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POSTER

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(Special Program Package)*)

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Application of Auto Logistic Regression Spatial Model using Variogram Based Weighting Matrix to Predict Poverty Village Category

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1063

The 3"* International Conference on Mathematics and Statistics (ICoMS-3) Institut Pertanian Bogor, Indonesia, 5-6 August 2008

SPECTRAL APPROACH FOR TIME SERIES ANALYSIS

Kusman Sadik

Department of Statistics, Institut Pertanian Bogor Jl. Meranti, Wing 22 Level 4, Kampus IPB Darmaga, Bogor 16680 – Indonesia

e-mail : kusmans(a),ipb.ac.id

Key Words

: Spectral analysis, seasonal adjustment procedures, cross spectral, fuzzy logic systems, input signals,

fourier transform, fractal space-time.

1.

Introduction

The first appearance of spectral analysis in the study of macroeconomic time series dates motivated by the

requirement of a more insightful knowledge of the series structure and supported by the contemporaneous

progress in spectral estimation and computation. The first works focused on the problem of seasonal

adjustment procedures and on the general spectral structure of economic data. Cross spectral methods were

pointed out from the outset as being important in discovering and interpreting the relationships between

economic variables. After the early years, the range of application of such analysis was extended to the study of

other econometric issues, among which the controversial trend-cycle separation, the related problem of

business cycles extraction and the analysis of co-movements among series, usefiil in the study of international

business cycles. In particular, cross spectral analysis allows a detailed study of the correlation among series. An

empirical investigation about the possibility that the market is in a self-organized critical state (SOC) show a

power law behaviour in the avalanche size, duration and laminar times during high activity period (Bartolozzi,

Leinweber and Thomas, 2005).

2.

Mathematical Models for Fractal Fluctuations and Deterministic

The larger scale fluctuations incorporate the smaller scale fluctuations as intemal fine scale structure and

may be visualized as a continuum of eddies (waves). Dynamical systems in nature are basically fiizzy logic

systems integrating a unified v^ole communicating network of input signals (perturbations) with

self-organized ordered two-way information transpot between the larger and smaller scales. Self-similar

structures are generated by iteration (repetition) of simple rules for growth processes on all scales of space and

time. Such iterative processes are simulated mathematically by numerical computations such as

X„+i

= F(^«)

where

X„+i

the value of the variable X at (« + i f " computational step is a fiinction F of its earlier value

X„.

The

spectrum of { x }

is defined to be the Fourier transform of

Yx(/^),

3. Model Concepts of Universal Spectrum of Fluctuations

A general systems theory was first developed to quantify the observed fractal space-time fluctuations in

turbulent fluid flows. The model concepts are independent of the exact details of the physical, chemical or

other properties and therefore applicable to all dynamical systems. The observed inverse power law form for

power spectra implies that the fi-actal fluctuations can be visualized to result from a hierarchical eddy

continuum structure for the overall pattern. Starting from this simple basic concept that large scale eddies

form as envelopes of intemal small scale eddy circulations the following important model predictions are

derived.

1064

Let Z

t

is time series stationer process with autocorrelation

γ

k

. Its Fourier

transformation is (Yujima, 1998) :

∑

∞ −∞ =

−

= k

iwk ke

w

f γ

π

2 1 ) (

Function

f(w)

is continu function non-negative that representative of spectrum for

autocorrelation function

γ

k

. Spectrum of

f(w)

and autocorrelation function

γ

k

are unique

Fourier transformation. Generally, spectrum of ARMA(p,q) process can be written

(West, 1999):

) ( ) ( 2

) ( ) ( ) (

2

iw p iw p

iw p iw q a

e e

e e w

f

φ πφ

θ θ σ

− −

[image:6.595.90.467.218.419.2]

=

Figure 1: Plots of the spectrum of MA(1) processes (0 = 0.5 for the left figure and 0 = - 0.5 for the right figure) – Simulated Data

Figure 1 plots the spectrum of MA(1) processes with positive and negative coefficients. When 0 > 0, we see that the spectrum is high for low frequencies and low for high frequencies. When 0 < 0, we observe the opposite. This is because when 0 is positive, we have positive one lag correlation which makes the series smooth with only small contribution from high frequency (say, day to day) components. When 0 is negative, we have negative one lag correlation, therefore the series fluctuates rapidly about its mean value.

Figure 3: Window Functions and Their Frequency Response (Simulated Data). _{The rectangular}

[image:6.595.73.473.535.712.2]

1065

respective Fourier transforms (right panel). For the latter, the number of points N = 15 has been chosen rather small to emphasize the differences. Note in the zoom (right panel, inset) the reduced side lobe amplitude and leakage of the Hanning and Hamming windows with respect to the rectangular one, the Hamming window performing better in the first side lobe.

The only way to prevent this effect, would be to choose T (or equivalently N) as a multiple of the largest period that is likely to occur. Unfortunately, this is feasible only if we have some idea of the frequencies involved in the process and would in any case entail some loss of data at one or both sample ends. As for the cutoff fr equencies ν_l= k_l/(N∆t) and ν_h= k_h/(N∆t), given the value of N,

they must be chosen to be multiples of T −1

,otherwise the filter does not completely remove the zero frequency component (i.e. the signal mean) and cannot help in eliminating unit roots.

4. Conclusion