Semi-metric portfolio optimization: a new algorithm reducing simultaneous asset shocks
SSemi-metric portfolio optimisation: a new algorithm reducing simultaneous assetshocks
Nick James ∗ , Max Menzies , Jennifer Chan School of Mathematics and Statistics, University of Sydney, NSW, Australia Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
Abstract
This paper proposes a new method for financialportfolio optimisation based on reducing simulta-neous asset shocks across a portfolio of assets. Weadopt the new semi-metrics of [James et al. , 2019]to determine the distance between two time series’structural breaks. We build on the optimal portfoliotheory of [Markowitz, 1952], but utilize distance be-tween asset structural breaks, rather than portfoliovariance, as our penalty function. Our experimentsare promising: on synthetic data, they indicate thatour proposed method does indeed diversify amongtime series with highly similar structural breaks. Onreal data, experiments illustrate that our proposedoptimisation method produces higher risk-adjustedreturns than mean variance portfolio optimisation.The predictive distribution is superior in every mea-sure, producing a higher mean, lower standard de-viation and less kurtosis. The main implication forthis method in portfolio management is reducing si-multaneous asset shocks and potentially sharp asso-ciated drawdowns, during periods of highly similarstructural breaks, such as a market crisis.
Modern portfolio theory provides an optimisation frameworkfor determining the optimal allocation of weights in an invest-ment portfolio, by optimising a specific objective function.The idea was first introduced by [Markowitz, 1952], and hasprogressed considerably since then.The fundamental contribu-tion of Markowitz was the concept of diversification amongstock portfolios, rather than analysing risk and return on anindividual security basis. One of the most notable advance-ments was the work of William Sharpe in [Sharpe, 1966], whoproposed a measure of risk-adjusted returns in financial port-folios, the Sharpe Ratio. The Sharpe Ratio is an indicationfor the potential reward in any candidate investment basedon the risk taken by the investor. The standard mathematicalrepresentation of the Sharpe ratio is the following optimisationproblem: ∗ [email protected] Given a collection of n assets, let R i be the historical returnsfor the i th asset in a collection, Σ be the matrix of historicalcovariances between stocks, R f the risk-free rate.Denote weights of a portfolio by w i . We seek to maximisethe Sharpe ratio: E ( R p ) = n (cid:88) i =1 w i R i (1) σ p = w T Σ w (2)Maximise Sharpe Ratio = E ( R p ) − R f σ p (3)(4)under typical constraints include ≤ w i ≤ , i = 1 , ..., n and (cid:80) ni =1 w i = 1 . Many domains in the physical and social sciences are inter-ested in the identification of structural breaks in various typesdata. [Ranshous et al. , 2015] and [Akoglu et al. , 2014] recentlyprovided an overview of anomaly detection methods withinthe context of network analysis, which can be used to identifyrelations among entities in high dimensional data. [Koutra etal. , 2016] determine change points in dynamic networks viagraph-based similarity measures.In the more statistical literature, focussed on time seriesdata, [Moreno and Neville, 2013], [Bridges et al. , 2015] and[Peel and Clauset, 2015] have developed change point modelsdriven by hypothesis tests, where p -values allow scientiststo quantify the confidence in their algorithm. Change pointalgorithms generally fall within statistical inference (namely,Bayesian) or hypothesis testing frameworks. Bayesian changepoint algorithms [Barry and Hartigan, 1993; Xuan and Mur-phy, 2007; Adams and Mackay, 2007] identify change pointwithin a Bayesian framework, but suffer from hyperparam-eter sensitivity and do not provide statistical error bounds( p -values), often leading to a lack of reliability.Within hypothesis testing, [Ross, 2015] outlines algorith-mic developments in various change point models initiallyproposed by Hawkins, Qiu and Kang in [Hawkins et al. , 2003].The framework for single and multiple change point detectionis explained clearly. Some of the more important develop-ments include the work of [Hawkins and Zamba, 2005], and a r X i v : . [ q -f i n . P M ] J a n oss et al. [Ross and Adams, 2011, 2012; Ross, 2014]. Rossrecently created the CPM package, which allows for flexibleimplementation of various change point models on time seriesdata. Given the package’s ease of use and flexibility, we buildour methodology on this suite of algorithms. The application of metric spaces has provided the groundworkfor research advancement in various areas of machine learning.In addition to more traditional metrics, such as the
Hausdorff and
Wasserstein metrics, semi-metrics , which may not satisfythe triangle inequality property of a metric, have been usedsuccessfully in various machine learning applications. Anoverview of such (semi-)metrics [Conci and Kubrusly, 2017]and applications was recently explored. The three primaryapplications include; image analysis [Baddeley, 1992; Dubuis-son and Jain, 1994; Gardner et al. , 2014], distance betweenfuzzy sets [Brass, 2002; Fujita, 2013; Gardner et al. , 2014;Rosenfeld, 1985] and computational methods [Eiter and Man-nila, 1997; Atallah, 1983; Atallah et al. , 1991; Shonkwiler,1989]. More recently, a review and computational analysis ofvarious (semi)-metrics was undertaken [James et al. , 2019] inmeasuring distance between sets of time series change points.
Markowitz portfolio optimisation is typically carried out withthe following set up. An objective function is generally ameasure of risk-adjusted return, such as the Sharpe Ratio: E ( R p ) − R f σ p . (5)Depending on the context, different assumptions, which mani-fest as constraints, will accompany the objective function. Inour analysis, we impose (cid:26) condition 1 , (cid:80) ni =1 w i = 1 condition 2 , ≤ w i ≤ The former condition states that all portfolio assets must beinvested; the latter prohibits short-selling in the portfolio. Thisobjective function is maximised with respect to the weights w i of the portfolio.Using the historical returns, expected return is calculatedas E ( R p ) = (cid:80) ni =1 w i R i . Portfolio variance is σ p = w T Σ w where Σ i,j is the covariance between historical returns ofassets i, j . Weights are selected to maximise the Sharpe Ratiosubject to conditions 1 and 2.This objective function will select an allocation of weightsbased on a trade-off between portfolio returns and variance.In many circumstances, variance is a suitable measure in afinancial securities context. However, it is not without itslimitations.There are a variety of reasons why a change point-relatedpenalty function may be a suitable alternative, or complement,to the covariance measure between two time series.1. Covariance is computed as an expectation E ( X, Y ) = E ( X − E X )( Y − E Y ) , which is an average (integral) over an entire probability space. In a financial context,this computes an average over time, which, in modernfinancial markets (and especially since the Global Fi-nancial Crisis), are disproportionately bull markets, withmost assets performing quite well together. As such, as-sets which rise together in a bull market but are actuallyof quite a different nature may be erroneously identifiedas similar.2. Covariance fails to capture dissimilarity between time se-ries during periods of market crisis and erratic behaviour.Investors are often concerned with how robust their port-folio is during such times. Portfolios that are optimisedusing covariance as a risk measure fail to determine theimpact of various asset combinations during times ofmarket crisis. For instance, if two assets are simultane-ously acting erratically, they may actually be negativelycorrelated during this time, and both included in a portfo-lio, which would increase, not reduce, erratic behaviour.Change points herald erratic behaviour, so using distancesbetween change points as an objective function may bet-ter separate out erratic behaviour in a portfolio.3. Investors are often interested in peak to trough measuresof asset performance. That is, how big is the drop incumulative returns from a local maximum to a local min-imum. Optimisation algorithms using covariance mea-sures fail to identify peak to trough behaviour and fail tominimise these behaviours. However, distances measuredbetween sets of change points (which denote structuralbreaks in the mean, variance and other stochastic quanti-ties) are better equipped to identify how similar two timeseries are with respect to peak to trough measures andallocate weights to minimise these precipitous drops.We formulate our new objective function to penalise struc-tural breaks and the associated erratic behaviour. We use theMJ . semi-metric of [James et al. , 2019], selected due toits good performance with outlier sensitivity, and the strongpossibility of outliers in this context. The MJ . distance iscalculated as follows: d . MJ ( S, T ) = (cid:32) (cid:80) t ∈ T d ( t, S ) . | T | + (cid:80) s ∈ S d ( s, T ) . | S | (cid:33) (6)Note d ( S, T ) = 0 if and only if S = T .Now, our MJ . distance matrix ( D . ) i,j is computed asfollows. Following a suitable change point algorithm, let asset i have set of change points S i , i = 1 , ..., n . Then form: ( D . ) i,j = d . MJ ( S i , S j ) (7)Next, we transform our distance matrix into an affinitymatrix, which mimics the properties of a covariance matrix.Recall two assets have correlation equal to if and only if theyare perfectly correlated. A i,j = 1 − D i,j max D ∀ i, j (8)Analogously, A ij = 1 if and only if d ( S i , S j ) = 0 , meaningthe two assets have identical change point sets.n the context of Markowitz portfolio optimisation, intro-ducing more stocks with lower correlation increases diversifi-cation and reduces systematic risk in the portfolio. Weights arechosen to maximise return while reducing total variance. Wemodify this insight, allocating weights which will maximisereturn while minimising affinity between sets of change points,that is, maximising the spread between change points and er-ratic behaviour. To do so, we substitute our adjusted affinitymatrix A for the original covariance matrix Σ , and optimiseour new risk-adjusted return measure - which we term the MJRatio objective function - with respect to portfolio weights. E ( R p ) = n (cid:88) i =1 w i R i (9) Ω p = w T A w (10)MJ Ratio = E ( R p ) − R f Ω p , (11)We retain the same constraints as equation 5 for the remainderof our experiments section, but our method is flexible enoughto vary such constraints. Experiment 1
In the first experiment, we generate 8 time series with candi-date change points. Time series 1-3 have identical numbers ofchange points with identical locations. Time series 4-6 alsohave identical numbers of change points with identical loca-tions. Time series 7 and 8 have one change point each. Theoptimisation results demonstrate that our optimisation frame-work is able to produce a more even distribution of changepoints across the portfolio. First, less weight - 6.9% - is al-located to time series 1, 2 and 3 - which have highly similarchange points. Similarly, time series 4, 5 and 6 are only allo-cated 5 % weight each. Time series 7 and 8, which possesshighly uncorrelated structural breaks are allocated significantlymore weight - (33.5% and 30.7% respectively. This experi-ment demonstrates that the algorithm provides diversificationwith regards to highly correlated structural breaks. Traditionalmean variance portfolio optimisation would be unable to doso. Optimisation Results: Synthetic Data ExperimentAsset Change points W ∗ Asset Table 1: Results for synthetic data experiments.
Note: Constraint on weights in optimiser - minimum of 5%invested in all assets. (a) Synthetic Time Series with Identical Change PointsFigure 1: Synthetic time series with change points. Mean returnsbetween time series are identical, so only change points are analysedin optimisation routine
Experiment 2
In our synthetic data experiment we generate 8 time seriesin a realistic scenario. Time series 1-4 have similar changepoints. Time series 5 and time series 6 have structural breaksthat are dissimilar to the rest of the time series. Time series7 and 8 both have identical change points. Accordingly, onewould hope that our optimisation algorithm allocates moreweight to assets 5 and 6, and less weight to the time serieswith highly similar change points (1,2,3,4 and 5, 6). Theresults of our experiment demonstrate that assets 1,2,3,4,5 and6 are all allocated a minimum allocation of 5%. Assets 5 and 6are allocated 32% and 38% respectively. Again, the algorithmproduces a candidate weight allocation one would have hopedfor in this scenario.Optimisation Results: Synthetic Data Experiment 2Asset Changepoints W ∗ Asset Table 2: Results for synthetic data experiments. ∗ indicates that bestperforming result may be excluded due to over-fitting Note: Constraint on optimisation weights: 5% minimum40% maximum.
Finally we apply our method to real financial data. We envis-age this method being suitable in an asset allocation context, sowe use indices and commodities as our underlying candidate a) Synthetic Time Series with Similar (but not identical) ChangePointsFigure 2: Synthetic Time Series with Change Points. Mean returnsbetween time series are identical, so only change points are analysedin optimisation routine
Optimisation Results: Real DataS&P500 ASX200 Oil Gold Nikkei IBOV Dow StoxxMVO 0.16 0.1 0.17 0.17 0.05 0.05 0.05 0.18CPO 0.05 0.25 0.05 0.25 0.05 0.075 0.05 0.225
Table 3: Results for real data experiments. MVO is the traditionalmean variance optimiser and CPO is our proposed change pointoptimiser investments. That is, we are simulating the role of an assetallocator, such as a pension fund or endowment, who wouldbe interested in macroeconomic asset allocation decisions. Weanalyse returns of the ASX 200 Index, S&P 500 Index, Oilspot price, Gold spot price, Nikkei Index, BOVESPA Index,Dow Jones Index and Stoxx 50 Index between January 2009and November 2019. There are several important details andassumptions in our experiments section on real data:1. We train our algorithm over a relatively long time periodin order to estimate the true dynamics between variousassets’ change points as precisely as possible. Trainingthe algorithm on longer time periods will provide a moreaccurate assessment of similarity in varying market dy-namics.2. However, there is a balance to be struck between goingback enough to learn appropriate dynamics between assetclasses and using too much history that relationshipsbetween assets no longer behave the way in which theywere estimated. The behaviour of individual asset classesand their relationships may change over time.3. The period from January 2018 - June 2019 is a suitableout of sample period to test the algorithm, due to thevaried market conditions. Most of 2018 provided rela-tively buoyant equity market returns, with a sharp drop inDecember 2018, followed by a prolonged recovery until (a) ASX 200 index returnsand change points (b) S&P 500 index returnsand change points(c) Oil spot price returns andchange points (d) Gold spot price returnsand change points(e) Nikkei index returns andchange points (f) Bovespa index returnsand change points(g) Dow Jones index returnsand change points (h) Stoxx 50 index returnsand change pointsFigure 3: Log returns time series with annotated change points inmean for each asset(a) MJ . dendrogram (b) MJ . transitivity analysisFigure 4: MJ . transitivity analysis for real asset portfolioa) CPO weighted asset trajectory(b) MVO weighted asset trajectoryFigure 5: CPO and MVO weighted asset trajectory. The surfacedisplays time on the x-axis, optimal portfolio weight on the y-axisand the cumulative return for each asset class (S&P 500 Index, Goldetc.) on the z-axis June 2019. We wish to examine how candidate portfolioswill perform in various market conditions, particularly inthe presence of large drawdowns. Thus, this is a suitableperiod to compare optimisation algorithms’ performance.4. The role of asset allocation is often guided by an in-vestment policy statement or mandate which providesupper and lower bounds for asset allocation decisions.This is captured by the constraints around the weightsin the portfolio optimisation. During pronounced bulland bear markets, institutional asset allocators may nothave the flexibility to implement global optimisation solu-tions. For example, if two asset classes had significantlyhigher returns and lower volatility than the remainder ofcandidate investments, the unconstrained solution wouldallocate all portfolio weight into these two assets. In-vestment weighting constraints prevent these contrivedscenarios from occurring.5. Our method provides an advantage over the simple cor-relation measure by addressing all three limitations insection 2.6. One possible drawback to our proposed method however,is that to learn meaningful relationships between assets’structural breaks, a long time series history is needed,preferably with many structural breaks.7.
Constraints:
We place a minimum 5 %, maximum 25 %of portfolio assets in any candidate investment.
We train the algorithm between January 2009 - December2017 and test performance on data between January 2018and June 2019. The training procedure learns the weightsallocated to each candidate investment using the objectivefunction, subject to constraints. We compare our change pointoptimisation method (CPO) with a more traditional meanvariance optimisation (MVO).First, the Mann-Whitney change point detection algorithmis applied to the training data (log returns between January2009 - December 2017), identifying the locations of structuralbreaks in the mean for each possible investment. This yields8 sets of change points, where each change point is indexedby time. We follow [James et al. , 2019] and apply the MJ . distance to determine the distance between candidate sets ofchange points. We analyse the distance matrix with hierar-chical clustering, and determine how badly the semi-metricfails the triangle inequality for all possible triples within thedistance matrix. Then, we optimise the MJ Ratio objectivefunction with respect to the weights, determining candidateweight allocations. Finally, we run an out of sample forecast-ing procedure using the weights estimated in our training data.We compare the predictive performance of the CPO and MVOalgorithms between January 2018 and June 2019.There are several interesting findings in our analysis:1. The distances measured between time series structuralbreaks indicates that there is a cluster of four highlysimilar assets (S&P 500, Stoxx 50, Dow Jones and Oil),a cluster of three moderately similar assets (BOVESPA,Nikkei 225 and ASX 200) and an outlier in gold. Thesemeasures confirm financial intuition and documentedrelationships between financial asset classes, in particulargold’s properties as a safe haven asset class. Both theDow Jones and S&P 500 Index are determined to be inthe same cluster, and accordingly quite similar. Giventhat there is significant overlap in the constituents of bothindices, this is a logical finding.2. The allocations of the CPO and MVO algorithms arequite different. In particular, the CPO algorithm allocatessignificantly more weight to gold and the ASX 200 Index,while allocating less weight to the S&P 500 Index, Oiland the Stoxx 50 Index.3. The MVO allocates 56.3% to the four indices in thehighly similar cluster (S&P 500, Stoxx 50, Dow Jonesand Oil). The CPO method allocates these indices a com-bined 21%. As expected, the change point optimisationis allocating less portfolio weight to assets with a highdegree of similarity regarding their change points. Giventhat our aim with this new objective function is to smoothout returns, particularly during periods of extreme volatil-ity such as market crises, the CPO does a superior job.4. CPO allocating 36% portfolio weight to gold, while MVOallocates 17.6%. This higher allocation to gold is ex-pected in this scenario, given the highly similar behaviour,and change points, of the other assets. It appears that theCPO method provides a more even distribution of weightacross different types of financial assets. a) CPO and MVO Performance(b) MVO and CPO Predictive DensitiesFigure 6: Out of sample performance and predictive densities (Jan2018- June 2019)
5. When considering portfolio risk in an optimisation frame-work, investors have a variety of measures they maychoose to optimise over. β , standard deviation, downsidedeviation and tracking error are just several of these. OurCPO model introduces a mathematical framework whichaddresses peak to trough (drawdown) losses as a measureof risk. Specifically, the model captures simultaneousasset shocks and aims to minimise the size of drawdownsby creating a uniform spread of change points across allportfolio holdings. To our knowledge, there are no exist-ing measures which provide a mathematical frameworkfor reduce the size of drawdowns and making the spreadof change points more uniform over time. After estimating the optimal allocation of weights for each as-set, we compare the performance and properties of the predic-tive distribution when using the MVO and CPO methods. Ta-ble 4 displays the results when allocating the selected weights w ∗ i to each asset i . First, the cumulative returns for the CPOmethod are higher - indicating that this method does not pro-duce a deterioration in performance. The CPO method endedthe period profitable, while the MVO method lost approxi- mately . over the period January 2018-June 2019. Inter-estingly, the CPO predictive distribution also produced a lowerstandard deviation, and a significant reduction in kurtosis. Thetwo predictive densities can be seen in Figure 6b, where theMVO predictive density clearly exhibits fatter tails.Perhaps most importantly, the drawdown experienced bythe CPO method is significantly smaller than that of the MVOmethod. Figure 6a shows the cumulative returns for eachasset allocation strategy. The period during December 2018displayed marked differences in performance. Although theMVO strategy was outperforming the CPO strategy until De-cember 2018, the CPO strategy had a significantly smallerdrawdown, as confirmed by Table 4. This is due to CPO’slarge position in gold, reducing the downward momentum ofthe total portfolio. This example demonstrates that the CPOalgorithm provided a superior risk-adjusted return to the MVOwith respect to all possible measures. Most importantly, thesize of the largest drawdown was reduced significantly.Predictive attributes and performanceOptimisation Method CPO MVOCumulative returns Table 4: Results for predictive performance on out-of-sample data
We have proposed a novel optimisation method, which utilisessemi-metrics as a distance measure to reduce simultaneousasset shocks across a portfolio of assets. Experiments on syn-thetic data confirm that we are able to detect similar time series- in terms of location of structural breaks - and accordinglyallocate these assets less portfolio weight. Experiments onreal data suggest that our method may significantly reduce thesize of portfolio drawdowns when compared to the traditionalmean variance framework. This novel optimisation frame-work may have significant implications for asset allocationand portfolio management professionals interested in alterna-tive measures of risk. Our method diversifies well away fromportfolio drawdown, and seeks to avoid the erratic behaviourof highly clustered change points. Our method is flexible, andalternative change point algorithms may be married with othersemi-metrics in the MJ p family, or other semi-metrics, foralternative approaches. eferences R. Adams and D. Mackay. Bayesian online changepoint de-tection. arXiv preprint arXiv:0710.3742 , 2007.L. Akoglu, H. Tong, and D. Koutra. Graph-based anomaly de-tection and description: a survey.
Data Mining and Knowl-edge Discovery , pages 223–247, 2014.M. J. Atallah, C. C. Ribeiro, and S. Lifschitz. Computing somedistance functions between polygons.
Pattern Recognition ,24:775–781, 1991.M. J. Atallah. A linear time algorithm for the Hausdorffdistance between convex polygons.
Inform. Process. Lett. ,17:207–209, 1983.A. J. Baddeley. Errors in binary images and an L p version ofthe Hausdorff metric. Nieuw Arch. Wisk , 10:157–183, 1992.D. Barry and J. Hartigan. A bayesian analysis for change pointproblems.
Journal of the American Statistical Association ,pages 309–319, 1993.P. Brass. On the nonexistence of Hausdorff-like metrics forfuzzy sets.
Pattern Recognition Lett. , 23:39–43, 2002.R. A. Bridges, J. P. Collins, E. M. Erragut, J. A. Laska, andB. D. Sullivan. Multi-level anomaly detection on time-varying graph data.
Proceedings of the 2015 IEEE/ACMInternational Conference of Advances in Social NetworksAnalysis and Mining 2015 , pages 579–583, 2015.A. Conci and C. Kubrusly. Distances between sets - a survey.
Advances in Mathematical Sciences and Applications , 26:1–18, 2017.M. P. Dubuisson and A. K. Jain. A modified Hausdorff dis-tance for object matching.
Proceedings of InternationalConference Pattern Recognition, Jerusalem , pages 566–568,1994.T. Eiter and H. Mannila. Distance measures for point sets andtheir computation.
Acta Inform. , 34:109–133, 1997.O. Fujita. Metrics based on average distance between sets.
Japan J. Indus. Appl, Math. , 30:1–19, 2013.A. Gardner, J. Kanno, C. A. Duncan, and R. Selmic. Measur-ing distance between unordered sets of different sizes.
Proc.IEEE Conf. Comput. Vis. Pattern Recognit. , pages 137–143,2014.D. M. Hawkins and K. D. Zamba. Statistical process controlfor shifts in mean or variance using a changepoint formula-tion.
Technometrics , 47:164–173, 2005.D. J. Hawkins, P. H. Qiu, and C. W. Kang. A change-pointmodel for a shift in variance.
Journal of Quality Technology ,42:355–366, 2003.N. James, M. Menzies, L. Azizi, and J. Chan. Novel semi-metrics for multivariate change point analysis and anomalydetection. arXiv e-prints , page arXiv:1911.00995, 2019.D. Koutra, N. Shah, J. T. Vogelstein, B. Gallagher, andC. Faloutsos. Delta-Con: Principled massive-graph similar-ity function with attribution.
ACM Transactions on Knowl-edge Discovery from Data (TKDD) , 28, 2016. H. Markowitz. The journal of finance.
Portfolio Selection ,pages 77–91, 1952.S. Moreno and J. Neville. Network hypothesis testing usingmixed kronecker product graph models.
Data Mining 2013IEEE 13th International Conference , pages 1163–1168,2013.L. Peel and A. Clauset. Detecting change points in the large-scale structure of evolving networks. 2015.S. Ranshous, S. Shen, D. Koutra, S. Harenberg, C. Falout-sos, and N. F Samatova. Anomaly detection in dynamicnetworks: a survey.
Wiley Interdisciplinary Reviews: Com-putational Statistics , pages 223–247, 2015.A. Rosenfeld. Distances between fuzzy sets.
Pattern Recogni-tion Lett. , 3:229–233, 1985.G. J. Ross and N. M. Adams. Sequential monitoring of aBernoulli sequence: When the pre-change parameter isunknown.
Computational Statistics , 28:463–479, 2011.G. J. Ross and N. M. Adams. Two nonparametric controlcharts for detecting arbitrary distribution changes.
Journalof Quality Technology , 44:102–116, 2012.G. J. Ross. Sequential change detection in the presence ofunknown parameters.
Statistics and Computing , 24:1017–1030, 2014.G. J. Ross. Parametric and nonparametric sequential changedetection in R: the cpm package.
Journal of StatisticalSoftware, Articles , 66(3):1–20, 2015.W. F. Sharpe. Mutual fund performance.
Journal of Business ,pages 119–138, 1966.R. Shonkwiler. An image algorithm for computing the Haus-dorff distance efficiently in linear time.
Inform. Process.Lett. , 30:87–89, 1989.X. Xuan and K. Murphy. Modeling changing dependencystructure in multivariate time series.