3 method for class 'kspm'
rstandard(model, ...)
## S3 method for class 'kspm'
rstandard(model, ...)

Arguments

`model`	an model of class "kspm", usually, a result of a call to `kspm`.
`...`	furter arguments passed to or from other methods (currently unused).

Details

Standardized residuals $t_i$ are obtained by $t_i = \frac{e_i}{\hat{\sigma} \sqrt{1 - h_{ii}}}$ where $e_i$ is the residual, $\hat{\sigma}$ is the estimated standard deviation of the errors and $h_{ii}$ is the leverage of subject i, i.e. the i th diagonal element of the Hat matrix.

Value

a vector containing the standardized residuals.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Optimisation to cumpute hyperparameter in Kernel Semi Parametric model

Description

internal function to optimize model for estimating hyperparameters

Usage

search.parameters(Y = NULL, X = NULL, kernelList = NULL, n = NULL,
  not.missing = NULL, compute.kernel = NULL, controlKspm = NULL)
search.parameters(Y = NULL, X = NULL, kernelList = NULL, n = NULL,
  not.missing = NULL, compute.kernel = NULL, controlKspm = NULL)

Arguments

`Y`	response matrix
`X`	X matrix
`kernelList`	of kernels
`n`	nb of samples
`not.missing`	nb of non missing samples
`compute.kernel`	boolean kernel computation
`controlKspm`	control parameters

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Extract residuals standard deviation

Description

Returns the residuals standard deviation (sigma) for object of class "kspm".

Usage

## S3 method for class 'kspm'
sigma(object, ...)
## S3 method for class 'kspm'
sigma(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`...`	additional optional argument (currently unused).

Details

The value returned by the method is $\sqrt{\frac{RSS}{edf}}$ where $RSS$ is the residual sum of squares and $edf$ is the effective degree of freedom.

Value

typically a number, the estimated standard deviation of the errors ("residual standard deviation")

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Choose a model by AIC or BIC in a Stepwise Algorithm

Description

Performs stepwise model selection for Kernel Semi Parametric Model by AIC or BIC.

Usage

stepKSPM(object, data = NULL, linear.lower = NULL,
  linear.upper = NULL, kernel.lower = NULL, kernel.upper = NULL,
  direction = "both", k = 2, kernel.param = "fixed", trace = TRUE)
stepKSPM(object, data = NULL, linear.lower = NULL,
  linear.upper = NULL, kernel.lower = NULL, kernel.upper = NULL,
  direction = "both", k = 2, kernel.param = "fixed", trace = TRUE)

Arguments

`object`	an object of class "kspm" with only one kernel.
`data`	data.
`linear.lower`	one side formula corresponding to the smallest set of variables that should be included in the linear part of the model.
`linear.upper`	one side formula corresponding to the largest set of variables that may be included in the linear part of the model.
`kernel.lower`	one side formula corresponding to the smallest set of variables that should be included in the kernel part of the model.
`kernel.upper`	one side formula corresponding to the largest set of variables that may be included in the kernel part of the model.
`direction`	the mode of stepwise search, can be one of "both" (default), "backward", or "forward".
`k`	type of information criteria used for the variable selection. If `k=2` AIC is used (default), if `k=log(n)`, BIC is used instead.
`kernel.param`	define if hyperparameters should be fixed (`"fixed"`) or reestimated at each iteration (`"change"`). Tu use the last option, hyperparameter of model provided in `object` should have been estimated by the model.
`trace`	integer. If positive, information is printed during the running of step.kspm. Larger values may give more information on the fitting process.

Details

This procedure may be done on kspm object defined with only one kernel part and for which a data frame including all variables was provided. Selection may be done on linear part only, on kernel part only or on both at the same time. To perform selection on linear (resp. kernel) part only, kernel.lower and kernel.upper (resp. linear.lower and linear.upper) should contain all the variables that should stay in the model for kernel (resp. linear) part.

Value

stepKSPM returns the selected model.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 4)
z3 <- rnorm(15, 6, 2)
z4 <- runif(15, -10, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
dfrm <- data.frame(x = x, z1 = z1, z2 = z2, z3 = z3, z4 = z4, y = y)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2 + z3 + z4,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0), data = dfrm)
stepKSPM(fit, k = 2, data = dfrm)

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 4)
z3 <- rnorm(15, 6, 2)
z4 <- runif(15, -10, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
dfrm <- data.frame(x = x, z1 = z1, z2 = z2, z3 = z3, z4 = z4, y = y)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2 + z3 + z4,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0), data = dfrm)
stepKSPM(fit, k = 2, data = dfrm)

Summarizing Kernel Semi parametric Model Fits

Description

summary method for an object of class "kspm"

Usage

## S3 method for class 'kspm'
summary(object, kernel.test = "all",
  global.test = FALSE, ...)
## S3 method for class 'kspm'
summary(object, kernel.test = "all",
  global.test = FALSE, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`kernel.test`	vector of characters indicating for which kernel a test should be performed. Default is `"all"`. If `"none"`, no test will be performed.
`global.test`	logical, if `TRUE`, a global test for kernel part is computed.
`...`	further arguments passed to or from other methods.

Details

the description of the model, including coefficients for the linear part and if asked for, test(s) of variance components associated with kernel part.

Value

Computes and returns the followimg summary statistics of the fitted kernel semi parametric model given in object

`residuals`	residuals
`coefficients`	a $p \times 4$ matrix with columns for the estimated coefficient, its standard error, t statistic and corresponding (two sided) p value for the linear part of the model.
`sigma`	the square root of the estimated variance of the random error $\sigma^2 = \frac{RSS}{edf}$ where $RSS$ is the residual sum of squares and $edf$ is the effective degree of freedom.
`edf`	effective degrees of freedom
`r.squared`	$R^2$ , the fraction of variance explained by the model, $1 - \frac{\sum e_i^2}{\sum(y_i - y^{\ast})^2}$ where $y^{\ast}$ is the mean of $y_i$ if there is an intercept and zero otherwise.
`adj.r.squared`	the above $R^2$ statistics, adjusted, penalizing for higher $p$ .
`score.test`	a $q \times 3$ matrix with colums for the estimated lambda, tau and p value for the q kernels for which a test should be performed.
`global.p.value`	p value from the score test for the global model.
`sample.size`	sample size (all: global sample size, inc: complete data sample size).

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.

Schweiger, Regev, et al. "RL SKAT: an exact and efficient score test for heritability and set tests." Genetics (2017): genetics 300395.

Li, Shaoyu, and Yuehua Cui. "Gene centric gene gene interaction: A model based kernel machine method." The Annals of Applied Statistics 6.3 (2012): 1134:1161.

Examples

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
summary(fit)

x <- 1:15
z1 <- runif(15, 1, 6)
z2 <- rnorm(15, 1, 2)
y <- 3*x + (z1 + z2)^2 + rnorm(15, 0, 2)
fit <- kspm(y, linear = ~ x, kernel = ~ Kernel(~ z1 + z2,
kernel.function = "polynomial", d= 2, rho = 1, gamma = 0))
summary(fit)

Score Tests for kernel part in kernel semi parametric model

Description

Perform score tests for kernel part in kernel semi parametric model

Usage

test.1.kernel(object)

test.global.kernel(object)

test.k.kernel(object, kernel.name)
test.1.kernel(object)

test.global.kernel(object)

test.k.kernel(object, kernel.name)

Arguments

`object`	an object of class "kspm"
`kernel.name`	vector of character listing names of kernels for which test should be performed

Value

p values

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Schweiger, Regev, et al. "RL SKAT: an exact and efficient score test for heritability and set tests." Genetics (2017): genetics 300395.

Li, Shaoyu, and Yuehua Cui. "Gene centric gene gene interaction: A model based kernel machine method." The Annals of Applied Statistics 6.3 (2012): 1134:1161.

Oualkacha, Karim, et al. "Adjusted sequence kernel association test for rare variants controlling for cryptic and family relatedness." Genetic epidemiology 37.4 (2013): 366:376.

Ge, Tian, et al. "A kernel machine method for detecting effects of interaction between multidimensional variable sets: An imaging genetics application." Neuroimage 109 (2015): 505:514.

Variable names of fitted models

Description

Simple utility returning names of variables involved in a kernel semi parametric model.

Usage

## S3 method for class 'kspm'
variable.names(object, ...)
## S3 method for class 'kspm'
variable.names(object, ...)

Arguments

`object`	an object of class "kspm", usually, a result of a call to `kspm`.
`...`	additional optional argument (currently unused).

Value

a list of character vectors. The first element correspond to the names of variables included in the linear part of the model. Then, a vector containing names of variables including in kernel part is provided for each kernel.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

Package 'KSPM'

Help Index

Case names of fitted models

Description

Usage

Arguments

Value

Author(s)

See Also

Extract Model Coefficients

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Confidence interavls for linear part of model parameters

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Cook's distance for a Kernel Semi Parametric Model Fit

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Conventional and Social media features of 187 movies.

Description

Usage

Format

Source

References

Computing kernel function derivatives

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Model deviance

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Energy consumption measuring hourly during 22 days

Description

Usage

Format

Source

Extract AIC from a Kernel Semi Parametric Model

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Extract Model Fitted values

Description

Usage

Arguments

Value

Author(s)