IAAT (Astronomy): Time Series Analysis

Institut für Astronomie und Astrophysik

Abteilung Astronomie

Sand 1, D-72076 Tübingen, Germany

Hinweis: Einige Seiten auf astro.uni-tuebingen.de können veraltet sein und werden nicht mehr aktualisiert.
Note: Some webpages at astro.uni-tuebingen.de may be outdated and will no longer be updated.

Time Series Analysis

This is the homepage of the time series analysis group at the Institute for Astronomy and Astrophysics, Sect. Astronomy, in Tübingen (Germany).

We are dealing with problems of the deterministic (periodic) and non-deterministic behavior of astronomical time series measured by X-ray satellites. The object of our work is to develop tools both for analysing and simulating time series in order to improve the understanding of the time dependent processes taking place in the objects observed by satellites (EXOSAT, Ginga, ROSAT, XTE, ASCA).

The Sunspot Number Time Series

One of the most fascinating time series is the sunspot number, which exhibits the well know 11y period and which additionally shows irregular behaviour on timescales from days to decades. The number of sunspots reaches a maximum about every 11 years, but successive maxima have spots with reversed magnetic polarity. Thus the whole cycle is 22 years long. At the Earth's surface, magnetometers measure the geomagnetic field magnitude and direction. The figure below (left) shows the AA Yearly Index, a way of quantifying the magnetic disturbances, over the last 120 years superimposed to the sunspot number.

At first glance, observations carried out over time periods of weeks/months suggest that the latter two phenomena are stochastic in nature, but observations over time periods of decades reveal an intriguing cyclic pattern of gradual increase and decrease in the average number of sunspots visible on the solar disk. This was first noted in 1843 by H. Schwabe, an amateur solar astronomer, and provided the first hint of the existence of the sunspot cycle, whose period Schwabe estimated to be about 10 years. Further work revealed that the length of successive sunspot cycles is not strictly constant but varies from 9 to 11.5 years, with an average cycle period of about 10.8 years. The plot shown above (right) is a historical reconstruction of yearly-averaged sunspot group counts (yellow curve), extending all the way back to the first telescopic sunspot observations in the early seventeenth century. The purple curve is the Zürich normalized sunspot number. Note how the amplitude of the cycle, or the peak average number of sunspots seen in a given year, varies from one cycle to the next. Note also how cycles are asymmetric, in that the rise from sunspot minimum to maximum occurs more rapidly than the subsequent fall from sunspot maximum to minimum. Another striking feature on this plot is the dramatically reduced number of sunspots observed in the time period spanning the years 1645-1715. This was first noticed by G. Spörer, and investigated more systematically by E.W. Maunder. This time period is now usually referred to as the Maunder minimum. Proxies of geomagnetic activity such as aurorae (green crosses) correlate well with the sunspot number, in the sense that lower auroral counts are associated with low amplitude sunspot cycles (e.g., 1940-1960), and high counts with high amplitude cycles (1800-1822).

The daily (yellow), monthly (blue) end monthly smoothed (red) sunspot numbers are plotted for the last cycle, together with the 12 months ahead predictions (red dots) and the uncertainty interval (green).

The first application of stochastic autoregressive process modelling was performed by Yule (1927) on the yearly sunspot number as introduced in 1848 by the Swiss astronomer Johann Rudolph Wolf. In addition to the well known periodic behaviour, such a stochastic model reveals the existence of an irregular component which can be described by an autoregressive process using a damped oscillator (period=10.91y, tau=34.56y) and a pure relaxator (tau=10.11y), which explains the observed maxima asymmetry in the sunspot cycles.

Time Series Database

Some of the points listed below are still under construction, but you will already find some real data.

Here you find some selected EXOSAT-ME time series from Active Galactic Nuclei.
The README-File contains more detailed information on the data available.
The following link is a compilation of 500 time series from agriculture, physics to tree-rings: Time Series Data Library
The FDM Freiburger Zentrum für Datenanalyse und Modellbildung in Freiburg (Germany) has a project group for Characterization und Modelling of Time Series. If you are interested in more details on this subject just contact Dr. Jens Timmer from the FDM in Freiburg (Germany).

Tools

In a collaboration with the Freiburger Zentrum für Datenanalyse und Modellbildung we have already developed some tools to simulate and to analyse continous time series.

Computing a periodogram with fast fourier techniques using the IDL library.
Estimation of power law noise parameters in the frequency state.
On generating power law noise.
- Information on the method used.
- IDL algorithm.
Autoregressive models in time series analysis.
- Simulate AR(q) time series.
- Estimation of AR(q) parameters.
State space modelling using the EM-algorithm.
NETLIB, a repository of mathematical software, data and documents.

Publications

Furthermore we have already published some of our results concerning time series analysis tools.

On generating power law noise
Timmer J., König M., 1995, A&A, 300, 707-710
Analyzing X-ray variability by Linear State Space Models
König M., Timmer J., 1997, A&A Suppl.Ser., 124 (3), 589
The Seyfert Galaxy NGC 6814 - a highly variable X-ray soure
König M., Friedrich S., Staubert R., Timmer J., 1997, A&A, 322, 747-750
Analyzing X-ray variability by Linear State Space Models (96k gzip'ed Postscript including figures)
König M., Timmer J., 1996, Astron.Ges. Abstract Series
A new period for the magnetic white dwarf KPD0253+5052
Friedrich S., König M., Schweizer W., 1997, A&A, 326, 218-220
Analyzing X-ray variability by State Space Models - Application to an EXOSAT AGN sample
König M., Timmer J., Staubert R. 1997, D.Maoz et al (eds.), Astronomical Time Series, 265-268, 1997 Kluwer Academic Publishers
Analyzing short-term X-ray variability of Cygnus X-1 - Application of Linear State Space Models
Pottschmidt K., König M. 1997, D.Maoz et al (eds.), Astronomical Time Series, 187-190, 1997 Kluwer Academic Publishers
Time Series Analysis of an EXOSAT AGN sample
König M., Wilms J., Staubert R., 1997, A&A, in preparation
Analyzing optical AGN lightcurves by Linear State Space Models
König M., Timmer J., 1997, A&A, in preparation
On the inclination and binarity of the pulsating pre-white dwarf PG2131+066
Paunzen E., König M., Dreizler S., 1998, A&A, 331, 162
Search for Variables in the TYCHO Data Stream
Friedrich S., König M., Wicenec A., TYCHO conference proceedings, Venice 1997
Correlation between variability timescale and X-ray spectral index in AGN
König M., Staubert R., Wilms J, 1997, A&A, 326, L25
A new bright X-ray galaxy
König M., Geckeler R., Staubert R., 1997, A&A, 329, 68
On a possible 13.81d period of the X-ray binary 4U1700-377
König M., Maisack M., 1997, A&A, 327, L33
The irregular temporal behaviour of the variable star R Scuti
König M., Paunzen E., Timmer J., 1997, MNRAS, accepted
Linear State Space Modeling of Gamma-Ray Burst Lightcurves (67k gzip'ed Postscript including figures)
Band D., König M., Chernenko A., 1997, GRB Symposium, Huntsville, USA
Analyzing short-term X-ray variability of Cygnus X-1 with Linear State Space Models
Pottschmidt K., König M., Wilms J., Staubert R. 1998, A&A, 334, 201
Zeitvariabilität in Aktiven Galaxien
Michael König, PhD thesis, 1997, University of Tübingen
Anwendung Linearer Zustandsraummodelle auf die Kurzzeitvariabilität des Schwarzlochkandidaten Cygnus X-1 (1.056k gzip'ed Postscript including figures)
Katja Pottschmidt, diploma thesis, 1997, University of Tübingen