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VCM.Rmd
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---
title: "VCM"
author: "linsq"
date: "2017年3月10日"
output: html_document
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
以下是会调用到但是不用发布的函数
```{r}
beta <- function(x,z,y,p,nknots,m,knots,lambda,sigma_error=diag(1,length(y)),Dm){
n <- length(y)
df1 <- nknots+p
Psi <- bs(z,knots = knots,degree = p)
Psi <- as.matrix(Psi[1:n,1:df1])
x1 <- diag(x)
coef <- solve(t(Psi) %*% x1 %*% solve(sigma_error) %*% x1 %*% Psi +
lambda*(nknots^(2*m-1))*t(Dm)%*%(Dm)) %*%t(Psi) %*% x1 %*% solve(sigma_error) %*% y
coef
}
valcv <- function(fold,x,z,y,p,nknots,m,knots,lambda,sigma_error=diag(1,length(y)),Dm){
df1 <- nknots+p
k = sample(rep(1:fold,length=n))
fit1 <- rep(NA,n)
for (j in 1:fold){
testset = (1:n)[k==j]
coef <- beta(x=x[-testset],z=z[-testset],y=y[-testset],
p=p,nknots = nknots,m=m,knots=knots,lambda=lambda,Dm=Dm)
l <- length(x[testset])
NPsi <- bs(z[testset],df=df1,degree = p)
Npsi <- as.matrix(NPsi[1:l,1:df1])
Ntheta <- Npsi %*% coef
fit1[testset] <- diag(x[testset]) %*% Ntheta
}
mse <- mean((y-fit1)^2)
return(mse)
}
```
## R Markdown
假设$y_t=x_t \theta(z_t)= x_t \cdot (z_t-0.25)^2+u_t$,其中$x_t=x_{t-1}+v_t$即$x_t$非平稳,$y_t$和$z_t$平稳，$u_t,v_t \sim N(0,1)$
也就是系数$\theta$是可变的，利用B样条估计各个变系数
估计系数,因为目前拟合结果是随lambda单调的，所以先给定lambda
参数说明：
y：因变量；x：自变量；z：辅助变量
p：B样条的次数；nknots：选取的节点个数
m：惩罚项阶数；lambda：惩罚项系数
cv=c("gcv","AIC","ocv","k-fold cv","MLE","REML"),选择最优lambda的方法
分别是按照gcv、aic、ocv和K折交叉验证来选，后两种方法还在开发中。
sigma_error：扰动项u的协差阵
knotsmethod=c("quantile,range"),样条取节点方法，"quantile"表示按数据的quantile取节点，"range"表示按数据的范围等距离取节点。
```{r}
require(splines)
val <- function(x,z,y,p=3,nknots,m=2,lambda=NULL,cv,sigma_error=diag(1,length(y)),knotsmethod="quantile"){
if(any(is.na(x)) | any(is.na(z)) | any(is.na(y)))
stop("NAs in data!")
n <- length(y)
df1 <- nknots+p
if(knotsmethod=="quantile"){
Psi <- bs(z,df=df1,degree = p)
knot <- attr(Psi,"knots")
}else{
knot<- approx(range(z),n=(nknots+2))$y[-c(1,nknots+2)]
Psi<- bs(z,df=df1,knots = knot)
}
Psi <- as.matrix(Psi[1:n,1:df1])
D <- diag(1,df1)
D[which(D==1)[-df1]+1] <- -1
Dm <- D
for(i in 1:(m-1)){Dm <- D %*% Dm}
Dm <- Dm[-(1:m),]
X <- diag(x) %*% Psi
u <- svd(X)$u
A <- diag(svd(X)$d)
v <- svd(X)$v
M <- solve(A) %*% t(v) %*% (nknots^(2*m-1)*t(Dm) %*% Dm) %*% v %*% solve(A)
c <- eigen(M)$values;w <- eigen(M)$vectors
if(is.null(lambda)){
lamb <- seq(-5,10,0.01)
crit <- rep(NA,length(lamb))
for(i in 1:length(lamb)){
Q <- diag(lamb[i]/(lamb[i]+1/c))
H <- u %*% t(u)- u%*%w %*% Q %*% t(u%*%w)
if(cv=="gcv"){
crit[i] <- sum((y-H %*% y)^2)/(n-sum(diag(H)))^2
}else if(cv=="ocv"){
crit[i] <- sum((y-H %*% y)/(1-diag(H))^2)
}else if(cv=="AIC"){
H0 <- X %*% solve(t(X) %*% X) %*% t(X)
var0 <- var(y-H0 %*% y)
crit[i] <- sum((y-H %*% y)^2)/var0+2*sum(diag(H))-2*n*log(var0)-n*log(2*pi)
}else if(cv=="MLE" | cv=="REML"){
stop("This method is still under development, please use other methods. ")
}else{
k <- as.numeric(strsplit(cv,split = "-")[[1]][1])
if(is.na(k)){
stop("Please choose the correct menthod!")
}else{
warning("Using this method may cost more time, please wait a moment.
And because of the uncertainty of the grouping may lead to different optimal lambda")
}
crit[i] <- valcv(fold=k,x=x,z=z,y=y,p=p,nknots=nknots,m=m,knots = knot,
lambda=lamb[i],sigma_error=diag(1,length(y)),Dm=Dm)
}
}
lambda <- lamb[which.min(crit)]
}
coef <- solve(t(X) %*% solve(sigma_error) %*% X +
lambda*(nknots^(2*m-1))*t(Dm)%*%(Dm)) %*% t(X) %*% solve(sigma_error) %*% y
theta <- Psi %*% coef
fit <- diag(x) %*% theta
residuals <- y-fit
H <- X %*% solve(t(X) %*% solve(sigma_error) %*% X +
lambda*(nknots^(2*m-1))*t(Dm)%*%(Dm)) %*% t(X) %*% solve(sigma_error)
gcv <- sum((y-fit)^2)/(n-sum(diag(H)))^2
var0 <- var(y- X %*% solve(t(X) %*% X) %*% t(X) %*% y)
AIC <- sum((y-fit)^2)/var0+2*sum(diag(H))-2*n*log(var0)-n*log(2*pi)
R-square <- 1- sum((y-fit)^2)/((n-1)*var(y))
Adjusted_R^2 <- 1- (sum((y-fit)^2)/(n-nknots))/((n-1)*var(y)/(n-1))
pvalue <- LRTval(x, z, y,nknots=nknots)$p.value
res <- list(coef=drop(coef),lambda=lambda,GCV=gcv,fit=fit,knots=knots,
theta=theta,residuals=residuals,degree=p,nknots=nknots,
m=m,sigma_error=sigma_error,R-square=R-square,Adjusted_R^2=Adjusted_R^2,
test_pvalue <- pvalue)
class(res) <- "val"
res
}
```
预测
```{r}
valpredict <- function(model,newx,newz){
if(class(model)!="val")
stop("the model must be varying-coffecient!")
n <- length(newx)
p <- model$degree
nknots <- model$nknots
df <- p+nknots
Psi <- bs(newz,df=df,degree = p)
Psi <- as.matrix(Psi[1:n,1:df])
theta <- Psi %*% model$coef
predict <- diag(newx) %*% theta
predict
}
```
检验是否应该用变系数，原假设是常系数
参数跟以上一致，需要注意的的是，这边的基函数是截断幂基函数（论文中证明了并没有影响），常用p=1就可以了。
```{r}
library(lme4)
library(RLRsim)
library(nlme)
LRTval=function(X, Z, Y, p=1,nknots=20,df=1,sigma.output=diag(rep(1, length(Y)))){
n=length(Y)
x1 <- diag(X)
z1 <- rep(1,n)
for (i in 1:p){z1=cbind(z1, z^i)}
myknots = quantile(unique(z),seq(0,1,length=(nknots+2))[-c(1,(nknots+2))])
z2 <- outer(z, myknots, FUN="-")
z3 <- z2*(z2>0)
if(p>1) {z3=z3^p}
A1 <- x1 %*% z1
A2 <- x1 %*% z3
Xnames = paste("X",1:ncol(A1),sep="")
Znames = paste("Z",1:ncol(A2),sep="")
fixed.model = as.formula(paste("DATA.temp ~ -1+",
paste(paste("X",1:ncol(A1),sep=""),collapse="+")))
fixed.model2 = as.formula(paste("DATA.temp ~ -1+",
paste(paste("X",1:(ncol(A1)-df),sep=""),collapse="+")))
random.model = as.formula(paste("~-1+",paste(paste("Z",1:ncol(A2),sep=""),collapse="+")))
DATA.output = as.vector( sigma.output %*% Y )
hat1 = sigma.output %*% A1 ;
hat2 = sigma.output %*% A2
DATA.temp= DATA.output
colnames(hat1)=Xnames
colnames(hat2)=Znames
subject<-rep(1,n)
ALLDATA = data.frame(cbind(subject,DATA.temp, hat1, hat2))
mA = lme(fixed=fixed.model, data=ALLDATA,
random=list(subject = pdIdent(random.model)), method="ML")
m0 = lm(fixed.model2, data=ALLDATA)
obs.LRT = as.numeric(2*(logLik(mA)-logLik(m0)))
pvalue = pchisq(obs.LRT,1,lower.tail = FALSE)
pvalue =ifelse(pvalue <1*10^-16,"<1e-16",pvalue)
res <- list(LRT =obs.LRT,p.value = pvalue)
res
}
```
#举个例子
```{r}
n=500
z <- runif(n)
v <- rnorm(n)
x <- cumsum(v)
theta=function(t){(t-0.25)^2}
u <- rnorm(n)
y <- x*theta(z)+u #实际的y
model <- val(x,z,y,p=3,nknots=20,lambda = 0.01) #指定lambda
model <- val(x,z,y,p=3,nknots=20,cv="gcv") #不指定lambda,根据gcv值选择最优lambda
model$lambda
model$coef #beta
model2 <- val(x,z,y,p=3,nknots=20,cv="AIC") #根据AIC选择
model2$lambda
model3 <- val(x,z,y,p=3,nknots=20,cv="ocv") #根据ocv选择
model3$lambda
model4 <- val(x,z,y,p=3,nknots=20,cv="5-fold cv") #根据5折交叉验证选择
model4$lambda
#想用10折交叉验证的话就用cv=“10-fold cv”
#耗时很长，不建议使用！！
#model$theta #x的变系数，也就是theta(z)
plot(theta(z),model$theta)
#检验是否应该用这模型
LRTval(x, z, y, p=1,nknots=20,df=1)
#预测
z2 <- runif(100)
v2 <- rnorm(100)
x2 <- cumsum(v2)
pre<- valpredict(model,newx = x2,newz = z2)
```