Classical split conformal prediction method

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,
  ...
)

Arguments

object: An object of class "cvforecast". It must have an argument x for the original univariate time series, an argument MEAN for the point forecasts and ERROR for the forecast errors on the validation set. See the results of a call to cvforecast.
alpha: A numeric vector of significance levels to achieve a desired coverage level $1-\alpha$. Defaults to 1 - 0.01 * object$level, the levels used in object.
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. Defaults to FALSE.
ncal: Length of the calibration set. If rolling = FALSE, it denotes the initial period of the calibration sets. Otherwise, it indicates the period of every rolling calibration set. Defaults to 10.
rolling: If TRUE, a rolling window strategy will be adopted to form the calibration set. Otherwise, an expanding window strategy will be used. Defaults to FALSE.
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. Defaults to 1.
weightfun: Function to return a vector of weights used for the sample quantile computation. Its first argument must be an integer indicating the number of observations for which weights are generated. Defaults to NULL, in which case equal weights are used. Currently, only non-data-dependent weights are supported.
kess: If TRUE, Kish's effective sample size is used for the sample quantile computation. Defaults to FALSE.
update: If TRUE, object already holds the results of a previous call and only the newly added time steps are computed; the prediction intervals produced earlier are carried over unchanged. Set by update.cpforecast and not normally set by hand. Defaults to FALSE.
na.rm: If TRUE, corresponding entries in the sample values and weights are removed if either is NA when calculating the sample quantile. Defaults to TRUE.
...: Other arguments are passed to weightfun.

Value

A list of class c("scp", "cpforecast", "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: An integer vector giving the number of conformal predictions performed in cross-validation for each forecast horizon.
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 information about the conformal prediction model: the resolved arguments in model$args, and the call and the arguments of the underlying cross-validation in model$cvforecast.

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.

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 $(1-\alpha)$-quantiles $\hat{q}_{t+h|t}$ 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 the same length ncal.

If symmetric is FALSE, $s_{t+h|t}^{u}=e_{t+h|t}$ for the upper interval bounds and $s_{t+h|t}^{l} = -e_{t+h|t}$ for the lower bounds. Instead of computing the $(1-\alpha)$-quantile, the $(1-\alpha/2)$-quantiles for the lower bound $\hat{q}_{t+h|t}^{l}$ and the 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

# Simulate time series from an AR(2) model
library(forecast)
set.seed(1)
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, window = 50)

# Classical conformal prediction with equal weights
scpfc <- scp(fc, ncal = 50, rolling = TRUE)
print(scpfc)
#> SCP 
#> 
#> Call:
#>  scp(object = fc, ncal = 50, rolling = TRUE) 
#> 
#>  cp_times (the forward step included): 101 (h=1), 100 (h=2), 99 (h=3)
#> 
#> Forecasts of the forward step:
#>     Point Forecast     Lo 95    Hi 95
#> 201      0.6271538 -1.023525 3.234253
#> 202      0.8607034 -1.291456 4.064625
#> 203      0.4935805 -1.675089 3.691253
summary(scpfc)
#> SCP 
#> 
#> Call:
#>  scp(object = fc, ncal = 50, rolling = TRUE) 
#> 
#>  cp_times (the forward step included): 101 (h=1), 100 (h=2), 99 (h=3)
#> 
#> Forecasts of the forward step:
#>     Point Forecast     Lo 95    Hi 95
#> 201      0.6271538 -1.023525 3.234253
#> 202      0.8607034 -1.291456 4.064625
#> 203      0.4935805 -1.675089 3.691253
#> 
#> Cross-validation error measures:
#>       ME   MAE   MSE RMSE    MPE    MAPE  MASE RMSSE Winkler_95 MSIS_95
#> CV 0.007 0.946 1.415 1.06 -3.933 269.763 0.992 0.882      6.123   6.568

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