Generalized Multiple Correlation Coefficient as a SimilarityMeasurement between Trajectories

Julen Urain1 Jan Peters1,2

Abstract— Similarity distance measure between two trajecto-ries is an essential tool to understand patterns in motion, forexample, in Human-Robot Interaction or Imitation Learning.The problem has been faced in many fields, from Signal Pro-cessing, Probabilistic Theory field, Topology field or Statisticsfield. Anyway, up to now, none of the trajectory similaritymeasurements metrics are invariant to all possible linear trans-formation of the trajectories (rotation, scaling, reflection, shearmapping or squeeze mapping). Also not all of them are robustin front of noisy signals or fast enough for real-time trajectoryclassification. To overcome this limitation this paper proposes asimilarity distance metric that will remain invariant in frontof any possible linear transformation. Based on Pearson’sCorrelation Coefficient and the Coefficient of Determination,our similarity metric, the Generalized Multiple CorrelationCoefficient (GMCC) is presented like the natural extension ofthe Multiple Correlation Coefficient. The motivation of thispaper is two-fold: First, to introduce a new correlation metricthat presents the best properties to compute similarities betweentrajectories invariant to linear transformations and compareit with some state of the art similarity distances. Second, topresent a natural way of integrating the similarity metric in anImitation Learning scenario for clustering robot trajectories.

I. INTRODUCTION

In the industry of tomorrow it is expected the robotsto collaborate with humans. The robots should not onlyimitate the humans [1], [2], but also interact with us. Theobjective of Human-Robot Interaction (HRI) is to understandintentions [3] and shape the interactions [4] between one ormore humans and one or more robots [5].

There is a lot of work done in human intention recognitionand generalization. In [6], Amor et al. developed a Human-robot solution based on PPCA. Their algorithm computes alow-dimensional projection to encapsule human-robot inter-actions in the training and then in testing, observing humanmotion, the robot was able to infer his movement. Theyfound out that for certain motions, even if the robot wastaught how to interact when human was moving hands up,the robot was able to generalize even when the human handswhere moving down. In [7], Interaction Primitives (IP) [8]were used to learn human robot interactions. ProbabilisticMotion Primitives (PROMP) [9] were used to model humanmotions for each skill and then the robot was able to guessthe skill human was doing through Bayes and then generalizeon the skill applying conditioning over PROMP. In [10],PROMP’s were also used to adapt robot movements to a

1Intelligent Autonomous Systems, TU Darmstadt2MPI for Intelligent Systems, TuebingenThe research leading to these results has received funding from the

European Communitys Framework Programme SHAREWORK{urain,peters}@ias.tu-darmstadt.de

Fig. 1: GMCC applied for clustering taught trajectories inImitation Learning

human-robot table tennis match. In [11], HMM were usedto recognize human gestures. The recognition was invariantto the starting position of the gestures. The sequences ofhuman motion were segmented into atomic components andclustered by HMM.

Anyway, up to now, the generalization on human gesturesrecognition has been applied on a few specific tasks andvery limited to small variations between the learned humanmotion and the observed one. The actual algorithms tends tofail recognizing gestures if these are similar shape to the onesalready in our gestures library but with different rotations.

This limitation can be overcome developing a good simi-larity measurement that can extract the relevant features fromthe learned human motions and then be able to recognizerotated or scaled new observations. In an ideal case of HRI,it should be enough teaching the robot how to interact withthe human for some particular cases and then, during thetesting, when the human is doing a similar shape motion, butmaybe rotated, the robot could find the similarities betweenthe previously learned motions and the new ones.

A. Problem Statement

The motion similarity has been studied in different fields.For example, In Probabilistic Theory field, the motion hasbeen represented as a probability distributions. This let thesimilarity distance to be robust with noisy motions. In SignalProcessing field the motion are represented as time series.Similarity metrics deals with time morphed time series.None of the commented similarity metrics consider spatialtransformations among trajectories and so, these similaritymetrics give poor performance measuring the similaritydistance among rotated or scaled trajectories.

Statistics field provides the best similarity metrics tocompare spatially transformed trajectories. In Statistics, twotrajectories similarity depends on how correlated are betweeneach other. A high correlation between trajectories will meana high similarity. For the case of unidimensional trajectoriesxt ∈ IR and yt ∈ IR where t is the time samples, Pearson’scorrelation coefficient is measured as follows

ρ(xt, yt) =cov(xt, yt)σxσy

. (1)

The correlation coefficient will be 1 if the trajectories aretotally positive correlated, 0 if there is no any linear corre-lation and -1 if there is a total negative linear correlation.When it comes multidimensional variables, several correla-tion coefficients exist. The first multidimensional coefficientwas proposed by Harold Hotelling in 1936 [12]. The Canon-ical Correlation Coefficient (CAC), measures the correlationbetween a linear combination of the first variable Xt ∈ IRn

and a linear combination of second variable Yt ∈ IRn

(a∗, b∗) = argmaxa,b

ρ(aᵀXt, bᵀYt),

CAC(Xt,Yt) = ρ(a∗ᵀXt, b∗ᵀYt).

(2)

Nowadays, the most famous multidimensional correlationcoefficients for considering spatial transformations are two:RV Coefficient and Distance Correlation (dCor). RV Coeffi-cient [13] is computed as follows

RV (Xt,Yt) =tr(XtX

ᵀt YtY

ᵀt )√

tr(XtXᵀt )

2tr(YtYᵀt )2

. (3)

As it can be seen RV coefficient is computed with XtXᵀt and

YtYᵀt . If we consider X2 = XQ where Q is any orthogonal

matrix, then, X2Xᵀ2 = XQQᵀXᵀ = XXᵀ. From here,

it is concluded that RV coefficient will be invariant to anyorthogonal transformation, so invariant to rotations.

On the other hand, Distance Correlation [14] is not onlyinvariant to linear transformations but also to some non lineartransformations also. Having two multidimensional vectorsXt ∈ IRn and Yt ∈ IRn; first, the euclidean distance iscomputed between different time samples

ak,j = ||Xj −Xk||, k, j = 0, . . . , T

bk,j = ||Yj − Yk||, k, j = 0, . . . , T

where || · || is the euclidean distance. Once computed, ak,jand ak,j are double normalized

Aj,k = ak,j − a·,k − aj,· + a·,·

Bj,k = bk,j − b·,k − bj,· + b·,·

and the obtained values used for computing the distancecovariance (dCov) and the distance correlation (dCor)

dCov2(X,Y ) =1

n2

n∑k=1

n∑j=1

Aj,kBj,k,

dCor(X,Y ) =dCov(X,Y )√

dVar(X)dVar(Y )

(4)

where, dVar2(X) = dCov2(X,X). dCor will only considerthe sign of the distances computed between different timesamples. If the normalized distances from k to j have thesame sign in both Akj and Bkj , the correlation will increaseand if the signs are different then decrease. So, dCor doesnot care if the relation between the variables is linear ornonlinear.

In this paper, we propose a new correlation measurementfor multidimensional variables. As far as we know, there is noany trajectory similarity measurement that remains invariantto any linear transformations and only linear transformations.Our algorithm is computed as an extension of the Multi-ple Correlation Coefficient for multidimensional variablesand it will be demonstrated that the Multiple CorrelationCoefficient can be considered as a particular case of thethe Generalized Multiple Correlation Coefficient(GMCC) forunidimensional variables.

The rest of the paper is organized as follows: In Section I-B, similarity measures from other fields are presented. InSection II, first the mathematical basis of our algorithmis briefly explained, to continue with the presentation ofthe novel similarity metric, GMCC. Section III is dedicatedto the experiments developed on the robot. First, GMCC’scharacteristics are studied and compared to RV Coefficientand dCor. Later, the GMCC is applied in a clustering taskof robot trajectories Fig. 1. Finally, the paper concludes inSection IV discussing ideas for future work.

B. Related Work

The similarity measurements presented in Section I-A arethe best similarity measurements when it comes to measurespatial invariant similarities. Nevertheless, there are also rel-evant similarity measurement in other fields: in 1) similaritydistances from Signal Processing field are presented, in 2)similarity distances from Topology field and in 3) similaritydistances from Probabilistic Theory field.

1) In Time Series field, the similarity has been facedfrom time shifting perspective. Dynamic Time Warp-ing (DTW) [15] is a well-known algorithm. Consid-ering two similar trajectories, DTW finds the bestmatching between them applying time shifting, any-way it doesn’t give any metric about how similartwo time series are. A similarity measure consider-ing the time shifting is the Edit Distance With RealPenalty (EDR) [16]. On it, Chen et al. propose asimilarity measurement based on string edit distance.The similarity distance between two trajectories willbe related with the number of elements we should addor delete to make them equal. None of these similaritymetrics are invariant to spatial transformations.

2) On Topology field, the similarity between trajectoriesis studied without considering the time informationand so, only spatial information is used. The twomost well-known similarity measurements are Frechetdistance [17] and Hausdorff distance [18]. Frechet dis-tance is presented as the bottleneck maximun distancebetween two trajectories. From a different approach,

Hausdorff distance is the greatest of all the distancesfrom a point in one set to the closest point in theother set. These distances doesn’t consider spatialtransformations of the trajectories.

3) On Probabilistic Theory field, Maximum Likelihoodis the most used algorithm for measuring the similaritybetween two trajectories. In [3], Maximum Likelihoodwas used to compute the similarity between learnedgestures and the observations. The benefits of usingprobabilistic approaches are several. In contrast withsimilarity metrics from other fields, Maximum Likeli-hood can not only give the probability of how similartwo trajectories are, but also, predict the trajectory frompartial observations of the trajectory. So, the algorithmscan be used for recognizing and predicting the humanmotions. The drawback is that no spatial transforma-tions are considered between learned motions and theobserved ones.

II. GENERALIZED MULTIPLE CORRELATIONCOEFFICIENT

We introduce a new correlation coefficient. This coefficientcan be used as a similarity measure between multidimen-sional time series. Having two multidimensional trajectoriesXt ∈ IRn and Yt ∈ IRn where t is the time samplesand n the dimensions, the Generalized Multiple CorrelationCoefficient (GMCC) is a measure of how well can be mappeda linear transformation between the trajectories Xt and Yt.As lot of correlation measurements, the GMCC takes valuesbetween 0 and 1. A coefficient of 1 means that there is acomplete correlation between Xt and Yt, and a value of 0means that there is no any possible linear relation betweenthe vectors Xt and Yt.

The invariability on linear transformation is the biggestcontributions of this new similarity metric. The coefficientwill remain invariant in front of any possible linear transfor-mation and only linear transformations, so the correlationcoefficient will decrease sharply when no linear correla-tions appear. This contribution is particularly interesting incomparison with other correlation coefficients such as RVcoefficient, that remains invariant with only with orthogonaltransformation matrices and dCor that remains invariant withnot only linear but also some nonlinear transformations.

As additional contributions, and similarly to other corre-lation coefficients, GMCC is robust when noisy trajectoriesare compared and the coefficient is obtained instantly incomparison with other Signal Processing field similaritymetrics such as EDR [16] or TWED [19] which computationtime is of O(t2) where t is the time samples of the comparedtrajectories Xt and Yt.

GMCC is an extension of the Multiple Correlation Coef-ficient. While the Multiple Correlation Coefficient computesthe correlation between an univariate dependant variable yand multiple independent variables X , the GMCC extendsit to multivariate dependant variables Y . Moreover, in Sec-tion II-B it will be demonstrated that Multiple Correlation

Coefficient can be computed as a particular case of theGMCC.

The Section II continues as follows: In Section II-A a briefintroduction to the mathematical basis for developing our co-efficient is presented. Once the Coefficient of Determinationand the Multiple Correlation Coefficients are introduced, inSection II-B our novel coefficient, GMCC is presented andcomputed.

A. Coefficient of Determination

Considering an univariate dependant variable yt ∈ IR andmultiple independent variables Xt ∈ IRn, the coefficientof determination, also know as R2 is the fraction of thevariance in the dependent variable y that is predictable fromthe independent variables X [20]

R2 =

∑Tt=0(yt − y)2∑Tt=0(y − y)2

(5)

where yt = AᵀX = a1x1t + a2x2t + · · · + anxnt withA ∈ Rn is the prediction model. yt can be expressed interms of predicted values yt and residual errors εt

yt = yt + εt

then, the parameters A that minimize the squared sum ofthe residual error εt can be obtained through an optimizationfunction

A∗ = argminA

(

T∑t=1

ε2t ) (6)

The solution of the optimization function from Eq. 6, A∗, iscomputed by Least-Squared regression and so, the optimalprediction model is y∗t = A∗ᵀXt. In the particular case ofhaving this optimal model, the Coefficient of Determinationcan be expressed in terms of the Pearson’s Correlationcoefficient [20]

R2 = cᵀR−1xx c

ᵀ (7)

where c = (ρyx1, . . . , ρxn

) is a vector with the correlationcoefficient between each individual independent variables xiand the dependent variable y. Rxx is a correlation matrixmeasuring the correlation between the independent variables

Rxx =

ρ11 ρ12 ρ13 . . . ρ1nρ21 ρ22 ρ23 . . . ρ2n

......

.... . .

...ρn1 ρn2 ρn3 . . . ρnn

.It is important to remark that in the case of having theindependent variables completely uncorrelated between eachother,Rxx becomes into the identity matrix

Rxx = Inxn (8)

and so, the coefficient of determination can be expressed asa sum of squared Pearson’s Correlation coefficients. MixingEq. 7 and Eq. 8

R2 =

n∑i=1

ρ2yxi.

Multiple Correlation Coefficient is the square root of Eq. 7and it represents how well we can predict the parametersof the dependant variable y with the multiple independentvariables X using the optimal linear model.

R =√cᵀR−1

xx cᵀ (9)

As remarked in [20], Multiple Correlation Coefficientwill increase when the number of the independent variablesincrease and so it will give a high correlation coefficientbetween any two variables. From regression perspective islogical. As we are applying Least-Square regression to fitX with y, when the independent variable increase, the noisecan be used to fit in y and so, we could have over-fittingproblems. Anyway, it is expected the noise not to affecta lot in low-dimensional cases such as three dimensionaleuclidean motions.

B. Generalized Multiple Correlation Coefficient

We propose a new correlation coefficient for the case ofmultiple dependant variables Yt ∈ IRn and multiple indepen-dent variables Xt ∈ IRn where t is the time sample and nis the dimension. The given name is Generalized MultipleCorrelation Coefficient (GMCC), as it can be introducedas the extension of the Multiple Correlation Coefficient formultivariate dependent variables. The most relevant propertyfrom GMCC in comparison with the state of the art multivari-ate correlation coefficients such as RV and dCor is that it willremain invariant to any possible linear transformations andonly linear transformations. For any nonsingular matrix H

GMCC(X,HX) = 1.

For Human-Robot Interaction framework or Imitation Learn-ing framework, a similarity measurement that remains in-variant between linearly transformed trajectories is a usefultool. This is the case of different scale, rotation, mirror,squeeze mapped or shear mapped trajectories. This similaritymeasurement can be applied from classication of humangestures in HRI environment to clustering of teached skillsin Imitation Learning as we will show in Section III.

GMCC lays between the RV coefficient that remainsinvariant in front of rotations and scaling and the distancecorrelation (dCor) that remains invariant for not only lineartransformations but also some nonlinear transformations. Inorder to obtain a scalar correlation coefficient between Yand X multidimensional trajectories, we propose a modifiedCoefficient of Determination. Based on Eq. 5

R2 =

∑Tt=0 ||Yt − Y ||2∑Tt=0 ||Yt − Y ||2

(10)

where || · || is the euclidean norm and Yt = HX whereH is any nonsingular matrix. The R2 coefficient fromEq. 10 is presented as the generalization of the Coefficient ofDetermination. Moreover, R2 coefficient from Eq. 5, can becomputed as a particular case of Eq. 10 for unidimensionalvariable y.

If Eq. 10 is extended

R2 =

∑Tt=0

√(y1t − y1)2 + · · ·+ (ynt − yn)2

2∑Tt=0

√(y1t − y1)2 + · · ·+ (ynt − yn)2

2

and now, if square root is eliminated by the power in boththe numerator and denominator and the numerator split

R2 =

∑Tt=0(y1t − y1)2

σ2Y

+ · · ·+∑T

t=0(ynt − yn)2

σ2Y

where σ2Y =

∑Tt=0(y1t − y1)2 + · · ·+

∑Tt=0(ynt − yn)2 =

σ2y1

+ · · · + σ2yn

. If now, each fraction is multiplied bytheir respective variance of the dependant variable σ2

yi, the

obtained equation is

R2 =

∑Tt=0(y1t − y1)2

σ2y1

σ2y1

σ2Y

+ · · ·+∑T

t=0(ynt − yn)2

σ2yn

σ2yn

σ2Y

.

For the particular case of the predicted linear models yit beenoptimal, based on Eq. 7, it can be rewritten as a weightedsum of the square of the Multiple Correlation Coefficient ofeach variable of Y

R2 =

n∑i=1

R2yi

σ2yi

σ2Y

(11)

General Multiple Correlation Coefficient (GMCC) is thesquare root of the general case of Coefficient of Determina-tion from Eq. 10 under the particular case of optimal linearprediction model computed in Eq. 11

R =

√√√√ n∑i=1

R2yi

σ2yi

σ2Y

. (12)

It can be observe, that for the particular case of Y being uni-dimensional, The General Multiple Correlation Coefficientbecomes into the Multiple Correlation Coefficient.

The Generalized Multiple Correlation Coefficient rep-resents how well a set of independent multidimensionalvariables can predict a set of dependent multidimensionalvariables by a linear model. GMCC will take values between0 and 1, where 1 means a complete prediction of Y t withXt. One of the drawbacks of GMCC in comparison with RVcoefficient and dCor coefficient is that it is not symmetric so,R(X,Y ) 6= R(Y ,X). In order to solve it, we can computethe symmetric GMCC

Rsym(X,Y ) =R(X,Y ) +R(Y,X)

2(13)

Traj. type dCor RV GMCCline 0.01 0.0006 0.0 0.001line 0.02 0.02 0.0 0.005line 0.04 0.01 0.0 0.02

circle 0.01 0.0 0.0 0.001circle 0.02 0.0 0.0 0.002circle 0.04 0.0035 0.0 0.0039

TABLE I: Similarity distance measurements for line andcircular trajectories with different noises for distance cor-relation(dCor), RV coefficient and GMCC

(a) Circular trajectory with artificial gaussian noise (b) Linear trajectory with artificial gaussian noise

(c) Circular trajectories artificially linearly transformed (d) Linear trajectories artificially linearly transformed

Fig. 2: Artificially generated trajectories for testing the experiments in Section III-A

This symmetric correlation coefficient from Eq. 13 is usedto compute a similarity distance measure

dX,Y = 1−Rsym(X,Y ). (14)

A similarity distance of 1 means that X and Y are com-pletely uncorrelated and the closer the distance is to 0, thehigher the linear correlation between X and Y .

In the experimental evaluation, this distance measure isgoing to be compared with the two most relevant multivariatecorrelation coefficients metrics RV and dCor.

III. EXPERIMENTAL EVALUATION

For the experimental evaluation of GMCC, the experi-ments have been split in two sections. In Section III-A, theGMCC characteristics will be studied in some predefinedtrajectories. This Section will be useful to facilitate the readerthe understanding of the capacities and limitations of thenovel metric.

In Section III-B, GMCC is applied for a trajectory cluster-ing problem and the obtained solution compared with RV anddCor solutions. Six different type of trajectories have beenrecorded with a KUKA LWR arm. The recorded trajectorieshave been clustered through a hierarchical clustering method.

A. GMCC characteristics

For this first evaluation, two simple 2D signals are goingto be used: a linear and a circular trajectory. To observethe robustness of GMCC in front of noisy signals, differentvariance gaussian noises were added in both the linearand circular trajectories. The obtained noisy trajectories arepresented in Fig. 2a and Fig. 2b. The similarity distancesbetween the original trajectories and the noisy signals werecomputed for RV, dCor and GMCC and the obtained valuesare presented in Table I.

From Table I, it can be observed that in terms of noise thethree similarity distances are pretty robust. GMCC is givingthe worst results in comparison with dCor and RV, but froma general perspective, three of them remains robust in frontof noisy signals as the obtained values are close to 0.

In a second experiment the selected two trajectories, a lineand a circle, were transformed by rotation, scaling, reflection,shear mapping and squeezing. The obtained trajectories arepresented in Fig. 2c and Fig. 2d. As the previous case,the similarity distances were computed between the originaltrajectory and the transformed ones for dCor, RV and GMCCand obtained results organized in Tables II and III.

These tables are representative of the real power ofGMCC. While the RV coefficient is measuring big distancesbetween the original line and the one transformed by shearmapping, rotation or squeezing, the GMCC remains robust innull distance. dCor’s distance is also pretty close to 0, but inshear mapping and squeezing gives a little bit worse results.GMCC is the coefficient that works better for similarityinvariant to linear transformations and will remain alwaysin lowest distance for any linear transformations.

GMCC is limited to linear transformations. GMCC dis-tance starts increasing when no linear mapping can be set be-tween the compared trajectories. Meanwhile, dCor is capable

Transf. type dCor RV GMCCrotation 0.0 0.01 0.0

scale 0.0 0.0 0.0reflection 0.0 0.009 0.0

shear 0.12 0.18 0.0squeeze 0.11 0.14 0.0

TABLE II: Similarity distance measurements for circulartrajectories with different transformations for distance cor-relation(dCor), RV coefficient and GMCC

(a) RV Coefficient distance matrix (b) dCor distance matrix (c) GMCC distance matrix

Fig. 3: Distance Matrices computed for RV, dCor and GMCC

of finding also nonlinear mappings between trajectories. Thedistance measured by dCor will be smaller than the distancewith GMCC for nonlinear correlations. In order to provethis, the similarity distance has been measured between theoriginal line and the original circle and the obtained distancespresented in Table IV.

The distance measured by GMCC is 0.87 while thedistance measured by dCor is 0.57. dCor is less sensibleto nonlinear transformations than GMCC. our correlationcoefficient is a worse tool for measuring nonlinear correlationbut at the same time is a better metric for measuring onlylinear correlations.

B. Clustering robot trajectories

Six different type of motions were recorded in a KUKALWR arm. The end effector’s cartesian position was recordedfor x, y and z. For each motion type five different demonstra-tions were recorded with different scales, different startingpoints and different orientations. The selected motions arelines, U-shape, S-shape, circular,triangle and hello trajecto-ries. All the demonstrations were fixed to the same timelength and same amount of time samples. Then, the similaritydistance matrix was computed among the demonstrations andthis distance matrix was used as the input for a hierarchicalclustering.

The similarity matrix of each 30 demonstrations ( sixgestures by five demonstrations) was computed for the three

Transf. type dCor RV GMCCrotation 0.0 0.82 0.0

scale 0.0 0.0 0.0reflection 0.0 0.819 0.0

shear 0.007 0.88 0.0squeeze 0.0029 0.16 0.0

TABLE III: Similarity distance measurements for lineartrajectories with different transformations for dCor, RV andGMCC

Transf. type dCor RV GMCCline Vs circle 0.5258 0.27378 0.871

TABLE IV: Similarity distance measurements for line Vscircle

correlation distances dCor, RV and GMCC. The obtainedsimilarity matrices are presented in the Fig. 3.

The first conclusion is extracted from Fig. 3b. dCordistance is measuring small distances not only between thedemonstrations of the same motion but also between thedemonstrations of different motions. From here, we canpredict that at least with the recorded motions, dCor willfail classifying or clustering them correctly. RV coefficientand GMCC are obtaining similar results. For some gesturesGMCC differentiates better. For example, between circulartrajectories and S-shape trajectories. Anyway, GMCC isgetting worse results discriminating linear movements andS-shape movements. It is also relevant to remark how badlythe hello trajectories are correlated. None RV coefficient orGMCC is giving good similarity distance between most ofthe hello trajectories.

The recorded demonstrations were clustered by hierarchi-cal clustering to observe the application of our algorithm ina real case scenario. During first evaluation, we consideredto cluster the demonstrations in six clusters. However, thehello trajectories were very dissimilar not only in comparisonwith other trajectories but also between each other as itcan be observed in the dendrograms from Fig. 4. This wasmaking the clustering to fail as the hello trajectories werenot clustered in the same cluster, but each one was a uniquecluster and then in a unique cluster we could find all the othergestures. In order to solve this, the hierarchical clusteringtries to cluster the no-hello motions in five different clusters.

The applied linkage criterion was the average distancebetween the clusters.The average linkage criterion will usethe average distance between all the elements in both clusterto decide if they should merge or not

d(u, v) =

n∑i=1

m∑j=1

d(u[i], v[j])

nm.

From the obtained dendrograms several conclusions can beextracted. Expected from the similarity matrix in Fig. 3b,dCor solution is not useful for clustering the recorded humangestures. In Fig. 4b, it can be observed how the clusters hasbeen built with dCor. There are two main clusters. The one inyellow, clustering in same cluster some triangle trajectories,

(a) RV’s dendrogram (b) dCor’s dendrogram (c) GMCC’s dendrogram

Fig. 4: Dendrograms computed for RV, dCor and GMCC

U-shape trajectories and S-shape trajectories. The other bigclustering some line trajectories, hello trajectories and circu-lar trajectories. This poor results can be understood as dCorfinds not only linear but also non-linear correlations betweentrajectories and so, it will find a small similarity distancebetween most of the trajectories as they are nonlinearlycorrelated.

The clustering obtained in GMCC’s dendrogram is betterthan the clustering obtained in RV’s dendrogram. Thereare less missclustered elements in GMCC than in RV.GMCC missclusters two triangle trajectories with the circulartrajectories and one U-shape trajectory with the S-shapetrajectories. Meanwhile, RV dendrogram is also creating aunique cluster between all U-shape and S-shape trajectories.Also, one of the found clusters by RV is a cluster withonly two elements containing one triangle trajectory and onecircular trajectory. We can conclude that the best tool forclustering three-dimensional motions is the GMCC.

IV. DISCUSSION AND FUTURE WORK

We presented a multidimensional correlation coefficientthat remains invariant for any linear transformation and onlylinear transformations. This novel multidimensional has beenproved to be a multidimensional extension of the MultipleCorrelation Coefficient. The validity of GMCC has beenproved in a real robot learning scenario and have showedbetter clustering capacities than state of art multivariatecorrelation coefficients such as RV and dCor.

While our similarity distance is invariant to any lineartransformation, it cannot deal with time morphing and timeshifting. A promising improvement of our algorithm willcome from the mixture between DTW and GMCC. Anotherinteresting improvement comes from extending GMCC toprobabilistic motions. PROMPs are a promising representa-tion of motions in Imitation Learning and we expect to mergePROMPs with GMCC.

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