Dichotomizd functional response regression (dfrr) model

Description

Implementing Function-on-Scalar Regression model, in which the response function is dichotomized and observed sparsely. This function fits the dichotomized functional response regression (dfrr) model as

Y_{i}(t)=I(\beta_0(t)+\beta_1(t)*x_{1i}+\ldots+\beta_{q-1}(t)*x_{(q-1)i}+\varepsilon_{i}(t)+\epsilon_{i}(t)\times\sigma^2>0),

where I(.) is the indicator function, \varepsilon_{i} is a Gaussian random function, and \epsilon_{i}(t) are iid standard normal for each i and t independent of \varepsilon_{i}. \beta_k and x_k for k=0,1,\ldots,q-1 are the functional regression coefficients and scalar covariates, respectively.

Details

@details Implementing Function-on-Scalar Regression model in which the response function is dichotomized and observed sparsely. This package provides smooth estimations of functional regression coefficients and principal components for the dichotomized funtional response regression (dfrr) model. The main function in the dfrr-package is dfrr().

Author(s)

Maintainer: Fatemeh Asgari fatemeh.asgari@medisin.uio.no

Authors:

• Saeed Hayati saeed.hayati@medisin.uio.no [contributor]

See Also

Useful links:

• https://github.com/asgari-fatemeh/dfrr

• Report bugs at https://github.com/asgari-fatemeh/dfrr/issues

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)

coefs<-coef(dfrr_fit)
  plot(coefs)

fitteds<-fitted(dfrr_fit)
  plot(fitteds)

resids<-residuals(dfrr_fit)
plot(resids)

fpcs<-fpca(dfrr_fit)
plot(fpcs,plot.contour=TRUE,plot.3dsurface = TRUE)

newdata<-data.frame(X=c(1,0))
  preds<-predict(dfrr_fit,newdata=newdata)
  plot(preds)


newdata<-data.frame(X=c(1,0))
newydata<-data.frame(.obs=rep(1,5),.index=c(0.0,0.1,0.2,0.3,0.7),.value=c(1,1,1,0,0))
preds<-predict(dfrr_fit,newdata=newdata,newydata = newydata)
plot(preds)

Get the basis functions from a dfrr-object

Description

Returns the basis functions employed in fitting a dfrr-object.

Usage

basis(object)

Arguments

Value

a basis object used in fitting the functional parameters. The basis object is the one created by the functions create.*.basis of the 'fda' package.

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
coefs<-coef(dfrr_fit,return.fourier.coefs=TRUE)

basis<-basis(dfrr_fit)
evaluated_coefs<-coefs%*%t(fda::eval.basis(time,basis))

#Plotting the regression coefficients
oldpar<-par(mfrow=c(1,2))

plot(time,evaluated_coefs[1,],'l',main="Intercept")
plot(time,evaluated_coefs[2,],'l',main="X")

par(oldpar)

Get estimated coefficients from a dfrr fit

Description

Returns estimations of the smooth functional regression coefficients \beta(t). The result is a matrix of either Fourier coefficients or evaluations. See Details.

Usage

## S3 method for class 'dfrr'
coef(
  object,
  standardized = NULL,
  unstandardized = !standardized,
  return.fourier.coefs = NULL,
  return.evaluations = !return.fourier.coefs,
  time_to_evaluate = NULL,
  ...
)

Arguments

Details

This function will return either the Fourier coefficients or the evaluation of estimated coefficients. Fourier coefficients which are reported are based on the a set of basis which can be determined by basis(dfrr_fit). Thus the evaluation of regression coefficients on the set of time points specified by vector time, equals to fitted(dfrr_fit)%*%t(eval.basis(time,basis(dfrr_fit))).

Consider that the unstandardized estimations are not identifiable. So, it is recommended to extract and report the standardized estimations.

Value

This function returns a matrix of dimension NxM or NxJ, depending the argument 'return.evaluations'. If return.evaluations=FALSE, the returned matrix is NxJ, where N denotes the number of functional regression coefficients, (the number of rows of the argument 'newData'), and J denotes the number of basis functions. Then, the NxJ matrix is the fourier coefficients of functional regression coefficients. If return.evaluations=TRUE, the returned matrix is NxM, where M is the length of the argument time_to_evaluate. Then, the NxM matrix is the functional regression coefficients evaluated at time points given in time_to_evaluate.

See Also

plot.coef.dfrr

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
coefs<-coef(dfrr_fit)
plot(coefs)

Dichotomized Functional Response Regression (dfrr)

Description

Implementing Function-on-Scalar Regression model, in which the response function is dichotomized and observed sparsely. This function fits the dichotomized functional response regression (dfrr) model as

Y_{i}(t)=I(\beta_0(t)+\beta_1(t)*x_{1i}+\ldots+\beta_{q-1}(t)*x_{(q-1)i}+\varepsilon_{i}(t)+\epsilon_{i}(t)\times\sigma^2>0),

where I(.) is the indicator function, \varepsilon_{i} is a Gaussian random function, and \epsilon_{i}(t) are iid standard normal for each i and t independent of \varepsilon_{i}. \beta_k and x_k for k=0,1,\ldots,q-1 are the functional regression coefficients and scalar covariates, respectively.

Usage

dfrr(
  formula,
  yind = NULL,
  data = NULL,
  ydata = NULL,
  method = c("REML", "ML"),
  rangeval = NULL,
  basis = NULL,
  times_to_evaluate = NULL,
  ...
)

Arguments

Value

The output is a dfrr-object, which then can be injected into other methods/functions to postprocess the fitted model, including: coef.dfrr,fitted.dfrr, basis, residuals.dfrr, predict.dfrr, fpca, summary.dfrr, model.matrix.dfrr, plot.coef.dfrr, plot.fitted.dfrr, plot.residuals.dfrr, plot.predict.dfrr, plot.fpca.dfrr

Examples

set.seed(2000)
N<-50;M<-24
X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)
#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
plot(dfrr_fit)

#Fitting dfrr model to the Madras Longitudinal Schizophrenia data
data(madras)
ids<-unique(madras$id)
N<-length(ids)

ydata<-data.frame(.obs=madras$id,.index=madras$month,.value=madras$y)

xdata<-data.frame(Age=rep(NA,N),Gender=rep(NA,N))
for(i in 1:N){
  dt<-madras[madras$id==ids[i],]
  xdata[i,]<-c(dt$age[1],dt$gender[1])
}
rownames(xdata)<-ids

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
#J is the number of basis functions that will be used in estimating the functional parameters.
madras_dfrr<-dfrr(Y~Age+Gender+Age*Gender, data=xdata, ydata=ydata, J=11,T_E=1)


coefs<-coef(madras_dfrr)
plot(coefs)

fpcs<-fpca(madras_dfrr)
plot(fpcs)
plot(fpcs,plot.eigen.functions=FALSE,plot.contour=TRUE,plot.3dsurface = TRUE)

oldpar<-par(mfrow=c(2,2))

fitteds<-fitted(madras_dfrr) #Plot first four fitted functions
  plot(fitteds,id=c(1,2,3,4))
par(oldpar)

resids<-residuals(madras_dfrr)
plot(resids)


newdata<-data.frame(Age=c(1,1,0,0),Gender=c(1,0,1,0))
  preds<-predict(madras_dfrr,newdata=newdata)
  plot(preds)


newdata<-data.frame(Age=c(1,1,0,0),Gender=c(1,0,1,0))
newydata<-data.frame(.obs=rep(1,5),.index=c(0,1,3,4,5),.value=c(1,1,1,0,0))
preds<-predict(madras_dfrr,newdata=newdata,newydata = newydata)
plot(preds)

Obtain fitted curves for a dfrr model

Description

Fitted curves refer to the estimations of latent functional response curves. The results can be either the Fourier coefficients or evaluation of the fitted functions. See Details.

Usage

## S3 method for class 'dfrr'
fitted(
  object,
  return.fourier.coefs = NULL,
  return.evaluations = !return.fourier.coefs,
  time_to_evaluate = NULL,
  standardized = NULL,
  unstandardized = !standardized,
  ...
)

Arguments

Details

This function will return either the Fourier coefficients or the evaluation of fitted curves to the binary sequences. Fourier coefficients which are reported are based on the a set of basis which can be determined by basis(dfrr_fit). Thus the evaluation of fitted latent curves on the set of time points specified by vector time, equals to fitted(dfrr_fit)%*%t(eval.basis(time,basis(dfrr_fit))).

Consider that the unstandardized estimations are not identifiable. So, it is recommended to extract and report the standardized estimations.

Value

This function returns a matrix of dimension NxM or NxJ, depending the argument return.evaluations. If return.evaluations=FALSE, the returned matrix is NxJ, where N denotes the sample size (the number of rows of the argument 'newData'), and J denotes the number of basis functions. Then, the NxJ matrix is the fourier coefficients of the fitted curves. If return.evaluations=TRUE, the returned matrix is NxM, where M is the length of the argument time_to_evaluate. Then, the NxM matrix is the fitted curves evaluated at time points given in time_to_evaluate.

See Also

plot.fitted.dfrr

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
fitteds<-fitted(dfrr_fit)
plot(fitteds)

Functional principal component analysis (fpca) of a dfrr fit

Description

fpca() returns estimations of the smooth principal components/eigen-functions and the corresponding eigen-values of the residual function in the dfrr model. The result is a named list containing the vector of eigen-values and the matrix of Fourier coefficients. See Details.

Usage

fpca(object, standardized = NULL, unstandardized = !standardized)

Arguments

Details

Fourier coefficients which are reported are based on the a set of basis which can be determined by basis(dfrr_fit). Thus the evaluation of pricipal component/eigen-function on the set of time points specified by vector time, equals to fpca(dfrr_fit)%*%t(eval.basis(time,basis(dfrr_fit))).

Consider that the unstandardized estimations are not identifiable. So, it is recommended to extract and report the standardized estimations.

Value

fpca(dfrr_fit) returns a list containtng the following components:

See Also

plot.fpca.dfrr

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
fpcs<-fpca(dfrr_fit)
plot(fpcs,plot.eigen.functions=TRUE,plot.contour=TRUE,plot.3dsurface = TRUE)

Madras Longitudinal Schizophrenia Study.

Description

Monthly records of presence/abscence of psychiatric symptom 'thought disorder' of 86 patients over the first year after initial hospitalisation for disease.

Usage

madras

Format

A data frame with 1032 observations and 5 variables

id

identification number of a patient

y

response 'thought disorder': 0 = absent, 1 = present

month

month since hospitalisation

age

age indicator: 0 = less than 20 years, 1 = 20 or over

gender

sex indicator: 0 = male, 1 = female

Source

Diggle PJ, Heagerty P, Liang KY, Zeger SL (2002). The analysis of Longitudinal Data, second ed., pp. 234-43. Oxford University Press, Oxford.
<http://faculty.washington.edu/heagerty/Books/AnalysisLongitudinal/datasets.html>

References

Jokinen J. Fast estimation algorithm for likelihood-based analysis of repeated categorical responses. Computational Statistics and Data Analysis 2006; 51:1509-1522.

Obtain model matrix for a dfrr fit

Description

Obtain model matrix for a dfrr fit

Usage

## S3 method for class 'dfrr'
model.matrix(object, ...)

Arguments

Details

'@return This function returns the model matrix.

Plot dfrr coefficients

Description

Plot a coef.dfrr object. The output is the plot of regression coefficients.

Usage

## S3 method for class 'coef.dfrr'
plot(x, select = NULL, ask.hit.return = TRUE, ...)

Arguments

Value

This function generates the plot of functional regression coefficients.

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
coefs<-coef(dfrr_fit)
plot(coefs)

Plot a dfrr fit

Description

Plot the regression coefficients, principal components, kernel function and residuals of a dfrr-object.

Usage

## S3 method for class 'dfrr'
plot(x, plot.kernel = TRUE, ...)

Arguments

Details

The contour plot of the kernel function is produced if the package ggplot2 is installed. Plotting the 3d surface of the kernel function is also depends on the package plotly. To produce the qq-plot, the package car must be installed.

Value

This function generates a set of plots, including functional regression coefficients, principal components, 2-d contour and 3d-surface of kernel function, and QQ-plot of residuals.

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
plot(dfrr_fit)

Plot dfrr fitted latent functions

Description

Plot a fitted.dfrr object.

Usage

## S3 method for class 'fitted.dfrr'
plot(
  x,
  id = NULL,
  main = NULL,
  col = "blue",
  lwd = 2,
  lty = "solid",
  cex.circle = 1,
  col.circle = "black",
  ylim = NULL,
  ...
)

Arguments

Details

The output is the plot of latent curves over the observed binary sequence. The binary sequence is illustrated with circles and filled circles for the values of zero and one, respectively.

Value

This function generates plot of fitted curves.

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
fitteds<-fitted(dfrr_fit)
plot(fitteds)

Plot dfrr functional principal components

Description

Plot a fpca.dfrr object.

Usage

## S3 method for class 'fpca.dfrr'
plot(
  x,
  plot.eigen.functions = TRUE,
  select = NULL,
  plot.contour = FALSE,
  plot.3dsurface = FALSE,
  plot.contour.pars = list(breaks = NULL, minor_breaks = NULL, n.breaks = NULL, labels =
    NULL, limits = NULL, colors = NULL, xlab = NULL, ylab = NULL, title = NULL),
  plot.3dsurface.pars = list(xlab = NULL, ylab = NULL, zlab = NULL, title = NULL, colors
    = NULL),
  ask.hit.return = TRUE,
  ...
)

Arguments

Details

This function plots the functional principal components, contour plot and 3d surface of the kernel function.

If ggplot2-package is installed, the contour plot of the kernel function is produced by setting the argument plot.contour=TRUE. Some graphical parameters of the contour plot can be modified by setting the (optional) argument plot.contour.pars.

If the package plotly is installed, the 3d surface of the kernel function is produced by setting the argument plot.3dsurface=TRUE. Some graphical parameters of the 3d surface can be modified by setting the (optional) argument plot.3dsurface.pars.

Value

This function generates the plot of principal components.

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

 #The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
fpcs<-fpca(dfrr_fit)
plot(fpcs,plot.eigen.functions=TRUE,plot.contour=TRUE,plot.3dsurface=TRUE)

Plot dfrr predictions

Description

Plot a predict.dfrr object.

Usage

## S3 method for class 'predict.dfrr'
plot(
  x,
  id = NULL,
  main = id,
  col = "blue",
  lwd = 2,
  lty = "solid",
  cex.circle = 1,
  col.circle = "black",
  ylim = NULL,
  ...
)

Arguments

Details

The output is the plot of predictions of latent functions given the new covariates. For the case in which newydata is also given, the predictions are plotted over the observed binary sequence. The binary sequence is illustrated with circles and filled circles for the values of zero and one, respectively.

Value

This function generates the plot of predictions.

References

Choi, H., & Reimherr, M. A geometric approach to confidence regions and bands for functional parameters . Journal of the Royal Statistical Society, Series B Statistical methodology 2018; 80:239-260.

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

 #The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)

newdata<-data.frame(X=c(1,0))
  preds<-predict(dfrr_fit,newdata=newdata)
  plot(preds)

newdata<-data.frame(X=c(1,0))
newydata<-data.frame(.obs=rep(1,5),.index=c(0.0,0.1,0.2,0.3,0.7),.value=c(1,1,1,0,0))
preds<-predict(dfrr_fit,newdata=newdata,newydata = newydata)
plot(preds)

QQ-plot for dfrr residuals

Description

The output gives the qq-plot of estimated measurment error.

Usage

## S3 method for class 'residuals.dfrr'
plot(x, ...)

## S3 method for class 'dfrr'
qq(x, ...)

Arguments

Value

This function generates the QQ-plot of residuals.

Examples

N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
resid<-residuals(dfrr_fit)
plot(resid)
# We can also use the qq function to draw the QQ-plot.

Prediction for dichotomized function-on-scalar regression

Description

Takes a dfrr-object created by dfrr() and returns predictions given a new set of values for a model covariates and an optional ydata-like data.frame of observations for the dichotomized response.

Usage

## S3 method for class 'dfrr'
predict(
  object,
  newdata,
  newydata = NULL,
  standardized = NULL,
  unstandardized = !standardized,
  return.fourier.coefs = NULL,
  return.evaluations = !return.fourier.coefs,
  time_to_evaluate = NULL,
  ...
)

Arguments

Details

This function will return either the Fourier coefficients or the evaluation of predictions. Fourier coefficients which are reported are based on the a set of basis which can be determined by basis(dfrr_fit). Thus the evaluation of predictions on the set of time points specified by vector time, equals to fitted(dfrr_fit,return.fourier.coefs=T)%*%t(eval.basis(time,basis(dfrr_fit))).

Value

This function returns a matrix of dimension NxM or NxJ, depending the argument 'return.evaluations'. If return.evaluations=FALSE, the returned matrix is NxJ, where N denotes the sample size (the number of rows of the argument 'newData'), and J denotes the number of basis functions. Then, the NxJ matrix is the fourier coefficients of the predicted curves. If return.evaluations=TRUE, the returned matrix is NxM, where M is the length of the argument time_to_evaluate. Then, the NxM matrix is the predicted curves evaluated at time points given in time_to_evaluate.

See Also

plot.predict.dfrr

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)

newdata<-data.frame(X=c(1,0))
  preds<-predict(dfrr_fit,newdata=newdata)
  plot(preds)

newdata<-data.frame(X=c(1,0))
newydata<-data.frame(.obs=rep(1,5),.index=c(0.0,0.1,0.2,0.3,0.7),.value=c(1,1,1,0,0))
preds<-predict(dfrr_fit,newdata=newdata,newydata = newydata)
plot(preds)

qq-plot Generic function

Description

This is a generic function for qq() method.

Usage

qq(x, ...)

Arguments

Value

This function generates the QQ-plot of residuals.

Obtain residuals for a dfrr model

Description

Returns the residuals of a fitted dfrr model. A dfrr model is of the form:

Y_{i}(t)=I(W_{i}(t)>0),

in which I(.) is the indicator function and W_{i}(t)=Z_{i}(t)+\epsilon_{i}(t)\times\sigma^2, where Z_{i}(t) is the functional part of the model and epsilon_{i}(t)\times\sigma^2 is the measurement error. The functional part of the model, consisting a location and a residual function of the form:

Z_{i}(t)=\sum_{j=1}^{q}\beta_{j}(t)*x_{ji}+\varepsilon_{i}(t),

and \epsilon_{i}(t) are iid standard normal for each i and t. The residuals reported in the output of this functions is the estimation of the measurement error of the model i.e. \epsilon_{i}(t)\times\sigma^2, which is estimated by:

E(W_{i}(t)-Z_{i}(t)\mid Y_{i}(t)).

Usage

## S3 method for class 'dfrr'
residuals(object, standardized = NULL, unstandardized = !standardized, ...)

Arguments

Value

This function returns either a matrix or a data.frame. If the argument ydata is specified, the return value is 'ydata' with a column added, namely 'residual'. Otherwise, the return value is a matrix of residuals of dimension NxM where N is the number of sample curves, and M is the length of argument 'yind' passed to the function dfrr.

See Also

plot.residuals.dfrr, qq.dfrr

Examples

set.seed(2000)
N<-50;M<-24

X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

#The argument T_E indicates the number of EM algorithm.
#T_E is set to 1 for the demonstration purpose only.
#Remove this argument for the purpose of converging the EM algorithm.
dfrr_fit<-dfrr(Y~X,yind=time,T_E=1)
resid<-residuals(dfrr_fit)


plot(resid)
# We can also use the qq function to draw the QQ-plot.

Simulating a Simple dfrr Model

Description

Simulation from a simple dfrr model:

Y_{i}(t)=I(\beta_0(t)+\beta_1(t)*x_{i}+\varepsilon_{i}(t)+\epsilon_{i}(t)\times\sigma^2>0),

where I(.) is the indicator function, \varepsilon_{i} is a Gaussian random function, and \epsilon_{i}(t) are iid standard normal for each i and t independent of \varepsilon_{i}. For demonstration purpose only.

Usage

simulate_simple_dfrr(
  beta0 = function(t) {
     cos(pi * t + pi)
 },
  beta1 = function(t) {
     2 * t
 },
  X = rnorm(50),
  time = seq(0, 1, length.out = 24),
  sigma2 = 0.2
)

Arguments

Value

This function returns a martix of binary values of dimension NxM where N denotes the length of X and M stands for the length of time.

Examples

N<-50;M<-24
X<-rnorm(N,mean=0)
time<-seq(0,1,length.out=M)
Y<-simulate_simple_dfrr(beta0=function(t){cos(pi*t+pi)},
                        beta1=function(t){2*t},
                        X=X,time=time)

Summary for a dfrr fit

Description

Summarise a fitted dfrr-object. Not implemented.

Usage

## S3 method for class 'dfrr'
summary(object, ...)

Arguments

Value

The function summary.dfrr computes and returns a list of summary statistics of the fitted dfrr model given in dfrr-object. Not implemented.