Chapter 12 Experimental Functionality
We have included some experimental functionality which we provide details of here.
12.1 Calibration of Predictive Distributions
Many time series models due to their parametric nature make strong assumptions about both the shape of the predictive distribution (shape) as well as the multi-step ahead evolution of the variance (scale). In practice we find that both the shape and form of the variance curve results in under or over sizing as measured by the % of exceedances falling within each quantile. In the additive ETS model without seasonality (AAN), the shape of the standardized error distribution is N(0,1) while the variance evolves based on the following formula:
\[ v_{t+h|t} = \sigma^2\left[1 + \left(h-1\right)\left\{\alpha^2 + \alpha\beta h + \frac{1}{6}\beta^2 h \left(2h - 1\right)\right\}\right] \] which is linear in h when the coefficient on the slope is zero and cubic otherwise. The issue of well calibrated distributions has been discussed in many papers, particularly in the ML domain for both regression and classification.
The purpose of the calibration function is to empirically adjust the multi-step variance based on the observed variance from a rolling prediction (backtest) as well as provide an alternative empirical sampling for the shape of the distribution.
12.2 Beyond Box-Cox : The Logit-Normal Assumption for Ratios
For series which are strongly bounded, for instance ratios or probabilities,
a simple approach is to assume that the log transformed odd ratios are normal,
leading to a logit-normal distributional
assumption, which is a simpler alternative to using the Beta distribution.
The tstransform function provides the option for using either the “box-cox”
or “logit” transforms, returning both the transform and inverse functions which
are used by a number of packages in our framework. The transformation argument
in the model specification across most of the internally developed packages will
now accept either “box-cox” or “logit” as valid options.