1 / 27

# R. Werner Solar Terrestrial Influences Institute - BAS - PowerPoint PPT Presentation

Time Series Analysis by descriptive statistic. R. Werner Solar Terrestrial Influences Institute - BAS. Def.: A time series is a sequence of data points measured at successive times (often) spaced in uniform time intervals.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' R. Werner Solar Terrestrial Influences Institute - BAS' - donoma

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

by

descriptive statistic

R. Werner

Solar Terrestrial Influences Institute - BAS

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!

Auto-correlation

symmetric for k

lag k

auto-covariance

used in practice:

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

often non-stationary (we have trends) andperiodicalvariations

Models:

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)

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:

Linear trend:

Polynomial trend:

recursive formulae

• 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 with trend

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

How to determine which is the better model approximation, additive or multiplicative?

Analysis of the variance:

- 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