Adaptive conformal prediction method

Compute prediction intervals and other information by applying the adaptive conformal prediction (ACP) method.

Usage

acp(
  object,
  alpha = 1 - 0.01 * object$level,
  gamma = 0.005,
  symmetric = FALSE,
  ncal = 10,
  rolling = FALSE,
  quantiletype = 1,
  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.
gamma: The step size parameter $\gamma>0$ for the $\alpha$ updating. Defaults to 0.005.
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.
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 are removed if they are NA when calculating the sample quantile. Defaults to TRUE.
...: Other arguments are passed to the weighted_quantile function for the quantile computation.

Value

A list of class c("acp", "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 "acp".
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$args, the call and the arguments of the underlying cross-validation in model$cvforecast, and the sequence of updated significance levels in model$alpha_update. The latter holds a single element alpha when symmetric = TRUE, and the two elements lower and upper when symmetric = FALSE; each is a list with one multivariate time series per confidence level, laid out like MEAN.

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

Details

The ACP method considers the online update: $$\alpha_{t+h|t}:=\alpha_{t+h-1|t-1}+\gamma(\alpha-\mathrm{err}_{t|t-h}),$$ for each individual forecast horizon h, respectively, where $\mathrm{err}_{t|t-h}=1$ if $s_{t|t-h}>q_{t|t-h}$, and $\mathrm{err}_{t|t-h}=0$ if $s_{t|t-h} \leq q_{t|t-h}$.

References

Gibbs, I., and Candes, E. (2021). "Adaptive conformal inference under distribution shift", Advances in Neural Information Processing Systems, 34, 1660–1672.

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)

# ACP with asymmetric nonconformity scores and rolling calibration sets
acpfc <- acp(fc, ncal = 50, rolling = TRUE)
print(acpfc)
#> ACP 
#> 
#> Call:
#>  acp(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 -1.706076 3.234253
#> 202      0.8607034 -1.825564      Inf
#> 203      0.4935805 -2.027163 3.691253
summary(acpfc)
#> ACP 
#> 
#> Call:
#>  acp(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 -1.706076 3.234253
#> 202      0.8607034 -1.825564      Inf
#> 203      0.4935805 -2.027163 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        Inf     Inf