State Space Model with Dynamic Coefficient $\alpha_t$ for Exponential Smoothing

This document outlines a state space model designed for analyzing time series data, incorporating a dynamic coefficient $\alpha_t$ to offer a flexible approach to exponential smoothing. This innovative model adapts to the evolving relationships between consecutive observations by allowing the smoothing coefficient to change over time.

Model Description

The state space model consists of two primary components:

1. Observation Equation: It illustrates the relationship between the current observation $y_t$ and the previous state $x_{t-1}$, alongside an error term $e_t$:

$$ y_t = x_{t-1} + e_t, \quad e_t \sim N(0, \sigma^2_e) $$

2. State Transition Equation: This equation defines the evolution of the state $x_t$ from $x_{t-1}$, incorporating the error term modified by the dynamically varying coefficient $\alpha_t$:

$$ x_t = x_{t-1} + \alpha_t e_t $$

The dynamic coefficient $\alpha_t$ is modeled at each time point using a Beta distribution, ensuring its values remain within the [0, 1] interval. This setup captures the essence of exponential smoothing, with the added benefit of dynamic adaptability.

Standard Deviation of Observation Error

The standard deviation of the difference between observation and state, $y_t - x_t$, is expressed as $(1 - \alpha_t)\sigma_e$. This formulation indicates how the uncertainty in the observation error scales with the value of $\alpha_t$, directly linking the model's adaptability to its predictive confidence.

Assumptions

• Observation Error: The error $e_t$ is assumed to be a Gaussian process with zero mean and known variance $\sigma^2_e$, reflecting the model's uncertainty about observations.
• Dynamic Coefficient $\alpha_t$: Varies over time within the [0, 1] interval, serving as an adaptive smoothing factor that modulates the influence of past observations on current state predictions.
• Independence: Error terms at different points in time are assumed to be independent, simplifying the model's structure.

Bayesian Estimation Process

Employing PyMC for Bayesian estimation, the model leverages prior distributions for unknown parameters, refining these priors into posterior distributions based on observed data:

1. Prior Definitions:

   • A Beta distribution is chosen for $\alpha_t$, suitable for coefficients that are naturally bounded between 0 and 1.
   • The standard deviation of the observation error, $\sigma_e$, is modeled using a HalfNormal distribution, emphasizing its positive value constraint.

2. Sampling:

   • Monte Carlo Markov Chain (MCMC) sampling is used to generate posterior distributions for the model parameters, facilitated by PyMC's efficient algorithms and diagnostic tools.

Usage of PyMC

PyMC, a powerful tool for Bayesian statistical modeling and probabilistic inference, provides:

• Intuitive model definition syntax.
• Efficient MCMC sampling.
• Diagnostic tools for assessing sampling convergence.
• Statistical summaries of sampling results.

This approach allows for a nuanced and adaptable estimation of complex time series models, highlighting the integration of uncertainty and dynamic adaptability in the context of exponential smoothing.