---
title: "Comparison with vanilla PELT"
output: rmarkdown::html_vignette
description: |
  Speed comparison with vanilla PELT algorithm.
vignette: >
  %\VignetteIndexEntry{Comparison with vanilla PELT}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


# Setup

## Logistic regression


``` r
#' Cost function for Logistic regression, i.e. binomial family in GLM.
#'
#' @param data Data to be used to calculate the cost values. The last column is
#'     the response variable.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return Cost value for the corresponding segment of data.
cost_glm_binomial <- function(data, family = "binomial") {
  data <- as.matrix(data)
  p <- dim(data)[2] - 1
  out <- fastglm::fastglm(
    as.matrix(data[, 1:p]), data[, p + 1],
    family = family
  )
  return(out$deviance / 2)
}

#' Implementation of vanilla PELT for logistic regression type data.
#'
#' @param data Data to be used for change point detection.
#' @param beta Penalty coefficient for the number of change points.
#' @param cost Cost function to be used to calculate cost values.
#' @keywords internal
#'
#' @noRd
#' @return A list consisting of two: change point locations and negative log
#'     likelihood values for each segment.
pelt_vanilla_binomial <- function(data, beta, cost = cost_glm_binomial) {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)
  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)
    for (i in 1:m)
    {
      k <- set[i] + 1
      cval[i] <- 0
      if (t - k >= p - 1) cval[i] <- suppressWarnings(cost(data[k:t, ]))
    }
    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    Fobj <- c(Fobj, min_val)
  }
  cp <- cp_set[[n + 1]]
  nLL <- 0
  cp_loc <- unique(c(0, cp, n))
  for (i in 1:(length(cp_loc) - 1))
  {
    seg <- (cp_loc[i] + 1):cp_loc[i + 1]
    data_seg <- data[seg, ]
    out <- fastglm::fastglm(
      as.matrix(data_seg[, 1:p]), data_seg[, p + 1],
      family = "binomial"
    )
    nLL <- out$deviance / 2 + nLL
  }

  output <- list(cp, nLL)
  names(output) <- c("cp", "nLL")
  return(output)
}

#' Function to update the coefficients using gradient descent.
#'
#' @param data_new New data point used to update the coeffient.
#' @param coef Previous coeffient to be updated.
#' @param cum_coef Summation of all the past coefficients to be used in
#'     averaging.
#' @param cmatrix Hessian matrix in gradient descent.
#' @param epsilon Small adjustment to avoid singularity when doing inverse on
#'     the Hessian matrix.
#' @keywords internal
#'
#' @noRd
#' @return A list of values containing the new coefficients, summation of
#'     coefficients so far and all the Hessian matrices.
cost_logistic_update <- function(
    data_new, coef, cum_coef, cmatrix, epsilon = 1e-10) {
  p <- length(data_new) - 1
  X_new <- data_new[1:p]
  Y_new <- data_new[p + 1]
  eta <- X_new %*% coef
  mu <- 1 / (1 + exp(-eta))
  cmatrix <- cmatrix + (X_new %o% X_new) * as.numeric((1 - mu) * mu)
  lik_dev <- as.numeric(-(Y_new - mu)) * X_new
  coef <- coef - solve(cmatrix + epsilon * diag(1, p), lik_dev)
  cum_coef <- cum_coef + coef
  return(list(coef, cum_coef, cmatrix))
}

#' Calculate negative log likelihood given data segment and guess of
#' coefficient.
#'
#' @param data Data to be used to calculate the negative log likelihood.
#' @param b Guess of the coefficient.
#' @keywords internal
#'
#' @noRd
#' @return Negative log likelihood.
neg_log_lik_binomial <- function(data, b) {
  p <- dim(data)[2] - 1
  X <- data[, 1:p, drop = FALSE]
  Y <- data[, p + 1, drop = FALSE]
  u <- as.numeric(X %*% b)
  L <- -Y * u + log(1 + exp(u))
  return(sum(L))
}

#' Find change points using dynamic programming with pruning and SeGD.
#'
#' @param data Data used to find change points.
#' @param beta Penalty coefficient for the number of change points.
#' @param B Initial guess on the number of change points.
#' @param trim Propotion of the data to ignore the change points at the
#'     beginning, ending and between change points.
#' @keywords internal
#'
#' @noRd
#' @return A list containing potential change point locations and negative log
#'     likelihood for each segment based on the change points guess.
segd_binomial <- function(data, beta, B = 10, trim = 0.025) {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)

  # choose the initial values based on pre-segmentation

  index <- rep(1:B, rep(n / B, B))
  coef.int <- matrix(NA, B, p)
  for (i in 1:B)
  {
    out <- fastglm::fastglm(
      as.matrix(data[index == i, 1:p]),
      data[index == i, p + 1],
      family = "binomial"
    )
    coef.int[i, ] <- coef(out)
  }
  X1 <- data[1, 1:p]
  cum_coef <- coef <- matrix(coef.int[1, ], p, 1)
  e_eta <- exp(coef %*% X1)
  const <- e_eta / (1 + e_eta)^2
  cmatrix <- array((X1 %o% X1) * as.numeric(const), c(p, p, 1))

  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)

    for (i in 1:(m - 1))
    {
      coef_c <- coef[, i]
      cum_coef_c <- cum_coef[, i]
      cmatrix_c <- cmatrix[, , i]
      out <- cost_logistic_update(data[t, ], coef_c, cum_coef_c, cmatrix_c)
      coef[, i] <- out[[1]]
      cum_coef[, i] <- out[[2]]
      cmatrix[, , i] <- out[[3]]
      k <- set[i] + 1
      cval[i] <- 0
      if (t - k >= p - 1) {
        cval[i] <-
          neg_log_lik_binomial(data[k:t, ], cum_coef[, i] / (t - k + 1))
      }
    }

    # the choice of initial values requires further investigation

    cval[m] <- 0
    Xt <- data[t, 1:p]
    cum_coef_add <- coef_add <- coef.int[index[t], ]
    e_eta_t <- exp(coef_add %*% Xt)
    const <- e_eta_t / (1 + e_eta_t)^2
    cmatrix_add <- (Xt %o% Xt) * as.numeric(const)

    coef <- cbind(coef, coef_add)
    cum_coef <- cbind(cum_coef, cum_coef_add)
    cmatrix <- abind::abind(cmatrix, cmatrix_add, along = 3)

    # Adding a momentum term (TBD)

    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    coef <- coef[, ind2, drop = FALSE]
    cum_coef <- cum_coef[, ind2, drop = FALSE]
    cmatrix <- cmatrix[, , ind2, drop = FALSE]
    Fobj <- c(Fobj, min_val)
  }

  # Remove change-points close to the boundaries

  cp <- cp_set[[n + 1]]
  if (length(cp) > 0) {
    ind3 <- (seq_len(length(cp)))[(cp < trim * n) | (cp > (1 - trim) * n)]
    cp <- cp[-ind3]
  }

  nLL <- 0
  cp_loc <- unique(c(0, cp, n))
  for (i in 1:(length(cp_loc) - 1))
  {
    seg <- (cp_loc[i] + 1):cp_loc[i + 1]
    data_seg <- data[seg, ]
    out <- fastglm::fastglm(
      as.matrix(data_seg[, 1:p]), data_seg[, p + 1],
      family = "binomial"
    )
    nLL <- out$deviance / 2 + nLL
  }

  output <- list(cp, nLL)
  names(output) <- c("cp", "nLL")
  return(output)
}
```

## Poisson regression


``` r
#' Cost function for Poisson regression.
#'
#' @param data Data to be used to calculate the cost values. The last column is
#'     the response variable.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return Cost value for the corresponding segment of data.
cost_glm_poisson <- function(data, family = "poisson") {
  data <- as.matrix(data)
  p <- dim(data)[2] - 1
  out <- fastglm::fastglm(as.matrix(data[, 1:p]), data[, p + 1], family = family)
  return(out$deviance / 2)
}

#' Implementation of vanilla PELT for poisson regression type data.
#'
#' @param data Data to be used for change point detection.
#' @param beta Penalty coefficient for the number of change points.
#' @param cost Cost function to be used to calculate cost values.
#' @keywords internal
#'
#' @noRd
#' @return A list consisting of two: change point locations and negative log
#'     likelihood values for each segment.
pelt_vanilla_poisson <- function(data, beta, cost = cost_glm_poisson) {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)
  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)
    for (i in 1:m)
    {
      k <- set[i] + 1
      if (t - k >= p - 1) cval[i] <- suppressWarnings(cost(data[k:t, ])) else cval[i] <- 0
    }
    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    Fobj <- c(Fobj, min_val)
    # if (t %% 100 == 0) print(t)
  }
  cp <- cp_set[[n + 1]]
  output <- list(cp)
  names(output) <- c("cp")

  return(output)
}

#' Function to update the coefficients using gradient descent.
#'
#' @param data_new New data point used to update the coeffient.
#' @param coef Previous coeffient to be updated.
#' @param cum_coef Summation of all the past coefficients to be used in
#'     averaging.
#' @param cmatrix Hessian matrix in gradient descent.
#' @param epsilon Small adjustment to avoid singularity when doing inverse on
#'     the Hessian matrix.
#' @param G Upper bound for the coefficient.
#' @param L Winsorization lower bound.
#' @param H Winsorization upper bound.
#' @keywords internal
#'
#' @noRd
#' @return A list of values containing the new coefficients, summation of
#'     coefficients so far and all the Hessian matrices.
cost_poisson_update <- function(data_new, coef, cum_coef, cmatrix, epsilon = 0.001, G = 10^10, L = -20, H = 20) {
  p <- length(data_new) - 1
  X_new <- data_new[1:p]
  Y_new <- data_new[p + 1]
  eta <- X_new %*% coef
  mu <- exp(eta)
  cmatrix <- cmatrix + (X_new %o% X_new) * min(as.numeric(mu), G)
  lik_dev <- as.numeric(-(Y_new - mu)) * X_new
  coef <- coef - solve(cmatrix + epsilon * diag(1, p), lik_dev)
  coef <- pmin(pmax(coef, L), H)
  cum_coef <- cum_coef + coef
  return(list(coef, cum_coef, cmatrix))
}

#' Calculate negative log likelihood given data segment and guess of
#' coefficient.
#'
#' @param data Data to be used to calculate the negative log likelihood.
#' @param b Guess of the coefficient.
#' @keywords internal
#'
#' @noRd
#' @return Negative log likelihood.
neg_log_lik_poisson <- function(data, b) {
  p <- dim(data)[2] - 1
  X <- data[, 1:p, drop = FALSE]
  Y <- data[, p + 1, drop = FALSE]
  u <- as.numeric(X %*% b)
  L <- -Y * u + exp(u) + lfactorial(Y)
  return(sum(L))
}

#' Find change points using dynamic programming with pruning and SeGD.
#'
#' @param data Data used to find change points.
#' @param beta Penalty coefficient for the number of change points.
#' @param B Initial guess on the number of change points.
#' @param trim Propotion of the data to ignore the change points at the
#'     beginning, ending and between change points.
#' @param epsilon Small adjustment to avoid singularity when doing inverse on
#'    the Hessian matrix.
#' @param G Upper bound for the coefficient.
#' @param L Winsorization lower bound.
#' @param H Winsorization upper bound.
#' @keywords internal
#'
#' @noRd
#' @return A list containing potential change point locations and negative log
#'     likelihood for each segment based on the change points guess.
segd_poisson <- function(data, beta, B = 10, trim = 0.03, epsilon = 0.001, G = 10^10, L = -20, H = 20) {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)

  # choose the initial values based on pre-segmentation

  index <- rep(1:B, rep(n / B, B))
  coef.int <- matrix(NA, B, p)
  for (i in 1:B)
  {
    out <- fastglm::fastglm(x = as.matrix(data[index == i, 1:p]), y = data[index == i, p + 1], family = "poisson")
    coef.int[i, ] <- coef(out)
  }
  X1 <- data[1, 1:p]
  cum_coef <- coef <- pmin(pmax(matrix(coef.int[1, ], p, 1), L), H)
  e_eta <- exp(coef %*% X1)
  const <- e_eta
  cmatrix <- array((X1 %o% X1) * as.numeric(const), c(p, p, 1))

  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)
    for (i in 1:(m - 1))
    {
      coef_c <- coef[, i]
      cum_coef_c <- cum_coef[, i]
      cmatrix_c <- cmatrix[, , i]
      out <- cost_poisson_update(data[t, ], coef_c, cum_coef_c, cmatrix_c, epsilon = epsilon, G = G, L = L, H = H)
      coef[, i] <- out[[1]]
      cum_coef[, i] <- out[[2]]
      cmatrix[, , i] <- out[[3]]
      k <- set[i] + 1
      cum_coef_win <- pmin(pmax(cum_coef[, i] / (t - k + 1), L), H)
      if (t - k >= p - 1) cval[i] <- neg_log_lik_poisson(data[k:t, ], cum_coef_win) else cval[i] <- 0
    }

    # the choice of initial values requires further investigation

    cval[m] <- 0
    Xt <- data[t, 1:p]
    cum_coef_add <- coef_add <- pmin(pmax(coef.int[index[t], ], L), H)
    e_eta_t <- exp(coef_add %*% Xt)
    const <- e_eta_t
    cmatrix_add <- (Xt %o% Xt) * as.numeric(const)
    coef <- cbind(coef, coef_add)
    cum_coef <- cbind(cum_coef, cum_coef_add)
    cmatrix <- abind::abind(cmatrix, cmatrix_add, along = 3)

    # Adding a momentum term (TBD)

    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    coef <- coef[, ind2, drop = FALSE]
    cum_coef <- cum_coef[, ind2, drop = FALSE]
    cmatrix <- cmatrix[, , ind2, drop = FALSE]
    Fobj <- c(Fobj, min_val)
  }

  # Remove change-points close to the boundaries

  cp <- cp_set[[n + 1]]
  if (length(cp) > 0) {
    ind3 <- (1:length(cp))[(cp < trim * n) | (cp > (1 - trim) * n)]
    if (length(ind3) > 0) cp <- cp[-ind3]
  }

  cp <- sort(unique(c(0, cp)))
  index <- which((diff(cp) < trim * n) == TRUE)
  if (length(index) > 0) cp <- floor((cp[-(index + 1)] + cp[-index]) / 2)
  cp <- cp[cp > 0]

  # nLL <- 0
  # cp_loc <- unique(c(0,cp,n))
  # for(i in 1:(length(cp_loc)-1))
  # {
  #   seg <- (cp_loc[i]+1):cp_loc[i+1]
  #   data_seg <- data[seg,]
  #   out <- fastglm(as.matrix(data_seg[, 1:p]), data_seg[, p+1], family="Poisson")
  #   nLL <- out$deviance/2 + nLL
  # }

  # output <- list(cp, nLL)
  # names(output) <- c("cp", "nLL")

  output <- list(cp)
  names(output) <- c("cp")

  return(output)
}

# Generate data from poisson regression models with change-points
#' @param n Number of observations.
#' @param d Dimension of the covariates.
#' @param true.coef True regression coefficients.
#' @param true.cp.loc True change-point locations.
#' @param Sigma Covariance matrix of the covariates.
#' @keywords internal
#'
#' @noRd
#' @return A list containing the generated data and the true cluster
#'    assignments.
data_gen_poisson <- function(n, d, true.coef, true.cp.loc, Sigma) {
  loc <- unique(c(0, true.cp.loc, n))
  if (dim(true.coef)[2] != length(loc) - 1) stop("true.coef and true.cp.loc do not match")
  x <- mvtnorm::rmvnorm(n, mean = rep(0, d), sigma = Sigma)
  y <- NULL
  for (i in 1:(length(loc) - 1))
  {
    mu <- exp(x[(loc[i] + 1):loc[i + 1], , drop = FALSE] %*% true.coef[, i, drop = FALSE])
    group <- rpois(length(mu), mu)
    y <- c(y, group)
  }
  data <- cbind(x, y)
  true_cluster <- rep(1:(length(loc) - 1), diff(loc))
  result <- list(data, true_cluster)
  return(result)
}
```

## Penalized linear regression


``` r
#' Cost function for penalized linear regression.
#'
#' @param data Data to be used to calculate the cost values. The last column is
#'     the response variable.
#' @param lambda Penalty coefficient.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return Cost value for the corresponding segment of data.
cost_lasso <- function(data, lambda, family = "gaussian") {
  data <- as.matrix(data)
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  out <- glmnet::glmnet(as.matrix(data[, 1:p]), data[, p + 1], family = family, lambda = lambda)
  return(deviance(out) / 2)
}

#' Implementation of vanilla PELT for penalized linear regression type data.
#'
#' @param data Data to be used for change point detection.
#' @param beta Penalty coefficient for the number of change points.
#' @param B Initial guess on the number of change points.
#' @param cost Cost function to be used to calculate cost values.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return A list consisting of two: change point locations and negative log
#'     likelihood values for each segment.
pelt_vanilla_lasso <- function(data, beta, B = 10, cost = cost_lasso, family = "gaussian") {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  index <- rep(1:B, rep(n / B, B))
  err_sd <- act_num <- rep(NA, B)
  for (i in 1:B)
  {
    cvfit <- glmnet::cv.glmnet(as.matrix(data[index == i, 1:p]), data[index == i, p + 1], family = family)
    coef <- coef(cvfit, s = "lambda.1se")[-1]
    resi <- data[index == i, p + 1] - as.matrix(data[index == i, 1:p]) %*% as.numeric(coef)
    err_sd[i] <- sqrt(mean(resi^2))
    act_num[i] <- sum(abs(coef) > 0)
  }
  err_sd_mean <- mean(err_sd) # only works if error sd is unchanged.
  act_num_mean <- mean(act_num)
  beta <- (act_num_mean + 1) * beta # seems to work but there might be better choices

  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)
  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)
    for (i in 1:m)
    {
      k <- set[i] + 1
      if (t - k >= 1) cval[i] <- suppressWarnings(cost(data[k:t, ], lambda = err_sd_mean * sqrt(2 * log(p) / (t - k + 1)))) else cval[i] <- 0
    }
    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    Fobj <- c(Fobj, min_val)
    if (t %% 100 == 0) print(t)
  }
  cp <- cp_set[[n + 1]]
  # nLL <- 0
  # cp_loc <- unique(c(0,cp,n))
  # for(i in 1:(length(cp_loc)-1))
  # {
  #  seg <- (cp_loc[i]+1):cp_loc[i+1]
  #  data_seg <- data[seg,]
  #  out <- glmnet(as.matrix(data_seg[, 1:p]), data_seg[, p+1], lambda=lambda, family=family)
  #  nLL <- deviance(out)/2 + nLL
  # }
  # output <- list(cp, nLL)
  output <- list(cp)
  names(output) <- c("cp")
  return(output)
}

#' Function to update the coefficients using gradient descent.
#' @param a Coefficient to be updated.
#' @param lambda Penalty coefficient.
#' @keywords internal
#'
#' @noRd
#' @return Updated coefficient.
soft_threshold <- function(a, lambda) {
  sign(a) * pmax(abs(a) - lambda, 0)
}

#' Function to update the coefficients using gradient descent.
#'
#' @param data_new New data point used to update the coeffient.
#' @param coef Previous coeffient to be updated.
#' @param cum_coef Summation of all the past coefficients to be used in
#'     averaging.
#' @param cmatrix Hessian matrix in gradient descent.
#' @param lambda Penalty coefficient.
#' @keywords internal
#'
#' @noRd
#' @return A list of values containing the new coefficients, summation of
#'     coefficients so far and all the Hessian matrices.
cost_lasso_update <- function(data_new, coef, cum_coef, cmatrix, lambda) {
  p <- length(data_new) - 1
  X_new <- data_new[1:p]
  Y_new <- data_new[p + 1]
  mu <- X_new %*% coef
  cmatrix <- cmatrix + X_new %o% X_new
  # B <- as.vector(cmatrix_inv%*%X_new)
  # cmatrix_inv <- cmatrix_inv - B%o%B/(1+sum(X_new*B))
  lik_dev <- as.numeric(-(Y_new - mu)) * X_new
  coef <- coef - solve(cmatrix, lik_dev)
  nc <- norm(cmatrix, type = "F") # the choice of norm affects the speed. Spectral norm is more accurate but slower than F norm.
  coef <- soft_threshold(coef, lambda / nc)
  cum_coef <- cum_coef + coef
  return(list(coef, cum_coef, cmatrix))
}

#' Calculate negative log likelihood given data segment and guess of
#' coefficient.
#'
#' @param data Data to be used to calculate the negative log likelihood.
#' @param b Guess of the coefficient.
#' @param lambda Penalty coefficient.
#' @keywords internal
#'
#' @noRd
#' @return Negative log likelihood.
neg_log_lik_lasso <- function(data, b, lambda) {
  p <- dim(data)[2] - 1
  X <- data[, 1:p, drop = FALSE]
  Y <- data[, p + 1, drop = FALSE]
  resi <- Y - X %*% b
  L <- sum(resi^2) / 2 + lambda * sum(abs(b))
  return(L)
}

#' Find change points using dynamic programming with pruning and SeGD.
#'
#' @param data Data used to find change points.
#' @param beta Penalty coefficient for the number of change points.
#' @param B Initial guess on the number of change points.
#' @param trim Propotion of the data to ignore the change points at the
#'     beginning, ending and between change points.
#' @param epsilon Small adjustment to avoid singularity when doing inverse on
#'    the Hessian matrix.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return A list containing potential change point locations and negative log
#'     likelihood for each segment based on the change points guess.
segd_lasso <- function(data, beta, B = 10, trim = 0.025, epsilon = 1e-5, family = "gaussian") {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)

  # choose the initial values based on pre-segmentation

  index <- rep(1:B, rep(n / B, B))
  coef.int <- matrix(NA, B, p)
  err_sd <- act_num <- rep(NA, B)
  for (i in 1:B)
  {
    cvfit <- glmnet::cv.glmnet(as.matrix(data[index == i, 1:p]), data[index == i, p + 1], family = family)
    coef.int[i, ] <- coef(cvfit, s = "lambda.1se")[-1]
    resi <- data[index == i, p + 1] - as.matrix(data[index == i, 1:p]) %*% as.numeric(coef.int[i, ])
    err_sd[i] <- sqrt(mean(resi^2))
    act_num[i] <- sum(abs(coef.int[i, ]) > 0)
  }
  err_sd_mean <- mean(err_sd) # only works if error sd is unchanged.
  act_num_mean <- mean(act_num)
  beta <- (act_num_mean + 1) * beta # seems to work but there might be better choices

  X1 <- data[1, 1:p]
  cum_coef <- coef <- matrix(coef.int[1, ], p, 1)
  eta <- coef %*% X1
  # c_int <- diag(1/epsilon,p) - X1%o%X1/epsilon^2/(1+sum(X1^2)/epsilon)
  # cmatrix_inv <- array(c_int, c(p,p,1))
  cmatrix <- array(X1 %o% X1 + epsilon * diag(1, p), c(p, p, 1))

  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)

    for (i in 1:(m - 1))
    {
      coef_c <- coef[, i]
      cum_coef_c <- cum_coef[, i]
      # cmatrix_inv_c <- cmatrix_inv[,,i]
      cmatrix_c <- cmatrix[, , i]
      k <- set[i] + 1
      out <- cost_lasso_update(data[t, ], coef_c, cum_coef_c, cmatrix_c, lambda = err_sd_mean * sqrt(2 * log(p) / (t - k + 1)))
      coef[, i] <- out[[1]]
      cum_coef[, i] <- out[[2]]
      # cmatrix_inv[,,i] <- out[[3]]
      cmatrix[, , i] <- out[[3]]
      if (t - k >= 2) cval[i] <- neg_log_lik_lasso(data[k:t, ], cum_coef[, i] / (t - k + 1), lambda = err_sd_mean * sqrt(2 * log(p) / (t - k + 1))) else cval[i] <- 0
    }

    # the choice of initial values requires further investigation

    cval[m] <- 0
    Xt <- data[t, 1:p]
    cum_coef_add <- coef_add <- coef.int[index[t], ]
    # cmatrix_inv_add <- diag(1/epsilon,p) - Xt%o%Xt/epsilon^2/(1+sum(Xt^2)/epsilon)

    coef <- cbind(coef, coef_add)
    cum_coef <- cbind(cum_coef, cum_coef_add)
    # cmatrix_inv <- abind::abind(cmatrix_inv, cmatrix_inv_add, along=3)
    cmatrix <- abind::abind(cmatrix, Xt %o% Xt + epsilon * diag(1, p), along = 3)

    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    coef <- coef[, ind2, drop = FALSE]
    cum_coef <- cum_coef[, ind2, drop = FALSE]
    cmatrix <- cmatrix[, , ind2, drop = FALSE]
    # cmatrix_inv <- cmatrix_inv[,,ind2,drop=FALSE]
    Fobj <- c(Fobj, min_val)
  }

  # Remove change-points close to the boundaries and merge change-points

  cp <- cp_set[[n + 1]]
  if (length(cp) > 0) {
    ind3 <- (1:length(cp))[(cp < trim * n) | (cp > (1 - trim) * n)]
    if (length(ind3) > 0) cp <- cp[-ind3]
  }

  cp <- sort(unique(c(0, cp)))
  index <- which((diff(cp) < trim * n) == TRUE)
  if (length(index) > 0) cp <- floor((cp[-(index + 1)] + cp[-index]) / 2)
  cp <- cp[cp > 0]

  # nLL <- 0
  # cp_loc <- unique(c(0,cp,n))
  # for(i in 1:(length(cp_loc)-1))
  # {
  #  seg <- (cp_loc[i]+1):cp_loc[i+1]
  #  data_seg <- data[seg,]
  #  out <- fastglm(as.matrix(data_seg[, 1:p]), data_seg[, p+1], family="binomial")
  #  nLL <- out$deviance/2 + nLL
  # }

  output <- list(cp)
  names(output) <- c("cp")
  return(output)
}

# Generate data from penalized linear regression models with change-points
#' @param n Number of observations.
#' @param d Dimension of the covariates.
#' @param true.coef True regression coefficients.
#' @param true.cp.loc True change-point locations.
#' @param Sigma Covariance matrix of the covariates.
#' @param evar Error variance.
#' @keywords internal
#'
#' @noRd
#' @return A list containing the generated data and the true cluster
#'    assignments.
data_gen_lasso <- function(n, d, true.coef, true.cp.loc, Sigma, evar) {
  loc <- unique(c(0, true.cp.loc, n))
  if (dim(true.coef)[2] != length(loc) - 1) stop("true.coef and true.cp.loc do not match")
  x <- mvtnorm::rmvnorm(n, mean = rep(0, d), sigma = Sigma)
  y <- NULL
  for (i in 1:(length(loc) - 1))
  {
    Xb <- x[(loc[i] + 1):loc[i + 1], , drop = FALSE] %*% true.coef[, i, drop = FALSE]
    add <- Xb + rnorm(length(Xb), sd = sqrt(evar))
    y <- c(y, add)
  }
  data <- cbind(x, y)
  true_cluster <- rep(1:(length(loc) - 1), diff(loc))
  result <- list(data, true_cluster)
  return(result)
}
```

# Logistic regression


``` r
set.seed(1)
p <- 5
x <- matrix(rnorm(300 * p, 0, 1), ncol = p)

# Randomly generate coefficients with different means.
theta <- rbind(rnorm(p, 0, 1), rnorm(p, 2, 1))

# Randomly generate response variables based on the segmented data and
# corresponding coefficients
y <- c(
  rbinom(125, 1, 1 / (1 + exp(-x[1:125, ] %*% theta[1, ]))),
  rbinom(300 - 125, 1, 1 / (1 + exp(-x[(125 + 1):300, ] %*% theta[2, ])))
)

segd_binomial(cbind(x, y), (p + 1) * log(300) / 2, B = 5)$cp
#> [1] 125

fastcpd.binomial(
  cbind(y, x),
  segment_count = 5,
  beta = "BIC",
  cost_adjustment = "BIC",
  r.progress = FALSE
)@cp_set
#> [1] 125

pelt_vanilla_binomial(cbind(x, y), (p + 1) * log(300) / 2)$cp
#> [1]   0 125

fastcpd.binomial(
  cbind(y, x),
  segment_count = 5,
  vanilla_percentage = 1,
  beta = "BIC",
  cost_adjustment = "BIC",
  r.progress = FALSE
)@cp_set
#> [1] 125
```

# Poisson regression


``` r
set.seed(1)
n <- 1500
d <- 5
rho <- 0.9
Sigma <- array(0, c(d, d))
for (i in 1:d) {
  Sigma[i, ] <- rho^(abs(i - (1:d)))
}
delta <- c(5, 7, 9, 11, 13)
a.sq <- 1
delta.new <-
  delta * sqrt(a.sq) / sqrt(as.numeric(t(delta) %*% Sigma %*% delta))
true.cp.loc <- c(375, 750, 1125)

# regression coefficients
true.coef <- matrix(0, nrow = d, ncol = length(true.cp.loc) + 1)
true.coef[, 1] <- c(1, 1.2, -1, 0.5, -2)
true.coef[, 2] <- true.coef[, 1] + delta.new
true.coef[, 3] <- true.coef[, 1]
true.coef[, 4] <- true.coef[, 3] - delta.new

out <- data_gen_poisson(n, d, true.coef, true.cp.loc, Sigma)
data <- out[[1]]
g_tr <- out[[2]]
beta <- log(n) * (d + 1) / 2

segd_poisson(
  data, beta, trim = 0.03, B = 10, epsilon = 0.001, G = 10^10, L = -20, H = 20
)$cp
#> [1]  380  751 1136 1251

fastcpd.poisson(
  cbind(data[, d + 1], data[, 1:d]),
  beta = beta,
  cost_adjustment = "BIC",
  epsilon = 0.001,
  segment_count = 10,
  r.progress = FALSE
)@cp_set
#> [1]  380  751 1136 1251

pelt_vanilla_poisson(data, beta)$cp
#> [1]    0  374  752 1133

fastcpd.poisson(
  cbind(data[, d + 1], data[, 1:d]),
  segment_count = 10,
  vanilla_percentage = 1,
  beta = beta,
  cost_adjustment = "BIC",
  r.progress = FALSE
)@cp_set
#> [1]  374  752 1133
```

# Penalized linear regression


``` r
set.seed(1)
n <- 1000
s <- 3
d <- 50
evar <- 0.5
Sigma <- diag(1, d)
true.cp.loc <- c(100, 300, 500, 800, 900)
seg <- length(true.cp.loc) + 1
true.coef <- matrix(rnorm(seg * s), s, seg)
true.coef <- rbind(true.coef, matrix(0, d - s, seg))
out <- data_gen_lasso(n, d, true.coef, true.cp.loc, Sigma, evar)
data <- out[[1]]
beta <- log(n) / 2 # beta here has different meaning

segd_lasso(data, beta, B = 10, trim = 0.025)$cp
#> [1] 100 300 520 800 901

fastcpd.lasso(
  cbind(data[, d + 1], data[, 1:d]),
  epsilon = 1e-5,
  beta = beta,
  cost_adjustment = "BIC",
  pruning_coef = 0,
  r.progress = FALSE
)@cp_set
#> [1] 100 300 520 800 901

pelt_vanilla_lasso(data, beta, cost = cost_lasso)$cp
#> [1] 100
#> [1] 200
#> [1] 300
#> [1] 400
#> [1] 500
#> [1] 600
#> [1] 700
#> [1] 800
#> [1] 900
#> [1] 1000
#> [1]   0 103 299 510 800 900

fastcpd.lasso(
  cbind(data[, d + 1], data[, 1:d]),
  vanilla_percentage = 1,
  epsilon = 1e-5,
  beta = beta,
  cost_adjustment = "BIC",
  pruning_coef = 0,
  r.progress = FALSE
)@cp_set
#> [1] 103 299 510 800 900
```

# Notes

This document is generated by the following code:

```shell
R -e 'knitr::knit("vignettes/comparison-pelt.Rmd.original", output = "vignettes/comparison-pelt.Rmd")'
```

# Appendix: all code snippets


``` r
knitr::opts_chunk$set(
  collapse = TRUE, comment = "#>", eval = TRUE, cache = FALSE,
  warning = FALSE, fig.width = 8, fig.height = 5
)
library(fastcpd)
#' Cost function for Logistic regression, i.e. binomial family in GLM.
#'
#' @param data Data to be used to calculate the cost values. The last column is
#'     the response variable.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return Cost value for the corresponding segment of data.
cost_glm_binomial <- function(data, family = "binomial") {
  data <- as.matrix(data)
  p <- dim(data)[2] - 1
  out <- fastglm::fastglm(
    as.matrix(data[, 1:p]), data[, p + 1],
    family = family
  )
  return(out$deviance / 2)
}

#' Implementation of vanilla PELT for logistic regression type data.
#'
#' @param data Data to be used for change point detection.
#' @param beta Penalty coefficient for the number of change points.
#' @param cost Cost function to be used to calculate cost values.
#' @keywords internal
#'
#' @noRd
#' @return A list consisting of two: change point locations and negative log
#'     likelihood values for each segment.
pelt_vanilla_binomial <- function(data, beta, cost = cost_glm_binomial) {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)
  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)
    for (i in 1:m)
    {
      k <- set[i] + 1
      cval[i] <- 0
      if (t - k >= p - 1) cval[i] <- suppressWarnings(cost(data[k:t, ]))
    }
    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    Fobj <- c(Fobj, min_val)
  }
  cp <- cp_set[[n + 1]]
  nLL <- 0
  cp_loc <- unique(c(0, cp, n))
  for (i in 1:(length(cp_loc) - 1))
  {
    seg <- (cp_loc[i] + 1):cp_loc[i + 1]
    data_seg <- data[seg, ]
    out <- fastglm::fastglm(
      as.matrix(data_seg[, 1:p]), data_seg[, p + 1],
      family = "binomial"
    )
    nLL <- out$deviance / 2 + nLL
  }

  output <- list(cp, nLL)
  names(output) <- c("cp", "nLL")
  return(output)
}

#' Function to update the coefficients using gradient descent.
#'
#' @param data_new New data point used to update the coeffient.
#' @param coef Previous coeffient to be updated.
#' @param cum_coef Summation of all the past coefficients to be used in
#'     averaging.
#' @param cmatrix Hessian matrix in gradient descent.
#' @param epsilon Small adjustment to avoid singularity when doing inverse on
#'     the Hessian matrix.
#' @keywords internal
#'
#' @noRd
#' @return A list of values containing the new coefficients, summation of
#'     coefficients so far and all the Hessian matrices.
cost_logistic_update <- function(
    data_new, coef, cum_coef, cmatrix, epsilon = 1e-10) {
  p <- length(data_new) - 1
  X_new <- data_new[1:p]
  Y_new <- data_new[p + 1]
  eta <- X_new %*% coef
  mu <- 1 / (1 + exp(-eta))
  cmatrix <- cmatrix + (X_new %o% X_new) * as.numeric((1 - mu) * mu)
  lik_dev <- as.numeric(-(Y_new - mu)) * X_new
  coef <- coef - solve(cmatrix + epsilon * diag(1, p), lik_dev)
  cum_coef <- cum_coef + coef
  return(list(coef, cum_coef, cmatrix))
}

#' Calculate negative log likelihood given data segment and guess of
#' coefficient.
#'
#' @param data Data to be used to calculate the negative log likelihood.
#' @param b Guess of the coefficient.
#' @keywords internal
#'
#' @noRd
#' @return Negative log likelihood.
neg_log_lik_binomial <- function(data, b) {
  p <- dim(data)[2] - 1
  X <- data[, 1:p, drop = FALSE]
  Y <- data[, p + 1, drop = FALSE]
  u <- as.numeric(X %*% b)
  L <- -Y * u + log(1 + exp(u))
  return(sum(L))
}

#' Find change points using dynamic programming with pruning and SeGD.
#'
#' @param data Data used to find change points.
#' @param beta Penalty coefficient for the number of change points.
#' @param B Initial guess on the number of change points.
#' @param trim Propotion of the data to ignore the change points at the
#'     beginning, ending and between change points.
#' @keywords internal
#'
#' @noRd
#' @return A list containing potential change point locations and negative log
#'     likelihood for each segment based on the change points guess.
segd_binomial <- function(data, beta, B = 10, trim = 0.025) {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)

  # choose the initial values based on pre-segmentation

  index <- rep(1:B, rep(n / B, B))
  coef.int <- matrix(NA, B, p)
  for (i in 1:B)
  {
    out <- fastglm::fastglm(
      as.matrix(data[index == i, 1:p]),
      data[index == i, p + 1],
      family = "binomial"
    )
    coef.int[i, ] <- coef(out)
  }
  X1 <- data[1, 1:p]
  cum_coef <- coef <- matrix(coef.int[1, ], p, 1)
  e_eta <- exp(coef %*% X1)
  const <- e_eta / (1 + e_eta)^2
  cmatrix <- array((X1 %o% X1) * as.numeric(const), c(p, p, 1))

  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)

    for (i in 1:(m - 1))
    {
      coef_c <- coef[, i]
      cum_coef_c <- cum_coef[, i]
      cmatrix_c <- cmatrix[, , i]
      out <- cost_logistic_update(data[t, ], coef_c, cum_coef_c, cmatrix_c)
      coef[, i] <- out[[1]]
      cum_coef[, i] <- out[[2]]
      cmatrix[, , i] <- out[[3]]
      k <- set[i] + 1
      cval[i] <- 0
      if (t - k >= p - 1) {
        cval[i] <-
          neg_log_lik_binomial(data[k:t, ], cum_coef[, i] / (t - k + 1))
      }
    }

    # the choice of initial values requires further investigation

    cval[m] <- 0
    Xt <- data[t, 1:p]
    cum_coef_add <- coef_add <- coef.int[index[t], ]
    e_eta_t <- exp(coef_add %*% Xt)
    const <- e_eta_t / (1 + e_eta_t)^2
    cmatrix_add <- (Xt %o% Xt) * as.numeric(const)

    coef <- cbind(coef, coef_add)
    cum_coef <- cbind(cum_coef, cum_coef_add)
    cmatrix <- abind::abind(cmatrix, cmatrix_add, along = 3)

    # Adding a momentum term (TBD)

    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    coef <- coef[, ind2, drop = FALSE]
    cum_coef <- cum_coef[, ind2, drop = FALSE]
    cmatrix <- cmatrix[, , ind2, drop = FALSE]
    Fobj <- c(Fobj, min_val)
  }

  # Remove change-points close to the boundaries

  cp <- cp_set[[n + 1]]
  if (length(cp) > 0) {
    ind3 <- (seq_len(length(cp)))[(cp < trim * n) | (cp > (1 - trim) * n)]
    cp <- cp[-ind3]
  }

  nLL <- 0
  cp_loc <- unique(c(0, cp, n))
  for (i in 1:(length(cp_loc) - 1))
  {
    seg <- (cp_loc[i] + 1):cp_loc[i + 1]
    data_seg <- data[seg, ]
    out <- fastglm::fastglm(
      as.matrix(data_seg[, 1:p]), data_seg[, p + 1],
      family = "binomial"
    )
    nLL <- out$deviance / 2 + nLL
  }

  output <- list(cp, nLL)
  names(output) <- c("cp", "nLL")
  return(output)
}
#' Cost function for Poisson regression.
#'
#' @param data Data to be used to calculate the cost values. The last column is
#'     the response variable.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return Cost value for the corresponding segment of data.
cost_glm_poisson <- function(data, family = "poisson") {
  data <- as.matrix(data)
  p <- dim(data)[2] - 1
  out <- fastglm::fastglm(as.matrix(data[, 1:p]), data[, p + 1], family = family)
  return(out$deviance / 2)
}

#' Implementation of vanilla PELT for poisson regression type data.
#'
#' @param data Data to be used for change point detection.
#' @param beta Penalty coefficient for the number of change points.
#' @param cost Cost function to be used to calculate cost values.
#' @keywords internal
#'
#' @noRd
#' @return A list consisting of two: change point locations and negative log
#'     likelihood values for each segment.
pelt_vanilla_poisson <- function(data, beta, cost = cost_glm_poisson) {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)
  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)
    for (i in 1:m)
    {
      k <- set[i] + 1
      if (t - k >= p - 1) cval[i] <- suppressWarnings(cost(data[k:t, ])) else cval[i] <- 0
    }
    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    Fobj <- c(Fobj, min_val)
    # if (t %% 100 == 0) print(t)
  }
  cp <- cp_set[[n + 1]]
  output <- list(cp)
  names(output) <- c("cp")

  return(output)
}

#' Function to update the coefficients using gradient descent.
#'
#' @param data_new New data point used to update the coeffient.
#' @param coef Previous coeffient to be updated.
#' @param cum_coef Summation of all the past coefficients to be used in
#'     averaging.
#' @param cmatrix Hessian matrix in gradient descent.
#' @param epsilon Small adjustment to avoid singularity when doing inverse on
#'     the Hessian matrix.
#' @param G Upper bound for the coefficient.
#' @param L Winsorization lower bound.
#' @param H Winsorization upper bound.
#' @keywords internal
#'
#' @noRd
#' @return A list of values containing the new coefficients, summation of
#'     coefficients so far and all the Hessian matrices.
cost_poisson_update <- function(data_new, coef, cum_coef, cmatrix, epsilon = 0.001, G = 10^10, L = -20, H = 20) {
  p <- length(data_new) - 1
  X_new <- data_new[1:p]
  Y_new <- data_new[p + 1]
  eta <- X_new %*% coef
  mu <- exp(eta)
  cmatrix <- cmatrix + (X_new %o% X_new) * min(as.numeric(mu), G)
  lik_dev <- as.numeric(-(Y_new - mu)) * X_new
  coef <- coef - solve(cmatrix + epsilon * diag(1, p), lik_dev)
  coef <- pmin(pmax(coef, L), H)
  cum_coef <- cum_coef + coef
  return(list(coef, cum_coef, cmatrix))
}

#' Calculate negative log likelihood given data segment and guess of
#' coefficient.
#'
#' @param data Data to be used to calculate the negative log likelihood.
#' @param b Guess of the coefficient.
#' @keywords internal
#'
#' @noRd
#' @return Negative log likelihood.
neg_log_lik_poisson <- function(data, b) {
  p <- dim(data)[2] - 1
  X <- data[, 1:p, drop = FALSE]
  Y <- data[, p + 1, drop = FALSE]
  u <- as.numeric(X %*% b)
  L <- -Y * u + exp(u) + lfactorial(Y)
  return(sum(L))
}

#' Find change points using dynamic programming with pruning and SeGD.
#'
#' @param data Data used to find change points.
#' @param beta Penalty coefficient for the number of change points.
#' @param B Initial guess on the number of change points.
#' @param trim Propotion of the data to ignore the change points at the
#'     beginning, ending and between change points.
#' @param epsilon Small adjustment to avoid singularity when doing inverse on
#'    the Hessian matrix.
#' @param G Upper bound for the coefficient.
#' @param L Winsorization lower bound.
#' @param H Winsorization upper bound.
#' @keywords internal
#'
#' @noRd
#' @return A list containing potential change point locations and negative log
#'     likelihood for each segment based on the change points guess.
segd_poisson <- function(data, beta, B = 10, trim = 0.03, epsilon = 0.001, G = 10^10, L = -20, H = 20) {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)

  # choose the initial values based on pre-segmentation

  index <- rep(1:B, rep(n / B, B))
  coef.int <- matrix(NA, B, p)
  for (i in 1:B)
  {
    out <- fastglm::fastglm(x = as.matrix(data[index == i, 1:p]), y = data[index == i, p + 1], family = "poisson")
    coef.int[i, ] <- coef(out)
  }
  X1 <- data[1, 1:p]
  cum_coef <- coef <- pmin(pmax(matrix(coef.int[1, ], p, 1), L), H)
  e_eta <- exp(coef %*% X1)
  const <- e_eta
  cmatrix <- array((X1 %o% X1) * as.numeric(const), c(p, p, 1))

  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)
    for (i in 1:(m - 1))
    {
      coef_c <- coef[, i]
      cum_coef_c <- cum_coef[, i]
      cmatrix_c <- cmatrix[, , i]
      out <- cost_poisson_update(data[t, ], coef_c, cum_coef_c, cmatrix_c, epsilon = epsilon, G = G, L = L, H = H)
      coef[, i] <- out[[1]]
      cum_coef[, i] <- out[[2]]
      cmatrix[, , i] <- out[[3]]
      k <- set[i] + 1
      cum_coef_win <- pmin(pmax(cum_coef[, i] / (t - k + 1), L), H)
      if (t - k >= p - 1) cval[i] <- neg_log_lik_poisson(data[k:t, ], cum_coef_win) else cval[i] <- 0
    }

    # the choice of initial values requires further investigation

    cval[m] <- 0
    Xt <- data[t, 1:p]
    cum_coef_add <- coef_add <- pmin(pmax(coef.int[index[t], ], L), H)
    e_eta_t <- exp(coef_add %*% Xt)
    const <- e_eta_t
    cmatrix_add <- (Xt %o% Xt) * as.numeric(const)
    coef <- cbind(coef, coef_add)
    cum_coef <- cbind(cum_coef, cum_coef_add)
    cmatrix <- abind::abind(cmatrix, cmatrix_add, along = 3)

    # Adding a momentum term (TBD)

    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    coef <- coef[, ind2, drop = FALSE]
    cum_coef <- cum_coef[, ind2, drop = FALSE]
    cmatrix <- cmatrix[, , ind2, drop = FALSE]
    Fobj <- c(Fobj, min_val)
  }

  # Remove change-points close to the boundaries

  cp <- cp_set[[n + 1]]
  if (length(cp) > 0) {
    ind3 <- (1:length(cp))[(cp < trim * n) | (cp > (1 - trim) * n)]
    if (length(ind3) > 0) cp <- cp[-ind3]
  }

  cp <- sort(unique(c(0, cp)))
  index <- which((diff(cp) < trim * n) == TRUE)
  if (length(index) > 0) cp <- floor((cp[-(index + 1)] + cp[-index]) / 2)
  cp <- cp[cp > 0]

  # nLL <- 0
  # cp_loc <- unique(c(0,cp,n))
  # for(i in 1:(length(cp_loc)-1))
  # {
  #   seg <- (cp_loc[i]+1):cp_loc[i+1]
  #   data_seg <- data[seg,]
  #   out <- fastglm(as.matrix(data_seg[, 1:p]), data_seg[, p+1], family="Poisson")
  #   nLL <- out$deviance/2 + nLL
  # }

  # output <- list(cp, nLL)
  # names(output) <- c("cp", "nLL")

  output <- list(cp)
  names(output) <- c("cp")

  return(output)
}

# Generate data from poisson regression models with change-points
#' @param n Number of observations.
#' @param d Dimension of the covariates.
#' @param true.coef True regression coefficients.
#' @param true.cp.loc True change-point locations.
#' @param Sigma Covariance matrix of the covariates.
#' @keywords internal
#'
#' @noRd
#' @return A list containing the generated data and the true cluster
#'    assignments.
data_gen_poisson <- function(n, d, true.coef, true.cp.loc, Sigma) {
  loc <- unique(c(0, true.cp.loc, n))
  if (dim(true.coef)[2] != length(loc) - 1) stop("true.coef and true.cp.loc do not match")
  x <- mvtnorm::rmvnorm(n, mean = rep(0, d), sigma = Sigma)
  y <- NULL
  for (i in 1:(length(loc) - 1))
  {
    mu <- exp(x[(loc[i] + 1):loc[i + 1], , drop = FALSE] %*% true.coef[, i, drop = FALSE])
    group <- rpois(length(mu), mu)
    y <- c(y, group)
  }
  data <- cbind(x, y)
  true_cluster <- rep(1:(length(loc) - 1), diff(loc))
  result <- list(data, true_cluster)
  return(result)
}
#' Cost function for penalized linear regression.
#'
#' @param data Data to be used to calculate the cost values. The last column is
#'     the response variable.
#' @param lambda Penalty coefficient.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return Cost value for the corresponding segment of data.
cost_lasso <- function(data, lambda, family = "gaussian") {
  data <- as.matrix(data)
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  out <- glmnet::glmnet(as.matrix(data[, 1:p]), data[, p + 1], family = family, lambda = lambda)
  return(deviance(out) / 2)
}

#' Implementation of vanilla PELT for penalized linear regression type data.
#'
#' @param data Data to be used for change point detection.
#' @param beta Penalty coefficient for the number of change points.
#' @param B Initial guess on the number of change points.
#' @param cost Cost function to be used to calculate cost values.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return A list consisting of two: change point locations and negative log
#'     likelihood values for each segment.
pelt_vanilla_lasso <- function(data, beta, B = 10, cost = cost_lasso, family = "gaussian") {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  index <- rep(1:B, rep(n / B, B))
  err_sd <- act_num <- rep(NA, B)
  for (i in 1:B)
  {
    cvfit <- glmnet::cv.glmnet(as.matrix(data[index == i, 1:p]), data[index == i, p + 1], family = family)
    coef <- coef(cvfit, s = "lambda.1se")[-1]
    resi <- data[index == i, p + 1] - as.matrix(data[index == i, 1:p]) %*% as.numeric(coef)
    err_sd[i] <- sqrt(mean(resi^2))
    act_num[i] <- sum(abs(coef) > 0)
  }
  err_sd_mean <- mean(err_sd) # only works if error sd is unchanged.
  act_num_mean <- mean(act_num)
  beta <- (act_num_mean + 1) * beta # seems to work but there might be better choices

  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)
  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)
    for (i in 1:m)
    {
      k <- set[i] + 1
      if (t - k >= 1) cval[i] <- suppressWarnings(cost(data[k:t, ], lambda = err_sd_mean * sqrt(2 * log(p) / (t - k + 1)))) else cval[i] <- 0
    }
    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    Fobj <- c(Fobj, min_val)
    if (t %% 100 == 0) print(t)
  }
  cp <- cp_set[[n + 1]]
  # nLL <- 0
  # cp_loc <- unique(c(0,cp,n))
  # for(i in 1:(length(cp_loc)-1))
  # {
  #  seg <- (cp_loc[i]+1):cp_loc[i+1]
  #  data_seg <- data[seg,]
  #  out <- glmnet(as.matrix(data_seg[, 1:p]), data_seg[, p+1], lambda=lambda, family=family)
  #  nLL <- deviance(out)/2 + nLL
  # }
  # output <- list(cp, nLL)
  output <- list(cp)
  names(output) <- c("cp")
  return(output)
}

#' Function to update the coefficients using gradient descent.
#' @param a Coefficient to be updated.
#' @param lambda Penalty coefficient.
#' @keywords internal
#'
#' @noRd
#' @return Updated coefficient.
soft_threshold <- function(a, lambda) {
  sign(a) * pmax(abs(a) - lambda, 0)
}

#' Function to update the coefficients using gradient descent.
#'
#' @param data_new New data point used to update the coeffient.
#' @param coef Previous coeffient to be updated.
#' @param cum_coef Summation of all the past coefficients to be used in
#'     averaging.
#' @param cmatrix Hessian matrix in gradient descent.
#' @param lambda Penalty coefficient.
#' @keywords internal
#'
#' @noRd
#' @return A list of values containing the new coefficients, summation of
#'     coefficients so far and all the Hessian matrices.
cost_lasso_update <- function(data_new, coef, cum_coef, cmatrix, lambda) {
  p <- length(data_new) - 1
  X_new <- data_new[1:p]
  Y_new <- data_new[p + 1]
  mu <- X_new %*% coef
  cmatrix <- cmatrix + X_new %o% X_new
  # B <- as.vector(cmatrix_inv%*%X_new)
  # cmatrix_inv <- cmatrix_inv - B%o%B/(1+sum(X_new*B))
  lik_dev <- as.numeric(-(Y_new - mu)) * X_new
  coef <- coef - solve(cmatrix, lik_dev)
  nc <- norm(cmatrix, type = "F") # the choice of norm affects the speed. Spectral norm is more accurate but slower than F norm.
  coef <- soft_threshold(coef, lambda / nc)
  cum_coef <- cum_coef + coef
  return(list(coef, cum_coef, cmatrix))
}

#' Calculate negative log likelihood given data segment and guess of
#' coefficient.
#'
#' @param data Data to be used to calculate the negative log likelihood.
#' @param b Guess of the coefficient.
#' @param lambda Penalty coefficient.
#' @keywords internal
#'
#' @noRd
#' @return Negative log likelihood.
neg_log_lik_lasso <- function(data, b, lambda) {
  p <- dim(data)[2] - 1
  X <- data[, 1:p, drop = FALSE]
  Y <- data[, p + 1, drop = FALSE]
  resi <- Y - X %*% b
  L <- sum(resi^2) / 2 + lambda * sum(abs(b))
  return(L)
}

#' Find change points using dynamic programming with pruning and SeGD.
#'
#' @param data Data used to find change points.
#' @param beta Penalty coefficient for the number of change points.
#' @param B Initial guess on the number of change points.
#' @param trim Propotion of the data to ignore the change points at the
#'     beginning, ending and between change points.
#' @param epsilon Small adjustment to avoid singularity when doing inverse on
#'    the Hessian matrix.
#' @param family Family of the distribution.
#' @keywords internal
#'
#' @noRd
#' @return A list containing potential change point locations and negative log
#'     likelihood for each segment based on the change points guess.
segd_lasso <- function(data, beta, B = 10, trim = 0.025, epsilon = 1e-5, family = "gaussian") {
  n <- dim(data)[1]
  p <- dim(data)[2] - 1
  Fobj <- c(-beta, 0)
  cp_set <- list(NULL, 0)
  set <- c(0, 1)

  # choose the initial values based on pre-segmentation

  index <- rep(1:B, rep(n / B, B))
  coef.int <- matrix(NA, B, p)
  err_sd <- act_num <- rep(NA, B)
  for (i in 1:B)
  {
    cvfit <- glmnet::cv.glmnet(as.matrix(data[index == i, 1:p]), data[index == i, p + 1], family = family)
    coef.int[i, ] <- coef(cvfit, s = "lambda.1se")[-1]
    resi <- data[index == i, p + 1] - as.matrix(data[index == i, 1:p]) %*% as.numeric(coef.int[i, ])
    err_sd[i] <- sqrt(mean(resi^2))
    act_num[i] <- sum(abs(coef.int[i, ]) > 0)
  }
  err_sd_mean <- mean(err_sd) # only works if error sd is unchanged.
  act_num_mean <- mean(act_num)
  beta <- (act_num_mean + 1) * beta # seems to work but there might be better choices

  X1 <- data[1, 1:p]
  cum_coef <- coef <- matrix(coef.int[1, ], p, 1)
  eta <- coef %*% X1
  # c_int <- diag(1/epsilon,p) - X1%o%X1/epsilon^2/(1+sum(X1^2)/epsilon)
  # cmatrix_inv <- array(c_int, c(p,p,1))
  cmatrix <- array(X1 %o% X1 + epsilon * diag(1, p), c(p, p, 1))

  for (t in 2:n)
  {
    m <- length(set)
    cval <- rep(NA, m)

    for (i in 1:(m - 1))
    {
      coef_c <- coef[, i]
      cum_coef_c <- cum_coef[, i]
      # cmatrix_inv_c <- cmatrix_inv[,,i]
      cmatrix_c <- cmatrix[, , i]
      k <- set[i] + 1
      out <- cost_lasso_update(data[t, ], coef_c, cum_coef_c, cmatrix_c, lambda = err_sd_mean * sqrt(2 * log(p) / (t - k + 1)))
      coef[, i] <- out[[1]]
      cum_coef[, i] <- out[[2]]
      # cmatrix_inv[,,i] <- out[[3]]
      cmatrix[, , i] <- out[[3]]
      if (t - k >= 2) cval[i] <- neg_log_lik_lasso(data[k:t, ], cum_coef[, i] / (t - k + 1), lambda = err_sd_mean * sqrt(2 * log(p) / (t - k + 1))) else cval[i] <- 0
    }

    # the choice of initial values requires further investigation

    cval[m] <- 0
    Xt <- data[t, 1:p]
    cum_coef_add <- coef_add <- coef.int[index[t], ]
    # cmatrix_inv_add <- diag(1/epsilon,p) - Xt%o%Xt/epsilon^2/(1+sum(Xt^2)/epsilon)

    coef <- cbind(coef, coef_add)
    cum_coef <- cbind(cum_coef, cum_coef_add)
    # cmatrix_inv <- abind::abind(cmatrix_inv, cmatrix_inv_add, along=3)
    cmatrix <- abind::abind(cmatrix, Xt %o% Xt + epsilon * diag(1, p), along = 3)

    obj <- cval + Fobj[set + 1] + beta
    min_val <- min(obj)
    ind <- which(obj == min_val)[1]
    cp_set_add <- c(cp_set[[set[ind] + 1]], set[ind])
    cp_set <- append(cp_set, list(cp_set_add))
    ind2 <- (cval + Fobj[set + 1]) <= min_val
    set <- c(set[ind2], t)
    coef <- coef[, ind2, drop = FALSE]
    cum_coef <- cum_coef[, ind2, drop = FALSE]
    cmatrix <- cmatrix[, , ind2, drop = FALSE]
    # cmatrix_inv <- cmatrix_inv[,,ind2,drop=FALSE]
    Fobj <- c(Fobj, min_val)
  }

  # Remove change-points close to the boundaries and merge change-points

  cp <- cp_set[[n + 1]]
  if (length(cp) > 0) {
    ind3 <- (1:length(cp))[(cp < trim * n) | (cp > (1 - trim) * n)]
    if (length(ind3) > 0) cp <- cp[-ind3]
  }

  cp <- sort(unique(c(0, cp)))
  index <- which((diff(cp) < trim * n) == TRUE)
  if (length(index) > 0) cp <- floor((cp[-(index + 1)] + cp[-index]) / 2)
  cp <- cp[cp > 0]

  # nLL <- 0
  # cp_loc <- unique(c(0,cp,n))
  # for(i in 1:(length(cp_loc)-1))
  # {
  #  seg <- (cp_loc[i]+1):cp_loc[i+1]
  #  data_seg <- data[seg,]
  #  out <- fastglm(as.matrix(data_seg[, 1:p]), data_seg[, p+1], family="binomial")
  #  nLL <- out$deviance/2 + nLL
  # }

  output <- list(cp)
  names(output) <- c("cp")
  return(output)
}

# Generate data from penalized linear regression models with change-points
#' @param n Number of observations.
#' @param d Dimension of the covariates.
#' @param true.coef True regression coefficients.
#' @param true.cp.loc True change-point locations.
#' @param Sigma Covariance matrix of the covariates.
#' @param evar Error variance.
#' @keywords internal
#'
#' @noRd
#' @return A list containing the generated data and the true cluster
#'    assignments.
data_gen_lasso <- function(n, d, true.coef, true.cp.loc, Sigma, evar) {
  loc <- unique(c(0, true.cp.loc, n))
  if (dim(true.coef)[2] != length(loc) - 1) stop("true.coef and true.cp.loc do not match")
  x <- mvtnorm::rmvnorm(n, mean = rep(0, d), sigma = Sigma)
  y <- NULL
  for (i in 1:(length(loc) - 1))
  {
    Xb <- x[(loc[i] + 1):loc[i + 1], , drop = FALSE] %*% true.coef[, i, drop = FALSE]
    add <- Xb + rnorm(length(Xb), sd = sqrt(evar))
    y <- c(y, add)
  }
  data <- cbind(x, y)
  true_cluster <- rep(1:(length(loc) - 1), diff(loc))
  result <- list(data, true_cluster)
  return(result)
}
set.seed(1)
p <- 5
x <- matrix(rnorm(300 * p, 0, 1), ncol = p)

# Randomly generate coefficients with different means.
theta <- rbind(rnorm(p, 0, 1), rnorm(p, 2, 1))

# Randomly generate response variables based on the segmented data and
# corresponding coefficients
y <- c(
  rbinom(125, 1, 1 / (1 + exp(-x[1:125, ] %*% theta[1, ]))),
  rbinom(300 - 125, 1, 1 / (1 + exp(-x[(125 + 1):300, ] %*% theta[2, ])))
)

segd_binomial(cbind(x, y), (p + 1) * log(300) / 2, B = 5)$cp

fastcpd.binomial(
  cbind(y, x),
  segment_count = 5,
  beta = "BIC",
  cost_adjustment = "BIC",
  r.progress = FALSE
)@cp_set

pelt_vanilla_binomial(cbind(x, y), (p + 1) * log(300) / 2)$cp

fastcpd.binomial(
  cbind(y, x),
  segment_count = 5,
  vanilla_percentage = 1,
  beta = "BIC",
  cost_adjustment = "BIC",
  r.progress = FALSE
)@cp_set
set.seed(1)
n <- 1500
d <- 5
rho <- 0.9
Sigma <- array(0, c(d, d))
for (i in 1:d) {
  Sigma[i, ] <- rho^(abs(i - (1:d)))
}
delta <- c(5, 7, 9, 11, 13)
a.sq <- 1
delta.new <-
  delta * sqrt(a.sq) / sqrt(as.numeric(t(delta) %*% Sigma %*% delta))
true.cp.loc <- c(375, 750, 1125)

# regression coefficients
true.coef <- matrix(0, nrow = d, ncol = length(true.cp.loc) + 1)
true.coef[, 1] <- c(1, 1.2, -1, 0.5, -2)
true.coef[, 2] <- true.coef[, 1] + delta.new
true.coef[, 3] <- true.coef[, 1]
true.coef[, 4] <- true.coef[, 3] - delta.new

out <- data_gen_poisson(n, d, true.coef, true.cp.loc, Sigma)
data <- out[[1]]
g_tr <- out[[2]]
beta <- log(n) * (d + 1) / 2

segd_poisson(
  data, beta, trim = 0.03, B = 10, epsilon = 0.001, G = 10^10, L = -20, H = 20
)$cp

fastcpd.poisson(
  cbind(data[, d + 1], data[, 1:d]),
  beta = beta,
  cost_adjustment = "BIC",
  epsilon = 0.001,
  segment_count = 10,
  r.progress = FALSE
)@cp_set

pelt_vanilla_poisson(data, beta)$cp

fastcpd.poisson(
  cbind(data[, d + 1], data[, 1:d]),
  segment_count = 10,
  vanilla_percentage = 1,
  beta = beta,
  cost_adjustment = "BIC",
  r.progress = FALSE
)@cp_set
set.seed(1)
n <- 1000
s <- 3
d <- 50
evar <- 0.5
Sigma <- diag(1, d)
true.cp.loc <- c(100, 300, 500, 800, 900)
seg <- length(true.cp.loc) + 1
true.coef <- matrix(rnorm(seg * s), s, seg)
true.coef <- rbind(true.coef, matrix(0, d - s, seg))
out <- data_gen_lasso(n, d, true.coef, true.cp.loc, Sigma, evar)
data <- out[[1]]
beta <- log(n) / 2 # beta here has different meaning

segd_lasso(data, beta, B = 10, trim = 0.025)$cp

fastcpd.lasso(
  cbind(data[, d + 1], data[, 1:d]),
  epsilon = 1e-5,
  beta = beta,
  cost_adjustment = "BIC",
  pruning_coef = 0,
  r.progress = FALSE
)@cp_set

pelt_vanilla_lasso(data, beta, cost = cost_lasso)$cp

fastcpd.lasso(
  cbind(data[, d + 1], data[, 1:d]),
  vanilla_percentage = 1,
  epsilon = 1e-5,
  beta = beta,
  cost_adjustment = "BIC",
  pruning_coef = 0,
  r.progress = FALSE
)@cp_set
```
