Joint work with

Yanfei Kang	Fotios Petropoulos	Feng Li
Beihang University	University of Bath	Central University of Finance and Economics

Outline

Motivation

Motivation

Motivation

Motivation

Introduction

• Point forecasting mainly forecasts the mean or the median of the distributions for future observations.
• Probabilistic forecasting can provide a comprehensive outlook of the expected future value and the future uncertainty.
• Time series features provide valuable information for decision makers.
• The superiority of forecast combinations over a single model.
  • No-free-lunch theorem (Wolpert & Macready, 1997).
  • Horses for courses (Petropoulos et al., 2014).
  • Merely tackling model uncertainty is sufficient to help (Petropoulos et al., 2018).

Challenges

• Previous literature mainly focuses on
  • point forecasting + forecast combinations.
• How do features affect the uncertainty estimation of forecasts?
• How to guarantee the effectiveness of the relationship in forecasting a newly given dataset?
• How to translate the relationship into an attempt to improve the forecasting performance?

General framework

GRATIS (Kang et al., 2020)

Dataset

Other components

• $42$ times series features (R package tsfeatures)

• Individual model pool

  

• Interval forecast evaluation $$\begin{array}{c}MSIS=\frac{1}{h}\frac{{\sum }_{t=n+1}^{n+h}(U_t-L_t)+\frac{2}{\alpha }(L_t-Y_t)1\{Y_t<L_t\}+\frac{2}{\alpha }(Y_t-U_t)1\{Y_t>U_t\}}{\frac{1}{n-m}{\sum }_{t=m+1}^n|Y_t-Y_{t-m}|}\end{array}$$

Linking features with performance

Why GAM?

GAM model for each individual model

• $\log \left({MSIS}_N\right)⟺F_{N\times P}$

Partial effect analysis

Partial effect analysis

Weight assignment

Adjusted softmax function

$$\begin{array}{c}{P}_{ij}=\frac{\exp \left\{\frac{{\mu }_i-\widehat{\log \left({MSIS}_{ij}\right)}}{{\sigma }_i}\right\}}{{\sum }_{k=1}^M\exp \left\{\frac{{\mu }_i-\widehat{\log \left({MSIS}_{ik}\right)}}{{\sigma }_i}\right\}},i=1,\dots ,N;j=1,\dots ,M\end{array}$$

• Negative values can be down-weighted to near-zero.
• $\log \left(MSIS\right)\uparrow ⟹\text{Accuracy}\downarrow ⟹P\downarrow$

Optimal threshold ratio search

For $i$th time series,

• calculate the ratio of weight $R_k=P_{ij}/max\left(P_{ik}\right)$.
• select individual models that satisfy $R_k>Tr$ $(0<Tr\le 1)$.

Combined forecasts

Combined prediction intervals

$$\begin{array}{cc}{f}_{wi}^l& =\frac{1}{{\sum }_{k=1}^S{P}_{ik}}\sum _{k=1}^S{P}_{ik}{f}_{ik}^l\\ f_{wi}^u& =\frac{1}{{\sum }_{k=1}^S{P}_{ik}}\sum _{k=1}^S{P}_{ik}{f}_{ik}^u\end{array}$$

Combined point forecasts

$$\begin{array}{c}{f}_{wi}=\frac{1}{2}(f_{wi}^l+f_{wi}^u)\end{array}$$

Optimal threshold ratio search

• Model combination $⟶$ Model selection.
• $Tr=0$ indicates that all the methods from the pool are selected.
• $Tr=1$ indicates that only the method with the minimal fitted $\log \left(MSIS\right)$ is selected.

Selection rates of each model

Performance for different confidence levels

Forecasting results

Conclusions

• Features are taken into account to estimate the uncertainty of forecasts (cross-learning).

• We propose an optimal threshold ratio searching algorithm to select an appropriate subset of models per time series for model combination.

• Our approach outperforms a variety of individual models with distinctions for both point forecasts and prediction intervals.

Thanks for your attention!