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PhD defense talk at Ecole Polytechnique. Funding: Hellebore Capital

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- 1. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Some contributions to the clustering of ﬁnancial time series and its applications to credit default swaps Gautier Marti November 10, 2017 Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 2. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Context of the PhD thesis PhD studies in hedge funds: Hellebore Capital Management, 63 Avenue des Champs-Elys´ees, Paris, France (April 1, 2014 - February 29, 2016) Hellebore Capital Limited, 81 Fulham Road, London, United Kingdom (March 1, 2016 - September 20, 2017) AXA IM Chorus, 18 Westlands Road, Hong Kong (October 1, 2017 - present) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 3. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Outline 1. Introduction to the credit default swap dataset Contributions: 2. About the consistency of clustering ﬁnancial time series 3. Improving standard distances between ﬁnancial time series i) a simple correlation + distribution distance ii) a geometrical approach to deﬁne dependence coeﬃcients from copulas 4. Perspectives Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 4. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Some movation for using clustering Statistical modelling is diﬃcult as the time series are non stationary, e.g. economic regimes are changing, it can be misleading to use data from a too distant past are near eﬃcient, i.e. behaving nearly like random walks (cf. the eﬃcient-market hypothesis (Fama, 1970) [5]) have a low signal-to-noise ratio, i.e. measure artifacts hide information in random ﬂuctuations are in an unfavorable statistical setting, too few relevant observations (length) wrt the number of variables (time series) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 5. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Some motivation for using clustering Clustering helps to reduce dimensionality, and thus can be used as a preprocessing for: Risk management, e.g. ﬁltering covariances, performance and risk attribution Investment, e.g. portfolio design, statarb, beta neutralization Data analysis, e.g. outliers detection and missing values imputation, exploration (www.datagrapple.com) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 6. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions 1 Introduction to credit default swaps 2 About the consistency of clustering ﬁnancial time series 3 Design of distances and alternative dependence coeﬃcients Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients 4 Summary and open questions Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 7. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Introduction to the credit default swap Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 8. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Introduction to the CDS market Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 9. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Introduction to the CDS raw dataset Putting Self-Supervised Token Embedding on the Tables [15] (ICMLA 2017) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 10. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions A ‘tick-by-tick’ dataset Autoregressive Convolutional Neural Networks for Asynchronous Time Series [1] (ICML Time Series WS 2017) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 11. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Historical daily time series of spread From the received ‘tick-by-tick’ prices, a synthetic order book is built. At 5pm London time, we save the mid-price of the best bid and best oﬀer in the order book for each entity. N ≈ 800 liquid CDSs, with T ≈ 3000. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 12. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions 1 Introduction to credit default swaps 2 About the consistency of clustering ﬁnancial time series 3 Design of distances and alternative dependence coeﬃcients Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients 4 Summary and open questions Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 13. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Clustering of Financial Time Series Stylized fact I: Financial time series correlations have a strong hierarchical block diagonal structure (Mantegna, 1999) [6] https://gmarti.gitlab.io/ml/2017/09/07/how-to-sort-distance-matrix.html Stylized fact II: Most correlations are spurious (Bouchaud, 1999) [3] Motivation for clustering ﬁnancial time series using correlation as a similarity measure: dimensionality reduction ≡ ﬁltering noisy correlations Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 14. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Challenge for the statistical practitioner The dilemma: the longer the time interval, the more precise the correlation estimates, but also the longer the time interval, the more unrealistic the stationarity hypothesis for these time series. Question: How does the clustering behave with statistical errors of the correlation estimates? Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 15. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions A ﬁrst theoretical approach - simpliﬁed setting We consider the following framework: ﬁnancial time series ≡ random walks they follow a joint elliptical distribution (e.g. Gaussian, Student) parameterized by a correlation matrix the correlation matrix has a hierarchical block structure: Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 16. Hierarchical clustering algorithms - A taxonomy We consider Hierarchical Agglomerative Clustering algorithms. Such as single linkage, average linkage, Ward. Space contracting vs. Space conserving vs. Space dilating [2] D(t+1) C (t) i ∪ C (t) j , C (t) k ≤ min D (t) ik , D (t) jk D(t+1) C (t) i ∪ C (t) j , C (t) k ∈ min D (t) ik , D (t) jk , max D (t) ik , D (t) jk D(t+1) C (t) i ∪ C (t) j , C (t) k ≥ max D (t) ik , D (t) jk Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 17. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Simulations in the simpliﬁed setting Some inﬂuential parameters: clustering algorithm number of observations T number of variables N relative to T contrast between the correlations, and their values correlation estimator (e.g. Pearson, Spearman) 100 200 300 400 500 Sample size 0.0 0.2 0.4 0.6 0.8 1.0 Score Empirical rates of convergence for Single Linkage Gaussian - Pearson Gaussian - Spearman Student - Pearson Student - Spearman 100 200 300 400 500 Sample size 0.0 0.2 0.4 0.6 0.8 1.0 Score Empirical rates of convergence for Average Linkage Gaussian - Pearson Gaussian - Spearman Student - Pearson Student - Spearman 100 200 300 400 500 Sample size 0.0 0.2 0.4 0.6 0.8 1.0 Score Empirical rates of convergence for Ward Gaussian - Pearson Gaussian - Spearman Student - Pearson Student - Spearman Ratio of the number of correct clustering obtained over the number of trials as a function of T Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 18. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions A consistency proof & ﬁrst convergence bounds A 2-step proof. First step: Which geometrical conﬁgurations lead to the true clustering? For space-conserving algorithms (e.g. Single, Complete, Average Linkage), a suﬃcient separability condition reads max Dintra := max 1≤i,j≤N C(i)=C(j) d(Xi , Xj ) < min 1≤i,j≤N C(i)=C(j) d(Xi , Xj ) =: min Dinter Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 19. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions A consistency proof & ﬁrst convergence bounds A 2-step proof. Second step: How long does it take for the estimates of the correlation coeﬃcients to be precise enough to be with high probability in a good conﬁguration for the clustering algorithm? Answer: Concentration inequalities for correlation coeﬃcients. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 20. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Convergence bounds Combining both steps, we get the following convergence rate: Convergence rate The probability of the clustering algorithm making an error is O log N T . Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 21. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Proof. Step 1 - A bit more details By induction. Let’s assume the separability condition is satisﬁed at step t, then min D (t) intra ≤ max D (t) intra < min D (t) inter ≤ max D (t) inter From the space-conserving property, we get: D (t+1) intra ∈ min D (t) intra, max D (t) intra and D (t+1) inter ∈ min D (t) inter, max D (t) inter . separability condition is satisﬁed at t+1, the clustering algorithm has not linked points from two diﬀerent clusters between step t and step t + 1. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 22. Proof. Step 2 - A bit more details Maximum statistical error - (Marti and Andler, IJCAI 2016) For space conserving algorithm the separability condition is met if ˆΣ − Σ ∞ < minρi ,ρj |ρi − ρj | 2 , where C(i) = C(j). This means that the statistical error has to be below the minimum correlation ‘contrast’ between the clusters. Weaker the ‘contrast’, more precise the correlation estimates have to be. N.B. From Cram´er–Rao lower bound, we get for Pearson correlation estimator: var(ˆρ) ≥ 1 I(ρ) = (ρ − 1)2 (ρ + 1)2 3(ρ2 + 1) . When correlation is high, it is easier to estimate. (Marti, SSP 2016) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 23. Correlation estimates concentration bounds number of variables N, observations T, minimum separation d Concentration bounds [4] If Σ and ˆΣ are the population and empirical Spearman correlation matrices respectively, then for N ≥ 24 log T + 2, we have with probability at least 1 − 1 T2 , ˆΣ − Σ ∞ ≤ 24 log N T . P(“correct clustering”) ≥ 1 − 2N2 e−Td2/24 Not sharp enough for reasonable values of N, T, d. For example, for N = 500, T = 2500, d = 0.2, we obtain ≈ −7750. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 24. Future developments & open questions Bounds are not sharp enough. We can try to reﬁne them using: (theoretical) Intrinsic dimension of the HCBM model [16]; (theoretical Use PSD-ness to reﬁne the bounds for the matrix (theoretical/empirical) A distance between dendrograms (instead of correct/incorrect) for a ﬁner analysis; (empirical) A study of ‘correctness’ isoquants: Precise convergence rates of clustering methodologies can provide a useful model selection criterion for practitioners! Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 25. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients 1 Introduction to credit default swaps 2 About the consistency of clustering ﬁnancial time series 3 Design of distances and alternative dependence coeﬃcients Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients 4 Summary and open questions Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 26. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Motivations Not only correlation! Diﬀerent kinds of returns distributions may exist in the data. We may want to reﬁne ‘correlation’ clusters with this information. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 27. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients A (too) naive distance and its pitfalls Applying L2 directly on the time series mixes correlation and volatility. We are looking for a better representation so that a L2 is meaningful. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 28. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients A (too) naive distance and its pitfalls Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 29. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Revisting Sklar theorem (1959)... Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 30. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients and Deheuvels empirical copulas (1981) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 31. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients A novel representation for time series Basically, we transform each time series of returns to a (normalized ranks, square root of marginal density) vector. Applying a L2 between two of these vectors is now equivalent to a distance in Spearman correlation + Hellinger between the densities. cf. (Marti et al., 2016) [13] (Pattern Recognition Letters) for more details on this representation and the associated distance. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 32. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Analysing the diﬀerences - Using the Sankey diagram cf. (Marti et al., 2015) [14] (ICMLA 2015) for guidelines on how to compare several clustering methodologies for ﬁnancial time series. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 33. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Exploring and improving dependence coeﬃcients – Motivations We want to use dependence measures which are: robust to noise (not Pearson then) and preserve as much information as possible, so that clusterings are more stable; can be tuned to look for speciﬁc dependencies, e.g. tail-dependence or more exotic ones. As copulas are a convenient way to capture all the dependence between two variables, we aim at leveraging them. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 34. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Minimum, Independence, Maximum copulas Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 35. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Relation to existing dependence coeﬃcients Some dependence coeﬃcients can be readily expressed as: deviation from Fr´echet-Hoeﬀding bounds Spearman’s ρS = 1 − 6 [0,1]2 (ui − uj )2 dC(ui , uj ), Gini’s γ, Kendall distribution distance, deviation from independence ui uj Spearman ρS = 12 [0,1]2 (C(ui , uj ) − ui uj )dui duj , Copula MMD, Schweizer-Wolﬀ’s σ, Hoeﬀding’s φ2 Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 36. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Idea: Relative position of empirical copula wrt ‘targets’ and ‘forgets’ In a classical setting, we choose the positive and negative dependence copulas as ‘targets’, and the independence one as a ‘forget’ dependence. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 37. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Optimal Transport between empirical copulas cf. (Marti et al., 2016) [12] (ICASSP 2016) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 38. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Why choosing Optimal Transport over f -divergences? Distances between Gaussian copulas C1, C2, C3: cf. (Marti et al., 2016) [7] (SSP 2016) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 39. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Standard setting: TFDC vs. Spearman cf. (Marti et al., 2017) [8] (NIPS Time Series WS 2016) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 40. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Power of TFDC and state-of-the-art dependence measures Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 41. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Some applications of the Target/Forget Dependence Coeﬃcient Applications in non-standard settings: We can look for particular associations between random variables. cf. (Marti et al., 2017) [8] (NIPS Time Series 2016) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 42. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients Impact of diﬀerent coeﬃcients on clustering Diﬀerent results... Stability and empirical convergence rates may help for choosing one over the others. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 43. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions 1 Introduction to credit default swaps 2 About the consistency of clustering ﬁnancial time series 3 Design of distances and alternative dependence coeﬃcients Alternative representation and correlation+distribution distance Copula-based dependence coeﬃcients 4 Summary and open questions Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 44. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Summary of contributions The contribution of the PhD thesis: bring a greater focus on statistical reliability (convergence rates and consistency) [9] (IJCAI 2016) consider alternative representation and distances [13] (Pattern Recognition Letters), [8] (NIPS Time Series 2016) visualizations [10] and a framework to test for clustering stability [14] (ICMLA 2015) an extensive and regularly updated survey of the literature: https://arxiv.org/pdf/1703.00485.pdf [11] (350+ references) Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 45. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Perspectives and open questions “How many clusters?” What is multivariate correlation? How to use it for hierarchical clustering? Using several time series representing a given entity, and dependence between random vectors? Riemannian geometry of correlation matrices (not a totally geodesic submanifold of the well-explored manifold of covariances) Entities switching clusters: noise or signal? More precise results for (empirical) convergence rates? Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 46. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Mikolaj Binkowski, Gautier Marti, and Philippe Donnat. Autoregressive convolutional neural networks for asynchronous time series. arXiv preprint arXiv:1703.04122, 2017. Zhenmin Chen and John W Van Ness. Space-conserving agglomerative algorithms. Journal of classiﬁcation, 13(1):157–168, 1996. Laurent Laloux, Pierre Cizeau, Marc Potters, and Jean-Philippe Bouchaud. Random matrix theory and ﬁnancial correlations. International Journal of Theoretical and Applied Finance, 3(03):391–397, 2000. Han Liu, Fang Han, Ming Yuan, John Laﬀerty, Larry Wasserman, et al. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 47. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions High-dimensional semiparametric gaussian copula graphical models. The Annals of Statistics, 40(4):2293–2326, 2012. Burton G Malkiel and Eugene F Fama. Eﬃcient capital markets: A review of theory and empirical work. The journal of Finance, 25(2):383–417, 1970. Rosario N Mantegna. Hierarchical structure in ﬁnancial markets. The European Physical Journal B-Condensed Matter and Complex Systems, 11(1):193–197, 1999. Gautier Marti, S´ebastien Andler, Frank Nielsen, and Philippe Donnat. Optimal transport vs. ﬁsher-rao distance between copulas for clustering multivariate time series. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 48. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions In Statistical Signal Processing Workshop (SSP), 2016 IEEE, pages 1–5. IEEE, 2016. Gautier Marti, S´ebastien Andler, Frank Nielsen, and Philippe Donnat. Exploring and measuring non-linear correlations: Copulas, lightspeed transportation and clustering. In NIPS 2016 Time Series Workshop, pages 59–69, 2017. Gautier Marti, S´ebastien Andler, Frank Nielsen, and Philippe Donnat. Clustering ﬁnancial time series: How long is enough? 2016. Gautier Marti, Philippe Donnat, Frank Nielsen, and Philippe Very. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 49. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions HCMapper: An interactive visualization tool to compare partition-based ﬂat clustering extracted from pairs of dendrograms. arXiv preprint arXiv:1507.08137, 2015. Gautier Marti, Frank Nielsen, Mikolaj Bi´nkowski, and Philippe Donnat. A review of two decades of correlations, hierarchies, networks and clustering in ﬁnancial markets. arXiv preprint arXiv:1703.00485, 2017. Gautier Marti, Frank Nielsen, and Philippe Donnat. Optimal copula transport for clustering multivariate time series. In Acoustics, Speech and Signal Processing (ICASSP), 2016 IEEE International Conference on, pages 2379–2383. IEEE, 2016. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 50. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Gautier Marti, Philippe Very, and Philippe Donnat. Toward a generic representation of random variables for machine learning. arXiv preprint arXiv:1506.00976, 2015. Gautier Marti, Philippe Very, Philippe Donnat, and Frank Nielsen. A proposal of a methodological framework with experimental guidelines to investigate clustering stability on ﬁnancial time series. In 14th IEEE International Conference on Machine Learning and Applications, ICMLA 2015, Miami, FL, USA, December 9-11, 2015, pages 32–37, 2015. Marc Szafraniec, Gautier Marti, and Philippe Donnat. Putting self-supervised token embedding on the tables. arXiv preprint arXiv:1708.04120, 2017. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 51. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Joel A Tropp. An introduction to matrix concentration inequalities. arXiv preprint arXiv:1501.01571, 2015. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 52. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - CRLB for correlation Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 53. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - CRLB for correlation - Proof Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 54. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - Fisher-Rao geodesic distance Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 55. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - Optimal Transport distances Other transportation distances: regularized discrete optimal transport, Sinkhorn distances, etc. Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 56. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - Geometry of covariances Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 57. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - The standard methodology: Pearson + MST Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 58. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - The Target/Forget Dependence Coeﬃcient Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 59. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - The Copula Transform Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 60. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - The correlation + distribution distance Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 61. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - The correlation + distribution distance Gautier Marti Some contributions to the clustering of ﬁnancial time series
- 62. Introduction to credit default swaps About the consistency of clustering ﬁnancial time series Design of distances and alternative dependence coeﬃcients Summary and open questions Appendix - Pearson correlation Gautier Marti Some contributions to the clustering of ﬁnancial time series

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