Testing for Lack of Dependence in Functional Linear Model Inga Maslova, Piotr Kokoszka Department of Mathematics and Statistics, Utah State University Introduction References Test ProcedureSimulation Study ResultsApplication ResultsConclusion  FLM allows to work directly with the measurements  Test statistic gives good approximation for sample sizes around 50  The empirical sizes are close to nominal sizes and are not affected by the choise of the number of PC’s  Test does not depend on the choise of the functional basis  Novel methodology that is relatively easy to apply, useful for finely sampled data analysis Objectives and Motivation We study the functional linear model, Y n = ΨX n + ε n, with functional response and explanatory variables. We propose a simple test of the nullity of Ψ based on the principal component decomposition. The test statistic has asymptotic chi-squared distribution, which is also an excellent approximation in finite samples. The methodology is applied to data from terrestrial magnetic observatories. Acknowlegment Contact Information  Bosq, D. (2000). Linear Processes in Function Spaces. Springer, New York.  Cardot, H., Ferraty, F., Mas, A. and Sarda, P. (2003). Testing hypothesis in the functional linear model. Scandinavian Journal of Statistics, 30,  Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis. Springer Verlag.  Research supported by NSF grant DMS  Data provided by the global network of observatories – INTERMAGNET data For further information please contact Inga Maslova Manuscript on which the poster is based is available upon request. Poster is available at  Finely sampled records available  Dimension reduction using functional principal components (FPC)  Magnetometer data analysis (Fig. 1). Testing whether auroral currents have impact on equatorial and mid- latitude currents Fig. 1 Horizontal intensities of the magnetic field measured at a high-, mid- and low-latitude stations during a substorm (left column) and a quiet day (right column). Note the different vertical scales for high-latitude records. Functional Linear Model: Test hypothesis: versus Introduce the operators: Denote their empirical counterparts by e.g. Define the eigenelements by Empirical eigenelements: implies Ψ practically vanishes onif Equivalently, in Hilbert space notation Identity Table 1. Results of the test for substorm days that occurred in 2001 from January to March (A), March to May (B), June to August (C). Test statistic: Under H 0 Fig. 2 Empirical size of the test for α = 1%, 5%, 10% (indicated by dotted lines) for different combinations of p and q. Here ε n and Y n, n = 1, 2,…, N are two independent Brownian Bridges. Fig. 3 Empirical power of the test for different combinations of principal components and different sample sizes N. Here X n and ε n are Brownian Bridges. In panels (a), (b) ||Ψ|| = 0.75; in panels (c), (d) ||Ψ|| = 0.5.  Size and power of the test do not depend on p and q.  Use R = 1000 replications of samples  Brownian bridge and motion processes, Fourier and spline bases give same results  Test if auroral geomagnetic activity reflected in the high-latitude curves has an effect on the processes in the equatorial belt reflected by the mid- and high-latitude curves  Use one-day high-latitude records during the substorms as X n. Let Y n be the records during the same substorm days and 1, 2, 3 days later. Check how long the effects of substorm last  Table 1 provides test results for the substorm effects during: January – March (40 substorms); March – May (42 substorms); June – August (42 substorms)  Selecting three month periods ensures that the data is identically distributed (distribution could be effected by the Earth rotation)  Substorm effect lasts for about 2 days Mid- latitude ABCABCABCABC BOU0BOU1BOU2BOU ?0? FRD0FRD1FRD2FRD ?0? THY0THY1THY2THY MMB0MMB1MMB2MMB ?0? 00? 00 Low-latitude HON0HON1HON2HON ?0?0?00000 SJG0SJG1SJG2SJG ? HER0HER1HER2HER KAK0KAK1KAK2KAK ?0?000?00