scp: Classical split conformal prediction method

Description

Compute prediction intervals and other information by applying the classical split conformal prediction (SCP) method.

Usage

scp(
  object,
  alpha = 1 - 0.01 * object$level,
  symmetric = FALSE,
  ncal = 10,
  rolling = FALSE,
  quantiletype = 1,
  weightfun = NULL,
  kess = FALSE,
  update = FALSE,
  na.rm = TRUE,
  ...
)

Value

A list of class c("scp", "cvforecast", "forecast")

with the following components:

x: The original time series.
series: The name of the series x.
xreg: Exogenous predictor variables used, if applicable.
method: A character string "scp".
cp_times: The number of times the conformal prediction is performed in cross-validation.
MEAN: Point forecasts as a multivariate time series, where the \(h\)th column holds the point forecasts for forecast horizon \(h\). The time index corresponds to the period for which the forecast is produced.
ERROR: Forecast errors given by \(e_{t+h|t} = y_{t+h}-\hat{y}_{t+h|t}\).
LOWER: A list containing lower bounds for prediction intervals for each level. Each element within the list will be a multivariate time series with the same dimensional characteristics as MEAN.
UPPER: A list containing upper bounds for prediction intervals for each level. Each element within the list will be a multivariate time series with the same dimensional characteristics as MEAN.
level: The confidence values associated with the prediction intervals.
call: The matched call.
model: A list containing detailed information about the cvforecast and conformal models.

If mean is included in the object, the components mean, lower, and upper will also be returned, showing the information about the test set forecasts generated using all available observations.

Arguments

object: An object of class "cvforecast". It must have an argument x for original univariate time series, an argument MEAN for point forecasts and ERROR for forecast errors on validation set. See the results of a call to cvforecast.
alpha: A numeric vector of significance levels to achieve a desired coverage level \(1-\alpha\).
symmetric: If TRUE, symmetric nonconformity scores (i.e. \(|e_{t+h|t}|\)) are used. If FALSE, asymmetric nonconformity scores (i.e. \(e_{t+h|t}\)) are used, and then upper bounds and lower bounds are produced separately.
ncal: Length of the calibration set. If rolling = FALSE, it denotes the initial period of calibration sets. Otherwise, it indicates the period of every rolling calibration set.
rolling: If TRUE, a rolling window strategy will be adopted to form the calibration set. Otherwise, expanding window strategy will be used.
quantiletype: An integer between 1 and 9 determining the type of quantile estimator to be used. Types 1 to 3 are for discontinuous quantiles, types 4 to 9 are for continuous quantiles. See the weighted_quantile function in the ggdist package.
weightfun: Function to return a vector of weights used for sample quantile computation. Its first argument must be an integer indicating the number of observations for which weights are generated. If NULL, equal weights will be used for sample quantile computation. Currently, only non-data-dependent weights are supported.
kess: If TRUE, Kish's effective sample size is used for sample quantile computation.
update: If TRUE, the function will be compatible with the update(update.cpforecast) function, allowing for easy updates of conformal prediction.
na.rm: If TRUE, corresponding entries in sample values and weights are removed if either is NA when calculating sample quantile.
...: Other arguments are passed to weightfun.

Details

Consider a vector \(s_{t+h|t}\) that contains the nonconformity scores for the \(h\)-step-ahead forecasts.

If symmetric is TRUE, \(s_{t+h|t}=|e_{t+h|t}|\). When rolling is FALSE, the \((1-\alpha)\)-quantile \(\hat{q}_{t+h|t}\) are computed successively on expanding calibration sets \(s_{1+h|1},\dots,s_{t|t-h}\), for \(t=\mathrm{ncal}+h,\dots,T\). Then the prediction intervals will be \([\hat{y}_{t+h|t}-\hat{q}_{t+h|t}, \hat{y}_{t+h|t}+\hat{q}_{t+h|t}]\). When rolling is TRUE, the calibration sets will be of same length ncal.

If symmetric is FALSE, \(s_{t+h|t}^{u}=e_{t+h|t}\) for upper interval bounds and \(s_{t+h|t}^{l} = -e_{t+h|t}\) for lower bounds. Instead of computing \((1-\alpha)\)-quantile, \((1-\alpha/2)\)-quantiles for lower bound (\(\hat{q}_{t+h|t}^{l}\)) and upper bound (\(\hat{q}_{t+h|t}^{u}\)) are calculated based on their nonconformity scores, respectively. Then the prediction intervals will be \([\hat{y}_{t+h|t}-\hat{q}_{t+h|t}^{l}, \hat{y}_{t+h|t}+\hat{q}_{t+h|t}^{u}]\).

Examples

Run this code

# Simulate time series from an AR(2) model
library(forecast)
series <- arima.sim(n = 200, list(ar = c(0.8, -0.5)), sd = sqrt(1))

# Cross-validation forecasting
far2 <- function(x, h, level) {
  Arima(x, order = c(2, 0, 0)) |>
    forecast(h = h, level)
}
fc <- cvforecast(series, forecastfun = far2, h = 3, level = 95,
                 forward = TRUE, initial = 1, window = 50)

# Classical conformal prediction with equal weights
scpfc <- scp(fc, symmetric = FALSE, ncal = 50, rolling = TRUE)
print(scpfc)
summary(scpfc)

# Classical conformal prediction with exponential weights
expweight <- function(n) {
  0.99^{n+1-(1:n)}
}
scpfc_exp <- scp(fc, symmetric = FALSE, ncal = 50, rolling = TRUE,
                 weightfun = expweight, kess = TRUE)