The Area of the Convex Hull of Sampled Curves:a Robust Functional Statistical Depth Measure

Guillaume Staerman* Pavlo Mozharovskyi* Stephan Clemencon**LTCI, Telecom Paris, Institut Polytechnique de Paris

Abstract

With the ubiquity of sensors in the IoT era,statistical observations are becoming increas-ingly available in the form of massive (multi-variate) time-series. Formulated as unsuper-vised anomaly detection tasks, an abundanceof applications like aviation safety manage-ment, the health monitoring of complex in-frastructures or fraud detection can now relyon such functional data, acquired and storedwith an ever finer granularity. The conceptof statistical depth, which reflects centralityof an arbitrary observation w.r.t. a statisti-cal population may play a crucial role in thisregard, anomalies corresponding to observa-tions with ’small’ depth. Supported by soundtheoretical and computational developmentsin the recent decades, it has proven to beextremely useful, in particular in functionalspaces. However, most approaches docu-mented in the literature consist in evaluat-ing independently the centrality of each pointforming the time series and consequently ex-hibit a certain insensitivity to possible shapechanges. In this paper, we propose a novelnotion of functional depth based on the areaof the convex hull of sampled curves, captur-ing gradual departures from centrality, evenbeyond the envelope of the data, in a nat-ural fashion. We discuss practical relevanceof commonly imposed axioms on functionaldepths and investigate which of them are sat-isfied by the notion of depth we promote here.Estimation and computational issues are alsoaddressed and various numerical experimentsprovide empirical evidence of the relevance ofthe approach proposed.

Proceedings of the 23rdInternational Conference on Artifi-cial Intelligence and Statistics (AISTATS) 2020, Palermo,Italy. PMLR: Volume 108. Copyright 2020 by the au-thor(s).

1 Introduction

Technological advances in data acquisition, manage-ment and warehousing (e.g. IoT, distributed plat-forms) enable massive data processing and are leadingto a wide variety of new applications in the digital-ized (service) industry. The need to design more andmore automated systems fed by ever more informativestreams of data manifests in many areas of human ac-tivity (e.g transportation, energy, health, commerce,finance/insurance). Monitoring the behavior/health ofcomplex systems offers a broad spectrum of machine-learning implementation as classification or anomalydetection. With the increasing industrial digitaliza-tion, data are more and more often collected in quasi-real time and naturally take the form of temporal seriesor functions. The case of functional data is thus of cru-cial interest in practice, refer to e.g. Ramsay and Sil-verman (2002, 2005) for an excellent account of Func-tional Data Analysis (FDA in abbreviated form) andof its applications. A functional dataset is typically aset of n ≥ 1 curves partially observed at different timepoints T1 = t1 < . . . < tp = T2 which can be seen as n(partially observed) realizations of a stochastic processX = (Xt)t∈[T1,T2]. Hence, the first step of FDA gener-ally consists in reconstruct the functional objects fromthese observations, by means of interpolation, smooth-ing or projection techniques. Whereas, with the designof many successful algorithms such as (deep) neuralnetworks, SVM’s or boosting procedures, the practiceof statistical learning has rapidly generalized these lastfew years, the automatic analysis of functional datato achieve complex (e.g. unsupervised) tasks such asanomaly detection is still a challenge, due to the hugevariety of possible patterns that may carry the relevantinformation for discrimination purposes. It is far fromstraightforward to generalize directly methods origi-nally introduced in the finite-dimensional case to thefunctional setup, unless preliminary filtering or dimen-sionality reduction techniques are used, see e.g Rossiand Villa (2006); Staerman et al. (2019). Such tech-niques essentially consist in projecting the observation,supposed to take their values in a certain Hilbert space,

ACH Depth

onto a subspace of finite dimensionality, generally de-fined by truncating their expansion in a Hilbertian ba-sis of reference or by means of a flexible dictionaryof functions/’atoms’. Next, one can apply any state-of-the-art algorithm tailored to the finite dimensioncase, based on the parsimonious representations thusobtained, cf Ferraty and Vieu (2006); Ramsay andSilverman (2002). The basis functions can be eitherchosen among a pre-selected dictionary (e.g. Fourier,wavelets, cosine packets, etc.) presumably capableof capturing the information carried by the curves orbuilt empirically using Functional Principal Compo-nent Analysis, retaining the most informative part ofthe (Kahrunen-Loeve) decomposition only, see Ram-say and Silverman (2005). Of course, the major draw-back of such FDA approaches lies in the fact that theyare dramatically conditioned by the finite-dimensionalrepresentation method chosen, the subsequent analy-sis of the data may be fully jeopardized if the lattermakes disappear some patterns relevant for the taskconsidered.

Originally introduced by J. Tukey to extend the notionof median/quantile to multivariate random variables(see Tukey (1975a)), a data depth is a function definedon the feature space and valued in [0, 1] used to deter-mine the ’representativeness’ of a point with respect toa statistical population and that should fulfill a vari-ety of desirable properties (ideally just like the quantilefunction in the univariate situation). Given a trainingset, a data depth function provides a score that mea-sures the centrality of any element w.r.t. a dataset andthen defines, notably, an ordering of all the elements ofthe dataset. In particular, it finds a very natural ap-plication in (unsupervised) anomaly detection, see e.gCuevas et al. (2007); Long and Huang (2016); Moslerand Mozharovskyi (2017) in supervised situations andHubert et al. (2015); Nagy et al. (2017) in the unsu-pervised context: an observation is considered all themore ’abnormal’, as its depth is small. This concepthas been extended to the functional data framework byintegrating univariate depth functions on the whole in-terval of time [T1, T2], see e.g. Claeskens et al. (2014);Fraiman and Muniz (2001); Hubert et al. (2015). Al-ternatively, various depth functions fully tailored tothe functional setup have been introduced in the sta-tistical literature, refer to e.g Chakraborty and Chaud-huri (2014); Dutta et al. (2011); Lopez-Pintado andRomo (2009, 2011). However, most of them fail to ful-fill certain desirable properties or face significant com-putational difficulties. It is the major purpose of thispaper to introduce a novel robust functional statisticaldepth measure dedicated to the analysis of functionaldata. Based on the area of the convex hull of col-lections of sampled curves, it is easy to compute andto interpret both at the same time. Given a curve

x lying in a certain functional space F (e.g. the Kol-mogorov space C([0, 1]), the space of real valued contin-uous functions on [0, 1]), the general idea is to quantifyits contribution, on average, to the area of the convexhull (ACH in short) of random curves in F with thesame probability law. Precisely, this function, referredto as the ACH depth throughout the article, is definedby the ratio of the ACH of the sample to that of thesample augmented by the curve x. We prove here thatit fulfills various properties desirable for depth func-tions. In particular, given its form, it exhibits sensi-tivity (i.e. the depth score of new/test curves that arefurther and further away from the training set of curvesdecreases smoothly), which property, quite desirableintuitively, is actually not satisfied by most statistical(functional) depth documented in the literature. Forinstance, the statistical Tukey depth assigns a score of0 to any element lying outside the convex hull of thetraining data, see Tukey (1975b); Dutta et al. (2011).In addition, the statistical depth we promote here isrobust to outliers: adding outliers to the training set ofcurves has little or no effect on the returned score andordering on a test set. For this reason, this functionaldepth is very well suited for unsupervised anomaly de-tection. In the functional setup, this task is extremelychallenging. Indeed, the richness of functional spacesleads to a huge diversity in the nature of possibly ob-served differences between curves. As dicussed in Hu-bert et al. (2015), three main types of anomaly canbe distinguished: shift, magnitude or shape anomalies.Anomalies can be either isolated/transient or persis-tent depending of their duration ; some of them beingmore difficult to detect (shape anomalies). Since thefunctional statistical depth measure we propose is re-lated to a whole batch of curves and do not reflectthe individual properties of a single curve, it enablesthe detection of a wide variety of anomaly shapes, asillustrated by the numerical experiments displayed inSection 4.

The paper is organized as follows. In Section 2, ba-sic concepts pertaining to the statistical depth theory,both for the multivariate framework and for the func-tional case, are briefly recalled for clarity’s sake. InSection 3 the functional statistical depth based on thearea of the convex hull of a batch of curves is intro-duced at length and its theoretical properties are in-vestigated, together with computational aspects. Sec-tion 4 presents numerical results in order to providestrong empirical evidence of the relevance of the noveldepth function proposed, for the purpose of unsuper-vised functional anomaly detection especially. Even-tually, concluding remarks are collected in Section 5.

Guillaume Staerman, Pavlo Mozharovskyi, Stephan Clemencon

2 Background and Preliminaries

For clarity, we start with recalling the concept of sta-tistical depth in the multivariate and functional frame-work. We next list the desirable properties it shouldfulfill and also briefly review recent advances in thisfield. Here and throughout, the Dirac mass at anypoint a is denoted by δa, the convex hull of any subsetA of Rd by conv(A).

2.1 Data Depth in a Multivariate Space

By data depth, one usually means a nonparametric sta-tistical function that determines the centrality of anyelement x ∈ Rd with respect to a statistical popula-tion. Given a dataset, a depth function provides acenter-outward ordering of the data points. Since itpermits to define a ranking of the (multivariate) ob-servations and local averages derived from it, a datadepth can be used for various tasks, including clas-sification (Lange et al. (2014)), clustering (Jornsten(2004)), anomaly detection (Serfling (2006)) or ranktests (Oja (1983)). In order to give a precise definition,some notations are needed. Let X be a random vari-able, defined on a certain probability space (Ω,A,P),taking its values in X ⊂ Rd with probability distri-bution P . Denote by P(X ) the set of all probabilitydistributions on X . A data depth is a function

D : Rd × P(X ) −→ [0, 1](x, P ) 7−→ D(x, P )

measurable with respect to its first argument x. It isinterpreted as follows: the closer the quantity D(x, P )to 1, the deeper (i.e. the more ’central’) the observa-tion x with respect to the distribution P is considered.As mentioned above, it naturally defines a predorderon the set X . In particular, medians of the multivari-ate distribution P corresponds to maximizers of thedepth function D and quantile regions are defined asdepth sublevel sets. A crucial example is the half-spacedepth DT (also called location depth sometimes) intro-duced in the seminal contribution Tukey (1975b). Itis defined as

DT (x, P ) = infP (H) : H closed half-space, x ∈ H,

for any x ∈ Rd and probability distribution on Rd.As the distribution P is generally unknown, a sta-tistical version can be built from independent copiesX1, . . . , Xn of the generic random vector X by meansof the plug-in principle, i.e. by replacing P by an em-pirical counterpart Pn, typically the raw empirical dis-tribution (1/n)

∑ni=1 δXi

(or a smooth/penalized ver-sion of the latter), yielding the empirical depth

Dn(x) = D(x, Pn). (1)

Empirical medians and quantile regions are then nat-urally defined as medians and quantile regions of theempirical depth (1). Of course, the relevance of adepth function regarding the measurement of central-ity in a multivariate space is guaranteed in the solecase where certain desirable properties are satisfied.We refer to Zuo and Serfling (2000) for an account ofthe statistical theory of multivariate data depth andmany examples.

2.2 Statistical Functional Depth

In this paper, we consider the situation where the r.v.X takes its values in a space of infinite dimension.Precisely, focus is on the case where the feature spaceis the vector space C([0, 1]) of real-valued continuousfunctions on [0, 1]:

X : (Ω, A, P) −→ C([0, 1])ω 7−→ X(ω) = (Xt(ω))t∈[0,1].

Recall that, when equipped with the sup norm ||.||∞,C([0, 1]) is a separable Banach space. We denote byP(C([0, 1])) the set of all probability laws on C([0, 1])and by Pt the 1-dimensional marginal of the law P ofthe stochastic process X at time point t ∈ [0, 1].

Depths in a functional framework have been first con-sidered in Fraiman and Muniz (2001), where it is pro-posed to define functional depths as simple integralsover time of a univariate depth function D, namely

(x, P ) ∈ C([0, 1]) 7→∫ 1

0D(xt, Pt)dt. Due to the aver-

aging effect, local changes for the curve x only induceslight modifications of the depth value, which makesanomaly detection approaches based on such ‘poorlysensitive’ functional depths ill-suited in general. Re-cently, alternative functional depths have been intro-duced, see Lopez-Pintado and Romo (2009, 2011) fordepths based on the geometry of the set of curves,Chakraborty and Chaudhuri (2014) for a notion ofdepth based on the L2 distance or Dutta et al. (2011)for a functional version of the Tukey depth. As dis-cussed in Nieto-Reyes and Battey (2016) and Gijbelsand Nagy (2018), the axiomatic framework introducedin Zuo and Serfling (2000) for multivariate depth is nolonger adapted to the richness of the topological struc-ture of functional spaces. Indeed, the vast majority ofthe functional depths documented in the literature donot fulfill versions of the most natural and elementaryproperties required for a depth function in a multivari-ate setup, cf Gijbels and Nagy (2018). However, thereis still no consensus about the set of desirable proper-ties that a functional depth should satisfy, beyond theform of sensitivity mentioned above. Those that ap-pear to be the most relevant in our opinion are listedbelow. By PX is meant the law of a functional r.v.taking its values in C([0, 1]).

ACH Depth

• (Non-degeneracy) For all non atomic distribu-tion P in P(C([0, 1])), we have

infx∈C([0,1])

D(x, P ) < supx∈C([0,1])

D(x, P ).

• (Affine invariance) The depth D is said to be(scalar-) affine invariant if for any x in C([0, 1])and all a, b in R, we have

D(x, PX) = D(ax+ b, PaX+b).

• (Maximality at the center) For any point-symmetric and non atomic distribution P withθ ∈ C([0, 1]) as center of symmetry, we have

D(θ, P ) = supx∈C([0,1])

D(x, P ).

• (Vanishing at ∞) For any non atomic distribu-tion P in P(C([0, 1])),

D(z, P ) −→||z||∞−→∞

infx∈C([0,1])

D(x, P ).

• (Decreasing w.r.t. the deepest point)For any P in P(C([0, 1])) such thatD(z, P ) = sup

x∈C([0,1])D(x, P ), D(x, P ) <

D(y, P ) < D(z, P ) holds for any x, y ∈ C([0, 1])such that mind(y, z), d(y, x) > 0 andmaxd(y, z), d(y, x) < d(x, z).

• (Continuity in x) For any non atomic distribu-tion P ∈ P(C([0, 1])), the function x 7→ D(x, P )is continuous w.r.t. the sup norm.

• (Continuity in P ) For all x in C([0, 1]), the map-ping P ∈ P(C([0, 1])) 7→ D(x, P ) is continuousw.r.t. the Levy-Prohorov metric.

Before introducing the ACH depth and investigatingits properties, a few remarks are in order. Thoughit obviously appears as mandatory to make the otherproperties meaningful, non-degeneracy, is actually notfulfilled by all the functional depths proposed, see e.gLopez-Pintado and Romo (2009, 2011); Dutta et al.(2011). The ’Maximality at center ’ and ’Decreasingw.r.t. the deepest point ’ properties permit to preservethe original center-outward ordering goal as well as thegoodness of fit for unimodal data of data depth in thefunctional framework. Many definition of the conceptof ”symmetry” in a functional space are detailed inthe Supplementary material for the sake of place. The’Continuity in x’ property extends a property fulfilledby cumulative distribution functions of multivariatecontinuous distributions. From a statistical perspec-tive, the ’Continuity in P ’ property is essential, inso-far as P must be replaced in practice by an estimator,cf Eq. (1), built from finite-dimensional observations,i.e. a finite number of sampled curves.

3 The Area of the Convex Hull of(Sampled) Curves

It is the purpose of this section to present at length thestatistical depth function we propose for path-valuedrandom variables. As shall be seen below, its defini-tion is based on very simple geometrical ideas and var-ious desirable properties can be easily checked from it.Statistical and computational issues are also discussedat length. By K2 is meant the collection of all com-pact subsets of R2 and λ denotes Lebesgue measure onthe plane R2. Consider an i.i.d. sample X1, . . . , Xn

drawn from P in P(C([0, 1])). The graph of any func-tion x in C([0, 1]) is denoted by

graph(x) = (t, y) : y = x(t), t ∈ [0, 1],

while we denote by graph(x1, . . . , xn) the set

n⋃i=1

graph(xi)

defined by a collection of n ≥ 1 functions x1, . . . , xnin C([0, 1]). We now give a precise definition of thestatistical depth measure we propose for random vari-ables valued in C([0, 1]).

Definition 3.1 Let J ≥ 1 be a fixed integer. TheACH depth of degree J is the function DJ : C([0, 1])×P(C([0, 1]))→ [0, 1] defined by: ∀x ∈ C([0, 1]),

DJ(x, P ) = E[

λ (conv (graph (X1, . . . , XJ)))λ (conv (graph (X1, . . . , XJ ∪ x)))

],

where X1, . . . , XJ are i.i.d. r.v.’s drawn from P . Itsaverage version DJ is defined by: ∀x ∈ C([0, 1]),

DJ(x, P ) =1

J

J∑j=1

Dj(x, P ).

The choice of J leads to various views of distributionP , the average variant permitting to combine all ofthem (up to degree J). When n ≥ J , an unbiasedstatistical estimation of DJ(., P ) can be obtained bycomputing the symmetric U -statistic of degree J , seeLee (1990): ∀x ∈ C([0, 1]),

DJ,n(x) = (2)

1(nJ

) ∑1≤i1<...<iJ≤n

λ (conv (graph (Xi1 , . . . , XiJ)))λ (conv (graph (Xi1 , . . . , XiJ , x)))

.

Considering the empirical average version given by

∀x ∈ C([0, 1]), DJ,n(x) =1

J

J∑j=1

Dj,n(x)

Guillaume Staerman, Pavlo Mozharovskyi, Stephan Clemencon

brings some ’stability’. However, the computationalcost rapidly increasing with J , small values of J arepreferred in practice. Moreover, as we illustrate inSection 4.1, J equal two already yields satisfactoryresults.

Approximation from sampled curves. In general,one does not observes the batch of continuous curvesX1, . . . , Xn on the whole time interval [0, 1] but atdiscrete time points only, the number p ≥ 1 of timepoints and the time points 0 ≤ t1 < t2 < . . . < tp ≤ 1themselves possibly varying depending on the curveconsidered. In such a case, the estimators above arecomputed from continuous curves reconstructed fromthe sampled curves available by means of interpolationprocedures or approximation schemes based on appro-priate basis. In practice, linear interpolation is usedfor this purpose with theoretical guarantees (refer toTheorem 3.2 below) facilitating significantly the com-putation of the empirical ACH depth, see subsection4.3.

3.1 Main Properties of the ACH Depth

In this subsection, we study theoretical properties ofthe population version of the functional depths intro-duced above and next establish the consistency of theirstatistical versions. The following result reveals that,among the properties listed in the previous subsection,five are fulfilled by the (average) ACH depth function.

Proposition 3.1 For all J ≥ 1, the depth functionDJ (respectively, DJ) fulfills the following properties:’non-degeneracy’, ’affine invariance’, ’vanishing at in-finity’, ’continuity in x’ and ’continuity in P ’. In addi-tion, the following properties are not satisfied: ’max-imality at center’ and ’decreasing w.r.t. the deepestpoint’.

Refer to the Appendix section for the technical proof.In a functional space, not satisfying maximality at cen-ter is not an issue. For instance, though the constanttrajectory y(t) ≡ 0 is a center of symmetry for theBrownian motion, it is clearly not representative ofthis distribution. In contrast, scalar-affine invarianceis relevant, insofar as it allows z-normalization of thefunctional data and continuity in P is essential to de-rive the consistency of DJ,n (respectively, of DJ,n), asstated below.

Theorem 3.1 Let J ≥ 1 and X1, . . . , Xn be n ≥ Jindependent copies of a generic r.v. X with distribu-tion P ∈ P(C([0, 1])). As n → ∞, we have, for anyx ∈ C([0, 1]), with probability one,

|DJ,n(x)−DJ(x, P )| → 0

and ∣∣DJ,n(x)− DJ(x, P )∣∣→ 0.

3.2 On Statistical/Computational Issues

As mentioned above, only sampled curves are availablein practice. Each random curve Xi being observed at

fixed time points 0 = t(i)1 < t

(i)2 < . . . < t

(i)pi = 1 (po-

tentially different for each Xi) with pi ≥ 1, we denotedby X ′1, . . . , X

′n the continuous curves reconstructed

from the sampled curves (Xi(t(i)1 ), . . . , Xi(t

(i)pi )),

1 ≤ i ≤ n, by linear interpolation. From a practi-cal perspective, one considers the estimator D′J,n(x)of DJ(x, P ) given by the approximation of DJ,n(x)obtained when replacing the Xi’s by the X ′i’s in (2).The (computationally feasible) estimator D′J,n(x) of

DJ(x, P ) is constructed in a similar manner. The re-sult stated below shows that this approximation stagepreserves almost-sure consistency.

Theorem 3.2 Let J ≤ n. Suppose that, as n→∞,

δ = max1≤i≤n

max2≤k≤pi

t(i)k+1 − t

(i)k

→ 0.

As n→∞, we have, for any x ∈ C([0, 1]), with proba-bility one, ∣∣D′J,n(x)−DJ(x, P )

∣∣→ 0

and ∣∣D′J,n(x)− DJ(x, P )∣∣→ 0.

Refer to the Appendix section for the technical proof.Given the batch of continuous and piecewise linearcurves X ′1, . . . , X

′n, although the computation cost

of the area of their convex hull is of order O(p log p)with p = maxi pi, that of the U-statistic D′J,n(x) (and

a fortiori that of D′J,n(x)) becomes very expensive

as soon as(nJ

)is large. As pointed out in Lopez-

Pintado and Romo (2009), even if the choice J = 2for statistics of this type, may lead to a computation-ally tractable procedure, while offering a reasonablerepresentation of the distribution, varying J permitsto capture much more information in general. For thisreason, we propose to compute an incomplete versionof the U -statistic D′J,n(x) using a basic Monte-Carloapproximation scheme with K ≥ 1 replications: ratherthan averaging over all

(nJ

)subsets of 1, . . . , n with

cardinality J to compute D′J,n(x), one averages overK ≥ 1 subsets drawn with replacement, forming anincomplete U -statistic, see Enqvist (1978). The sameapproximation procedure can be applied (in a random-ized manner) to each of the U -statistics involved in theaverage D′J,n(x), as described in the SupplementaryMaterial.

ACH Depth

4 Numerical Experiments

From a practical perspective, this section explores cer-tain properties of the functional depth proposed us-ing simulated data. It also describes its performancecompared with the state-of-the-art methods on (real)benchmark datasets. As a first go, we focus on the im-pact of the choice of the tuning parameter K, whichrules the trade-off between approximation accuracyand computational burden and parameter J . Pre-cisely, it is investigated through the stability of theranking induced by the corresponding depths. We nextinvestigate the robustness of the ACH depth (ACHDin its abbreviated form), together with its ability to de-tect abnormal observations of various types. Finally,the ACH depth is benchmarked against alternativedepths standing as natural competitors in the func-tional setup using real datasets. A simulation-basedstudy of the variance of the ACH depth is postponedto the Supplementary Material.

n 2n 3n 4n 5n 10n 20n 35n 50n 75nK

0.88

0.90

0.92

0.94

0.96

0.98

1.00

Scor

e J = 2J = 3J = 4

n 2n 3n 4n 5n 10n 20n 35n 50n 75nK

0.45

0.50

0.55

0.60

0.65

0.70

Score

Figure 1: Boxplots of the approximations of DJ,n(x0)(top) and DJ,n(x3) (bottom) over different size of K.The black crosses correspond to the exact depth mea-sure DJ,n for each J respectively.

For the sake of simplicity, the two same simulateddatasets, represented in Figure 3, are used through-out the section. The dataset (a) corresponds to sam-ple path segments of the geometric Brownian motionwith mean 2 and variance 0.5, a stochastic processwidely used in statistical modeling. The dataset (b)consists of smooth curves given by x(t) = a cos(2πt) +b sin(2πt), t ∈ [0, 1], where a and b are independentlyand uniformly distributed on [0, 0.05], as proposed

by Claeskens et al. (2014). Four curves xi : i ∈0, 1, 2, 3 have been incorporated to each dataset: adeep curve and three atypical curves (anomalies), withexpected depth-induced ranking DJ(x3) < DJ(x2) ≈DJ(x1) < DJ(x0).

4.1 Choosing Tuning Parameters K and J

Parameter K reflects the trade-off between statisticalperformance and computational time. In order to in-vestigate its impact on the stability of the method,we compute depths of the deepest and most atypicalcurves (x0 and x3) for dataset (b), taking J = 2, 3, 4.Figure 1 presents boxplots of the approximated ACHD(together with the exact values of ACHD) over 100repetitions. Note that, as expected, depth values growwith J . The variance of the depth decreases takingsufficiently small values for K = 5n and almost disap-pearing for K ≥ 20n, while decreasing pattern remainsthe same for different values of K. For these reasons,we keep K = 5n in what follows.

The choice of J is less obvious, and clearly when de-scribing an observation in a functional space a sub-stantial part of information is lost anyway. Never-theless, one observes that computational burden in-creases exponentially with J and thus smaller valuesare preferable. Figure 2 shows the rank-rank plots ofdatasets (a) and (b) for small values of J = 2, 3, 4 andindicates, that depth-induced ranking does not changemuch with J . Thus, for saving computational time, weuse value J = 2 in all subsequent experiments.

0 20 40 60 80 1000

20

40

60

80

100

Rank

2 vs 3

0 20 40 60 80 1000

20

40

60

80

100

3 vs 4

0 20 40 60 80 1000

20

40

60

80

100

2 vs 4

0 20 40 60 80 100

Rank0

20

40

60

80

100

Rank

0 20 40 60 80 100

Rank0

20

40

60

80

100

0 20 40 60 80 100

Rank0

20

40

60

80

100

Figure 2: Rank-Rank plot for different values of J (2,3 and 4). The first line represents the rank over thedataset (a) while the second line represents the dataset(b).

Guillaume Staerman, Pavlo Mozharovskyi, Stephan Clemencon

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

x(t)

x0x1x2x3

0.0 0.2 0.4 0.6 0.8 1.0−0.08

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

Figure 3: Datasets (a) (left) and (b) (right) containing 100 paths with four selected observations. The colors arethe same for the four selected observations of both datasets (a) and (b).

dτ (σ0, σα)(×10−2)α 0 5 10 15 25 30

ACHDLocation 0 0.6 1.3 2.2 4.3 5.2Isolated 0 0.3 1.3 0.9 1.6 2.4Shape 0 0.9 2 2.6 4.2 4.7

FSDOLocation 0 3.6 7.3 10 16 20Isolated 0 0.8 3.6 3.2 7.2 9.4Shape 0 1.6 2.9 4.2 6.6 7.4

FTLocation 0 5.1 9.5 13 20 23Isolated 0 0.7 2.7 2.7 5.9 7.2Shape 0 1.7 2.9 4.3 6.6 7.7

FIFLocation 0 7 8.2 7.3 7.3 8.9Isolated 0 9.3 12 11 10 12Shape 0 7.4 7.9 10 14 14

Table 1: Kendall’s tau distances between the rank returned with normal data (σ0) and contamined data (σα,over different portion of contamination α with location, isolated and shape anomalies) for ACHD and threestate-of-the-art methods. Bold numbers indicate best stability of the rank over the contaminated datasets.

4.2 Robustness

Under robustness of a statistical estimator on under-stands its ability not to be “disturbed” by atypical ob-servations. We explore robustness of ACHD in the fol-lowing simulation study: between the original datasetand the same dataset contaminated with anomalies,we measure (averaged over 10 random repetitions)Kendall’s τ distance of two depth-induced rankings σand σ′, respectively, of the original data:

dτ (σ, σ′) =n(n− 1)

2

∑i<j

I(σ(i)−σ(j))(σ′(i)−σ′(j))<0 .

In their overview work, Hubert et al. (2015) introducetaxonomy for atypical observations, focusing on Lo-cation, Isolated, and Shape anomalies. Here, we addLocation anomalies to dataset (a) and Isolated andShape anomalies to dataset (b); other types of anoma-lies for both datasets can be found in the Supple-

mentary Material. The abnormal functions are con-structed as follows. Location anomalies for dataset(a) are x(t) = x(t) + ax(t) with a drawn uniformlyon [0, 1]. Isolated anomalies for dataset (b) are con-structed by adding a peak at t0 (drawn uniformly on[0, 1]) of amplitude b (drawn uniformly on [0.03, 0.06])such that y(t0) = y(t0) + b and y(t) = y(t) forany t 6= t0. Shape anomalies for dataset (b) arez(t) = z(t) + 0.01× cos(2πtf) + 0.01× sin(2πtf) withf drawn uniformly from 1, 2, ..., 10. By varying thepercentage of abnormal observations α, we compareACHD to several of the most know in the literaturedepth approaches: the functional Stahel-Donoho depth(FSDO) (Hubert et al., 2015) and the functional Tukeydepth (FT) (Claeskens et al., 2014), and also to thefunctional isolation forest (FIF) algorithm (Staermanet al., 2019) which proves satisfactory anomaly de-tection; see Table 1. One can observe that ACHDconsistently preserves depth-induced ranking despiteinserted abnormal observation, even if their fractionα reaches 30%. FSDO behaves competitively giving

ACH Depth

slightly better results than ACHD for shape anoma-lies.

4.3 Applications to Anomaly Detection

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a2

4

6

8

10

12

14

Num

ber o

f ano

mal

ies d

etec

ted Location Anomalies

ACHDFSDOFTFIF

0 2 4 6 8 10

b2

4

6

8

10

12

14

Isolated Anomalies

1.0 1.2 1.4 1.6 1.8 2.0

e

2

4

6

8

10

12

Shape Anomalies

Figure 4: Number of anomalies detected over a gridof parameters for three types of anomalies (location,isolated, and shape) for ACHD and three further state-of-the-art methods.

Further, we explore the ability of ACHD to detectatypical observations. For this, we conduct an ex-periment in settings similar to those in Section 4.2,while changing degree of abnormality gradually for 15(out of 100 curves) in dataset (a). Thus, we alter ain [0, 3] for location anomalies, b in [0, 10] for isolatedanomalies, and e in [1, 2] for shape anomalies to am-plify the ”spikes” of oscillations such that z(t) = ez(t).(For an illustration of abnormal curves the reader isreferred to the Supplementary Material.) Figure 4illustrates number of anomalies detected by ACHD,FSDO, FT, and FIF for different parameters of abnor-mality. While it is difficult to find the general winner,ACHD behaves favorably in all the considered casesand clearly outperforms the two other depths whenthe data is contaminated with isolated anomalies.

We conclude this section with a real-world data bench-mark based on three datasets: Octane (Esbensen,2001), Wine (Larsen et al., 2006), and EOG (Chenet al., 2015). The Wine dataset consists of 397measurements of proton nuclear magnetic resonance(NMR) spectra of 40 different wine samples, the Oc-tane dataset are 39 near infrared (NIR) spectra ofgasoline samples with 226 measurements, while theEOG dataset represents the electrical potential be-tween electrodes placed at points close to the eyes with1250 measurements. (Graphs of the three datasets canbe found in the Supplementary Material.) As pointedout by Hubert et al. (2015), it is difficult to detectanomalies in the first two datasets, while they are eas-ily seen during the human eye inspection. For the EOGdataset, we assign smaller of the two classes to be ab-normal. To the existing state-of-the-art methods, weadd here Isolation Forest (IF) (Liu et al., 2008) andthe One-Class SVM (OC) (Scholkopf et al., 2001)—multivariate methods applied after a proper dimen-sion reduction (to the dimension 10) using Functional

Principal Component Analysis (FPCA).(Ramsay andSilverman, 2002). Portions of detected anomalies (byall the considered methods), indicated in Table 2, hinton very competitive performance of ACHD in the ad-dressed benchmark.

ACHD FSDO FT FIF IF OCOctane 1 0.5 0.33 1 0.5 0.5Wine 1 0 0 1 0 1EOG 0.73 0.55 0.48 0.43 0.63 0.6

Table 2: Portion of detected anomalies of benchmarkmethods for the Octane, Wine, and EOG datasets.

5 Conclusion

In this paper, we have introduced a novel functionaldepth function on the space C([0, 1]) of real valuedcontinuous curves on [0, 1] that presents various ad-vantages. Regarding interpretability first, the depthcomputed at a query curve x in C([0, 1]) takes theform of an expected ratio, quantifying the relativeincrease of the area of the convex hull of i.i.d. ran-dom curves when adding x to the batch. We haveshown that this depth satisfies several desirable prop-erties and have explained how to solve approxima-tion issues, concerning the sampled character of ob-servations in practice and scalability namely. Nu-merical experiments on both synthetic and real datahave highlighted a number of crucial benefits: re-duced variance of its statistical versions, robustnesswith respect to the choice of tuning parameters andto the presence of outliers in the training sample,capacity of detecting (possibly slight) anomalies ofvarious types, surpassing competitors such as depthsof integral type, for isolated anomalies in particu-lar. The open-source implementation of the method,along with all reproducing scripts, can be accessed athttps://github.com/GuillaumeStaermanML/ACHD.

Acknowlegments

The authors thank Stanislav Nagy for his helpful re-marks. This work has been funded by BPI Francein the context of the PSPC Project Expresso (2017-2021).

References

Chakraborty and Chaudhuri (2014). The spatial dis-tribution in infinite dimensional spaces and relatedquantiles and depths. The annals of statistics.

Chen, Y., Keogh, E., Hu, B., Begum, N., Bagnall, A.,Mueen, A., and Batista, G. (2015). The ucr timeseries classification archive.

Guillaume Staerman, Pavlo Mozharovskyi, Stephan Clemencon

Claeskens, G., Hubert, M., Slaets, L., and Vak-ili, K. (2014). Multivariate functional halfspacedepth. Journal of American Statistical Association,109(505):411–423.

Cuevas, A., Febrero, M., and Fraiman, R. (2007). Ro-bust estimation and classification for functional datavia projection-based depth notions. ComputationalStatistics, 22(3):481–496.

Dutta, Ghosh, and Chaudhuri (2011). Some intriguingproperties of tukey’s half-space depth. Bernouilli.

Enqvist, E. (1978). On sampling from sets of randomvariables with application to incomplete U -statistics.PhD thesis, Lund University.

Esbensen, K. (2001). Multivariate data analysis-inpractice. Camo Software.

Ferraty, F. and Vieu, P. (2006). Nonparametric Func-tional Data Analysis. Springer-Verlag, New York.

Fraiman, R. and Muniz, G. (2001). Trimmed meansfor functional data. Test, 10(2):419–440.

Gijbels, I. and Nagy, S. (2018). On a general definitionof depth for functional data. Statistical Science.

Hubert, M., Rousseeuw, P. J., and Segaert, P. (2015).Multivariate functional outlier detection. StatisticalMethods & Applications, 24(2):177–202.

Jornsten, R. (2004). Clustering and classificationbased on the l1 data depth. Journal of MultivariateAnalysis, 90(1):67 – 89.

Lange, T., Mosler, K., and Mozharovskyi, P. (2014).Fast nonparametric classification based on datadepth. Statistical Papers, 55(1):49–69.

Larsen, F., Berg, F., and Engelsen, S. (2006). Anexploratory chemometric study of 1 h nmr spectraof table wine. Journal of Chemometrics, 20:198 –208.

Lee, A. J. (1990). U -statistics: Theory and practice.Marcel Dekker, Inc., New York.

Liu, F. T., Ting, K. M., and Zhou, Z. (2008). Isolationforest. In 2008 Eighth IEEE International Confer-ence on Data Mining, pages 413–422.

Long and Huang (2016). A study of functional depths.preprint.

Lopez-Pintado, S. and Romo, J. (2009). On the con-cept of depth for functional data. Journal of theAmerican Statistical Association.

Lopez-Pintado, S. and Romo, J. (2011). A half-regiondepth for functional data. Computational Statisticsand Data Analysis.

Mosler, K. and Mozharovskyi, P. (2017). Fast dd-classification of functional data. Statistical Papers,58(4):1055–1089.

Nagy, S., Gijbels, I., and Hlubinka, D. (2017).Depth-based recognition of shape outlying func-tions. Journal of Computational and GraphicalStatistics, 26(4):883–893.

Nieto-Reyes, A. and Battey, H. (2016). A topologi-cally valid definition of depth for functional data.Statistical Science.

Oja (1983). Descriptive statistics for multivariate dis-tributions. Statistics and Probability Letters.

Ramsay, J. O. and Silverman, B. W. (2002). AppliedFunctional Data Analysis: Methods and Case Stud-ies. Springer-Verlag, New York.

Ramsay, J. O. and Silverman, B. W. (2005). Func-tional Data Analysis. Springer-Verlag, New York.

Rossi, F. and Villa, N. (2006). Support vector machinefor functional data classification. Neurocomputing,69(7):730 – 742.

Scholkopf, B., Platt, J., Shawe-Taylor, J., Smola, A.,and Williamson, R. (2001). Estimating the supportof a high-dimensional distribution. Neural Compu-tation, 13(7):1443–1471.

Serfling, R. (2006). Depth functions in nonparametricmultivariate inference. DIMACS Series in DiscreteMathematics and Theoretical Computer Science, 72.

Staerman, G., Mozharovskyi, P., Clemencon, S., andd’Alche Buc, F. (2019). Functional isolation for-est. In Proceedings of The 11th Asian Conferenceon Machine Learning.

Tukey (1975a). Mathematics and the picturing ofdata. In Proceedings of the International Congressof Mathematicians.

Tukey, J. W. (1975b). Mathematics and the picturingof data. In James, R., editor, Proceedings of the In-ternational Congress of Mathematicians, volume 2,pages 523–531. Canadian Mathematical Congress.

Zuo, B. and Serfling, R. (2000). General notions ofstatistical depth function. The Annals of Statistics,28(2):461–482.