Forecasting: principles and practice. Rob J Hyndman 4 Hierarchical forecasting

Transcription

1 Forecasting: principles and practice Rob J Hyndman 4 Hierarchical forecasting 1

2 Outline 1 Hierarchical and grouped time series 2 Optimal forecast reconciliation 3 hts package for R 4 Application: Australian tourism 5 Lab Session 5 2

3 Forecasting the PBS 3

4 ATC drug classification A Alimentary tract and metabolism B Blood and blood forming organs C Cardiovascular system D Dermatologicals G Genito-urinary system and sex hormones H Systemic hormonal preparations, excluding sex hormones and insulins J Anti-infectives for systemic use L Antineoplastic and immunomodulating agents M Musculo-skeletal system N Nervous system P Antiparasitic products, insecticides and repellents R Respiratory system S Sensory organs V Various 4

5 ATC drug classification 14 classes A Alimentary tract and metabolism 84 classes A10 Drugs used in diabetes A10B Blood glucose lowering drugs A10BA Biguanides A10BA02 Metformin 5

6 Australian tourism 6

7 Australian tourism Quarterly data on visitor night from 1998:Q1 2013:Q4 From: National Visitor Survey, based on annual interviews of 120,000 Australians aged 15+, collected by Tourism Research Australia. Split by 7 states, 27 zones and 76 regions (a geographical hierarchy) Also split by purpose of travel Holiday Visiting friends and relatives (VFR) Business Other 304 bottom-level series 6

8 Spectacle sales Monthly UK sales data from Provided by a large spectacle manufacturer Split by brand (26), gender (3), price range (6), materials (4), and stores (600) About 1 million bottom-level series 7

9 Hierarchical time series A hierarchical time series is a collection of several time series that are linked together in a hierarchical structure. Total A B C AA AB AC BA BB BC CA CB CC 8

10 Hierarchical time series A hierarchical time series is a collection of several time series that are linked together in a hierarchical structure. Total A B C AA AB AC BA BB BC CA CB CC Examples Pharmaceutical sales Tourism demand by state and region 8

11 Grouped time series A grouped time series is a collection of time series that can be grouped together in a number of non-hierarchical ways. Total Total A B X Y AX AY BX BY AX BX AY BY 9

12 Grouped time series A grouped time series is a collection of time series that can be grouped together in a number of non-hierarchical ways. Total Total A B X Y AX AY BX BY AX BX AY BY Examples Spectacle sales by brand, gender, stores, etc. Tourism by state and purpose of travel 9

13 The problem 1 How to forecast time series at all nodes such that the forecasts add up in the same way as the original data? 2 Can we exploit relationships between the series to improve the forecasts? 10

14 The problem 1 How to forecast time series at all nodes such that the forecasts add up in the same way as the original data? 2 Can we exploit relationships between the series to improve the forecasts? The solution 1 Forecast all series at all levels of aggregation using an automatic forecasting algorithm. (e.g., ets, auto.arima,...) 2 Reconcile the resulting forecasts so they add up correctly using least squares optimization (i.e., find closest reconciled forecasts to the original forecasts). 3 This is available in the hts package in R. 10

15 Outline 1 Hierarchical and grouped time series 2 Optimal forecast reconciliation 3 hts package for R 4 Application: Australian tourism 5 Lab Session 5 11

16 Hierarchical time series Total A B C 12

17 Hierarchical time series Total A B C y t : observed aggregate of all series at time t. y X,t : observation on series X at time t. b t : vector of all series at bottom level in time t. 12

18 Hierarchical time series Total A B C y t : observed aggregate of all series at time t. y X,t : observation on series X at time t. b t : vector of all series at bottom level in time t. y t = y t y A,t y B,t y C,t = Y A,t y B,t y C,t 12

19 Hierarchical time series Total A B C y t : observed aggregate of all series at time t. y X,t : observation on series X at time t. b t : vector of all series at bottom level in time t. y t = y t y A,t y B,t y C,t = } {{ } S y A,t y B,t y C,t }{{} b t y t = Sb t 12

20 Hierarchical time series Total A B C AX AY AZ BX BY BZ CX CY CZ 13

21 Hierarchical time series Total A B C y t = AX AY AZ BX BY BZ CX CY CZ y t y A,t y B,t y AX,t y C,t y AX,t y AY,t y AZ,t y AY,t y BX,t y AZ,t = y y BX,t BY,t y BZ,t y BY,t y y BZ,t CX,t y CY,t y CX,t y CZ,t y CY,t }{{} y CZ,t b t }{{} 13 S

22 Hierarchical time series Total A B C y t = AX AY AZ BX BY BZ CX CY CZ y t y A,t y B,t y AX,t y C,t y AX,t y AY,t y AZ,t y AY,t y BX,t y AZ,t = y y BX,t BY,t y BZ,t y BY,t y y BZ,t CX,t y CY,t y CX,t y CZ,t y CY,t }{{} y t = Sb t y CZ,t b t }{{} 13 S

23 Grouped time series AX AY A BX BY B X Y Total 14

24 Grouped time series AX AY A BX BY B y t = X Y Total y t y A,t y B,t y X,t y AX,t y Y,t = y AY,t y AX,t y BX,t y AY,t y BY,t }{{} y BX,t b t y BY,t }{{} S 14

25 Grouped time series AX AY A BX BY B y t = X Y Total y t y A,t y B,t y X,t y AX,t y Y,t = y AY,t y AX,t y BX,t y AY,t y BY,t }{{} y BX,t b t y BY,t }{{} S y t = Sb t 14

26 Hierarchical and grouped time series Every collection of time series with aggregation constraints can be written as where y t = Sb t y t is a vector of all series at time t b t is a vector of the most disaggregated series at time t S is a summing matrix containing the aggregation constraints. 15

27 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. 16

28 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. (In general, they will not add up.) 16

29 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. (In general, they will not add up.) Reconciled forecasts must be of the form: ỹ n (h) = SPŷ n (h) for some matrix P. 16

30 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. (In general, they will not add up.) Reconciled forecasts must be of the form: ỹ n (h) = SPŷ n (h) for some matrix P. P extracts and combines base forecasts ŷ n (h) to get bottom-level forecasts. S adds them up 16

31 Optimal combination forecasts Main result The best (minimum sum of variances) unbiased forecasts are obtained when P = (S Σ 1 h S) 1 S Σ 1 h, where Σ h is the h-step base forecast error covariance matrix. 17

32 Optimal combination forecasts Main result The best (minimum sum of variances) unbiased forecasts are obtained when P = (S Σ 1 h S) 1 S Σ 1 h, where Σ h is the h-step base forecast error covariance matrix. ỹ n (h) = S(S Σ 1 h S) 1 S Σ 1 ŷn(h) Problem: Σ h hard to estimate, especially for h > 1. Solutions: Ignore Σ h (OLS) Assume Σ h diagonal (WLS) [Default in hts] Try to estimate Σ h (GLS) 17 h

33 Features Covariates can be included in initial forecasts. Adjustments can be made to initial forecasts at any level. Very simple and flexible method. Can work with any hierarchical or grouped time series. Conceptually easy to implement: regression of base forecasts on structure matrix. 18

34 Outline 1 Hierarchical and grouped time series 2 Optimal forecast reconciliation 3 hts package for R 4 Application: Australian tourism 5 Lab Session 5 19

35 hts package for R hts: Hierarchical and Grouped Time Series Methods for analysing and forecasting hierarchical and grouped time series Version: Depends: R ( 3.2.0), forecast ( 8.1) Imports: SparseM, Matrix, matrixcalc, parallel, utils, methods, graphics, grd LinkingTo: Rcpp ( ), RcppEigen Suggests: testthat, knitr, rmarkdown Published: Author: Rob J Hyndman, Alan Lee, Earo Wang, Shanika Wickramasuriya Maintainer: Rob J Hyndman <Earo.Wang at gmail.com> BugReports: License: GPL ( 2) URL: 20

36 Example using R library(hts) # bts is a matrix containing the bottom level time series # nodes describes the hierarchical structure y <- hts(bts, nodes=list(2, c(3,2))) 21

37 Example using R library(hts) # bts is a matrix containing the bottom level time series # nodes describes the hierarchical structure y <- hts(bts, nodes=list(2, c(3,2))) Total A B AX AY AZ BX BY 21

38 Example using R library(hts) # bts is a matrix containing the bottom level time series # nodes describes the hierarchical structure y <- hts(bts, nodes=list(2, c(3,2))) Total A B AX AY AZ BX BY # Forecast 10-step-ahead using WLS combination method # with auto.arima() used at each node fc <- forecast(y, h=10, fmethod='arima') 22

39 forecast.gts() function Usage forecast(object, h, method = c("comb", "bu", "mo", "tdgsf", "tdgsa", "tdfp"), fmethod = c("ets", "rw", "arima"), weights = c("wls", "ols", "mint", "nseries"), covariance = c("shr","sam"), positive = TRUE, parallel = FALSE, num.cores = 2,...) Arguments object Hierarchical time series object of class gts. h Forecast horizon method Method for distributing forecasts within the hierarchy. fmethod Forecasting method to use weights Weights used for "optimal combination" method. covariance Shrinkage estimator or sample estimator for GLS covariance. positive If TRUE, forecasts are forced to be strictly positive parallel If TRUE, allow parallel processing num.cores If parallel = TRUE, specify how many cores to be used 23

40 Outline 1 Hierarchical and grouped time series 2 Optimal forecast reconciliation 3 hts package for R 4 Application: Australian tourism 5 Lab Session 5 24

41 Australian tourism 25

42 Australian tourism Domestic visitor nights Quarterly data: From: National Visitor Survey, based on annual interviews of 120,000 Australians aged 15+, collected by Tourism Research Australia. 25

43 Australian tourism Hierarchy: States (7) Zones (27) Regions (82) 25

44 Australian tourism Hierarchy: States (7) Zones (27) Regions (82) Base forecasts ETS (exponential smoothing) models 25

45 Base forecasts Domestic tourism forecasts: Total Visitor nights Year 26

46 Base forecasts Domestic tourism forecasts: NSW Visitor nights Year 26

47 Base forecasts Domestic tourism forecasts: VIC Visitor nights Year 26

48 Base forecasts Domestic tourism forecasts: Nth.Coast.NSW Visitor nights Year 26

49 Base forecasts Domestic tourism forecasts: Metro.QLD Visitor nights Year 26

50 Base forecasts Domestic tourism forecasts: Sth.WA Visitor nights Year 26

51 Base forecasts Domestic tourism forecasts: X201.Melbourne Visitor nights Year 26

52 Base forecasts Domestic tourism forecasts: X402.Murraylands Visitor nights Year 26

53 Base forecasts Domestic tourism forecasts: X809.Daly Visitor nights Year 26

54 Reconciled forecasts 27

55 Reconciled forecasts 27

56 Reconciled forecasts 27

57 Forecast evaluation Select models using all observations; Re-estimate models using first 12 observations and generate 1- to 8-step-ahead forecasts; Increase sample size one observation at a time, re-estimate models, generate forecasts until the end of the sample; In total 24 1-step-ahead, 23 2-steps-ahead, up to 17 8-steps-ahead for forecast evaluation. 28

58 Forecast evaluation Training sets Test sets h = 1 time 29

59 Forecast evaluation Training sets Test sets h = 1 time 29

60 Forecast evaluation Training sets Test sets h = 1 time 29

61 Forecast evaluation Training sets Test sets h = 1 time 29

62 Forecast evaluation Training sets Test sets h = 1 time 29

63 Forecast evaluation Training sets Test sets h = 1 time 29

64 Forecast evaluation Training sets Test sets h = 1 time 29

65 Forecast evaluation Training sets Test sets h = 1 time 29

66 Forecast evaluation Training sets Test sets h = 1 time 29

67 Forecast evaluation Training sets Test sets h = 1 time 29

68 Forecast evaluation Training sets Test sets h = 1 time 29

69 Forecast evaluation Training sets Test sets h = 1 time 29

70 Forecast evaluation Training sets Test sets h = 1 time 29

71 Forecast evaluation Training sets Test sets h = 1 time 29

72 Forecast evaluation Training sets Test sets h = 1 time 29

73 Forecast evaluation Training sets Test sets h = 1 time 29

74 Forecast evaluation Training sets Test sets h = 1 time 29

75 Forecast evaluation Training sets Test sets h = 1 time 29

76 Forecast evaluation Training sets Test sets h = 1 time 29

77 Forecast evaluation Training sets Test sets h = 1 time 29

78 Forecast evaluation Training sets Test sets h = 2 time 29

79 Forecast evaluation Training sets Test sets h = 3 time 29

80 Forecast evaluation Training sets Test sets h = 4 time 29

81 Forecast evaluation Training sets Test sets h = 5 time 29

82 Forecast evaluation Training sets Test sets h = 6 time 29

83 Hierarchy: states, zones, regions Forecast horizon RMSE h = 1 h = 2 h = 3 h = 4 h = 5 h = 6 Ave Australia Base Bottom WLS GLS States Base Bottom WLS GLS Regions Base Bottom WLS GLS

84 Outline 1 Hierarchical and grouped time series 2 Optimal forecast reconciliation 3 hts package for R 4 Application: Australian tourism 5 Lab Session 5 31

85 Lab Session 5 32