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.
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).
Using methods of descriptive Statistic of quantitative cross-section analysis, important measures are:
For the time series
meaningful only for
symmetric for k
used in practice:
Correspondence of the cross-correlation to thequantitative cross-section analysis
Relation of two time series, co-variance:
with lag k
It is not known which series is the leading series
non-symmetric for k
often non-stationary (we have trends) andperiodicalvariations
R: rest, noise
by logarithmizing → transition to additive model
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:
Not to be used for prognoses,
(increasing with p)
logistic trend functions
Local trend:movingaverage (running mean), to remove oscillations (seasonality)
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:
FFT of the basic period with trend
A very simple method for constant seasonalvariations
Assumption: no trend!
i is the month
k is the number of years
the perfect case:
Standardized phase average
1 if the month number i
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
The entire time interval is used for T1
Harmonic analysis - non-equidistant time intervals
- choice of the basic period
r2 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 Tj
Plot of the intensities against the frequencies fj
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
line parallel to x-axis → additive model
if the SLP linear line → multiplicative model
no decision → mixture model
Box/Cox Transformation additive or multiplicative?
for λ ≠ 0
for λ = 0
or in a simpler form
for λ ≠ 0
for λ = 0
λ = 0 multiplicative model
λ = 1 additive model
Use simple coeff. λ 1/4;1/3; 1/2;...
Acknowledgement additive or multiplicative?
I want to acknowledge to the Ministery of Education
and Science to support this work under the contract