R. Werner Solar Terrestrial Influences Institute - BAS Time Series Analysis by descriptive statistic

Def.: A time series is a sequence of data points measured at successive times (often) spaced in uniform time intervals. Time series analysis comprises methods that attempt to understand time series, often either to understand the underlying context of the data points - where did they come from, what generated them, or to make forecasts (predictions). From Wikipedia.org

Using methods of descriptive Statistic of quantitative cross-section analysis, important measures are: - arithmetic mean - variance - correlation coefficient Do not forget visualization scatter plots example: histogramms For the time series meaningful only for stationarity!

Auto-correlation For the time series auto-covariance used in practice: symmetric for k lag k

It is not known which series is the leading series Correspondence of the cross-correlation to the quantitative cross-section analysis with lag k or Relation of two time series, co-variance: cross-correlation: non-symmetric for k

Time series decomposition into components often non-stationary (we have trends) and periodical variations Models: additive : multiplicative: mixed: T: trend S: seasonal R: rest, noise by logarithmizing → transition to additive model

Step by step: 1.Trend determination 2.Trend subtraction from the series and determination of the seasonal component 3. After removing the seasonal component, the rest remains After this: analysis of the rest, correlation, seasonality or other periodicities or a trend

Determination of the trend Global trend (over the entire observation interval) or polynomial regression model of order p, splines Square sum of errors: F-test Not to be used for prognoses, (increasing with p)

Other linear models: exponential model logistic trend functions A>0 C>0 Local trend: moving average (running mean), to remove oscillations (seasonality) odd: even: point numbers

How does the variance change? 2q+1 is the number of sampling points b i are the weights where Besides, for removing the seasonal means, we have to calculate the running mean over 13 months, with b i = 1/24 for the first and the last month, otherwise b i =1/12 ! For the given examples:

Trend removing by calculation of differences Linear trend: Polynomial trend: recursive formulae

Problems related to the trend determination 1.For short time series, the determined trend will not be equal to the long time trend, and will not be distinguishable from the longer periodicities 2.By smoothing the reversal points of the time series are shifted 3.The production of autocorrelations by smoothing with running averages (quasi-periodicities – Slutzky effect)

FFT of the basic period,without trend FFT of the basic period with trend

Determination of the seasonal component Phase average Assumption: no trend! i is the month k is the number of years also: the perfect case: in practice: A very simple method for constant seasonal variations Standardized phase average

Or dummy regression with: 1 if the month number i 0 else 12 equations ! or together with a polynomial trend For a multiplicative model:

Periodogram analysis Strategies: - Step by step determination of the period T p - Test of a theoretical hypothesis Fourier frequencies (n odd) The entire time interval is used for T 1 Harmonic analysis - non-equidistant time intervals - choice of the basic period

Harmonic series oIf j/n are Fourier frequencies, the regressor functions are orthogonal. All coefficients can be calculated together and they are not changed by the choice of a new m oIf j/n are not Fourier frequencies, then we have to calculate all coefficients again by changing m oIf the data number is equal to the calculated coefficients, then we have no degree of freedom, the calculated series is not an estimation. The error term is zero! → filter

It can be proven that r 2 is the determination coefficient, the part of the explained sum of the squared deviations, besides is the explained sum of the squared deviations Plot of the intensities against the periods T j Periodogram Spectrogram Plot of the intensities against the frequencies f j

Other methods are:  Lomb-Scargle Periodogram  Wavelet

How to determine which is the better model approximation, additive or multiplicative? Analysis of the variance: spread versus level plot (SLP-diagram) - splitting the time series in to intervals, - determination of the standard deviations in the intervals - plotting the stand. dev. against the means line parallel to x-axis → additive model if the SLP linear line → multiplicative model no decision → mixture model

Box/Cox Transformation for λ ≠ 0 or in a simpler form for λ ≠ 0 for λ = 0 λ = 0 multiplicative model λ = 1 additive model for λ = 0 Determination of λ: - stand. dev. plot against logarithms of the mean time interval points - combination with SLP Use simple coeff. λ 1/4;1/3; 1/2;...

I want to acknowledge to the Ministery of Education and Science to support this work under the contract DVU01/0120 Acknowledgement