Adaptive Forecasting Model Of The Number Of Marriages In Samara Region

Abstract

In the analysis of socio - economic systems and processes, a priori information on quantitative patterns inherent in causal relationships between economic indicators necessary for the formation of models is often missing. One of the important demographic indicators is the number of marriages concluded. The number of marriages concluded by months shows a pronounced seasonality. The paper considers adaptive predictive models of the number of marriages in Samara region. The paper proposes an adaptive seasonal model. As a learning algorithm, the recursive least squares method with forgetting coefficients is used. It is shown that the forecast can be explained on the basis of a three-parameter autoregressive model. The undoubted advantage of the resulting model is the ability to build an accurate forecast for the year ahead on a monthly basis. Test cases showed the advantage of an adaptive forecasting model based on autoregression. The construction of the model according to the SARIMA methodology (seasonal autoregressive integrated moving average) confirmed the single root hypothesis. However, taking the seasonal first difference does not allow constructing, it does not allow constructing a high-quality autoregressive model of residues. This fact may indicate the presence of a long memory in the time series. In the future, it is planned to create a model based on autoregression with fractional order differences.

Keywords: Recursive least squarefractional calculusestimation of parametersfractional white noise

Introduction

Under the socio-economic system usually understand a complex probabilistic dynamic system, covering the processes of production, exchange, distribution and consumption of material and other goods. The main method for studying socio-economic systems is the modeling method, i.e. a method of theoretical and practical action aimed at the development and use of models.

In the analysis of socio-economic systems and processes, a priori information on quantitative patterns inherent in causal relationships between economic indicators necessary for the formation of models is often missing. In connection with the role, the role of methods for constructing models from experimental data is increasing.

Demographic forecasting is an integral part of the analysis of socio-economic processes in general. Many demographic time series are not stationary, have a complex structure, and their elements are formed under the influence of various factors, like many other phenomena of our society.

One of the important demographic indicators is the number of marriages concluded. The number of marriages concluded by months shows a pronounced seasonality. In May Russia, three to four times fewer marriages are registered in May than in August. Relatively few marriages also take place in January ‒ March, and most of all in August and September, the decrease in the frequency of marriages in December and April is associated with the spread of church registration and the ceremony of marriage (these months usually have posts).

Typically, additive static models ( Greene, 2017; Magnus, 2017) or the SARIMA (seasonal autoregressive integrated moving average) models ( Arlt, Trcka, & Arltová, 2017; Arlt & Trcka, 2019; Oliveira, Steffen, & Cheung, 2017) are used to model seasonal components.

In this paper, an adaptive autoregressive model with custom coefficients using the recursive least squares method is proposed.

Problem Statement

There is a time series of the number of marriages by month for the period from 2006 to 2016. We assume that the mathematical model of the number of marriages is described by the autoregression equation with time-variable parameters

$$\begin{array}{}\text{(1)}& y_i-\underset{m=\mathrm{1}}{\overset{r}{\sum }}{b}_{\mathrm{0}}^{\left(m\right)}\left(i\right)y_{i-m}=e_i,\end{array}$$

It is necessary to create an adaptive predictive model based on autoregression with the optimal number of parameters according to the observed sequence of marriages. $\left\{y_i\right\}$.

Research Questions

The paper is devoted to the issue of constructing an adaptive model for predicting the monthly number of marriages. The paper determines the optimal order of autoregressive models according to the Akaike criterion. To determine the coefficients, the recursive least squares method with forgetting coefficients is used. Test cases showed the advantage of an adaptive forecasting model based on autoregression.

Purpose of the Study

The aim of the study is to develop a model for predicting the number of marriages based on the autoregressive model. To achieve this goal, the article solved the problem of choosing a model of minimal complexity according to the Akaik criterion and training the resulting model according to the least squares criterion. The resulting allows for the forecast for 11 months in advance.

Research Methods

Findings

Figure 01 presents an estimate of the autocorrelation function of the number of marriages with a 95% significance level. As follows from Figure 01 , the autocorrelation function decreases rather slowly; therefore, the choice of the order of autoregression is not a trivial task. The Matlab Identification toolbox extension has a function that allows you to determine the structure of autoregression from the minimum of Akaike information criterion (AIC) and Rissanen (MDL) criteria, as well as from the minimum condition for the error function. In our case, the autoregression of the order of 13 will be optimal by the Akaike criterion and close to optimal by the criterion of the minimum of the error function. An analysis of the obtained autoregressive model shows that the coefficients $b_{}^{\left(\mathrm{1}\right)}\left(i\right),\dots ,b_{\mathrm{0}}^{\left(\mathrm{11}\right)}\left(i\right)$are close to zero.

We will use the model of autoregression of the type for forecasting

$$\begin{array}{}\text{(5)}& y_i-\underset{m=\mathrm{11}}{\overset{\mathrm{13}}{\sum }}{b}_{\mathrm{0}}^{\left(m\right)}\left(i\right)y_{i-m}=e_i.\end{array}$$

The identification results of model ( 5 ) using LS and RLS are presented in Figure 02 .

The determination coefficients for the LS model are $R_{LS}^{\mathrm{2}}=\mathrm{0.87}$ and for the RLS model $R_{RLS}^{\mathrm{2}}=\mathrm{0.89}.$

It is worth noting that an attempt to create a model according to the SARIMA methodology led to a simple model with a coefficient of determination $R_{SARIMA}^{\mathrm{2}}=\mathrm{0.83}.$ An increase in the number of parameters resulted in no improvement in this indicator. This fact may indicate that the time series has a long memory ( Baillie, 1996).

A further area of research may be the construction of models of the number of marriages based on autoregressions with differences of fractional order ( Guegan & Ferrara, 2008; Ivanov, Engelgardt, & Sandler, 2018) and fractional SARIMA model ( Ray, 1993; Liu, Chen, & Zhang, 2017).

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Conclusion

The paper proposes an adaptive seasonal model of the number of marriages in Samara region. As a learning algorithm, the recursive least squares method with forgetting coefficients is used. It is shown that the forecast can be explained on the basis of a three-parameter autoregressive model.

The construction of the model according to the SARIMA methodology (seasonal autoregressive integrated moving average) confirmed the single root hypothesis. However, taking the seasonal first difference does not allow constructing, it does not allow constructing a high-quality autoregressive model of residues. This fact may indicate the presence of a long memory in the time series. In the future, it is planned to create a model based on autoregression with fractional order differences.

Acknowledgments

The authors gratefully acknowledge the contributions of prof. O.A. Katsyuba in improving the paper.

References

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