Correlation Analysis on Live Data Streams
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The document discusses the evolution and applications of correlation analysis in various domains such as finance, biology, and smart cities, highlighting the significance of live data streaming and multimodal correlation methods. It addresses challenges such as data inconsistency, noise, and scalability while presenting various correlation techniques like Pearson and Spearman methods. Additionally, it covers advancements in self-learning machines and the importance of online anomaly detection within streaming data.
![CORRELATION MATRIX
16
* Figure borrowed from [Mueen et al. 2010]
*
Thresholded(=0.5)CorrelationMatrix
350x350CorrelationMatrix](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-16-320.jpg)
![STOCHASTICITY
20
Why bother? Multiple flavors
If X, Y are independent, then 𝜌(X, Y) = 0
However, the converse is not true, as only
the first two moments are considered
e.g., 𝜌(X, Y) = 0 even when Y = X2
Not invariant under nonlinear strictly
increasing transformations
Pearson’s correlation only recognizes linear
dependence
var(X) and var(Y) have to be finite
Non-stationary random processes
Stochastic correlation process
Applications
Finance (Brownian motion), biology
Time-varying correlation
Rolling correlation - lagged indicator
Dynamic Conditional Correlation [Engle’02]
Local Correlation [Langnau’09]
Wishart autoregressive process [Gourieroux’09]
Transformations [van Emmerich’06]
arctan and
Modified Jacobi process [Ma’09]
tanh transformation [Teng’16]](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-20-320.jpg)
![SPURIOUS CORRELATION
* Figure borrowed from [Anscombe 1973]
*
WHY CONTEXT IS IMPORTANT?
22
Identical r=0.816
Clearly spurious
Linear correlation is perhaps not the right metic
Identical summary statistics](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-22-320.jpg)
![24
ROBUSTNESS
*
* Figure borrowed from “Introduction to Robust Estimation and Hypothesis Testing” by Rand. R. Wilcox
Sensitivity to anomalies
Linear relationship between x & y but r = -0.21
PEARSON CORRELATION
Quadrant (signum) correlation coefficient#
# [Blomqvist, 1950] ^[Pasman and Shevlyakov, 1987]
Correlation median estimator^](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-24-320.jpg)
![STREAMING CORRELATION
Wide Spectrum of Approaches
Sliding Windows, Damped Windows
Reduction
Smoothing
Down sampling
DFT [Agrawal et al. 1993, Zhu and Shasha 2002,
Qiu et al. 2018]
DWT [Chan and Fu 1999, Popivanov & Miller 2002]
PCA, SVD
PAA [Faloutsos and Yi 2000]
APAC [Chakrabarti et al. 2001]
Random projections [Grellmann et al. 2016]
LSH [Sundaram et al. 2013]
DATA SKETCHES
31](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-31-320.jpg)
![Bursts
Page views, Clicks, Retweets
Correlated burstiness (e.g., data center operations)
Root-cause analysis
STREAMING CORRELATION
* Figure borrowed from [Sakurai et al. 2005]
*
Lag between time series
Lagged correlation/Cross-correlation
ACTIONABLE INSIGHTS
32
[Zhu and Shasha 2002, Levine et al. 2016, Wu et al. 2017]
[Vlachos et al. 2008, Kotov et al. 2011, Shafer et al. 2012, Kusmierczyk and Norvag 2015]](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-32-320.jpg)
![PREDICTION MOTION
35
Human Motion/Robotics
[Bütepage et al. 2017, Martinez et a al. 2017, Byravan & Fox]
Traffic Prediction
[Hermes et al. 2010, Walker et al. 2014, Yu et al. 2017]
360-Degree Video (AR/VR)
[Bao et al. 2017, Vishwanath et al., 2017]
Potpourri: Deep Learning Based Approaches
[Oh et al. 2015, Mathieu et al. 2016, Liang et al. 2017]](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-35-320.jpg)
![40
AUDIO-VIDEO-TEXT
Speaker identification in multi-speaker scenarios
[Huang & Kingsbury 2013, Chung & Zisserman 2016, Torfi et al. 2017]
Cross-Modal Correlation Learning in Audio and Lyrics
[Yu et al. 2017, Tang et al. 2017]
Speech enhancement
[Xu et al. 2014, Hou et al. 2017, Kolbæk et al. 2017]
Action Recognition and Video Highlight Detection
[Wu et al. 2013, Sun et al. 2013, Takahashi et al. 2017]
DEEP LEARNING WAY
Emotion Recognition
[Tzirakis et al. 2017, Pini et al. 2017]](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-40-320.jpg)
![43
HETEROSCEDASTICITY
Methods
(Adjusted) Percentile Bootstrap [Wilcox 1996]
Nested Bootstrap [DiCiccio et al. 1992]
Challenge: Detecting non-linear correlation
2001](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-43-320.jpg)
![44
Potential sources: Event based monitoring via sensors, Occlusion in a video
Techniques: Resampling
Lomb-Scargle Fourier Transform [Andersson 2007, Rehfeld et al. 2011]
Kernel - such as Laplacian, Gaussian - based methods
IRREGULARLY SPACED TIME SERIES](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-44-320.jpg)
![TIME VARYING CORRELATION
45
* Figure borrowed from [Fu et al. 2013]
TVCC: Time Varying Correlation Coefficient
fMRI time-courses of 4 Regions of Interests (ROI)
✦ Time varying joint distribution
✦ Parameter non-constancy/Instability
F Test [Chow’60]
SupF Test [Quandt’60]
Lagrange Multiplier Test
[Nabeya and Tanaka’88, Nylom’89]
✦ Co-integrated processes
I(1) [Hansen’90]
✦ Stochastic vs. Deterministic
✦ Dynamic Conditional Correlation (DCC)
[Engle’02]
*
Post stimulus period with significant difference
(p<0.05) w.r.t. pre-stimulus interval](https://image.slidesharecdn.com/correlationanalysisstratanyc2018-180914172532/85/Correlation-Analysis-on-Live-Data-Streams-45-320.jpg)
Correlation Analysis on Live Data Streams
- 1. 1 CORRELATION ANALYSIS ON LIVE DATA STREAMS ARUN KEJARIWAL
- 2. 2 STREAMING Serve data at rest LIVE Serve data as it is being generated STREAMING vs. LIVE
- 3. MOBILE DATA TRAFFIC 17% of total IP traffic by 2021 VR/AR Traffic CAGR of 82% from 2016-2021 ANNUAL GLOBAL IP TRAFFIC 3.3 ZB by 2021 BROADBAND SPEEDS Reach 53 Mbps by 2021 INTERNET VIDEO SURVEILLANCE 3.4% of all Internet video traffic by 2021 LIVE INTERNET VIDEO 13% of Internet video traffic by 2021 THE NUMBERS 3 https://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/complete-white-paper-c11-481360.html
- 4. 4 DATA FUSION CHARACTERISTICS DISTRIBUTED HETEROGENEOUS NON-LINEAR INTERNET OF THINGS Sensors and Actuators SMART CITIES SMART HEALTH CARE POLLUTON MONITORING INSIGHTS PROBABILISTIC METHODS THEORY OF EVIDENCE MACHINE LEARNING FUZZY LOGIC CHALLENGES DATA: INCONSISTENCY MISSINGNESS, NOISE LATENCY BATTERY CONSUMPTION INTELLIGENT DECISION MAKING
- 5. 5 TRAFFIC SURVEILLANCE CONGESTION DETECTION ACCIDENT DETECTION HEALTH EMERGENCY RISK ALERT RAIN WATER CLOGGING, HYDROPLANING MUDSLIDE AN EXAMPLE APPLICATION OF DATA FUSION
- 7. CORRELATION ANALYSIS A LONG HISTORY! 7
- 8. EARLY FLAVOR DATES BACK to 1885! 8
- 9. APPLICATION DOMAINS 9 BIOLOGY ANTHROPLOGY AGRICULTURE PHILOSOPHY PSYCHOLOGY FINANCE ECONOMETRICS STATISTICS NETWORKING OPERATIONS
- 10. Common representation learning Example: Across audio and video modalities in a single live stream Multimodal Linear/non-linear correlation Example: Across univariate time series of operations data Unimodal CORRELATION ANALYSIS FLAVORS: AT A HIGH LEVEL 10
- 11. CROSS-MODAL ANALYSIS Joint Representation of Text- Audio-Video INTERPRETABILITY Comparative analysis of deep representations AUTHENTICITY Live streams vs. Recording OBJECT IDENTIFICATION Multiple cameras MULTI-STREAM SYNCHRONIZATION Different vantage points Unreliable timestamps LOCALIZATION Video ⟷ Audio Audio ⟷ Text CORRELATION ANALYSIS WHY BOTHER? 11
- 12. Pearson Spearman 𝜌 Kendall 𝜏 Goodman and Kruskal COMMON TYPES OF CORRELATION 12 Ties are dropped
- 13. CORRELATION ANALYSIS A REAL LIFE EXAMPLE 13 Root cause analysis Expose investment avenues Surface optimization opportunities Risk minimization Medical diagnosis Learning
- 15. 15 Symmetric Lack of context Spurious correlations Non-actionable Network topology CORRELATION MATRIX Scalability Hundreds of millions of time series Use cases Multiple Regression Discriminant Analysis Mahalanobis Distance
- 16. CORRELATION MATRIX 16 * Figure borrowed from [Mueen et al. 2010] * Thresholded(=0.5)CorrelationMatrix 350x350CorrelationMatrix
- 17. MULTIPLE CORRELATION 1957 MULTIVARIATE CASE 17 Proportion of variance of xj that cannot be explained by other independent variables Depends on the choice of the dependent variable Correlation Matrix 2-D case
- 18. 18 Time Varying
- 19. ROLLING CORRELATION CONSTANTLY EVOLVING DATA 19
- 20. STOCHASTICITY 20 Why bother? Multiple flavors If X, Y are independent, then 𝜌(X, Y) = 0 However, the converse is not true, as only the first two moments are considered e.g., 𝜌(X, Y) = 0 even when Y = X2 Not invariant under nonlinear strictly increasing transformations Pearson’s correlation only recognizes linear dependence var(X) and var(Y) have to be finite Non-stationary random processes Stochastic correlation process Applications Finance (Brownian motion), biology Time-varying correlation Rolling correlation - lagged indicator Dynamic Conditional Correlation [Engle’02] Local Correlation [Langnau’09] Wishart autoregressive process [Gourieroux’09] Transformations [van Emmerich’06] arctan and Modified Jacobi process [Ma’09] tanh transformation [Teng’16]
- 22. SPURIOUS CORRELATION * Figure borrowed from [Anscombe 1973] * WHY CONTEXT IS IMPORTANT? 22 Identical r=0.816 Clearly spurious Linear correlation is perhaps not the right metic Identical summary statistics
- 23. * ROBUSTNESS r =0.8 r =0.2 r is highly sensitive to slight change, as measured by Kolmogorov distance, in one of the marginal distributions Bivariate Normal Distribution Contaminated Normal Distribution PEARSON CORRELATION 23 * Figure borrowed from “Introduction to Robust Estimation and Hypothesis Testing” by Rand. R. Wilcox Influence Function (IF) of Pearson’s Correlation Unbounded Pearson’s correlation does not have infinitesimal robustness x x y y zz Recall, first order approximation of sample IF of r is: r-i : correlation coefficient based on all but the ith observation
- 24. 24 ROBUSTNESS * * Figure borrowed from “Introduction to Robust Estimation and Hypothesis Testing” by Rand. R. Wilcox Sensitivity to anomalies Linear relationship between x & y but r = -0.21 PEARSON CORRELATION Quadrant (signum) correlation coefficient# # [Blomqvist, 1950] ^[Pasman and Shevlyakov, 1987] Correlation median estimator^
- 25. ROBUSTNESS PEARSON CORRELATION 25 Based on Robust Principal Variables Alternatives
- 26. ROBUSTNESS PEARSON CORRELATION 26 Percentage Bend Correlation
- 28. Live Data
- 29. STREAMING CORRELATION Incremental, One pass CHARACTERISTICS 29 Other correlation measures are not amenable to incremental computation Applications Security Correlation power analysis
- 31. STREAMING CORRELATION Wide Spectrum of Approaches Sliding Windows, Damped Windows Reduction Smoothing Down sampling DFT [Agrawal et al. 1993, Zhu and Shasha 2002, Qiu et al. 2018] DWT [Chan and Fu 1999, Popivanov & Miller 2002] PCA, SVD PAA [Faloutsos and Yi 2000] APAC [Chakrabarti et al. 2001] Random projections [Grellmann et al. 2016] LSH [Sundaram et al. 2013] DATA SKETCHES 31
- 32. Bursts Page views, Clicks, Retweets Correlated burstiness (e.g., data center operations) Root-cause analysis STREAMING CORRELATION * Figure borrowed from [Sakurai et al. 2005] * Lag between time series Lagged correlation/Cross-correlation ACTIONABLE INSIGHTS 32 [Zhu and Shasha 2002, Levine et al. 2016, Wu et al. 2017] [Vlachos et al. 2008, Kotov et al. 2011, Shafer et al. 2012, Kusmierczyk and Norvag 2015]
- 34. PREDICTION MOTION 1952 “… prediction motion is the continuation of tracking motion after a target disappears from view.” 1955 1962 Unlike duration of target presentation, target speed exerts an influence on prediction accuracy. EXPLORED >5 DECADES BACK! 34
- 35. PREDICTION MOTION 35 Human Motion/Robotics [Bütepage et al. 2017, Martinez et a al. 2017, Byravan & Fox] Traffic Prediction [Hermes et al. 2010, Walker et al. 2014, Yu et al. 2017] 360-Degree Video (AR/VR) [Bao et al. 2017, Vishwanath et al., 2017] Potpourri: Deep Learning Based Approaches [Oh et al. 2015, Mathieu et al. 2016, Liang et al. 2017]
- 36. AUDIO-VIDEO CORRELATION A LONG HISTORY: EXPERIMENTAL PSYCHOLOGY -> DEEP LEARNING 36
- 37. AUDIO-VIDEO CORRELATION 1952 People utilize visual and postural experiences in perceiving the position of an object in the field, of the whole field. 1897 Localization of sounds varied, being different when the source of sound was in sight from what it was when this was out of sight, and also in the latter case differing with different directions of attention, or with different suggestions as to the direction from which the sound came. 1941 SOUND LOCALIZATION: AN APPLICATION 37
- 38. AUDIO-VIDEO CORRELATION 1977 1960 1976 LIP READING: AN APPLICATION 38 Demonstrated the influence of vision on speech perception McGurk and MacDonald Established the relationship of the visually perceived symbols to the underlying linguistic system Use visual information as an aid when white noise made speech difficult to hear 1954 There’s a great opportunity for the visual contribution at low speech-to-noise ratios
- 39. AUDIO-VIDEO CORRELATION 2007 2004 Combined acoustic and visual feature vectors to distinguish live synchronous audio-video recordings from replay attacks that use audio with a still photo. LIVENESS AND SYNCHRONIZATION: AN APPLICATION 39 Extracted the correlated components of audio and lip features based on Canonical Correlation Analysis. 2009 Showed that there exists a relationship between perception of video presented in screen and accompanying audio signals, both stereo and spatial
- 40. 40 AUDIO-VIDEO-TEXT Speaker identification in multi-speaker scenarios [Huang & Kingsbury 2013, Chung & Zisserman 2016, Torfi et al. 2017] Cross-Modal Correlation Learning in Audio and Lyrics [Yu et al. 2017, Tang et al. 2017] Speech enhancement [Xu et al. 2014, Hou et al. 2017, Kolbæk et al. 2017] Action Recognition and Video Highlight Detection [Wu et al. 2013, Sun et al. 2013, Takahashi et al. 2017] DEEP LEARNING WAY Emotion Recognition [Tzirakis et al. 2017, Pini et al. 2017]
- 41. 41 OPEN PROBLEMS
- 42. 42 Online anomaly detection: speed-accuracy trade-off On the Runtime-Efficacy Trade-off of Anomaly Detection Techniques for Real-Time Streaming Data* by Choudhary et al. 2017 Breakouts/Changepoints Skew in location of anomalies SUSCEPTIBILITY TO ANOMALIES * https://arxiv.org/pdf/1710.04735.pdf
- 43. 43 HETEROSCEDASTICITY Methods (Adjusted) Percentile Bootstrap [Wilcox 1996] Nested Bootstrap [DiCiccio et al. 1992] Challenge: Detecting non-linear correlation 2001
- 44. 44 Potential sources: Event based monitoring via sensors, Occlusion in a video Techniques: Resampling Lomb-Scargle Fourier Transform [Andersson 2007, Rehfeld et al. 2011] Kernel - such as Laplacian, Gaussian - based methods IRREGULARLY SPACED TIME SERIES
- 45. TIME VARYING CORRELATION 45 * Figure borrowed from [Fu et al. 2013] TVCC: Time Varying Correlation Coefficient fMRI time-courses of 4 Regions of Interests (ROI) ✦ Time varying joint distribution ✦ Parameter non-constancy/Instability F Test [Chow’60] SupF Test [Quandt’60] Lagrange Multiplier Test [Nabeya and Tanaka’88, Nylom’89] ✦ Co-integrated processes I(1) [Hansen’90] ✦ Stochastic vs. Deterministic ✦ Dynamic Conditional Correlation (DCC) [Engle’02] * Post stimulus period with significant difference (p<0.05) w.r.t. pre-stimulus interval
- 46. STREAMING CORRELATION Missing data Packets being dropped owing to, say, unexpected high traffic Data collection: Every, say, 5 seconds How to scale analysis to milli-seconds granularity? Unequal length time series Different sampling rates Small samples Bootstrapping Low SNR (Signal to Noise Ratio) 46
- 47. NEXT FRONTIER LEVERAGING MULTIMODAL & CONTINUOUSLY EVOLVING CORRELATION FOR SELF-LEARNING 47
- 48. SELF-LEARNING MACHINES A LONG HISTORY!
- 49. 49 SELF-LEARNING EARLY WORK: GAMES 19501914 constructed a device which played an end game of king and rook against king. The machine played the side with king and rook and would force checkmate in a few moves however its human opponent played. Gerald Tesauro 199519592002
- 50. 2002 2017 Amongst the first and most famous was the chess-playing automaton constructed in 1769 by Baron Kempelen … 1953 Alan Turing 2012 1970 GAME PLAYING POTPOURRI 50 19962007 The game of checkers has roughly 500 billion billion possible positions (5 × 1020)
- 51. SELF-LEARNING 2016 REINFORCEMENT LEARNING 51 2016201720172018 AlphaGo Mastering the game of Go with deep neural networks and tree search by Silver et al. AlphaGo Zero Mastering the game of Go without Human knowledge by Silver et al. Alpha Zero Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm by Silver et al. Libratus Superhuman AI for heads-up no-limit poker: Libratus beats top professionals by Brown and Sandholm
- 54. 54 Rank Correlation Methods by Kendall and Gibbons READINGS Correlation and Regression by Bobko Correlation by Garson Robust Correlation: Theory and Applications by Shevlyakov and Oja A Mathematical Theory of Evidence by Shafer BOOKS
- 55. 55 Fast Approximate Correlation of Massive Time-series Data by Queen et al., 2010 READINGS Local Correlation Detection with Linearity Enhancement in Streaming Data by Xie et al., 2013 Fast Distributed Correlation Discovery Over Streaming Time-Series Data by Guo et al., 2015 Random Projection of Fast and Efficient Multivariate Correlation Analysis of High-Dimensional Data: A New Approach by Grellmann et al., 2016 Detection of Highly Correlated Live Data Streams by Alseghayer et al., 2017 STREAMING CORRELATION
- 56. 56 Perception of body position and of the position of the visual field by Witkin, 1949 READINGS Autocorrelation, a principle for the evaluation of sensory information by the nervous system by Reichardt, 1961 EARLY RESEARCH IN AUDIO-VISUAL CORRELATION Binocular cross-correlation in time and space by Tyler and Julesz, 1978 Neurontropy. an entropy like measure of neural correlation by Julesz and Tyler, 1976 Cross correlation of sensory stimuli and electroencephalogram by Morgan, 1969
- 57. 57 Patch to the future: Unsupervised visual prediction by Walker et al., 2014 READINGS Motion-Prediction-based Multicast for 360-Degree Video Transmissions by Bao et al., 2017 Dual Motion GAN for Future-Flow Embedded Video Prediction by Liang et al., 2017 Deep representation learning for human motion prediction and classification by Bütepage et al., 2017 On human motion prediction using recurrent neural networks by Martinez, 2017 RECENT WORKS IN PREDICTION MOTION
- 58. 58 On Deep Multi-View Representation Learning by Wang et al., 2015 READINGS Correlational Neural Networks by Chandar et al., 2017 Common Representation Learning Using Step-based Correlation Multi-Modal CNN by Bhatt et al., 2017 Objects that Sound by Arandjelović and Zisserman, 2017 Deep Correlation Feature Learning for Face Verification in the Wild by Deng et al., 2017 COMMON REPRESENTATION LEARNING
- 59. 59 READINGS Audiovisual Synchronization and Fusion Using Canonical Correlation Analysis by Sargin et al., 2007 The predictive power of trajectory motion by Watamaniuk, 2005 Seeing motion behind occluders by Watamaniuk and McKee, 1995 Temporal Coherence Theory For The Detection And Measurement Of Visual Motion by Grzywacz et al.,1995 Probabilistic Motion Estimation Based On Temporal Coherence by Burgi et al., 2000 AUDIO-VISUAL RESEARCH
- 60. 60 A New Method of Audio-Visual Correlation Analysis by Kunka and Kostek, 2009 READINGS Uncertainty in ontologies: Dempster–Shafer theory for data fusion applications by Bellenger and Gatepaille, 2011 City Data Fusion: Sensor Data Fusion in the Internet of Things by Wang et al., 2015 Data Fusion and IoT for Smart Ubiquitous Environments: A Survey by Alam et al., 2017 Correlation Analysis of Audio and Video Contents: A Metadata based Approach by Algur et al., 2015 POTPOURRI
- 61. 61 READINGS POTPOURRI Correlation detection as a general mechanism for multisensory integration by Parise and Ernst, 2016 Origin of information-limiting noise correlations by Kanitscheidera et al., 2015 Temporal structure and complexity affect audio-visual correspondence detection by Denison et al., 2013 Correlation versus causation in multisensory perception by Mitterer, Jesse, 2010
- 62. 62 READINGS MATHEMATICS A Bayesian approach to problems in stochastic estimation and control by Ho and Lee, 1964 A Generalization of Bayesian Inference by Dempster, 1968 Multidimensional Scaling by Kruskal and Wish, 1978