Abstract
This article analyzes how Big Data is changing the way we learn from observations. We describe the changes in statistical methods in seven areas that have been shaped by the Big Data-rich environment: the emergence of new sources of information; visualization in high dimensions; multiple testing problems; analysis of heterogeneity; automatic model selection; estimation methods for sparse models; and merging network information with statistical models. Next, we compare the statistical approach with those in computer science and machine learning and argue that the convergence of different methodologies for data analysis will be the core of the new field of data science. Then, we present two examples of Big Data analysis in which several new tools discussed previously are applied, as using network information or combining different sources of data. Finally, the article concludes with some final remarks.
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References
Aghabozorgi S, Shirkhorshidi AS, Wah TY (2015) Time-series clustering—a decade review. Inform Syst 53:16–38
Akaike H (1973) Information theory and an extension of the maximum likelihood method. In: Petrov N, Caski F (eds) Proceeding of the 2nd symposium on information theory. Academiai Kiado, Budapest, pp 267–281
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723
Alonso A, Peña D (2018) Clustering time series by linear dependency. Stat Comput. https://doi.org/10.1007/s11222-018-9830-6
Ando T, Bai J (2017) Clustering huge number of financial time series: a panel data approach with high-dimensional predictors and factor structures. J Am Stat Assoc 112(519):1182–1198
Arlot S, Celisse A (2010) A survey of cross-validation procedures for model selection. Stat Surv 4:40–79
Arribas-Gil A, Romo J (2014) Shape outlier detection and visualization for functional data: the outliergram. Biostatistics 15(4):603–619
Asimov D (1985) The grand tour: a tool for viewing multidimensional data. SIAM J Sci Stat Comp 6:128–143
Bai J, Ng S (2002) Determining the number of factors in approximate factor models. Econometrica 70(1):191–221
Bailey TC, Sapatinas T, Powell KJ, Krzanowski WJ (1998) Signal detection in underwater sound using wavelets. J Am Stat Assoc 93:73–83
Banfield JD, Raftery AE (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics 49:803–821
Barabási AL (2016) Network Science. Cambridge University Press, Cambridge
Barber RF, Candès EJ (2015) Controlling the false discovery rate via knockoffs. Ann Stat 43(5):2055–2085
Basu S, Michailidis G (2015) Regularized estimation in sparse high-dimensional time series models. Ann Stat 43:1535–1567
Benito M, García-Portugués E, Marron JS, Peña D (2017) Distance-weighted discrimination of face images for gender classification. Stat 6(1):231–240
Benjamini Y (2010) Discovering the false discovery rate. J R Stat Soc B 72(4):405–416
Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc B 57(1):289–300
Bergmeir C, Benítez JM (2012) On the use of cross-validation for time series predictor evaluation. Inf Sci 191:192–213
Bertini E, Tatu A, Keim D (2011) Quality metrics in high-dimensional data visualization: an overview and systematization. IEEE Trans Vis Comput Graph 17:2203–2212
Besag J (1986) On the statistical analysis of dirty pictures. J R Stat Soc B 48(3):259–302
Bickel PJ, Levina E (2008) Regularized estimation of large covariance matrices. Ann Stat 36(1):199–227
Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exp. https://doi.org/10.1088/1742-5468/2008/10/P10008
Bouveyron C, Brunet-Saumard C (2014) Model-based clustering of high-dimensional data: a review. Comput Stat Data Anal 71:52–78
Box GEP, Tiao GC (1968) A bayesian approach to some outlier problems. Biometrika 55(1):119–129
Breiman L (2001) Statistical modeling: the two cultures (with comments and a rejoinder by the author). Stat Sci 16:199–231
Breiman L, Friedman J, Olshen R, Stone C (1984) Classification and regression trees. Chapman and Hall/CRC, New York
Brockwell SE, Gordon IR (2001) A comparison of statistical methods for meta-analysis. Stat Med 20:825–840
Bühlmann P, van de Geer S (2011) Statistics for high-dimensional data: methods, theory and applications. Springer, Berlin, Heidelberg
Bühlmann P, van de Geer S (2018) Statistics for big data: a perspective. Stat Prob Lett 136:37–41
Bühlmann P, Drineas P, Kane M, van der Laan M (2016) Handbook of big data. Chapman and Hall/CRC, Boca Raton
Cai TT (2017) Global testing and large-scale multiple testing for high-dimensional covariance structures. Annu Rev Stat Appl 4:423–446
Cai TT, Liu W (2011) Adaptive thresholding for sparse covariance matrix estimation. J Am Stat Assoc 106:672–684
Cai TT, Liu W (2016) Large-scale multiple testing of correlations. J Am Stat Assoc 111:229–240
Cai TT, Zhuo HH (2012) Optimal rates of convergence for sparse covariance matrix estimation. Ann Stat 40(5):2389–2420
Cai TT, Liu W, Luo X (2011) A constrained \(\ell _{1}\) minimization approach to sparse precision matrix estimation. J Am Stat Assoc 106:594–607
Caiado J, Maharaj EA, D’urso P (2015) Time series clustering. In: Handbook of cluster analysis, CRC Press, pp 241–264
Cairo A (2016) The truthful art: data, charts, and maps for communication. New Riders
Candès E, Tao T (2006) Near-optimal signal recovery from random projections: universal encoding strategies. IEEE Trans Inf Theory 52:5406–5425
Candès E, Romberg JK, Tao T (2006) Stable signal recovery from incomplete and inaccurate measurements. Commun Pure Appl Math 52:1207–1223
Candès E, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J ACM 58(3):11
Candès EJ, Fan Y, Janson L, Lv J (2016) Panning for gold: model-free knockoffs for high-dimensional controlled variable selection. Technical report, May 2016, Department of Statistics, Stanford University
Cao R (2017) Ingenuas reflexiones de un estadístico en la era del big data. Bol de Estad e Investig Oper 33(3):295–321
Carmichael I, Marron JS (2018) Data science vs. statistics: two cultures? Jpn J Stat Data Sci 1(1):117–138
Cerioli A, Farcomeni A, Riani M (2013) Robust distances for outlier-free goodness-of-fit testing. Comput Stat Data Anal 65:29–45
Chen CP, Zhang CY (2014) Data-intensive applications, challenges, techniques and technologies: a survey on big data. Inform Sci 275:314–347
Chen H, De P, Hu YJ, Hwang BH (2014) Wisdom of crowds: the value of stock opinions transmitted through social media. Rev Financ Stud 27(5):1367–1403
Chen J, Chen Z (2008) Extended Bayesian information criteria for model selection with large model spaces. Biometrika 95(3):759–771
Chernozhukov V, Galichon A, Hallin M, Henry M (2017) Monge–Kantorovich depth, quantiles, ranks and signs. Ann Stat 45(1):223–256
Cook RD (2018) An introduction to envelopes: dimension reduction for efficient estimation in multivariate statistics. Wiley, New York
Cook D, Buja A, Cabrera J, Hurley C (1995) Grand tour and projection pursuit. J Comput Graph Stat 4:155–172
Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20(3):273–297
Cover TM, Hart PE (1967) Nearest neighbour pattern classification. IEEE Trans Inform Theory 13:21–27
Cuesta-Albertos JA, Gordaliza A, Matrán C (1997) Trimmed k-means: an attempt to robustify quantizers. Ann Stat 25(2):553–576
Cuevas A (2014) A partial overview of the theory of statistics with functional data. J Stat Plan Inference 147:1–23
Domingos P, Pazzani M (1997) On the optimality of the simple Bayesian classifier under zero-one loss. Mach Learn 29:103–130
Donoho D (2006a) Compressed sensing. IEEE Trans Inf Theory 52:1289–1306
Donoho D (2006b) For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution. Commun Pure Appl Math 59:797–829
Donoho D (2017) 50 years of data science. J Comput Graph Stat 26(4):745–766
Dryden IL, Hodge DJ (2018) Journeys in big data statistics. Stat Prob Lett 136:121–125
Efron B, Hastie T (2016) Computer age statistical inference. Cambridge University Press, Cambridge
Evergreen SDH (2016) Effective data visualization: the right chart for the right data. SAGE Publications
Faith J, Mintram R, Angelova M (2006) Targeted projection pursuit for visualizing gene expression data classifications. Bioinformatics 22:2667–2673
Fan J, Han F, Liu H (2014) Challenges of big data analysis. Natl Sci Rev 1(2):293–314
Forni M, Hallin M, Lippi M, Reichlin L (2005) The generalized dynamic factor model: one-sided estimation and forecasting. J Am Stat Assoc 100:830–840
Fraiman R, Justel A, Svarc M (2008) Selection of variables for cluster analysis and classification rules. J Am Stat Assoc 103:1294–1303
Friedman J, Hastie T, Tibshirani R (2008) Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3):432–441
Frühwirth-Schnatter S (2006) Finite mixture and Markov switching models. Springer, New York
Galeano P, Peña D (2019) Outlier detection in high-dimensional time series (Unpublished manuscript)
Galeano P, Peña D, Tsay RS (2006) Outlier detection in multivariate time series by projection pursuit. J Am Stat Assoc 101:654–669
Galimberti G, Manisi A, Soffritti G (2017) Modelling the role of variables in model-based cluster analysis. Stat Comput 28(1):1–25
Gandomi A, Haider M (2015) Beyond the hype: big data concepts, methods, and analytics. Int J of Inf Manage 35(2):137–144
García-Ferrer A, Highfield RA, Palm F, Zellner A (1987) Macroeconomic forecasting using pooled international data. J Bus Econ Stat 5:53–67
Geisser S (1975) The predictive sample reuse method with applications. J Am Stat Assoc 70:320–328
Genton MG (2001) Classes of kernels for machine learning: a statistics perspective. J Mach Learn Res 2:299–312
Genton MG, Johnson C, Potter K, Stenchikov G, Sun Y (2014) Surface boxplots. Stat 3(1):1–11
Genton MG, Castruccio S, Crippa P, Dutta S, Huser R, Sun Y, Vettori S (2015) Visuanimation in statistics. Stat 4(1):81–96
Giannone D, Reichlin L, Small D (2008) Nowcasting: the real-time informational content of macroeconomic data. J Monet Econ 55:665–676
Gómez V, Maravall A (1996) Programas tramo and seats. Documento de Trabajo, Banco de España SGAPE-97001
Guhaniyogi R, Dunson DB (2015) Bayesian compressed regression. J Am Stat Assoc 110:1500–1514
Hall P, Marron JS, Neeman A (2005) Geometric representation of high dimension, low sample size data. J R Stat Soc B 67(3):427–444
Härdle WK, Lu HHS, Shen X (2018) Handbook of big data analytics. Springer
Hastie T, Pregibon D (1992) Generalized linear models. In: Chambers JM, Hastie TJ (eds) Statistical models in S, Chap 6. Wadsworth & Brooks/Cole
Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning: data mining, inference, and prediction. Springer, New York
Hastie T, Tibshirani R, Wainwright M (2015) Statistical learning with sparsity: the lasso and generalizations. Chapman and Hall/CRC, Boca Raton
Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12:55–67
Hornik K (1991) Approximation capabilities of multilayer feedforward networks. Neural Netw 4:251–257
Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35(1):73–101
Hyvärinen A, Oja E (2000) Independent component analysis: algorithms and applications. Neural Netw 13:411–430
Irizarry RA (2001) Local harmonic estimation in musical sound signals. J Am Stat Assoc 96:357–367
Jain AK (1989) Fundamentals of digital image processing. Prentice Hall, Englewood Cliffs, NJ
James W, Stein C (1961) Estimation with quadratic loss. In: Proceedings of 4th Berkeley symposium on mathematical statistics and probability, vol I, University of California Press, pp 361–379
Johnstone IM, Titterington DM (2009) Statistical challenges of high-dimensional data. Philos Trans R Soc A 367:4237–4253
Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53:457–481
Kaufman L, Rousseeuw PJ (1990) Finding groups in data: an introduction to cluster analysis. Wiley, New York
Kokoszka P, Reimherr M (2017) Introduction to functional data analysis. Chapman and Hall/CRC, Boca Raton
Kolaczyk ED (2009) Statistical analysis of network data. Springer, New York
Kriegel HP, Kröger P, Zimek A (2009) Clustering high-dimensional data: a survey on subspace clustering, pattern-based clustering, and correlation clustering. ACM Trans Knowl Discov Data 3(1):1
Lam XY, Marron JS, Sun D, Toh KC (2018) Fast algorithms for large-scale generalized distance weighted discrimination. J Comput Graph Stat 27(2):368–379
Lauritzen SL (1996) Graphical Models. Oxford University Press Inc., New York
LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521:436–444
Liu W (2013) Gaussian graphical model estimation with false discovery rate control. Ann Stat 41(6):2948–2978
López-Pintado S, Romo J (2009) On the concept of depth for functional data. J Am Stat Assoc 104:718–734
Lu X, Marron JS, Haaland P (2014) Object-oriented data analysis of cell images. J Am Stat Assoc 109:548–559
MacQueen J (1967) Some methods for classification and analysis of multivariate observations. Proceedings of the 5th Berkeley symposium on mathematical statistics and probability vol 1, pp 281–297
Majumdar A (2009) Image compression by sparse PCA coding in curvelet domain. Signal Image Video Process 3:27–34
Maronna RA, Martin RD, Yohai V, Salibián-Barrera M (2019) Robust statistics: theory and methods (with R), 2nd edn. Wiley, Hoboken, NJ
Meinshausen N, Bühlmann P (2006) High dimensional graphs and variable selection with the lasso. Ann Stat 34(3):1436–1462
Mosteller F, Wallace DL (1963) Inference in an authorship problem: a comparative study of discrimination methods applied to the authorship of the disputed federalist papers. J Am Stat Assoc 58:275–309
Munzner T (2014) Visualization analysis and design. Chapman and Hall/CRC, Boca Raton
Norets A (2010) Approximation of conditional densities by smooth mixtures of regressions. Ann Stat 38(3):1733–1766
de Oliveira MF, Levkowitz H (2003) From visual data exploration to visual data mining: a survey. IEEE Trans Vis Comput Graph 9:378–394
Pan W, Shen X (2007) Penalized model-based clustering with application to variable selection. J Mach Learn Res 8:1145–1164
Pang B, Lee L (2008) Opinion mining and sentiment analysis. Found Trends Inf Retr 2:1–135
Paradis L, Han Q (2007) A survey of fault management in wireless sensor networks. J Netw Syst Manag 15:171–190
Peña D (2014) Big data and statistics: trend or change. Bol de Estad e Investig Oper 30:313–324
Peña D, Box GEP (1987) Identifying a simplifying structure in time series. J Am Stat Assoc 82:836–843
Peña D, Poncela P (2004) Forecasting with nonstationary dynamic factor models. J Econom 119(2):291–321
Peña D, Prieto FJ (2001a) Cluster identification using projections. J Am Stat Assoc 96:1433–1445
Peña D, Prieto FJ (2001b) Robust covariance matrix estimation and multivariate outlier detection. Technometrics 43:286–310
Peña D, Sánchez I (2005) Multifold predictive validation in armax time series models. J Am Stat Assoc 100:135–146
Peña D, Tiao GC, Tsay RS (2001) A course in time series analysis. Wiley, Hoboken, NJ
Peña D, Viladomat J, Zamar R (2012) Nearest-neighbors medians clustering. Stat Anal Data Min 5(4):349–362
Peña D, Smucler E, Yohai VJ (2019a) Forecasting multiple time series with one-sided dynamic principal components. J Am Stat Assoc. https://doi.org/10.1080/01621459.2018.1520117
Peña D, Tsay RS, Zamar R (2019b) Empirical dynamic quantiles for visualization of high-dimensional time series. Technometrics. https://doi.org/10.1080/00401706.2019.1575285
Pigoli D, Hadjipantelis PZ, Coleman JS, Aston JAD (2018) The statistical analysis of acoustic phonetic data: exploring differences between spoken romance languages (with discussion). J R Stat Soc C 67:1–27
Quijano-Sánchez L, Liberatore F (2017) The big chase: a decision support system for client acquisition applied to financial networks. Decis Support Syst 98:49–58
Rabiner LR (1989) A tutorial on hidden Markov models and selected applications in speech recognition. Proc IEEE 77:257–286
Radke RJ, Andra S, Al-Kofahi O, Roysam B (2005) Image change detection algorithms: a systematic survey. IEEE Trans Image Process 14:294–307
Raftery AE, Dean N (2006) Variable selection for model-based clustering. J Am Stat Assoc 101:168–178
Ramsay JO, Silverman BW (2005) Functional data analysis, 2nd edn. Springer, New York
Ren Z, Sun T, Zhang CH, Zhou HH (2015) Asymptotic normality and optimalities in estimation of large gaussian graphical model. Ann Stat 43(3):991–1026
Riani M, Atkinson AC, Cerioli A (2009) Finding an unknown number of multivariate outliers. J R Stat Soc B 71(2):447–466
Riani M, Atkinson AC, Cerioli A (2012) Problems and challenges in the analysis of complex data: static and dynamic approaches. In: di Ciaccio A, Coli M, Angulo JM (eds) Advanced statistical methods for the analysis of large data-sets. Springer, Berlin, Heidelberg, pp 145–157
Rosenblatt F (1958) The perceptron: a probabilistic model for information storage and organization in the brain. Psychol Rev 65(6):386–408
Rousseeuw P, van den Bossche W (2018) Detecting deviating data cells. Technometrics 60(2):135–145
Ryan TP, Woodall WH (2005) The most-cited statistical papers. J Appl Stat 32(5):461–474
Samuel AL (1959) Some studies in machine learning using the game of checkers. IBM J Res Dev 3:210–229
Schölkopf B, Smola A, Müller KR (1997) Kernel principal component analysis. In: Gerstner W, Germond A, Hasler M, Nicoud JD (eds) Artificial Neural Networks ICANN’97, vol 1327. Lecture Notes in Computer Science, pp 583–588
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464
Sesia M, Sabatti C, Candès EJ (2018) Gene hunting with knockoffs for hidden Markov models. Biometrika. https://doi.org/10.1093/biomet/asy033
Shao J (1993) Linear model selection by cross-validation. J Am Stat Assoc 88:486–494
Shen H, Huang JZ (2008) Sparse principal component analysis via regularized low rank matrix approximation. J Multivariate Anal 99(6):1015–1034
Shi JQ, Choi R (2011) Gaussian process regression analysis for functional data. CRC Press, Boca Raton
Small C (1990) A survey of multidimensional medians. Int Stat Rev 58:263–277
Stock JH, Watson MW (2002) Forecasting using principal components from a large number of predictors. J Am Stat Assoc 97:1167–1179
Stone M (1974) Cross-validatory choice and assessment of statistical predictions. J R Stat Soc B 36(2):111–147
Stone M (1977) An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. J R Stat Soc B 39(1):44–47
Sun Y, Genton MG (2011) Functional boxplots. J Comput Graph Stat 20(2):316–334
Tausczik YR, Pennebaker JW (2010) The psychological meaning of words: Liwc and computerized text analysis methods. J Lang Soc Psychol 29:24–54
Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc B 12:267–288
Tong H (2012) Threshold models in non-linear time series analysis. Springer, New York
Tong H, Lim KS (1980) Threshold autoregression, limit cycles and cyclical data (with discussion). J R Stat Soc B 42(3):245–292
Torrecilla JL, Romo J (2018) Data learning from big data. Stat Prob Lett 136:15–19
Tsay RS, Chen R (2018) Nonlinear time series analysis. Wiley, Hoboken, NJ
Tukey JW (1970) Exploratory data analysis. Addison-Wesley Pub, Co, Reading, MA
Tzeng JY, Byerley W, Devlin B, Roeder K, Wasserman L (2003) Outlier detection and false discovery rates for whole-genome DNA matching. J Am Stat Assoc 98:236–246
Vidal R (2011) Subspace clustering. IEEE Signal Proc Mag 28:52–68
Wang S, Zhu J (2008) Variable selection for model-based high-dimensional clustering and its application to microarray data. Biometrics 64:440–448
Wei F, Tian W (2018) Heterogeneous connection effects. Stat Prob Lett 133:9–14
Witten DM, Tibshirani R (2010) A framework for feature selection in clustering. J Am Stat Assoc 105:713–726
Witten DM, Tibshirani R, Hastie T (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10(3):515–534
Xia Y, Cai T, Cai TT (2016) Testing differential networks with applications to detecting gene-by-gene interactions. Biometrika 102:247–266
Yang Y (2005) Can the strengths of aic and bic be shared? A conflict between model identification and regression estimation. Biometrika 92:937–950
Zhang P (1993) Model selection via multifold cross validation. Ann Stat 21(1):299–313
Zhao SD, Cai TT, Li H (2014) Direct estimation of differential networks. Biometrika 101:253–268
Zhou Z, Wu WB (2009) Local linear quantile estimation for nonstationary time series. Ann Stat 37:2696–2729
Zhu X, Pan R, Li G, Liu Y, Wang H (2017) Network vector autoregression. Ann Stat 45(3):1096–1123
Acknowledgements
The invitation to write this article came from the editor Jesús López-Fidalgo and we are very grateful to him for his encouragement. The applications presented in this paper were carried out with Federico Liberatore, Lara Quijano-Sánchez and Carlo Sguera, post-docs at the UC3M-BS Institute of Financial Big Data. Iván Blanco and Jose Luis Torrecilla, also post-docs in the Institute, have also contributed with useful discussions. The ideas in this article have been clarified with the comments of Andrés Alonso, Anibal Figueiras, Rosa Lillo, Juan Romo and Rubén Zamar. To all them, our gratitude.
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This research has been supported by Grant ECO2015-66593-P of MINECO/FEDER/UE.
This invited paper is discussed in comments available at: https://doi.org/10.1007/s11749-019-00639-5, https://doi.org/10.1007/s11749-019-00640-y, https://doi.org/10.1007/s11749-019-00641-x, https://doi.org/10.1007/s11749-019-00642-w, https://doi.org/10.1007/s11749-019-00643-9, https://doi.org/10.1007/s11749-019-00644-8, and https://doi.org/10.1007/s11749-019-00646-6, https://doi.org/10.1007/s11749-019-00647-5, https://doi.org/10.1007/s11749-019-00648-4.
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Galeano, P., Peña, D. Data science, big data and statistics. TEST 28, 289–329 (2019). https://doi.org/10.1007/s11749-019-00651-9
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DOI: https://doi.org/10.1007/s11749-019-00651-9
Keywords
- Machine learning
- Sparse model selection
- Statistical learning
- Network analysis
- Multivariate data
- Time series