Body size has a positive, linear association with the area traversed by an individual in its normal activities of food gathering, mating and caring for the young   . As expected, the area traversed in nature is larger and more variable in male than in female gracile opossum . Therefore, we expect that activity patterns will differ between male and female gracile opossum. Specifically, we will test the null hypothesis of homogeneity of mean curves of activity patterns from male and female gracile opossum. To the best of our knowledge this is the first attempt to integrate wavelets to model curves derived from activity data with functional data analysis in the framework of functional analysis of variance.
Ramsay et al. Wavelets are especially built to provide regular esti- mates through multiscale shrinkage . We refer to Kist and Pinheiro  for a detailed development of the wavelet functional data analysis for de- pendent errors. Wavelets are basically elements of some specially built basis of the space of square-integrable functions.
This means the following. There are many different wavelet bases.
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Therefore, wavelet analysis is immediately equipped with a fast transform algorithm. In what follows, we give a general presentation of this procedure as needed for the animal temporal activity data set. Suppose f belongs to a convenient Besov space . We write the non- linear wavelet estimators as. The hypotheses of interest are.
The dual relationship between Besov spaces and wavelet bases leads to a na- tural change in the hypotheses being tested.
Instead of testing H 0 vs H 1 , a slight formal change is made, as proposed in  for independent errors. A two- step procedure is developed by Kist and Pinheiro  for dependent errors. The test statistic is compared to the cut-off point and the decision reject or not H 0 can then be made. Although these tests and hypotheses are mathematically different from the aforementioned hypotheses, for all applied purposes, they all yield the same inter- pretation as follows.
Thus, rejecting H 0 means that the average time-curves are taken with respect to observations of female and male behaviors are statistically diffe- rent. On the other hand, whenever H 0 is not rejected, one understands that there is not enough empirical evidence for each group to have different underlying functions. Again for our case, this can be interpreted as the data not providing statistically significant evidence that male and female specimens differ in their temporal activity behavior. The specimens of the gracile mouse opossum G.
Individuals captured were marked with a numbered ear tag and their sex and age were recorded. To assess spontaneous locomotor activity of the gracile mouse opossum we used an automated motor activity monitor Acti-Track v2. Instrument, Barcelona, Spain . Thirty-two infrared beam breaks, 16 each on perpendicular walls, were mounted 3 cm above the box frame floor and connected to an interface LE , LSI Letica Scientific In- struments, Barcelona, Spain , and data were sent to a computer.
Thence, loco- motor activity was assessed as a rate of light beam breaks during the period of the experiment . This means that whenever the mouse opossum was moving vertically or horizontally the rate of light breaks would be recorded as the activity variable. Hence, the higher the rate of light beam breaks the higher the activity of the individual.
Functional ANOVA using INLA
At the beginning of the experiment, each gracile mouse opossum was placed in the Perspex box and allowed to freely explore for 24 hours to habituate the individuals before conducting the experiments. Testing in the actimeter was done in an isolated room between and , which corresponds to the activity of gracile mouse opossums in the wild. Experimental settings such as those used in our study provide relevant infor- mation not only for the study of animal temporal activity pattern, but also for several areas of ecology and behavior including for example the association be- tween social and sexual preferences and genetic variation at microsattelite loci  , modulation of vocalization by hormones  , and the link between heri- table neuroendocrine variation and male sexual behavior .
The data analysis was performed as follows.
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Twelve hours were selected from the continuous observed curves. Three families of wavelet bases were employed on the data: Symmlets, Coiflets and Daubechies. Preliminary analyses led to one smoothness parameter for each family: Symmlets 8, Coiflets 3 and Daubechies 6.
These are different wavelet bases. The aforementioned data set was composed of temporal activity curves from 12 hours of data acquisition, in which data were taken every second for 6 males and 7 females of G. We then estimated f 1 and f 2. Figure 1 shows three estimators for each gender, based on three different wave- let bases: Symmlets 8, Coiflets 3, and Daubechies 6.
The data is shown in gray. With no loss of generality, time is transformed to [ 0 , 1 ]. Moreover, the proposed dependent estimators are much more regularized then the previously available wavelet estimators. Numerical results did not differ much among the bases chosen shown in Figure 1. However, the visual results for Coifflets were in general coarser than the other two bases. We should point out that there are no wavelets bases that are both compactly supported and symmetrical. Figure 1. Temporal activity curve curves analyzed using the proposed wavelet model. Curves obtained for three estimators for each gender, each onE based on a different wavelet base: Symmlets 8, Coiflets 3, and Daubechies 6.
The robust mean absolute deviation MAD or the standard deviation may be employed for the estimates of the measure of the noise variability. Standard deviation is in general superior to MAD, since the latter yields less regular estimated curves . The choice of the wavelet basis is usually quite unim- portant. The use of any such basis leads to the same inferential results. Some local characteristics of the estimated curves are highlighted or shadowed by each basis, but the results are the same.
The data were comprised of a total of 43, observations for each specimen, which means one observation at every second for 12 hours of experiment. The individual autocorrelation estimates varied from 0. The maximum difference between any estimates for the same data set given by the choice of the wavelet basis was not greater than.
One should note that, as expected, the estimated curve with independent errors was much less regular than its dependent counterpart. This happens considering each pair of curves for fixed wavelet basis and thresholding pro- cedure. Figure 1 displays the preliminary results for three families of bases.
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RPubs - Functional Data Analysis using zertbucaturb.tk R package
Hung-Kung Liu. Although the methodologies presented are designed for curve data, they can be extended to surface data. The author provides a number of simple methods for functional hypothesis testing. He discusses pointwise, L 2-norm-based, F -type, and bootstrap tests.
Assuming only basic knowledge of statistics, calculus, and matrix algebra, the book explains the key ideas at a relatively low technical level using real data examples. Each chapter also includes bibliographical notes and exercises. Neem contact met mij op over Events Sprekers Incompany. Welkom terug. Uw account. Agenda Seminars Masterclasses e-learning Sprekers Incompany.
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