Abstract
The autoregressive model is a tool used in time series analysis to describe and model time series data. Its main structure is a linear equation using the previous values to compute the next time step; i.e., the short time relationship is the core component of the autoregressive model. Therefore, short-term effects can be modeled in an easy way, but the global structure of the model is not obvious. However, this global structure is a crucial aid in the model selection process in data analysis. If the global properties are not reflected in the data, a corresponding model is not compatible. This helpful knowledge avoids unsuccessful modeling attempts. This article analyzes the global structure of the autoregressive model through the derivation of a closed form. In detail, the closed form of an autoregressive model consists of the basis functions of a fundamental system of an ordinary differential equation with constant coefficients; i.e., it consists of a combination of polynomial factors with sinusoidal, cosinusoidal, and exponential functions. This new insight supports the model selection process.
Highlights
The increasing digitalization of all areas of society and the economy is leading to ever greater volumes of data
As a rule of thumb, domain-specific models are preferable for corresponding problems. This rule corresponds to the principle of using existing knowledge, especially if this information is already reflected in the mathematical model
The selection of a suitable model is a critical task in data analysis
Summary
The increasing digitalization of all areas of society and the economy is leading to ever greater volumes of data. When modeling radioactive decay [1] or the endemic development of a pandemic [2], a domain-specific model is a good starting point for data analysis If this background knowledge about the domain and the use-case is not available or if this knowledge should not be used, general purpose models can be used for prediction. If the data contradict the character of a model, the quality of the model-based prediction suffers; e.g., if the model has a linear relationship, the data rather correspond to a parabola, a contradiction is given and the model is at most to be used with caution or is to be rejected This decision assumes that all essential characteristics of a model are known and are not a black box.
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