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R. Werner Solar Terrestrial Influences Institute - BAS

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Def.: A time series is a sequence of data points

measured at successive times (often)

spaced in uniform time intervals.

Time series analysiscomprises 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 tomake 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 forgetvisualization
- scatter plots
- example: histogramms

For the time series

meaningful only for

stationarity!

Correspondence of the cross-correlation to thequantitative cross-section analysis

Relation of two time series, co-variance:

with lag k

or

It is not known which series is the leading series

cross-correlation:

non-symmetric for k

Time series decomposition into components

often non-stationary (we have trends) andperiodicalvariations

Models:

additive:

T: trend

S: seasonal

R: rest, noise

multiplicative:

by logarithmizing → transition to additive model

mixed:

- Trend determination
- 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

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:movingaverage (running mean), to remove oscillations (seasonality)

odd:

point numbers

even:

where

2q+1 is the number of sampling points

bi are the weights

Besides, for removing the seasonal means, we have to calculate the running mean over 13 months, with bi = 1/24 for the first and the last month, otherwise bi=1/12 !

For the given examples:

Problems related to the trend determination

- 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
- By smoothing the reversal points of the time series are shifted
- 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

A very simple method for constant seasonalvariations

Assumption: no trend!

also:

Phaseaverage

i is the month

k is the number of years

the perfect case:

in practice:

Standardized phase average

1 if the month number i

0 else

12 equations !

or together with a polynomial trend

For a multiplicative model:

Strategies: - Step by stepdetermination of the period Tp

- Test of a theoretical hypothesis

Fourier frequencies

(n odd)

The entire time interval is used for T1

Harmonic analysis - non-equidistant time intervals

- choice of the basic period

- If 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
- If j/n are not Fourier frequencies, then we have to calculate all coefficients again by changing m
- If 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

r2 is the determination coefficient, the part of the explained sum of the squared deviations,

besides is the explained sum of the squared deviations

Periodogram

Plot of the intensities against the periods Tj

Spectrogram

Plot of the intensities against the frequencies fj

- 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

for λ ≠ 0

for λ = 0

or in a simpler form

for λ ≠ 0

for λ = 0

- Determination of λ:
- stand. dev. plot against logarithms of the mean time interval points
- combination with SLP

λ = 0 multiplicative model

λ = 1 additive model

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

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