Time Series Analysis and Mining with R

1. Time Series Analysis with R Yanchang Zhao http://www.RDataMining.com 30 September 2014 1 / 39

2. Outline Introduction Time Series Decomposition Time Series Forecasting Time Series Clustering Time Series Classi

3. cation Online Resources 2 / 39

4. Time Series Analysis with R 1 I time series data in R I time series decomposition, forecasting, clustering and classi

5. cation I autoregressive integrated moving average (ARIMA) model I Dynamic Time Warping (DTW) I Discrete Wavelet Transform (DWT) I k-NN classi

6. cation 1Chapter 8: Time Series Analysis and Mining, in book R and Data Mining: Examples and Case Studies. http://www.rdatamining.com/docs/RDataMining.pdf 3 / 39

7. R I a free software environment for statistical computing and graphics I runs on Windows, Linux and MacOS I widely used in academia and research, as well as industrial applications I 5,800+ packages (as of 13 Sept 2014) I CRAN Task View: Time Series Analysis http://cran.r-project.org/web/views/TimeSeries.html 4 / 39

8. Time Series Data in R I class ts I represents data which has been sampled at equispaced points in time I frequency=7: a weekly series I frequency=12: a monthly series I frequency=4: a quarterly series 5 / 39

9. Time Series Data in R a <- ts(1:20, frequency = 12, start = c(2011, 3)) print(a) ## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec ## 2011 1 2 3 4 5 6 7 8 9 10 ## 2012 11 12 13 14 15 16 17 18 19 20 str(a) ## Time-Series [1:20] from 2011 to 2013: 1 2 3 4 5 6 7 8 9 10... attributes(a) ## $tsp ## [1] 2011 2013 12 ## ## $class ## [1] "ts" 6 / 39

12. What is Time Series Decomposition To decompose a time series into components: I Trend component: long term trend I Seasonal component: seasonal variation I Cyclical component: repeated but non-periodic uctuations I Irregular component: the residuals 8 / 39

13. Data AirPassengers Data AirPassengers: monthly totals of Box Jenkins international airline passengers, 1949 to 1960. It has 144(=1212) values. plot(AirPassengers) Time AirPassengers 1950 1952 1954 1956 1958 1960 100 200 300 400 500 600 9 / 39

14. Decomposition apts - ts(AirPassengers, frequency = 12) f - decompose(apts) plot(f$figure, type = b) # seasonal figures 2 4 6 8 10 12 −40 −20 0 20 40 60 Index f$figure 10 / 39

15. Decomposition plot(f) observed 150 250 350 450 100 300 500 trend seasonal −40 0 20 40 60 −40 0 20 40 60 2 4 6 8 10 12 random Time Decomposition of additive time series 11 / 39

18. Time Series Forecasting I To forecast future events based on known past data I For example, to predict the price of a stock based on its past performance I Popular models I Autoregressive moving average (ARMA) I Autoregressive integrated moving average (ARIMA) 13 / 39

19. Forecasting # build an ARIMA model fit - arima(AirPassengers, order = c(1, 0, 0), list(order = c(2, 1, 0), period = 12)) fore - predict(fit, n.ahead = 24) # error bounds at 95% confidence level U - fore$pred + 2 * fore$se L - fore$pred - 2 * fore$se ts.plot(AirPassengers, fore$pred, U, L, col = c(1, 2, 4, 4), lty = c(1, 1, 2, 2)) legend(topleft, col = c(1, 2, 4), lty = c(1, 1, 2), c(Actual, Forecast, Error Bounds (95% Confidence))) 14 / 39

20. Forecasting 1950 1952 1954 1956 1958 1960 1962 Time 100 200 300 400 500 600 700 Actual Forecast Error Bounds (95% Confidence) 15 / 39

23. Time Series Clustering I To partition time series data into groups based on similarity or distance, so that time series in the same cluster are similar I Measure of distance/dissimilarity I Euclidean distance I Manhattan distance I Maximum norm I Hamming distance I The angle between two vectors (inner product) I Dynamic Time Warping (DTW) distance I ... 17 / 39

24. Dynamic Time Warping (DTW) DTW

25. nds optimal alignment between two time series. library(dtw) idx - seq(0, 2 * pi, len = 100) a - sin(idx) + runif(100)/10 b - cos(idx) align - dtw(a, b, step = asymmetricP1, keep = T) dtwPlotTwoWay(align) Index Query value 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 18 / 39

26. Synthetic Control Chart Time Series I The dataset contains 600 examples of control charts synthetically generated by the process in Alcock and Manolopoulos (1999). I Each control chart is a time series with 60 values. I Six classes: I 1-100 Normal I 101-200 Cyclic I 201-300 Increasing trend I 301-400 Decreasing trend I 401-500 Upward shift I 501-600 Downward shift I http://kdd.ics.uci.edu/databases/synthetic_control/synthetic_ control.html 19 / 39

27. Synthetic Control Chart Time Series # read data into R sep='': the separator is white space, i.e., one # or more spaces, tabs, newlines or carriage returns sc - read.table(./data/synthetic_control.data, header = F, sep = ) # show one sample from each class idx - c(1, 101, 201, 301, 401, 501) sample1 - t(sc[idx, ]) plot.ts(sample1, main = ) 20 / 39

28. Six Classes 1 24 26 28 30 32 34 36 101 15 20 25 30 35 40 45 25 30 35 40 45 0 10 20 30 40 50 60 201 Time 301 0 10 20 30 401 25 30 35 40 45 10 15 20 25 30 35 0 10 20 30 40 50 60 501 Time 21 / 39

29. Hierarchical Clustering with Euclidean distance # sample n cases from every class n - 10 s - sample(1:100, n) idx - c(s, 100 + s, 200 + s, 300 + s, 400 + s, 500 + s) sample2 - sc[idx, ] observedLabels - rep(1:6, each = n) # hierarchical clustering with Euclidean distance hc - hclust(dist(sample2), method = ave) plot(hc, labels = observedLabels, main = ) 22 / 39

30. Hierarchical Clustering with Euclidean distance 20 40 60 80 100 120 140 4 6 66 1 22 11 11 11 1 11 66 6 6 6 66 4 4 44 44 4 44 33 33 33 5 5 55 55 33 33 55 55 22 2 22 2 22 hclust (*, average) dist(sample2) Height 23 / 39

31. Hierarchical Clustering with Euclidean distance # cut tree to get 8 clusters memb - cutree(hc, k = 8) table(observedLabels, memb) ## memb ## observedLabels 1 2 3 4 5 6 7 8 ## 1 10 0 0 0 0 0 0 0 ## 2 0 3 1 1 3 2 0 0 ## 3 0 0 0 0 0 0 10 0 ## 4 0 0 0 0 0 0 0 10 ## 5 0 0 0 0 0 0 10 0 ## 6 0 0 0 0 0 0 0 10 24 / 39

32. Hierarchical Clustering with DTW Distance myDist - dist(sample2, method = DTW) hc - hclust(myDist, method = average) plot(hc, labels = observedLabels, main = ) # cut tree to get 8 clusters memb - cutree(hc, k = 8) table(observedLabels, memb) ## memb ## observedLabels 1 2 3 4 5 6 7 8 ## 1 10 0 0 0 0 0 0 0 ## 2 0 4 3 2 1 0 0 0 ## 3 0 0 0 0 0 6 4 0 ## 4 0 0 0 0 0 0 0 10 ## 5 0 0 0 0 0 0 10 0 ## 6 0 0 0 0 0 0 0 10 25 / 39

33. Hierarchical Clustering with DTW Distance 0 200 400 600 800 1000 3 33 3 33 55 33 33 5 55 5 55 55 66 6 46 4 44 44 44 44 66 66 66 11 111 11 111 22 22 22 22 22 hclust (*, average) myDist Height 26 / 39

36. Time Series Classi

37. cation Time Series Classi

38. cation I To build a classi

39. cation model based on labelled time series I and then use the model to predict the lable of unlabelled time series Feature Extraction I Singular Value Decomposition (SVD) I Discrete Fourier Transform (DFT) I Discrete Wavelet Transform (DWT) I Piecewise Aggregate Approximation (PAA) I Perpetually Important Points (PIP) I Piecewise Linear Representation I Symbolic Representation 28 / 39

40. Decision Tree (ctree) ctree from package party classId - rep(as.character(1:6), each = 100) newSc - data.frame(cbind(classId, sc)) library(party) ct - ctree(classId ~ ., data = newSc, controls = ctree_control(minsplit = 20, minbucket = 5, maxdepth = 5)) 29 / 39

41. Decision Tree pClassId - predict(ct) table(classId, pClassId) ## pClassId ## classId 1 2 3 4 5 6 ## 1 100 0 0 0 0 0 ## 2 1 97 2 0 0 0 ## 3 0 0 99 0 1 0 ## 4 0 0 0 100 0 0 ## 5 4 0 8 0 88 0 ## 6 0 3 0 90 0 7 # accuracy (sum(classId == pClassId))/nrow(sc) ## [1] 0.8183 30 / 39

42. DWT (Discrete Wavelet Transform) I Wavelet transform provides a multi-resolution representation using wavelets. I Haar Wavelet Transform { the simplest DWT http://dmr.ath.cx/gfx/haar/ I DFT (Discrete Fourier Transform): another popular feature extraction technique 31 / 39

43. DWT (Discrete Wavelet Transform) # extract DWT (with Haar filter) coefficients library(wavelets) wtData - NULL for (i in 1:nrow(sc)) f a - t(sc[i, ]) wt - dwt(a, filter = haar, boundary = periodic) wtData - rbind(wtData, unlist(c(wt@W, wt@V[[wt@level]]))) g wtData - as.data.frame(wtData) wtSc - data.frame(cbind(classId, wtData)) 32 / 39

44. Decision Tree with DWT ct - ctree(classId ~ ., data = wtSc, controls = ctree_control(minsplit=20, minbucket=5, maxdepth=5)) pClassId - predict(ct) table(classId, pClassId) ## pClassId ## classId 1 2 3 4 5 6 ## 1 98 2 0 0 0 0 ## 2 1 99 0 0 0 0 ## 3 0 0 81 0 19 0 ## 4 0 0 0 74 0 26 ## 5 0 0 16 0 84 0 ## 6 0 0 0 3 0 97 (sum(classId==pClassId)) / nrow(wtSc) ## [1] 0.8883 33 / 39

45. plot(ct, ip_args = list(pval = F), ep_args = list(digits = 0)) 1 V57 £ 117 117 2 W43 £ −4 −4 3 W5 £ −8 −8 Node 4 (n = 68) 123456 1 0.8 0.6 0.4 0.2 0 Node 5 (n = 6) 123456 1 0.8 0.6 0.4 0.2 0 6 W31 £ −6 −6 Node 7 (n = 9) 123456 1 0.8 0.6 0.4 0.2 0 Node 8 (n = 86) 123456 1 0.8 0.6 0.4 0.2 0 9 V57 £ 140 140 Node 10 (n = 31) 123456 1 0.8 0.6 0.4 0.2 0 11 V57 £ 178 178 12 W22 £ −6 −6 Node 13 (n = 80) 123456 1 0.8 0.6 0.4 0.2 0 14 W31 £ −13 −13 Node 15 (n = 9) 123456 1 0.8 0.6 0.4 0.2 0 Node 16 (n = 99) 123456 1 0.8 0.6 0.4 0.2 0 17 W31 £ −15 −15 Node 18 (n = 12) 123456 1 0.8 0.6 0.4 0.2 0 19 W43 £ 3 3 Node 20 (n = 103) 123456 1 0.8 0.6 0.4 0.2 0 Node 21 (n = 97) 123456 1 0.8 0.6 0.4 0.2 0 34 / 39

46. k-NN Classi

47. cation I

48. nd the k nearest neighbours of a new instance I label it by majority voting I needs an ecient indexing structure for large datasets k - 20 newTS - sc[501, ] + runif(100) * 15 distances - dist(newTS, sc, method = DTW) s - sort(as.vector(distances), index.return = TRUE) # class IDs of k nearest neighbours table(classId[s$ix[1:k]]) ## ## 4 6 ## 3 17 35 / 39

49. k-NN Classi

50. cation I

51. nd the k nearest neighbours of a new instance I label it by majority voting I needs an ecient indexing structure for large datasets k - 20 newTS - sc[501, ] + runif(100) * 15 distances - dist(newTS, sc, method = DTW) s - sort(as.vector(distances), index.return = TRUE) # class IDs of k nearest neighbours table(classId[s$ix[1:k]]) ## ## 4 6 ## 3 17 Results of majority voting: class 6 35 / 39

52. The TSclust Package I TSclust: a package for time seriesclustering 2 I measures of dissimilarity between time series to perform time series clustering. I metrics based on raw data, on generating models and on the forecast behavior I time series clustering algorithms and cluster evaluation metrics 2http://cran.r-project.org/web/packages/TSclust/ 36 / 39

55. Online Resources I Chapter 8: Time Series Analysis and Mining, in book R and Data Mining: Examples and Case Studies http://www.rdatamining.com/docs/RDataMining.pdf I R Reference Card for Data Mining http://www.rdatamining.com/docs/R-refcard-data-mining.pdf I Free online courses and documents http://www.rdatamining.com/resources/ I RDataMining Group on LinkedIn (7,000+ members) http://group.rdatamining.com I RDataMining on Twitter (1,700+ followers) @RDataMining 38 / 39

56. The End Thanks! Email: yanchang(at)rdatamining.com 39 / 39

Time Series Analysis and Mining with R

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