# 多項式近似メモ

```# Function to estimate polynomial coefficients and coordinates estimated
# 多項式近似の係数と、推定座標を返す関数
glmPolynomial<-function(y,x,dg,X){# y:dependent values (従属値列),x:descriptive values (説明変数),dg:degrees of polynomials (多項式の次数)
# Xs: Matrix consisted of x^1,x^2,...x^{dg}
# Xs: xの1,2,...,dg 乗の値でできた行列
Xs<-matrix(0,length(X),dg)
xs <- matrix(0,length(x),dg)
for(i in 1:dg){
Xs[,i]<-X^i
xs[,i] <- x^i
}
# glm() is generalized linear regression function in R
# glm() はＲの一般化線形回帰関数
fit<-glm(y~xs[,1:dg])
# beta: coefficients estimated
# beta: 推定係数ベクトル
beta<-fit\$coefficients
# Xs2 has an additional column for x^0
# Xs2 には、x^0のための列を追加
Xs2<-cbind(rep(1,length(X)),Xs)
# predY is a matrix of values for individual degrees of x
# predY は xの各次数の項
predY<-t(t(Xs2)*beta)
# apply(): Sum up all columns corresponding to 0 to dg degrees
# apply()関数で0:dg次数のすべてを足し合わせる
sumupPredY<-apply(predY,1,sum)
list(beta=beta,x=X,predY=sumupPredY)
}

# Generate data
N <- 7
x<-seq(from=0,to=1,length=N) + rnorm(N)
y<-runif(N)+x^(N+2)*2+sin(x*6*pi)

# GLM is to be applied to multiple degrees (dgs)
# GLMを複数の次数に適用
dgs<-1:(N-1)
X <- seq(from=min(x)-0.2,to=max(x)+0.2,length=1000)
Ys <- matrix(0,length(dgs),length(X))
for(i in 1:length(dgs)){
outglm<-glmPolynomial(y,x,dgs[i],X)
Ys[i,] <- outglm\$predY
}

xlim<-c(min(X),max(X))
ylim<-c(min(y)-10,max(y)+10)
par(mfcol=c(2,3))
for(i in 1:length(dgs)){
plot(X,Ys[i,],xlim=xlim,ylim=ylim,col=dgs,type="l",main=i)
points(x,y,col=2,pch=20,cex=2)
}
par(mfcol=c(1,1))
```