Multifactor analysis of the time series

 Contents

• Introduction
• Description of algorithm (pdf) and software (zip)
• References
• Analysis of the time series

The birth of the multifactor analysis of the time series (singular-spectrum analysis [1]) is usually associated with the first papers of Broomhead and King [2,3]. At present the list of publications is more than hundred. Around sixty references can be found in the book [1] together with many examples of application of such approach to analysis of the time series and description of various theoretical and practical tasks.

We call the represented algorithm as multifactor analysis of the time series (MAS), since the algorithm has likeness to one of variants of the factor analysis, because in MAS the time series are represented as the sample values of a random vector (of given length M) with the subsequent application of singular value decomposition (SVD) to a sample correlation matrix of such vector and evaluation of the principal components. On the other hand, it is possible to interpret MAS as decomposition of the time series using the system of orthogonal basis functions, which are obtained on the basis of the original time series.

Such algorithm is ‘essentially a model-free technique; it is more an exploratory, model building tool than a confirmatory procedure. It aims at a decomposition of the original series into a sum of a small number of interpretable components such as a slow varying trend, oscillatory components and a 'structureless' noise’ [1]. Possible application of MAS is very wide and include climatology, meteorology, signal processing and physics. This approach can be applicable to other fields of sciences and human activity.

Thus, using MAS, the interpreter can extract slowly varying components, oscillating components, noise components, i.e. to implement the denoise procedure. The choice of number of basis functions for decomposition of initial time series formally is not given and is determined by the interpreter according to a type of problem.

Description of MAS with examples of processing of the time series can be downloaded as pdf file. Software modules written in MATLAB language can be downloaded as zip file.

References

[1] Golyandina, N., V. Nekrutkin and A. Zhigljavsky (2001) Analysis of Time Series Structure: SSA and Related Techniques, Chapman & Hall/CRC.

[2] Broomhead, D. S. and G. P. King (1986) Extracting qulitative dynamics from experimental data, Physica D, 20,217-236.

[3] Broomhead, D. S. and G. P. King (1986) On the qualitative analysis of experimental dynamical system, In S. Sarkar (Ed.), Nonlinear Phenomena and Chaose, pp. 113-144, Adam Higer, Bristol.

Contents

• Analysis of the temperature time series
• Analysis of the rivers run-off

The multifactor analysis is implemented for processing of time series from 37 meteorological stations (monthly data). Slowly (or "like slowly") components are recovered with the use of decomposition on 132 principal components. The results of recovery are represented in graphic form, including 15 first principal components and diagrams for 5 first principal components, which demonstrate the structure of the time series.

Number	Latitude.N	Longitude	Station	Period
04360	65.37	322.21	ANGMAGSSALIK	1895	2000
04063	65.41	341.55	AKUREYRI	1882	2000
01001	70.59	351.20	JAN MAYEN	1921	2000
01316	60,24	5,19	BERGEN/FREDRIKSBERG	1816	2000
01152	67.16	14.22	BODO	1868	2000
01028	74.31	19.01	BJORNOVA	1920	2000
02196	65.50	24.09	HAPARANDA	1860	1999
22113	68.58	33.03	MURMANCK	1918	1999
22550	64.35	40.30	ARKHANGEL'SK	1813	2000
22165	68.39	43.18	KANIN NOS M.	1915	2000
23405	65.27	52.16	UST'-TSIL'MA	1889	1999
20744	72.23	52.44	MALYE KARAMAKULY	1897	2000
23330	66.32	66.32	SALEKHARD	1883	2000
23840	61.15	73.30	SURGUT	1885	1996
20674	73.30	80.14	DIKSON	1916	2000
23472	65.47	87.57	TURUKHANSK	1881	2000
24507	64.10	100.04	TURA	1928	1999
20891	71.59	102.27	KHATANGA	1929	1999
24641	63.46	121.37	VILYUISK	1898	1999
21921	70.41	127.24	KYUSYUR	1918	1998
24266	67.33	133.23	VERKHOYANSK	1891	2000
21647	73.11	143.56	SHALAUROVA	1928	2000
25399	66.10	169.50	UELEN	1928	1999
25551	64.68	170.02	MARKOVO	1894	2000
25563	64.47	177.34	ANADYR'	1898	2000
21982	70.97	181.28	VRANGELYA	1926	1999
70200	64.30	194.44	NOME	1907	2000
70219	60.47	198.12	BETHEL	1923	2000
70026	71.18	203.32	BARROW	1882	2000
70261	64.49	212.08	FAIRBANKS	1904	2000
71938	67.49	244.52	COPPERMINE	1937	2000
71934	60.00	248.02	FT SMITH	1914	1998
71916	63.20	269.17	CHESTERFIELD	1921	1991
71917	80.00	274.04	EUREKA	1947	1999
71090	70.27	291.23	CLYDE	1942	1999
04210	72,47	303,56	UPERNAVIK	1873	2000
04250	64,10	308,57	GOTHAAB	1866	2000

Analysis of the rivers run-off

The multifactor analysis is implemented for processing of river run-off for 7 Siberia rivers(monthly and annual data). In the case of monthly data, slowly (or "like slowly") components are recovered with the use of decomposition on 132 principal components. The results of recovery are represented in graphic form, including 40 first principal components and diagrams for 5 first principal components, which demonstrate the structure of the time series. In the case of annual data for decomposition of the time series 11 principal components is used.

River name	Period
Dvina (monthly and annual data)	1881	1988
Enisei (monthly data)	1936	1992
Indigirka (monthly data)	1937	1993
Kolima (monthly data)	1927	1989
Lena (monthly data)	1935	1993
Ob (monthly data)	1930	1993
Pechora (monthly data)	1932	1987