Adaptive conformal prediction method — acp • conformalForecast

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 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 $1-\alpha$.
gamma: The step size parameter $\gamma>0$ for $\alpha$ updating.
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.
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 are removed if it is NA when calculating sample quantile.
...: Other arguments are passed to the weighted_quantile function for quantile computation.

Value

A list of class c("acp", "cpforecast", "forecast") with the following components:

x: The original time series.
series: The name of the series x.
method: A character string "acp".
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 information abouth the conformal prediction model.

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)
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)

# ACP with asymmetric nonconformity scores and rolling calibration sets
acpfc <- acp(fc, symmetric = FALSE, gamma = 0.005, ncal = 50, rolling = TRUE)
print(acpfc)
#> ACP 
#> 
#> Call:
#>  acp(object = fc, gamma = 0.005, symmetric = FALSE, ncal = 50,  
#>      rolling = TRUE) 
#> 
#>  cp_times = 99 (the forward step included) 
#> 
#> Forecasts of the forward step:
#> Cross-validation
#> 
#> Call:
#>  acp(object = fc, gamma = 0.005, symmetric = FALSE, ncal = 50,  
#>      rolling = TRUE) 
#> 
#>  fit_times =  (the forward step included) 
#> 
#> Forecasts of the forward step:
#>     Point Forecast     Lo 95    Hi 95
#> 201      0.1969684 -1.055741 1.795689
#> 202     -0.5275402 -2.526663 1.438041
#> 203     -0.6180594 -2.402479 1.119214
summary(acpfc)
#> ACP 
#> 
#> Call:
#>  acp(object = fc, gamma = 0.005, symmetric = FALSE, ncal = 50,  
#>      rolling = TRUE) 
#> 
#>  cp_times = 99 (the forward step included) 
#> 
#> Forecasts of the forward step:
#> Cross-validation
#> 
#> Call:
#>  acp(object = fc, gamma = 0.005, symmetric = FALSE, ncal = 50,  
#>      rolling = TRUE) 
#> 
#>  fit_times =  (the forward step included) 
#> 
#> Forecasts of the forward step:
#>     Point Forecast     Lo 95    Hi 95
#> 201      0.1969684 -1.055741 1.795689
#> 202     -0.5275402 -2.526663 1.438041
#> 203     -0.6180594 -2.402479 1.119214
#> 
#> Cross-validation error measures:
#>        ME  MAE   MSE RMSE      MPE    MAPE  MASE RMSSE Winkler_95 MSIS_95
#> CV -0.038 0.93 1.341 1.04 -220.016 425.573 0.958 0.875      6.023   6.002