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# A closer look at correlations

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# A closer look at correlations

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You may have already read many times that the job of a Data Scientist is to skim through a huge amount of data searching for correlations between some variables of interest. And also, that one of his worst enemies (besides correlation doesn't imply causation) is spurious correlation. But what really is correlation? Are there several types of correlations? Some "good", some "bad"? What about their estimation? This talk will be a very visual presentation around the notion of correlation and dependence. I will first illustrate how the standard linear correlation is estimated (Pearson coefficient), then some more robust alternative: the Spearman coefficient. Building on the geometric understanding of their nature, I will present a generalization that can help Data Scientists to explore, interpret, and measure the dependence (not necessarily linear or comonotonic) between the variables of a given dataset. Financial time series (stocks, credit default swaps, fx rates), and features from the UCI datasets are considered as use cases.

You may have already read many times that the job of a Data Scientist is to skim through a huge amount of data searching for correlations between some variables of interest. And also, that one of his worst enemies (besides correlation doesn't imply causation) is spurious correlation. But what really is correlation? Are there several types of correlations? Some "good", some "bad"? What about their estimation? This talk will be a very visual presentation around the notion of correlation and dependence. I will first illustrate how the standard linear correlation is estimated (Pearson coefficient), then some more robust alternative: the Spearman coefficient. Building on the geometric understanding of their nature, I will present a generalization that can help Data Scientists to explore, interpret, and measure the dependence (not necessarily linear or comonotonic) between the variables of a given dataset. Financial time series (stocks, credit default swaps, fx rates), and features from the UCI datasets are considered as use cases.

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### A closer look at correlations

1. 1. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion A closer look at correlations Paris Machine Learning Meetup #3 Season 4 G. Marti, S. Andler, F. Nielsen, P. Donnat HELLEBORECAPITAL November 9, 2016 Gautier Marti A closer look at correlations
2. 2. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
3. 3. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion What is correlation? E[Xi Xj ] − E[Xi ]E[Xj ] (E[X2 i ] − E[Xi ]2)(E[X2 j ] − E[Xj ]2) ∈ [−1, 1] N k=1(xik − xi )(xjk − xj ) N k=1(xik − xi )2 N k=1(xjk − xj )2 ∈ [−1, 1] import numpy as np np.corrcoef(x_i,x_j) Gautier Marti A closer look at correlations
4. 4. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
5. 5. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
6. 6. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Pearson correlation Gautier Marti A closer look at correlations
7. 7. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Pearson correlation Gautier Marti A closer look at correlations
8. 8. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Pearson correlation Gautier Marti A closer look at correlations
9. 9. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Pearson correlation Gautier Marti A closer look at correlations
10. 10. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Pearson correlation Gautier Marti A closer look at correlations
11. 11. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Pearson correlation Gautier Marti A closer look at correlations
12. 12. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Pearson correlation with outliers Gautier Marti A closer look at correlations
13. 13. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
14. 14. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Spearman correlation: Pearson on ranks Gautier Marti A closer look at correlations
15. 15. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Spearman correlation: Pearson on ranks Gautier Marti A closer look at correlations
16. 16. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Spearman correlation: Pearson on ranks Gautier Marti A closer look at correlations
17. 17. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Spearman correlation: Pearson on ranks Gautier Marti A closer look at correlations
18. 18. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Spearman correlation: Pearson on ranks Gautier Marti A closer look at correlations
19. 19. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Spearman correlation: Pearson on ranks Gautier Marti A closer look at correlations
20. 20. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Pearson correlation coeﬃcient Spearman correlation coeﬃcient Spearman correlation with outliers Gautier Marti A closer look at correlations
21. 21. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
22. 22. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
23. 23. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC From ranks to empirical copula Sklar’s Theorem  For (Xi , Xj ) having continuous marginal cdfs FXi , FXj , its joint cumulative distribution F is uniquely expressed as F(Xi , Xj ) = C(FXi (Xi ), FXj (Xj )), where C is known as the copula of (Xi , Xj ). Gautier Marti A closer look at correlations
24. 24. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC Minimum, Independence, Maximum copulas Fr´echet–Hoeﬀding copula bounds For any copula C : [0, 1]2 → [0, 1] and any (u, v) ∈ [0, 1]2 the following bounds hold: W(u, v) ≤ C(u, v) ≤ M(u, v), where W is the copula for counter-monotonic random variables, and M is the copula for co-monotonic random variables. 0 0.5 1 ui 0 0.5 1 uj w(ui,uj) 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0 0.5 1 ui 0 0.5 1 uj W(ui,uj) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.5 1 ui 0 0.5 1 uj π(ui,uj) 0.00036 0.00037 0.00038 0.00039 0.00040 0.00041 0.00042 0.00043 0.00044 0 0.5 1 ui 0 0.5 1 uj Π(ui,uj) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.5 1 ui 0 0.5 1 uj m(ui,uj) 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0 0.5 1 ui 0 0.5 1 uj M(ui,uj) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Gautier Marti A closer look at correlations
25. 25. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
26. 26. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC A metric space for copulas Gautier Marti A closer look at correlations
27. 27. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC A metric space for copulas Gautier Marti A closer look at correlations
28. 28. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC Which metric? (Regularized) Optimal Transport Distance is the minimum cost of transportation to transform one pile of dirt into another one, i.e. the amount of dirt moved times the distance by which it is moved. EMD = |x1 − x2| EMD = 1 6|x1 − x3| + 1 6|x2 − x3| Gautier Marti A closer look at correlations
29. 29. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC Which metric? (Regularized) Optimal Transport Its geometry has good properties in general , and for copulas . 0 0.5 1 0 0.5 1 0.0000 0.0015 0.0030 0.0045 0.0060 0.0075 0.0090 0.0105 0.0120 0 0.5 1 0 0.5 1 0.0000 0.0015 0.0030 0.0045 0.0060 0.0075 0.0090 0.0105 0.0120 0 0.5 1 0 0.5 1 0.0000 0.0015 0.0030 0.0045 0.0060 0.0075 0.0090 0.0105 0.0120 0 0.5 1 0 0.5 1 0.0000 0.0015 0.0030 0.0045 0.0060 0.0075 0.0090 0.0105 0.0120 0 0.5 1 0 0.5 1 Bregman barycenter copula 0.0000 0.0008 0.0016 0.0024 0.0032 0.0040 0.0048 0.0056 0 0.5 1 0 0.5 1 Wasserstein barycenter copula 0.0000 0.0004 0.0008 0.0012 0.0016 0.0020 0.0024 0.0028 0.0032 Gautier Marti A closer look at correlations
30. 30. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC A metric space for copulas Gautier Marti A closer look at correlations
31. 31. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC A metric space for copulas Gautier Marti A closer look at correlations
32. 32. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC A metric space for copulas Gautier Marti A closer look at correlations
33. 33. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC A metric space for copulas Gautier Marti A closer look at correlations
34. 34. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
35. 35. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC The Target/Forget Dependence Coeﬃcient (TFDC) Gautier Marti A closer look at correlations
36. 36. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC The Target/Forget Dependence Coeﬃcient (TFDC) Now, we can deﬁne our bespoke dependence coeﬃcient: Build the forget-dependence copulas {CF l }l Build the target-dependence copulas {CT k }k Compute the empirical copula Cij from xi , xj TFDC(Cij ) = minl D(CF l , Cij ) minl D(CF l , Cij ) + mink D(Cij , CT k ) Gautier Marti A closer look at correlations
37. 37. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC TFDC Power 0.00.20.40.60.81.0 xvals power.cor[typ,] xvals power.cor[typ,] 0.00.20.40.60.81.0 xvals power.cor[typ,] xvals power.cor[typ,] cor dCor MIC ACE MMD CMMD RDC TFDC 0.00.20.40.60.81.0 xvals power.cor[typ,] xvals power.cor[typ,] 0 20 40 60 80 100 0.00.20.40.60.81.0 xvals power.cor[typ,] 0 20 40 60 80 100 xvals power.cor[typ,] Noise Level Power Figure: Power of several dependence coeﬃcients as a function of the noise level in eight diﬀerent scenarios. Insets show the noise-free form of each association pattern. The coeﬃcient power was estimated via 500 simulations with sample size 500 each. Gautier Marti A closer look at correlations
38. 38. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
39. 39. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
40. 40. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC Clustering of empirical copulas Gautier Marti A closer look at correlations
41. 41. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC Financial correlations - Stocks CAC 40 Figure: Stocks: More mass in the bottom-left corner, i.e. lower tail dependence. Stock prices tend to plummet together. Gautier Marti A closer look at correlations
42. 42. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC Financial correlations - Credit Default Swaps Figure: Credit default swaps: More mass in the top-right corner, i.e. upper tail dependence. Insurance cost against entities’ default tends to soar in stressed market. Gautier Marti A closer look at correlations
43. 43. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC Financial correlations - FX rates Figure: FX rates: Empirical copulas show that dependence between FX rates are various. For example, rates may exhibit either strong dependence or independence while being anti-correlated during extreme events. Gautier Marti A closer look at correlations
44. 44. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC Associations between features in UCI datasets Dependence patterns (= clustering centroids) found between features in UCI datasets Breast Cancer (wdbc) 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 Libras Movement 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 Parkinsons 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 Gamma Telescope 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 Gautier Marti A closer look at correlations
45. 45. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
46. 46. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Explore the correlations with clustering Query your dataset about correlations with TFDC The Art of formulating questions about correlations Encode your dependence hypothesis as a copula, and your query as a “k-NN search”. Gautier Marti A closer look at correlations
47. 47. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion 1 Introduction 2 Standard correlation coeﬃcients Pearson correlation coeﬃcient Spearman correlation coeﬃcient 3 A metric space for copulas On the importance of the normalization Which metric? (Regularized) Optimal Transport A customizable dependence coeﬃcient: TFDC 4 Applications Explore the correlations with clustering Query your dataset about correlations with TFDC 5 Conclusion Gautier Marti A closer look at correlations
48. 48. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Summary Designing data-driven tailored correlation coeﬃcients Gautier Marti A closer look at correlations
49. 49. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Take Home Message Gautier Marti A closer look at correlations
50. 50. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Internships at Hellebore If you are interested by an internship at Hellebore in applied machine learning for Finance (NLP, Text Classiﬁcation, Information Extraction), please contact: stage@helleboretech.com in ML/Finance research (copulas, bayesian inference, clustering, time series analysis), please contact: gmarti@helleborecapital.com Gautier Marti A closer look at correlations
51. 51. HELLEBORECAPITAL Introduction Standard correlation coeﬃcients A metric space for copulas Applications Conclusion Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, pages 2292–2300, 2013. Gautier Marti, S´ebastien Andler, Frank Nielsen, and Philippe Donnat. Optimal transport vs. ﬁsher-rao distance between copulas for clustering multivariate time series. In IEEE Statistical Signal Processing Workshop, SSP 2016, Palma de Mallorca, Spain, June 26-29, 2016, pages 1–5, 2016. A Sklar. Fonctions de r´epartition `a n dimensions et leurs marges. Universit´e Paris 8, 1959. Gautier Marti A closer look at correlations