ARMA Model Estimation Using Analytical Derivatives and State Space
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Solution Overview
Problem
Conventional methods for estimating autoregressive moving average (ARMA) models using numerical derivatives are inefficient, especially for models with multiple parameters, and can produce incorrect results when used with the Shumway-Stoffer algorithm due to non-conformance with its assumptions.
Innovation Solution
An alternative state space representation is generated, placing AR parameters in the first row of the state transition matrix and MA parameters in the observation matrix, allowing for the use of analytical derivatives with the Shumway-Stoffer algorithm to estimate ARMA models more efficiently and accurately.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If numerical derivatives are used to estimate ARMA models, then the method can be applied generally, but the computing time and resources increase significantly especially for models with multiple parameters
Solution Approach 1:
The patent replaces the mechanical numerical approximation process with an analytical mathematical derivation. Instead of using numerical methods to estimate derivatives of the likelihood function, the patent derives closed-form analytical expressions for these derivatives, eliminating the need for iterative numerical computation and significantly reducing computing time while maintaining accuracy.
Solution Approach 2:
The patent performs preliminary analytical derivation of the gradient and Hessian matrices before the actual estimation process. By pre-computing the analytical formulas for derivatives and storing them in a optimized form, the patent eliminates the need for repeated numerical calculations during model estimation, thereby reducing computational burden.
2Productivity
If numerical derivatives are used with the Shumway-Stoffer algorithm, then model estimation can proceed, but incorrect results are produced due to non-conformance with algorithm assumptions
Solution Approach 1:
The patent changes the parameter representation in the state-space model from the conventional form to an alternative formulation where the state transition matrix and observation matrix are redefined. This parameter transformation ensures compatibility with the Shumway-Stoffer algorithm assumptions, allowing correct application of analytical derivatives while maintaining the ability to estimate ARMA models accurately.
3Measurement precision
If multiple ARMA models are generated for a given set of data, then forecasting accuracy may improve, but the computing resources required become prohibitive
Solution Approach 1:
The patent replaces computationally intensive numerical derivative calculations with efficient analytical derivative formulas. This substitution reduces the computational complexity from requiring multiple iterative numerical approximations to a single analytical computation, enabling the generation of multiple ARMA models with the same computational cost as generating one model numerically.
Data Source
AI summary
Systems are provided to estimate autoregressive moving average (ARMA) models using maximum likelihood estimation and analytical derivatives, and to use such models for forecasting. The evaluation of the analytical derivatives during estimation of the model parameters may be performed using a state space representation with certain characteristics. An ARMA model estimated using maximum likelihood estimation, analytical derivatives, and the state space representation with certain characteristics can be used to forecast/predict values that are likely to occur in the future, given some set of previously-occurring values.