Autocorrelated multistep-ahead conformal prediction method

Compute prediction intervals and other information by applying the Autocorrelated Multistep-ahead Conformal Prediction (AcMCP) method. The method can only deal with asymmetric nonconformity scores, i.e., forecast errors.

Usage

acmcp(
  object,
  alpha = 1 - 0.01 * object$level,
  ncal = 10,
  rolling = FALSE,
  integrate = TRUE,
  scorecast = TRUE,
  lr = 0.1,
  Tg = NROW(object$ERROR),
  delta = 0.01,
  Csat = NULL,
  KI = max(abs(object$ERROR), na.rm = TRUE),
  update = FALSE,
  ma_method = c("CSS-ML", "CSS"),
  ...
)

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 \(1-\alpha\).
ncal: Length of the burn-in period for training the scorecaster. If rolling = TRUE, it is also used as the length of the trailing windows for learning rate calculation and the windows for the calibration set. If rolling = FALSE, it is used as the initial period of calibration sets and trailing windows for learning rate calculation.
rolling: If TRUE, a rolling window strategy will be adopted to form the trailing window for learning rate calculation and the calibration set for scorecaster if applicable. Otherwise, expanding window strategy will be used.
integrate: If TRUE, error integration will be included in the update process.
scorecast: If TRUE, scorecasting will be included in the update process.
lr: A positive initial learning rate used for quantile tracking.
Tg: The time that is set to achieve the target absolute coverage guarantee before this. It must be greater than 1 when Csat is not supplied. Defaults to the number of cross-validation periods in object.
delta: A number in \((0, 1)\). The target absolute coverage guarantee is set to \(1-\alpha-\delta\).
Csat: A positive constant ensuring that by time Tg, an absolute guarantee is of at least \(1-\alpha-\delta\) coverage. Derived from Tg and delta when not supplied.
KI: A non-negative constant to place the integrator on the same scale as the scores. Defaults to the largest absolute forecast error in object.
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.
ma_method: Estimation method for the MA\((h-1)\) scorecaster. "CSS-ML" uses conditional sum of squares for starting values followed by maximum likelihood. "CSS" uses conditional sum of squares only and may be faster, especially for longer forecast horizons, but can produce different estimates.
...: Not used.

Value

A list of class c("acmcp", "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 "acmcp".
cp_times: An integer vector giving the number of conformal predictions performed in cross-validation for each forecast horizon.
scorecast_times: An integer vector giving the number of successful scorecasts for each forecast horizon. Returned when scorecast = TRUE.
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.

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

Similar to the PID method, the AcMCP method also integrates three modules (P, I, and D) to form the final iteration. However, instead of performing conformal prediction for each individual forecast horizon h separately, AcMCP employs a combination of an MA\((h-1)\) model and a linear regression model of \(e_{t+h|t}\) on \(e_{t+h-1|t},\dots,e_{t+1|t}\) as the scorecaster. This allows the AcMCP method to capture the relationship between the \(h\)-step ahead forecast error and past errors. Scorecasts are constructed recursively, so longer forecast horizons require more history before all scorecaster inputs are available. For horizon \(h\), the first scorecast can be computed at cross-validation error index ncal + \(h(h-1)/2\). The number of successful scorecasts is reported in scorecast_times.

References

Wang, X., and Hyndman, R. J. (2024). "Online conformal inference for multi-step time series forecasting", arXiv preprint arXiv:2410.13115.

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,
                 forward = TRUE, initial = 1, window = 50)

# AcMCP setup
Tg <- 200; delta <- 0.01
Csat <- 2 / pi * (ceiling(log(Tg) * delta) - 1 / log(Tg))
KI <- 2
lr <- 0.1

# AcMCP with integrator and scorecaster
acmcpfc <- acmcp(fc, ncal = 50, rolling = TRUE,
             integrate = TRUE, scorecast = TRUE,
             lr = lr, Tg = Tg, KI = KI, Csat = Csat)
print(acmcpfc)
#> ACMCP 
#> 
#> Call:
#>  acmcp(object = fc, ncal = 50, rolling = TRUE, integrate = TRUE,  
#>      scorecast = TRUE, lr = lr, Tg = Tg, Csat = Csat, KI = KI) 
#> 
#>  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 -1.013545 2.673790
#> 202      0.8607034 -2.090441 4.044708
#> 203      0.4935805 -2.055964 3.526965
summary(acmcpfc)
#> ACMCP 
#> 
#> Call:
#>  acmcp(object = fc, ncal = 50, rolling = TRUE, integrate = TRUE,  
#>      scorecast = TRUE, lr = lr, Tg = Tg, Csat = Csat, KI = KI) 
#> 
#>  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 -1.013545 2.673790
#> 202      0.8607034 -2.090441 4.044708
#> 203      0.4935805 -2.055964 3.526965
#> 
#> 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      6.618   7.132
acmcpfc$scorecast_times
#> [1] 101 100  98