Homogeneity test for functional data

arXiv:1507.01835v1 [stat.ME] 7 Jul 2015 HOMOGENEITY TEST FOR FUNCTIONAL DATA RAM´ ON FLORES, ROSA LILLO AND JUAN ROMO Ram´on Flores Department of Mathematics, Universidad Aut´onoma de Madrid, Spain e-mail: [email protected] Rosa Lillo Department of Statistics, Universidad Carlos III de Madrid, Spain e-mail: [email protected] Juan Romo Department of Statistics, Universidad Carlos III de Madrid, Spain e-mail: [email protected] Abstract. In the context of functional data analysis, we propose new two sample tests for homogeneity. Based on some well-known depth measures, we construct four different statistics in order to measure distance between the two samples. A simulation study is performed to check the efficiency of the tests when confronted with shape and magnitude perturbation. Finally, we apply these tools to measure the homogeneity in some samples of real data, obtaining good results using this new method. Keywords: functional depth, homogeneity, FDA. 1. Introduction In the recent years functional data analysis (FDA) has become a very active domain of research in Statistics, because of its own interest and also for its applications in a number of contexts such as medical science, biology, chemistry and social sciences. In essence, the objects of study in FDA are real functions which are assumed to be generated by means of a stochastic process. The functions are observed in a Date: July 8, 2015. 1 2 RAM ´ ON FLORES, ROSA LILLO AND JUAN ROMO certain number of fixed points or time instants, but instead of being treated as multivariate data, they are smoothed using appropriate tools. Nevertheless, a number of techniques of multivariate data have been adapted or generalized to the FDA context. The main references in this field are Ramsay-Silverman [RS05] and Ferraty-Vieu [FV06]. In this paper, we address the problem of homogeneity between samples of functions; that is, given two samples of curves, we need to decide whether these two samples have been produced or not by the same process so that they have equal probability distributions. This problem has been recently considered. Our test fits in the framework of the rank test of L´opez-Pintado and Romo [LR09] (based in turn on [LS93]) to establish the homogeneity of two functional samples. Other authors have confronted related problems, but more from the point of view of comparing operators rather than testing homogeneity. Benko et al. [BHK09] present methods for testing equality of means between functional data that rely, respectively, on bootstrap and asymptotic procedures. Horv´ath-Kokoszka ([HK12] and [HK13]) also describe tests to compare the equality of covariance operators. Cluster algorithms have also been proposed in [ACMM03], for example. Finally, a different point of view is developed by Cuevas-Febrero-Fraiman in [CFF04], where an F-test for analysis of variance based on functional distances was proposed. In a similar way, the approach we take in this paper is related to distances between the two functional samples which are based on depth measures, and it is important to remark that we intend to test homogeneity rather than means or covariances. Consider an interval T⊂R, and a finite sample F={x1. . . xn}of real functions defined over the interval. We will always assume that the functions lie in C1(T). The concepts of distance between samples that we introduce in this paper will be based on the statistical depth, a concept originating in the statistical analysis of multivariate data and then extended to functional data. In our context, a depth functional with respect to the sample Fwill be a functional d:C1(T)→R, whose value should depend in a certain way on the sample Fand also on a depth measure defined a priori. In this way, the value of dover the function will constitute a measure of how deep the function f“inside” the sample Fis. By means of these functionals, we construct four families of statistics which are shown to be useful to decide if two samples of functions are homogeneous or heterogeneous. In order to understand the behaviour of the measures with respect to differences of magnitude and shape in HOMOGENEITY TEST FOR FUNCTIONAL DATA 3 the samples, we have tested our methods on several samples of simulated functions. Moreover, we discuss homogeneity in some real contexts, such as Ramsay height data, the tecator sample and the mitochondrial data MCO. See the last section for details. The structure of the paper is as follows. In Section 2 we present the concept of depth, introduce the notion of depth with regard to a sample and describe our statistics. Section 3 reviews the measures of depth that are used in order to construct the different homogeneity tests. Section 4 is devoted to the analysis of some samples of simulated data, whereas in the last section we perform the test for the real data examples. 2. Distances between functional samples In the growing field of FDA, where functions are data, a crucial general goal is to define concepts which mimic or transport the usual notions in multivariate data analysis. The depth of functions was defined -in any of its versionsin this sense, as a generalization to this context of a notion of centrality, with the deepest function of a certain sample being an adequate definition of the “median” of the data. In order to introduce intuitive statistics that indicate in some way distance between two samples, we propose the definition of depth of a sample with respect to another one. Given a certain measure of functional depth d, and given a sample of functions Fand another function gnot necessarily in F, we denote by dF(g) the depth of g with regard to the sample F ∪ {g}. We define henceforth the notion of the deepest function of a sample with respect to another: Definition 2.1. Let Fand Gbe two finite samples of continuous functions defined in an interval T. The deepest function of Gwith regard to Fis the function gof the sample Gwhich maximizes dF(g)among g∈ G. We will denote this function by DF(G), or simply D(G)if the base sample Fis understood. If there is more than one function in Gfor which the depth is reached, we can choose any of them as DF(G), or else we can consider the whole set as the deepest subsample of Gwith regard to F. Observe that if the samples Fand Gare large enough, the probability of finding two functions of Gwhich maximize the depth decreases, so usually we can talk about the deepest function. 4 RAM ´ ON FLORES, ROSA LILLO AND JUAN ROMO The definition of the deepest function of Gwith respect to Fis addressed to propose a solution to the problem of homogeneity in the context of functional data. Given two or more samples of functions, we say that the functions are homogeneous if they come from the same experiment, and then have equal probability distributions. In our framework, explicitly determining the distributions is usually a very difficult problem, so we are forced to design different strategies to test homogeneity. We offer hence a different approach to the problem, by using the depth measures to perform an analysis which, by nature, may include the distance between functions, their magnitude and their shape. Our starting point are two samples of functional data, Fand G. The functions DF(G) or DG(F) may supply interesting information about homogeneity. In this sense the concept of deepest function can be used in different ways. In the following, we propose several possible statistics which depend on the notion of deepest function and allow us to undertake the analysis of homogeneity. We define the first statistic P1as P1(F,G) = dFDGG. Probably this is the more natural approach to the homogeneity problem, since roughly speaking, the function DGGis the most representative element of the experiment which produces the sample G. Hence, it is reasonable to compute how deep this estimator is with respect to F. The greater this depth, the less likely the two samples come from different experiments. The second statistic is defined as a variation of the previous one: P2(F,G) = |P1(F,G)−P1(F,F)|. This definition may be considered a kind of normalization of the previous one. It could happen that the nature of the experiment which originates the sample Fmakes impossible for any datum of the experiment to reach the value 1 (for example, if the experiment produces two well-defined “bands” of functions, or if some deep functions cross each other in close points). In this case, P1(F,F) would give a good estimation of the maximum of these depths, and the difference |P1(F,G)−P1(F,F)|would be more informative than the value P1(F,G) alone; see the computations section for interesting questions about this issue. In this case, the samples are likely to come HOMOGENEITY TEST FOR FUNCTIONAL DATA 5 from the same experiment as the statistic gets closer to zero. It would probably be equivalent to consider the quotient instead of the absolute value of the difference. A different approach is given by the statistic P3(F,G) = dF(DFG), which identifies the F-depth of the deepest function of Gwith respect to F. This is the function of the sample Gwhich is more likely to come from the experiment that generates the sample F, and then it is relevant from the point of view of the classification. In particular, |P3(F,F)|=P1(F,F) for any sample F. Observe that the function DFGcould not be a good estimator for the result of the experiment that generates F. Then, if we intend to use it for the classification of experiments, it would also be interesting to produce a measure that controls simultaneously the F-depth and the G-depth of DGG. One possible option is to define a measure in [0,1]×[0,1] whose values are the F-depth and the G-depth of DFG; in this context, the first number would be the measure of the depth itself, while the second would be interpreted as a control number of how sharp the measure is. However, this approach is bivariant, so we propose instead an alternative univariant version that avoids that disadvantage and captures essentially the same information: P4(F,G) = |P3(F,G)−P1(F,F)||P3(F,G)−P1(G,G)|. The greater this number, the less likely the two samples come from the same experiment. Once the statistics are defined, we propose the following method for testing the null hypothesis of equality of distributions of the two functional samples. We use a bootstrap approach to test the null hypothesis of homogeneity. 1. Select a functional depth measure dFand a statistic P=Pifor some i∈ {1,2,3,4}, which will depend on the previous concrete choice. In this paper, to deal with dFwe will use Fraiman-Muniz depth, h-modal depth, random depth RPD, band depth BD and modified band depth mBD, but there are other possible choices for the depth measure. 2. Now consider the samples Fand G, and propose as a null hypothesis H0that Fand Gcome from the same experiment. We perform then a hypothesis test to reject (or not) H0. 6 RAM ´ ON FLORES, ROSA LILLO AND JUAN ROMO 3. Define the sample Has the union F ∪ G, and obtain Nbootstrap samples of Hof size |H|. For any 1 ≤j≤N, let Sjbe the corresponding sample, denote by S1 jthe sample of the first |F| functions and by S2 jthe sample of the last |G|. Then compute Pj=P(S1 j,S2 j). 4. For an appropriate size α, compute a confidence interval for the values Pj. 5. The null hypothesis will be rejected if and only if the functional P(F,G) does not belong to the interval, and in this case we will assume that the groups are not homogeneous. The nature of the computations suggests using one-sided confidence intervals. To obtain the critical value at 95% of confidence, we trim five percent of the data in the appropriate side of the interval: in measures P1and P3there should be smaller values, while in the normalized one values will be larger. Below we present our results with both simulated data and real data, but we first review the functional depths we use. 3. Functional depths The concept of depth in the context of functional data analysis generalizes the same notion for multivariate data. While the multivariate measures are mainly addressed to explore a certain centrality of a point in some real vector space, the different nature of the functional data forces the statistics to consider other features of the functions involved, such as the shape of the functions or the amount of time they spend in a certain range of real numbers. In this sense, we have chosen different depth measures which in turn explore different features of the functions inside the samples. We start with the pioneering work of Fraiman-Muniz, whose goal is to measure how much time every function is deep inside the sample. Fraiman-Muniz depth. Consider a sample of curves {x1(t),...xn(t)}defined on the interval [0,1]. Denote by I(−) the indicator function and consider, for every i∈1. . . n, the function: Fn,t(xi(t)) = 1 n n X k=1 I(xk(t)≤xi(t)), and also the univariate depth Dn(xi(t)) = 1 − |1 2−Fn,t(xi(t))|. HOMOGENEITY TEST FOR FUNCTIONAL DATA 7 Then, the Fraiman-Muniz depth of the function xi(t) is defined in [FM01] as the integral: FM(xi(t)) = Z1 0 Dn(xi(t))dt. h-modal depth. This measure was first defined by Cuevas et al. [CFF06] and is addressed to identify the functional mode of the sample. Consider again a sample of curves {x1(t),...xn(t)}, select a value hwhich should be interpreted as a bandwidth, and also consider a kernel function defined on the real positive numbers. Then the h-modal depth of the function xi(t) with respect to Kand his defined as: hDn(xi, h) = n X k=1 K(kxi−xkk) h. In this paper, as recommended by the aforementioned authors, we take the norm L2,has the 15th percentile of the empirical distribution of the norms kxi−xkk, and Kbeing a convenient truncated Gaussian kernel. Random projection depths. These two versions of depth were proposed by Cuevas et al. [CFF07], and combine random projections of the functions of the sample in different directions with a bivariate data depth which is used to order the corresponding results. More precisely, given a sample of functions {x1(t),...xn(t)} and ν, a realization of a stochastic process whose values are random directions, we define the projection of xialong the direction νas Ti,ν =Z1 0 ν(t)xi(t)dt, and analogously, T′ i,ν =Z1 0 ν(t)x′ i(t)dt considering the derivatives instead of the trajectories of the function. If we select a bivariate data depth Dand assume Prealizations of V, we may define the following two versions of the random projection depth: RPD1(xi) = 1/P P X p=1 D(Ti,ν, Ti,ν), 8 RAM ´ ON FLORES, ROSA LILLO AND JUAN ROMO which takes into account only the trajectories of the functions, and RPD2(xi) = 1/P P X p=1 D(Ti,ν, T′ i,ν) which considers the functions and their derivatives. On this note, the role of Dto compute depths will be played by the bivariate version of h-modal depth. Moreover, we checked that the results obtained in our contexts using RPD1 and RPD2 were similar, and as the second one was computationally harder, in this paper we use only the first version, which we will denote simply by RPD. Band depth. In [LR09], L´opez-Pintado and Romo define two different versions of a new depth of essentially geometric nature. It is based on the concept of band, understood as a portion of the plane that is delimited by the sample of curves. More precisely, fix the sample F, and given a continuous function defined in T, denote by G(x) the graph of x. Then, for every jsuch that 2 ≤j≤n, the n-th band depth is defined by: BD(j) n(x) = n j−1 X 1≤i1≤i2≤...≤ij≤n I{G(x)⊆B(xi1, xi2,...,xij)}. Here xi1, xi2,...,xijare functions in the sample and B(xi1, xi2,...,xij) is: Bj(x) = Bj(x;xi1, . . . , ij) = {(t, y)∈T×R: mink=11,...ijxk(t)≤y≤maxk=11,...ijxk(t)≤x(t)}. Here Istands, as usual, for the indicator function. Note that BD(j) n(x) measures the proportion of j-uplas (xi1, xi2,...,xij) in Fsuch that xbelongs to the band determined by them. Next we review the global band depth, that compiles all the previous measures. Given a sample Fas above and a value Jsuch that 2 ≤J≤n, the band depth of a function xis defined as BDn,J (x) = J X j=2 BD(j) n(x). Of course, from an analytic point of view, the most logical choice for Jis n, so we collect all the posible information given by the curves in the sample F. However, if |F| is big, the depth can become computationally intractable. The authors prove HOMOGENEITY TEST FOR FUNCTIONAL DATA 9 that the value is quite stable in J, so in this paper we will use J= 2. In this case, the depth depends generally on non-degenerate bands. The authors also define a modified version of the band depth, by considering bands in the interval T, instead of bands in the plane: Aj(x) = Aj(x;xi1, . . . , ij) = {t∈T: mink=11,...ijxk(t)≤x(t)≤maxk=11,...ijxk(t)≤x(t)}. Now the authors consider a Lebesgue measure λon the interval (usually the standard one), and define as in the previous case: mBD(j) n(x) = n j−1 X 1≤i1≤i2≤...≤ij≤n λ(Aj(x)) λ(T), again with 2 6=j6=n. Now the definition of the modified band depth is analogous to the previous one: mBDn,J (x) = J X j=2 BD(j) n(x), for 2 ≤J≤n. 4. Simulation study In order to describe the characteristics and features of our procedures, we perform a simulation study using the four different statistics defined in Section 2 and the five depth measures defined in the previous section: Fraiman-Muniz, h-modal, random measure, band depth and modified band-depth. We consider six functional populations defined in [0,1], which are considered as the realizations of a stochastic process X(−) which has continuous trajectories in the interval [0,1]. Sample 0. This is the reference set, generated by a Gaussian process X(t) = E(t) + e(t) with mean function E(t) = E(X(t)) = 30t3/2(1 −t), and e(t) is a centered Gaussian process, whose covariance matrix is given by Cov(ei, ej) = 0.3∗exp(−|ti−tj| 0.3). The remaining sets are produced by perturbing the generation process in two ways. The first three suffer magnitude contamination in the mean, while the covariant matrix does not change. 16 RAM ´ ON FLORES, ROSA LILLO AND JUAN ROMO both methods show weakness when combined with Fraiman-Muniz depth, which seem not quite appropriate to confront these kind of observations. Figure 3. Mitochondrial calcium data 5.2. MCO data. Now we apply our measures to the mitochondrial calcium overload dataset ([RM03]), previously studied from a statistical point of view in [CFF06] and [BCC11]. The functional variable measures the level of mitochondrial calcium in mouse cardiac cells, as high levels of this element usually imply good protection of these cells in the event of ischemia process. The ultimate goal of the study is to test the power of the drug Cariporide to increase the levels of calcium in the cells. The dataset consists of control group of 45 observations and a treated group of 44. The levels of MCO are measured every ten seconds during an hour, so each function is observed in principle at 360 points; however, the data which correspond to the first three minutes are eliminated from the sample, as they show a high variability which depend on factors that are hard to control. In Table 5 we present the results of our computations for the mitochondrial data MCO. Different from the case of Ramsay data, we do not know a priori if the data are naturally split into two samples or not. Again the measure P2offers the greatest evidence for the splitting hypothesis, as it shows heterogeneity in all the cases. The measure P1also offers support to that hypothesis, as it only fails to make a difference when combining with the random depths. Measure P3only rejects homogeneity in HOMOGENEITY TEST FOR FUNCTIONAL DATA 17 P1CV Rej. P2CV Rej. P3CV Rej. P4CV Rej. FM 0.689 0.814 Yes 0.263 0.127 Yes 0.922 0.894 No 0.0002 0.002 No dmode 4.425 5.865 Yes 2.649 1.392 Yes 6.854 6.799 Yes 0.046 0.416 No RPD 0.227 0.205 No 0.0366 0.033 Yes 0.244 0.239 No 0.000007 0.00004 No BD 0.047 0.07 Yes 0.096 0.088 Yes 0.078 0.111 Yes 0.008 0.002 Yes mBD 0.338 0.449 Yes 0.181 0.077 Yes 0.502 0.498 No 0.00026 0.00034 No Table 5. Measures for MCO data half of the cases, and P4just one. From the point of view of the depth, banddepth shows again difference in all the samples. For these data, the rank test shows heterogeneity in two cases, when it is carried out with the h-modal and band depths. Figure 4. Tecator data 5.3. Tecator. The tecator dataset has been intensively studied in recent years, see for example [FV06], [LY08] and [MLR13]. Tecator is a commercial name for a Infracted Food Analyzer, which in this case is used to measure the infrared absorbance spectrum of meat samples. These absorbances are given as functions of the intensity of the light measured just before and just after passing through the sample. The observations measure the contents of moisture, protein and fat in every sample of meat, and the goal is to separate two samples according to their different levels of fat. The discrete observations consist of 100 channel-absorbance spectrum for a given wavelength, which are made continuous using a B-spline basis of order 6. The dataset is divided into data with high fat content (77 observations) and data with low content (the remaining 158). Following the approach of the aforementioned papers of Ferraty-Vieu and Li-Yu, we have computed our homogeneity measures also for the 18 RAM ´ ON FLORES, ROSA LILLO AND JUAN ROMO P1CV Rej. P2CV Rej. P3CV Rej. P4CV Rej. FM 0.946 0.866 No 0.032 0.094 No 0.983 0.959 No 0.00001 0.0002 No dmode 9.301 7.686 No 0.144 2.062 No 9.446 8.91 No 2.008 3.173 No RPD 0.243 0.233 No 0.007 0.016 No 0.249 0.248 No 0.0000004 0.000003 No BD 0.382 0.344 No 0.085 0.095 No 0.457 0.417 No 0.0002 0.001 No mBD 0.511 0.476 No 0.007 0.037 No 0.518 0.514 No 0.003 0.00002 Yes Table 6. Measures for tecator data spectrometric data and for the second derivative of it. Recall that the discrete derivative is defined by means of the differences between subsequent points where the values for the functions are taken. The results for the first case appear in Table 5: Our computations support the widespread impression that the meat samples of the tecator data may proceed from the same sample. As just one out of our 24 measures is able to separate the data (concretely P4combined with modified band-depth), it is quite likely that this is an outlier instead of a genuine difference. Moreover, it can be seen that the critical values are usually quite far from the extremes of the corresponding interval. More evidence is extracted from the rank test, which shows homogeneity in the five cases, and always in a quite robust way. The evidence then suggests that we cannot reject the hypothesis of equality between the two samples. Figure 5. Tecator second derivatives HOMOGENEITY TEST FOR FUNCTIONAL DATA 19 P1CV Rej. P2CV Rej. P3CV Rej. P4CV Rej. FM 0.835 0.859 Yes 0.091 0.067 Yes 0.919 0.887 No 0.0001 0.0008 Yes dmode 10.581 12.032 Yes 2′729 0.722 Yes 13′413 12.402 No 0.874 5.663 No RPD 0.215 0.225 Yes 0.025 0.017 Yes 0.237 0.233 No 0.00003 0.00008 No BD 0.121 0.089 No 0.056 0.1 No 0.179 0.121 No 0.00006 0.00437 No mBD 0.448 0.461 Yes 0.054 0.036 Yes 0.498 0.482 No 0.101 0.0003 Yes Table 7. Measures for tecator data (second derivatives) Rank test Ramsay MCO Tecator Tecator 2 FM 1733 2140 8427 7737 h-modal 1233 1625 8553 8490 RPD 1721 2051 8296 7768 BD 1159 1482 8136 6989 mBD 1703 2140 8427 7757 CV 1623.095 1781.395 7595.08 7595.08 Table 8. Rank test in real data 5.4. Tecator second derivatives. There is quite a lot more evidence of heterogeneity in the sample of the second derivatives, as we may check in Table 7. As in the previous sample MCO, both P1and P2are able to separate, in this case four out of five cases, and again the other two measures do not seem too powerful in this case. The scheme is very similar to that case, except for the fact that band-depth gives no difference in any of the four cases. It is also remarkable that P4only separates when combining with modified band-depth, just as it happens in the possible outlier case described above. In this case the rank-test supports the hypothesis of non homogeneity, as it is shown in all of the five observations. 6. Discussion In this paper, we have defined some new measures of distances between samples of functions to solve the problem of homogeneity in the context of functional data 20 RAM ´ ON FLORES, ROSA LILLO AND JUAN ROMO analysis. Combining these measures with the depth functions defined by Fraiman- Muniz, Cuevas-Fraiman-Muniz and L´opez-Pintado-Romo, we propose a hypothesis test based on the bootstrap methology and apply it to a number of simulated and real functional data. Our measures shows their effectiveness in detecting differences of magnitudes and shape in some samples generated by Gaussian processes, and moreover are able to show heterogeneity for Ramsay data, mitochondrial data and the second derivatives tecator data. It is significant that our methods show homogeneity in the tecator data without differentation, a phenomenon widely dealt with in the literature. It is also noteworthy that our method improves the rank-test in some cases. Once the concept of depth of a function with regard to a sample is defined, several generalizations appear to be possible. For example, the sample of tecator data discussed above shows that there is information about homogeneity hidden in the derivatives that cannot be directly extracted from the original functions. 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