 # Algorithms of Parametric Estimation of Polynomial Trend Models of Time Series on Discrete Transforms

## Article excerpt

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

Time series analysis and forecasting are the actual problems of finance, marketing, strategic planning and management. In the strategic management the most actual problem is forecasting strategic competitiveness of company. In the aim of solving this problem analytics need to forecasting financial and economic indicators of companies and markets. The main researches use one of two ways for analysis and forecasting time series: parametric and nonparametric econometrics. The parametric approach often bases on one-dimensional time series models as trend and seasonal trend models, ARMA and ARIMA. The nonparametric approaches use modern ways of analysis and forecasting as neural network, fuzzy logic etc. These methods also show high efficiency on financial markets.

An effective solution to a wide range of applications of time series analysis (TSA) is bases on the use of discrete transformations (DT). These transformations are the acknowledged tools for creating computationally efficient algorithms for solving problems of time series analysis (fast algorithms, algorithms with a reduced computational complexity of algorithms with balanced computational complexity).

In TSA, the discrete orthogonal transformations (DOT) are widely used in the different orderings of discrete Walsh functions (DWF). We propose a number of generalizations of DWF systems including the oblique ways. Among the oblique DWF should be noted discrete systems of inclined Walsh functions and piecewise exponential functions. These generalizations of oblique discrete Walsh systems are bases on the systems of orthogonal and non-orthogonal Rademacher functions. In introduced the oblique systems of discrete functions entitled "discrete bases of piecewise exponential functions". However, these can be build using the shifted inclined Rademacher functions, and we presume should be call as «discrete bases of shifted inclined Walsh functions". For brevity, we will use the name for these bases "oblique discrete Walsh bases".

In the frame of this work, we realize under the DT is the transformation introduced by multiplying a vector by a matrix whose rows are the basis vectors of an orthogonal or oblique discrete basis with N = 2n dimension:

...

where f = {f (i), i = 0, N - 1}T - N- dimensional time series vector; F = {F(i), i = 0, N - 1}T - Ndimensional vector of coefficients DT.

In this work, we develop efficient (computationally) algorithms for parametric estimation of polynomial trend models of time series based on the oblique DT.

TREND MODELS OF TIME SERIES

Effective application DT is in the creating time series models for solving problems of analysis and forecasting. Many problems of TSA have deal with regular time series (TS). In the analysis of time series structures often take the typical model in the form of an additive sum of four components: trend, seasonal fluctuations, cyclical component and random component. Frequently, depending on the particular situation the truncated models are used. Thus, some applications limited to using the form model:

...

where p(j) - trend (function deterministic trend); s(j) - stationary random component.

The task of allocation trend, in which it reduces to identify the class of adequate models, is important in many TSA applications. Among these time series processing tasks may be noted the problem of identification, smoothing, extrapolation, components extraction and spectral analysis of time series. This is because in TSA tasks the time series is divided to the components and further the components are studied. Even when a trend in itself is not of interest, it is necessary to study the allocation of the spectrum at higher frequencies. Many researchers have noted the importance of taking into account the trend in the spectral analysis of time series. In recommend taking into account the trend at all stages of the analysis. …

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