R/utils.R

In exdqlm: Extended Dynamic Quantile Linear Models

#
log.g<-function(gam){	log(2)+stats::pnorm(-abs(gam),log=T)+0.5*gam^2 }
L.fn<-function(p0){ stats::uniroot(function(gam) exp(log.g(gam))-(1-p0), c(-1000,0))$root }
U.fn<-function(p0){ stats::uniroot(function(gam) exp(log.g(gam))-p0, c(0,1000))$root }
p.fn<-function(p0,gam){ (p0-as.numeric(gam<0))/exp(log.g(gam))+as.numeric(gam<0)}
A.fn<-function(p0,gam){ temp.p = p.fn(p0,gam); return((1-2*temp.p)/(temp.p*(1-temp.p))) }
B.fn<-function(p0,gam){ temp.p = p.fn(p0,gam); return((2)/(temp.p*(1-temp.p))) }
C.fn<-function(p0,gam){ temp.p = p.fn(p0,gam); return((as.numeric(gam>0)-temp.p)^(-1)) }
#
CheckLossFn = function(p0,diff){diff*p0 - diff*as.numeric(diff<0)}
#
dlm_df = function(y, model, df, dim.df, s.priors = list(l0=1,S0=10), just.lik=FALSE){
  ### Gets the Time Series Length / Replicate number
  y = check_ts(y)
  TT = nrow(y)
  ### Gets the State Parameter dimension and Prior Distribution Parameters
  m0 = model$m0
  C0 = model$C0
  l0 = s.priors$l0
  S0 = s.priors$S0
  n = length(m0)
  ### Constructs F and G
  FF = model$FF
  GG = model$GG
  ### Variable Saving
  ### Posterior Distribution
  m = matrix(0,TT,n)
  C = array(0,c(TT,n,n))
  ### Predictive State Distribution
  a = matrix(0,TT,n)
  R = array(0,dim = c(TT,n,n))
  P = array(0,dim = c(TT,n,n))
  W = array(0,dim = c(TT,n,n))
  ### One-Step Ahead Forecast
  f = matrix(0,TT,1)
  Q = array(0,c(TT,1,1))
  inv.Q = array(0,c(TT,1,1))
  ### Regression Variables
  e = matrix(0,TT,1)
  A = array(0,c(TT,n,1))
  ### Sample Variance
  S = vector("numeric",TT)
  l = vector("numeric",TT)

  # Prior Dim Check
  m0 = matrix(m0,n,1)
  C0 = matrix(C0,n,n)
  ### Discount Factor Blocking
  df.mat = make_df_mat(df,dim.df,n)

  ### First Update
  ### One-step state forecast
  a[1,]  = GG[,,1] %*% m0
  P[1,,] = GG[,,1] %*% C0 %*% t(GG[,,1])
  W[1,,] = df.mat * P[1,,]
  R[1,,] = P[1,,] + W[1,,]
  ### One-step ahead forecast
  f[1,] = t(FF[,1]) %*% a[1,]
  Q[1,,] = as.matrix(1 + t(FF[,1]) %*% R[1,,] %*% FF[,1],1,1)
  inv.Q[1,,] = chol2inv(chol(Q[1,,]))
  ### Auxilary Variables
  e[1,]  = as.matrix(y[1,] - f[1,],1,1)
  A[1,,] = R[1,,] %*% FF[,1] %*% inv.Q[1,,]
  ### Variance update
  l[1] = l0 + 1
  S[1] = l0 * S0 / l[1] + (t(e[1,]) %*% inv.Q[1,,] %*% e[1,] / l[1])
  ### Posterior Distribution
  m[1,]  = a[1,] + as.matrix(A[1,,],n,1) %*% e[1,]
  C[1,,] = R[1,,] - as.matrix(A[1,,],n,1) %*% Q[1,,] %*% t(A[1,,])
  C[1,,] = (C[1,,] + t(C[1,,]))/2

  for(i in 2:TT){
    ### One-step state forecast
    a[i,]  = GG[,,i] %*% m[i-1,]
    P[i,,] = GG[,,i] %*% C[i-1,,] %*% t(GG[,,i])
    W[i,,] = df.mat * P[i,,]
    R[i,,] = P[i,,] + W[i,,]
    ### One-step ahead forecast
    f[i,] = t(FF[,i]) %*% a[i,]
    Q[i,,] = matrix(1 + t(FF[,i])%*% R[i,,]%*% FF[,i],1,1)
    inv.Q[i,,] = chol2inv(chol(Q[i,,]))
    ### Auxilary Variables
    e[i,]  = as.matrix(y[i,] - f[i,],1,1)
    A[i,,] = as.matrix(R[i,,] %*% FF[,i] %*% inv.Q[i,,],n,1)
    ### Variance update
    l[i] = l[i-1] + 1
    S[i] = l[i-1] * S[i-1] / l[i] + (t(e[i,]) %*% inv.Q[i,,] %*% e[i,] / l[i])
    ### Posterior Distribution
    m[i,]  = a[i,] + as.matrix(A[i,,],n,1) %*% e[i,]
    C[i,,] = R[i,,] - as.matrix(A[i,,],n,1) %*% Q[i,,] %*% t(as.matrix(A[i,,],n,1))
    C[i,,] = (C[i,,] + t(C[i,,]))/2
  }

  ### Adjust By Variance
  R[1,,] = S0 * R[1,,]
  Q[1,,]   = S0 * Q[1,,]
  C[1,,]   = S[1] * C[1,,]
  for(i in 2:TT){
    R[i,,] = S[i-1] * R[i,,]
    Q[i,,]   = S[i-1] * Q[i,,]
    C[i,,]   = S[i] * C[i,,]
  }

  # Calculate Log-Likelihood
  det.Q = log(abs(Q[1,,])) ; llik = lgamma((l0+1)/2)-lgamma(l0/2)-log(pi*l0)/2-det.Q/2-(l0+1)*log(1+t(e[1,])%*%inv.Q[1,,]%*%e[1,]/l0)/2
  for(t in 2:TT){
    det.Q = log(abs(Q[t,,]))
    llik = llik + lgamma((l[t-1]+1)/2)-lgamma(l[t-1]/2)-log(pi*l[t-1])/2-det.Q/2-(l[t-1]+1)*log(1+t(e[t,])%*%inv.Q[t,,]%*%e[t,]/l[t-1])/2
  }
  if(just.lik){
    return(list(llik = llik))
  }

  ## SMOOTHING
  ### Initializes recursive relations
  sa = matrix(0,TT,n)
  sR = array(0, dim = c(TT,n,n))
  ### Runs the recursive equations
  sa[TT,]  = m[TT,]
  sR[TT,,] = C[TT,,]
  for(k in 1:(TT-1)){
  ### Computes the Auxilary recursion Variable B
    B = C[TT-k,,] %*% t(GG[,,i]) %*% solve(R[TT-k+1,,])
    sa[TT-k,] = m[TT-k,] + B %*% (sa[TT-k+1,] - a[TT-k+1,])
    sR[TT-k,,] = C[TT-k,,] + B %*% (sR[TT-k+1,,] - R[TT-k+1,,]) %*% t(B)
  }
  ### Adjusts the variance update
  for(k in 1:TT){
    sR[TT-k,,] = S[TT] * sR[TT-k,,] / S[TT-k]
  }
  return(list(fm = m, fC = C, m = sa, C = sR,model = model, s = S, n = l))
}
#
make_df_mat = function(df,dim.df,n){
  if(sum(dim.df)!=n){ stop("sum of component dimensions given in dim.df does not match m0") }
  if(length(df)!=length(dim.df)){ stop("length of component discount factors does not match length of component dimensions") }
  dfs = rep(df,dim.df)
  n.dfs = length(dim.df)
  ind.dfs = c(0,sapply(1:length(dim.df),function(x){sum(dim.df[1:x])}),n)
  df.mat = matrix(0,n,n)
  for(j in 1:n.dfs){
    df.mat[(ind.dfs[j]+1):ind.dfs[(j+1)],(ind.dfs[j]+1):ind.dfs[(j+1)]] = (1-dfs[ind.dfs[(j+1)]])/dfs[ind.dfs[(j+1)]]
  }
  return(df.mat)
}
#
check_mod = function(model){
  if(dlm::is.dlm(model)){
    model = dlmMod(model)
  }
  if(!is.vector(model$m0)){
    if(ncol(model$m0) != 1){
      stop("m0 must be a vector or a matrix with 1 column")
      }
    }
  p = length(model$m0)
  model$C0 = as.matrix(model$C0)
  if(p != dim(model$C0)[1] & p != dim(model$C0)[2]){
    stop("C0 must be a square matrix matching the dimension of m0")
    }
  if(!all.equal(model$C0, t(model$C0)) | !all(eigen(model$C0)$values >= 0)){
    stop("C0 must be a covariance matrix")
  }
  if(!is.vector(model$FF)){
    if(nrow(model$FF) != p){
      stop("FF must be a vector of length matching the dimension of m0, or a matrix with number of rows matching the dimension of m0")
    }
  }else{
    if(length(model$FF) != p){
      stop("FF must be a vector of length matching the dimension of m0, or a matrix with number of rows matching the dimension of m0")
    }
  }
  if(is.null(dim(model$GG)[3])){
    model$GG = as.matrix(model$GG)
  }else{
    if(is.na(dim(model$GG)[3])){
      model$GG = as.matrix(model$GG)
    }else{
      model$GG = as.array(model$GG)
    }
  }
  if(p != dim(model$GG)[1] & p != dim(model$GG)[2]){
    stop("GG must be a square matrix matching the dimension of m0, or an array with first two dimensions matching the dimension of m0")
  }
  model$m0 = as.matrix(model$m0)
  model$FF = as.matrix(model$FF)
  return(model)
}
#
check_logics = function(gam.init,sig.init,fix.gamma,fix.sigma,dqlm.ind){
  retval <- NULL
  retval$gam.init = gam.init
  retval$fix.gamma = fix.gamma
  retval$dqlm.ind = dqlm.ind
  if(dqlm.ind){
    if(gam.init!=0 | !fix.gamma){
      retval$gam.init <- gam.init <- 0
      retval$fix.gamma <- fix.gamma <- TRUE
    }
  }else{
    if(gam.init==0 && fix.gamma==TRUE){
      retval$dqlm.ind = TRUE
    }
  }
  if(fix.gamma & is.na(gam.init)){ stop("when fix.gamma = TRUE, gam.init must be specified") }
  if(fix.sigma & is.na(sig.init)){ stop("when fix.sigma = TRUE, sig.init must be specified") }
  return(retval)
}
#
check_ts = function(dat){
  dat = as.matrix(dat)
  if(all(dim(dat)>1)){
    stop("data must be univariate time-series")
  }
  if(dim(dat)[1]<dim(dat)[2]){
    dat = t(dat)
  }
  return(invisible(dat))
}
#
is.exdqlm = function(m){ return(inherits(m,"exdqlm")) }