# Custom logistic regression model

    for (package in c("ggplot2", "mvtnorm")) {
      if (!requireNamespace(package, quietly = TRUE)) utils::install.packages(
        package, repos = "https://cloud.r-project.org", quiet = TRUE
      )
    }

We try to reproduce the logistic regression models with custom cost functions and show that the results are similar to the built-in logistic regression models.

The built-in logistic regression model in `fastcpd()` is implemented with the help of the fastglm package. The fastglm package utilizes the iteratively reweighted least squares with the step-halving approach to help safeguard against convergence issues. If a custom cost function is used with gradient descent, we should expect the results will be similar to the built-in logistic regression model.

Specifying the `cost`, `cost_gradient` and `cost_hessian` parameters below, we can obtain similar results as the built-in logistic regression model.

    set.seed(1)
    x <- matrix(rnorm(1500, 0, 1), ncol = 5)
    theta <- rbind(rnorm(5, 0, 1), rnorm(5, 2, 1))
    y <- c(
      rbinom(125, 1, 1 / (1 + exp(-x[1:125, ] %*% theta[1, ]))),
      rbinom(175, 1, 1 / (1 + exp(-x[126:300, ] %*% theta[2, ])))
    )
    binomial_data <- data.frame(y = y, x = x)

    result <- fastcpd.binomial(cbind(y, x), r.progress = FALSE, cost_adjustment = NULL)
    #> Warning: fit_glm: fitted probabilities numerically 0 or 1 occurred

    #> Warning: fit_glm: fitted probabilities numerically 0 or 1 occurred

    #> Warning: fit_glm: fitted probabilities numerically 0 or 1 occurred

    #> Warning: fit_glm: fitted probabilities numerically 0 or 1 occurred
    summary(result)
    #>
    #> Call:
    #> fastcpd.binomial(data = cbind(y, x), r.progress = FALSE, cost_adjustment = NULL)
    #>
    #> Change points:
    #> 126
    #>
    #> Cost values:
    #> 56.90525 30.76875
    #>
    #> Parameters:
    #>    segment 1 segment 2
    #> 1  0.7259293  1.878525
    #> 2 -1.0294802  2.704376
    #> 3  1.0576503  3.702310
    #> 4 -0.8812767  2.258796
    #> 5  0.2419351  2.524173

    logistic_loss <- function(data, theta) {
      x <- data[, -1]
      y <- data[, 1]
      u <- x %*% theta
      nll <- -y * u + log(1 + exp(u))
      nll[u > 10] <- -y[u > 10] * u[u > 10] + u[u > 10]
      sum(nll)
    }
    logistic_gradient <- function(data, theta) {
      x <- data[nrow(data), -1]
      y <- data[nrow(data), 1]
      c(-(y - 1 / (1 + exp(-x %*% theta)))) * x
    }
    logistic_hessian <- function(data, theta) {
      x <- data[nrow(data), -1]
      prob <- 1 / (1 + exp(-x %*% theta))
      (x %o% x) * c((1 - prob) * prob)
    }
    result <- fastcpd(
      y ~ . - 1, binomial_data, epsilon = 1e-5, cost = logistic_loss,
      cost_gradient = logistic_gradient, cost_hessian = logistic_hessian,
      r.progress = FALSE
    )
    summary(result)
    #>
    #> Call:
    #> fastcpd(formula = y ~ . - 1, data = binomial_data, cost = logistic_loss,
    #>     cost_gradient = logistic_gradient, cost_hessian = logistic_hessian,
    #>     epsilon = 1e-05, r.progress = FALSE)
    #>
    #> Change points:
    #> 22 125
    #>
    #> Parameters:
    #>   segment 1  segment 2 segment 3
    #> 1 -59.20045  0.8170446  1.902379
    #> 2 -34.56676 -0.9600438  2.751578
    #> 3 216.53373  0.9353306  3.734179
    #> 4 -80.96420 -0.7393653  2.247423
    #> 5  51.25224  0.1390591  2.535372

Note that the result obtained through custom cost functions is inferior compared to the one obtained through built-in models. We remark that the results can be improved with extra parameters already provided in the package. The detailed discussion of several advanced usages of the package can be found in Advanced examples.

# Appendix: all code snippets

    knitr::opts_chunk$set(collapse = TRUE, comment = "#>", eval = FALSE)
    library(fastcpd)
    for (package in c("ggplot2", "mvtnorm")) {
      if (!requireNamespace(package, quietly = TRUE)) utils::install.packages(
        package, repos = "https://cloud.r-project.org", quiet = TRUE
      )
    }
    set.seed(1)
    x <- matrix(rnorm(1500, 0, 1), ncol = 5)
    theta <- rbind(rnorm(5, 0, 1), rnorm(5, 2, 1))
    y <- c(
      rbinom(125, 1, 1 / (1 + exp(-x[1:125, ] %*% theta[1, ]))),
      rbinom(175, 1, 1 / (1 + exp(-x[126:300, ] %*% theta[2, ])))
    )
    binomial_data <- data.frame(y = y, x = x)

    result <- fastcpd.binomial(cbind(y, x), r.progress = FALSE, cost_adjustment = NULL)
    #> Warning: fit_glm: fitted probabilities numerically 0 or 1 occurred

    #> Warning: fit_glm: fitted probabilities numerically 0 or 1 occurred

    #> Warning: fit_glm: fitted probabilities numerically 0 or 1 occurred

    #> Warning: fit_glm: fitted probabilities numerically 0 or 1 occurred
    summary(result)
    #>
    #> Call:
    #> fastcpd.binomial(data = cbind(y, x), r.progress = FALSE, cost_adjustment = NULL)
    #>
    #> Change points:
    #> 126
    #>
    #> Cost values:
    #> 56.90525 30.76875
    #>
    #> Parameters:
    #>    segment 1 segment 2
    #> 1  0.7259293  1.878525
    #> 2 -1.0294802  2.704376
    #> 3  1.0576503  3.702310
    #> 4 -0.8812767  2.258796
    #> 5  0.2419351  2.524173
    logistic_loss <- function(data, theta) {
      x <- data[, -1]
      y <- data[, 1]
      u <- x %*% theta
      nll <- -y * u + log(1 + exp(u))
      nll[u > 10] <- -y[u > 10] * u[u > 10] + u[u > 10]
      sum(nll)
    }
    logistic_gradient <- function(data, theta) {
      x <- data[nrow(data), -1]
      y <- data[nrow(data), 1]
      c(-(y - 1 / (1 + exp(-x %*% theta)))) * x
    }
    logistic_hessian <- function(data, theta) {
      x <- data[nrow(data), -1]
      prob <- 1 / (1 + exp(-x %*% theta))
      (x %o% x) * c((1 - prob) * prob)
    }
    result <- fastcpd(
      y ~ . - 1, binomial_data, epsilon = 1e-5, cost = logistic_loss,
      cost_gradient = logistic_gradient, cost_hessian = logistic_hessian,
      r.progress = FALSE
    )
    summary(result)
    #>
    #> Call:
    #> fastcpd(formula = y ~ . - 1, data = binomial_data, cost = logistic_loss,
    #>     cost_gradient = logistic_gradient, cost_hessian = logistic_hessian,
    #>     epsilon = 1e-05, r.progress = FALSE)
    #>
    #> Change points:
    #> 22 125
    #>
    #> Parameters:
    #>   segment 1  segment 2 segment 3
    #> 1 -59.20045  0.8170446  1.902379
    #> 2 -34.56676 -0.9600438  2.751578
    #> 3 216.53373  0.9353306  3.734179
    #> 4 -80.96420 -0.7393653  2.247423
    #> 5  51.25224  0.1390591  2.535372