Conformal Time Series Forecasting

Conformal time series forecasting is a probabilistic forecasting method using [[Conformal Prediction]] Conformal Prediction Conformal prediction is a method to sequentially predict consistent confidence intervals using nonconformity measures. .

For any given model $\mathcal M$, conformal time series forecasting trains on a training dataset $\mathcal D_{\text{Train}}$ then calculates a [[Confidence Interval]] Confidence Interval Estimates from a sample can be entitled a confidence interval using a calibration dataset $\mathcal D_{\text{Calibration}}$. The confidence interval is directly used for inference. This framework is called the inductive conformal prediction (ICP).

How to Forecast the Confidence Interval

For a dataset $\mathcal D$, we split it, e.g., $\mathcal D_{\text{Training}}$ and $\mathcal D_{\text{Calibration}}$, where

$$ \begin{align} \lvert \mathcal D_{\text{Training}} \rvert &= n \\ \lvert \mathcal D_{\text{Calibration}} \rvert &= m. \end{align} $$

After training the model, we use the calibration dataset $\mathcal D_{\text{Calibration}}$ to find the distribution of a predefined nonconformity measure, e.g.,

$$ R_i = \Delta( \mathcal M(x^{(i)}\vert \mathcal D_{\text{Training}}) , y^{(i)} ). $$

By looking at the distribution, we find the interval that leads to $P(R_i > R_{th}) \sim 0.05$. In practice, we can simply use quantiles. Using the $R_{th}$, we can find the corresponding deviation of predictions, $\epsilon$.

For inference, we simply use $[\hat y - \epsilon, \hat y + \epsilon]$ as our predicted range.

Multihorizon Forecasting

In a multihorizon forecasting problem, we need to correct our ranges using the [[Bonferroni Correction]] Bonferroni Correction Bonferroni correction is very useful in a multiple comparison problem .