Kalman Filter R code reference

[BikashPandey17]

Seems like the Kalman filter has been implemented in R, is it possible to give reference to the mathematical formulae used while implementing it.

Especially in the following code

date[nrow(date) + 1,1] <-all[nrow(all),1]+1
pred_all<-NULL
  for (n in 2:ncol(all)-1) {
    Y<-ts(data = all[n+1], start = 1, end =nrow(all)+1)  
    sig_w<-0.01
    w<-sig_w*randn(1,100) # acceleration which denotes the fluctuation (Q/R) rnorm(100, mean = 0, sd = 1)
    sig_v<-0.01
    v<-sig_v*randn(1,100)   
    t<-0.45
    phi<-matrix(c(1,0,t,1),2,2)
    gama<-matrix(c(0.5*t^2,t),2,1)
    H<-matrix(c(1,0),1,2)
    #Kalman
    x0_0<-p0_0<-matrix(c(0,0),2,1)
    p0_0<-matrix(c(1,0,0,1),2,2)
    Q<-0.01
    R<-0.01
    X<-NULL
    X2<-NULL
    pred<-NULL
    for (i in 0:nrow(all)) {
      namp <-paste("p", i+1,"_",i, sep = "")
      assign(namp, phi%*%(get(paste("p", i,"_",i, sep = "")))%*%t(phi)+gama%*%Q%*%t(gama))
      namk <- paste("k", i+1, sep = "")
      assign(namk,get(paste("p", i+1,"_",i, sep = ""))%*%t(H)%*%(1/(H%*%get(paste("p", i+1,"_",i, sep = ""))%*%t(H)+R)))
      namx <- paste("x", i+1,"_",i, sep = "")
      assign(namx,phi%*%get(paste("x", i,"_",i, sep = "")))
      namE <- paste("E", i+1, sep = "")
      assign(namE,Y[i+1]-H%*%get(paste("x", i+1,"_",i, sep = "")))
      namx2 <- paste("x", i+1,"_",i+1, sep = "")
      assign(namx2,get(paste("x", i+1,"_",i, sep = ""))+get(paste("k", i+1, sep = ""))%*%get(paste("E", i+1, sep = "")))
      namp2 <- paste("p", i+1,"_",i+1, sep = "")
      assign(namp2,(p0_0-get(paste("k", i+1, sep = ""))%*%H)%*%get(paste("p", i+1,"_",i, sep = "")))
      X<-rbind(X,get(paste("x", i+1,"_",i,sep = ""))[1])
      X2<-rbind(X2,get(paste("x", i+1,"_",i,sep = ""))[2])
      if(i>2){
        remove(list=(paste("p", i-1,"_",i-2, sep = "")))
        remove(list=(paste("k", i-1, sep = "")))
        remove(list=(paste("E", i-1, sep = "")))
        remove(list=(paste("p", i-2,"_",i-2, sep = "")))
        remove(list=(paste("x", i-1,"_",i-2, sep = "")))
        remove(list=(paste("x", i-2,"_",i-2, sep = "")))}
    }
    pred<-NULL
    pred<-cbind(Y,X,round(X2,4))
    pred<-as.data.frame(pred)
    pred$region<-colnames(all[,n+1])
    pred$date<-date$date
    pred$actual<-rbind(0,(cbind(pred[2:nrow(pred),1])/pred[1:nrow(pred)-1,1]-1)*100)
    pred$predict<-rbind(0,(cbind(pred[2:nrow(pred),2])/pred[1:nrow(pred)-1,2]-1)*100)
    pred$pred_rate<-(pred$X/pred$Y-1)*100
    pred$X2_change<-rbind(0,(cbind(pred[2:nrow(pred),3]-pred[1:nrow(pred)-1,3])))
    pred_all<-rbind(pred_all,pred)
  }

Thank you