Adaptive Conformal Time Series Forecasting under Structural Breaks

1. Motivation & Background

Multi-horizon time series forecasting is critical across diverse domain applications, including energy grid load management, micro-climate sensing, financial risk management, and healthcare monitoring. Traditional temporal forecasting models—ranging from classical autoregressive models ($\text{ARIMA}$, $\text{GARCH}$, $\text{State-Space Models}$) to modern deep architectures ($\text{DeepAR}$, $\text{Temporal Fusion Transformer}$, $\text{N-BEATS}$)—primarily focus on point predictions or rely on Gaussian error distributions.

However, real-world continuous data streams exhibit two fundamental challenges:

1. Non-Stationary Regime Shifts: Structural breaks, seasonal shifts, and macroeconomic policy interventions disrupt temporal exchangeability.
2. Heavy-Tailed & Heteroscedastic Noise: Empirical residual distributions frequently deviate from Gaussian assumptions, leading to severely under-calibrated prediction intervals.

To tackle these challenges, this project develops Adaptively Weighted Conformal Time Series Forecasting (AW-CTSF), a distribution-free framework providing finite-sample predictive coverage guarantees without assuming data exchangeability or parametric noise structures.

2. Problem Formulation

Let ${X_t, Y_t}{t=1}^T$ denote a multivariate time series process where $X_t \in \mathbb{R}$ represents exogenous covariates and $Y_t \in \mathbb{R}$ is the target variable. Given historical observation trajectory $\mathcal{H}_T = {(X_t, Y_t)}{t=1}^T$, our goal is to forecast the future sequence over prediction horizon $H$:

Rather than emitting point predictions $\hat{Y}{T+h}$, we construct marginal prediction intervals $\mathcal{C}{1-\alpha}^{(h)}(X_{T+h})$ at significance level $\alpha \in (0, 1)$ such that:

Under structural breaks, the joint data generating process shifts over time: $P_t(X, Y) \neq P_{t+1}(X, Y)$. Standard conformal prediction breaks down because past calibration errors are no longer exchangeable with future residuals.

3. Proposed Method: AW-CTSF Framework

Our proposed framework integrates a Deep State-Space Base Forecaster, a Maximum Mean Discrepancy (MMD) Change-Point Estimator, and Dynamic Kernel-Weighted Quantile Calibration.

[ Historical Series H_T ] ---> [ Deep State-Space Model ] ---> [ Point Estimate & Dispersion ]
                                                                      |
[ MMD Drift Detector ]    ---> [ Temporal Weighting w_t ] ---> [ Weighted Quantile q_(1-α) ]
                                                                      |
                                                               [ Conformal Interval C_(1-α) ]

Step 1: Deep State-Space Point & Variance Estimation

We utilize a recurrent latent state transition function $\psi_\theta$ to update the dynamic system state $h_t$:

The base model outputs multi-step conditional location predictions $\hat{Y}{t+h} = \mu\theta(h_t, h)$ and conditional dispersion scale estimates $\hat{\sigma}{t+h} = \sigma\theta(h_t, h)$. We define normalized non-conformity scores:

Step 2: Change-Point Aware Temporal Weighting

To adjust for distribution drift, calibration time points $t \in {1, \dots, T-h}$ are assigned non-uniform weights $w_t$ combining exponential decay and dynamic MMD distance:

where $\gamma \in (0, 1]$ is the temporal memory factor, $\lambda > 0$ controls drift sensitivity, and $\widehat{D}{\text{MMD}}^2$ measures kernel discrepancy between historical sliding window $\mathcal{P}{t-\Delta:t}$ and recent window $\mathcal{P}_{T-\Delta:T}$.

Step 3: Weighted Conformal Calibration

The adaptive non-conformity quantile $q_{1-\alpha}^{(h)}$ is computed from empirical distribution function $\sum_{t} \tilde{w}t \delta{S_t^{(h)}}$:

The resulting conformal prediction band at horizon $h$ is given by:

4. Theoretical Guarantees

Under bounded distribution total variation drift $\sum_{t=1}^T |\mathcal{P}t - \mathcal{P}{t+1}|_{\text{TV}} \le \Delta_T$, the coverage gap satisfies finite-sample bounds:

• Property 1 (Exchangeable Reduction): When the process is stationary ($\Delta_T = 0$) and $\gamma = 1$, the bound collapses to exact finite-sample valid conformal prediction $1 - \alpha$.
• Property 2 (Adaptivity to Regime Shifts): When structural breaks occur, MMD down-weights pre-break calibration samples, shrinking interval over-expansion while maintaining coverage.

5. Implementation & Python Architecture

Below is a modular implementation of the non-conformity calibration and temporal quantile estimation module:

import numpy as np
from scipy.spatial.distance import cdist

class AdaptiveConformalForecaster:
    """
    Adaptively Weighted Conformal Time Series Forecaster (AW-CTSF).
    Computes distribution-free prediction intervals under temporal drift.
    """
    def __init__(self, alpha: float = 0.1, gamma: float = 0.98, lambda_mmd: float = 2.0):
        self.alpha = alpha
        self.gamma = gamma
        self.lambda_mmd = lambda_mmd

    def _kernel_mmd(self, X_past: np.ndarray, X_recent: np.ndarray, sigma: float = 1.0) -> float:
        """Computes RBF kernel MMD between two windowed trajectories."""
        K_pp = np.exp(-cdist(X_past, X_past, 'sqeuclidean') / (2 * sigma**2))
        K_rr = np.exp(-cdist(X_recent, X_recent, 'sqeuclidean') / (2 * sigma**2))
        K_pr = np.exp(-cdist(X_past, X_recent, 'sqeuclidean') / (2 * sigma**2))
        return np.mean(K_pp) + np.mean(K_rr) - 2 * np.mean(K_pr)

    def compute_weights(self, residuals: np.ndarray, window_size: int = 20) -> np.ndarray:
        """Calculates combined exponential and MMD temporal weights."""
        T = len(residuals)
        weights = np.zeros(T)
        recent_window = residuals[-window_size:] if T >= window_size else residuals

        for t in range(T):
            exp_weight = self.gamma ** (T - 1 - t)
            if t >= window_size:
                past_window = residuals[t - window_size:t]
                mmd_dist = self._kernel_mmd(past_window, recent_window)
            else:
                mmd_dist = 0.0
            weights[t] = exp_weight * np.exp(-self.lambda_mmd * mmd_dist)

        return weights / np.sum(weights)

    def calibrate_interval(self, y_true: np.ndarray, y_pred: np.ndarray, scale: np.ndarray):
        """Computes adaptive non-conformity score quantile."""
        scores = np.abs(y_true - y_pred) / (scale + 1e-8)
        weights = self.compute_weights(scores[:, None])
        
        # Sort scores and accumulate normalized weights
        sort_idx = np.argsort(scores)
        sorted_scores = scores[sort_idx]
        cum_weights = np.cumsum(weights[sort_idx])

        # Pick threshold where cumulative weight reaches 1 - alpha
        target_quantile = 1.0 - self.alpha
        q_idx = np.searchsorted(cum_weights, target_quantile)
        q_idx = min(q_idx, len(sorted_scores) - 1)
        return sorted_scores[q_idx]

6. Benchmark Evaluation

We evaluated AW-CTSF against competitive baseline forecasting models across standard non-stationary time series datasets (Electricity Transformer Dataset, Weather, Macroeconomic FRED-MD).

References

• Barber, R. F., Candès, E. J., Ramdas, A., & Tibshirani, R. J. (2023). Conformal prediction beyond exchangeability. Annals of Statistics, 51(2), 816-845.
• Xu, C., & Xie, Y. (2021). Conformal prediction interval for dynamic time-series. Advances in Neural Information Processing Systems (NeurIPS), 34, 11559-11569.
• Lim, B., Arık, S. Ö., Loeff, N., & Pfister, T. (2021). Temporal fusion transformers for interpretable multi-horizon time series forecasting. International Journal of Forecasting, 37(4), 1748-1764.