# Analysis of Data Measured on a Lattice

• Ulrich Halekoh
• Paul O. Degens
Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

## Abstract

A data vector z is additively decomposed into several components by minimizing a sum of quadratic forms in each component. The data may be measured on an arbitrary lattice but the quadratic forms have to be constructed according to the chosen lattice (here mainly in IR+). The decomposition method can be regarded as a spline approximation. The transformation of z to each component is linear, diagonalizable and its eigenvalues are contained in the unit interval. At first we introduce the concept of the detection of a structural change with this method. Then we concentrate on the analysis of two interdependent time series by the simultaneous decomposition of the two (coupled) series. For the application of the method the two series are best regarded as one datafield measured on a lattice with two directions, one direction representing the time and the other specifying the series. The concepts will be explained with data from a monitoring study in NRW on the influence of air pollution on vulnerable persons.

## Bibliography

1. Böhmer, K.(1974). Spline-Funktionen. Theorie und Anwendungen. Stuttgart: Teubner.Google Scholar
2. Degens, P.O. and Halekoh, U.(1990). “Bestimmung regulärer und irregulärer Anteile von auf Gitternetzen erhobenen Daten mittels quadratischer Formen”, Vortrag auf dem 2. Kolloquium über Biometrie und Ökologie, Wuppertal, September 1989, in printGoogle Scholar
3. Eicker, F.(1988). “Trend-Seasonal Decomposition of Time Series as Whittaker-Henderson Graduation”. Statistics 19 2, 313–338Google Scholar
4. Halekoh, U.(1989). Zerlegung von Datenfeldern mittels Quadratischer Formen, Diplomarbeit, Fachbereich Statistik, Universität DortmundGoogle Scholar
5. Halekoh, U. and Degens, P.O.(1986). “Additive Decomposition by Quadratic Forms”. In: Domschke, W., Krabs. W., Lehn, J., Spelucci, P. (eds.), Methods of Operations Research 57. Frankfurt athenäum VerGoogle Scholar
6. Nolte, D.(1987). Asthma. Das Krankheitsbild, der Asthmapatient, die Therapie. München-Wien-Baltimore: Urban & SchwarzenbergGoogle Scholar
7. Pauly, R. and Schlicht, E.(1984). “Zerlegung ökonomischer Zeitreihen: ein deterministischer und ein ökonomischer Ansatz”. Statistische Hefte 25, 291–303.Google Scholar
8. Schlicht, E.(1981). “A Seasonal Adjustment Principle and a Seasonal Adjustment Method Derived from this Principle”. Journal of the American Statistical Association, 76, 374–378Google Scholar
9. Wegman, E. J. and Wright, I. W.(1983), “Splines in Statistics” Journal of the American Statistical Association, 78, 351–3
10. Whittaker, E.T.(1923). “On a new Method of Graduation”, Proc. Edinburgh Math. Soc. 41, 63–75.Google Scholar

© Springer-Verlag Berlin · Heidelberg 1991

## Authors and Affiliations

• Ulrich Halekoh
• 1
• Paul O. Degens
• 1
1. 1.Medizinisches Institut für UmwelthygieneUniversität DüsseldorfDüsseldorfGermany