Optimal forecast reconciliation for big time series data

Transcription

1 Rob J Hyndman Optimal forecast reconciliation for big time series data Total A AA AB AC B BA BB BC C CA CB CC

2 Follow along using R Requirements Install the hts package and its dependencies. Optimal forecast reconciliation for big time series data 2

3 Outline 1 Hierarchical and grouped time series 2 BLUF: Best Linear Unbiased Forecasts 3 Application: Australian tourism 4 Temporal hierarchies 5 References Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 3

4 Labour market participation Australia and New Zealand Standard Classification of Occupations 8 major groups 43 sub-major groups 97 minor groups 359 unit groups * 1023 occupations Example: statistician 2 Professionals 22 Business, Human Resource and Marketing Professionals 224 Information and Organisation Professionals 2241 Actuaries, Mathematicians and Statisticians Statistician Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 4

5 Labour market participation Australia and New Zealand Standard Classification of Occupations 8 major groups 43 sub-major groups 97 minor groups 359 unit groups * 1023 occupations Example: statistician 2 Professionals 22 Business, Human Resource and Marketing Professionals 224 Information and Organisation Professionals 2241 Actuaries, Mathematicians and Statisticians Statistician Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 4

6 Australian tourism demand Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 5

7 Australian tourism demand 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 Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 5

8 Hierarchical time series A hierarchical time series is a collection of several time series that are linked together in a hierarchical structure. Total A AA AB AC B BA BB BC C CA CB CC Examples Labour turnover by occupation Tourism by state and region Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 6

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 AA AB AC B BA BB BC C CA CB CC Examples Labour turnover by occupation Tourism by state and region Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 6

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 AA AB AC B BA BB BC C CA CB CC Examples Labour turnover by occupation Tourism by state and region Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 6

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 Examples Labour turnover by occupation and state Tourism by state and purpose of travel Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 7

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 Labour turnover by occupation and state Tourism by state and purpose of travel Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 7

13 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 Labour turnover by occupation and state Tourism by state and purpose of travel Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 7

14 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. Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 8

15 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. Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 8

16 Hierarchical time series Total A B C y t = [y t, y A,t, y B,t, y C,t ] = 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 A,t y B,t y C,t Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 8

17 Hierarchical time series Total A B C y t = [y t, y A,t, y B,t, y C,t ] = 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 A,t y B,t y C,t }{{} S Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 8

18 Hierarchical time series Total A B C y t = [y t, y A,t, y B,t, y C,t ] = 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 A,t y B,t y C,t }{{}}{{} b t S Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 8

19 Hierarchical time series Total A B C y t = [y t, y A,t, y B,t, y C,t ] = yt = Sb t 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 A,t y B,t y C,t }{{}}{{} b t S Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 8

20 Hierarchical time series Total A B C y t = AX AY AZ y t y A,t y B,t y C,t y AX,t y AY,t y AZ,t y BX,t y BY,t y BZ,t y CX,t y CY,t y CZ,t = BX BY BZ } {{ } S y AX,t y AY,t y AZ,t y BX,t y BY,t y BZ,t y CX,t y CY,t y CZ,t }{{} b t CX CY CZ Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 9

21 Hierarchical time series Total A B C y t = AX AY AZ y t y A,t y B,t y C,t y AX,t y AY,t y AZ,t y BX,t y BY,t y BZ,t y CX,t y CY,t y CZ,t = BX BY BZ } {{ } S y AX,t y AY,t y AZ,t y BX,t y BY,t y BZ,t y CX,t y CY,t y CZ,t }{{} b t CX CY CZ Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 9

22 Hierarchical time series Total A B C y t = AX AY AZ y t y A,t y B,t y C,t y AX,t y AY,t y AZ,t y BX,t y BY,t y BZ,t y CX,t y CY,t y CZ,t = BX BY BZ } {{ } S y AX,t y AY,t y AZ,t y BX,t y BY,t y BZ,t y CX,t y CY,t y CZ,t }{{} b t CX CY CZ y t = Sb t Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 9

23 Grouped data AX AY A BX BY B y t = y t y A,t y B,t y X,t y Y,t y AX,t y AY,t y BX,t y BY,t = X Y Total }{{} S y AX,t y AY,t y BX,t y BY,t } {{ } b t Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 10

24 Grouped data AX AY A BX BY B y t = y t y A,t y B,t y X,t y Y,t y AX,t y AY,t y BX,t y BY,t = X Y Total }{{} S y AX,t y AY,t y BX,t y BY,t } {{ } b t Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 10

25 Grouped data AX AY A BX BY B y t = y t y A,t y B,t y X,t y Y,t y AX,t y AY,t y BX,t y BY,t = X Y Total }{{} S y AX,t y AY,t y BX,t y BY,t } {{ } b t y t = Sb t Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 10

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. Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 11

27 hts package for R hts: Hierarchical and grouped time series Methods for analysing and forecasting hierarchical and grouped time series Version: 4.5 Depends: forecast ( 5.0), SparseM Imports: parallel, utils Published: Author: Rob J Hyndman, Earo Wang and Alan Lee with contributions from Shanika Wickramasuriya Maintainer: Rob J Hyndman <Rob.Hyndman at monash.edu> BugReports: License: GPL ( 2) Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 12

28 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))) Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 13

29 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 Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 13

30 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 summary(y) smatrix(y) Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 14

31 Example using R Example 1 illustrating the usage of the "groups" argument abc <- ts(5 + matrix(sort(rnorm(1600)), ncol = 16, nrow = 100)) sex <- rep(c("female", "male"), each = 8) state <- rep(c("nsw","vic","qld","sa","wa","nt","act","tas"), 2) gc <- rbind(sex, state) # a matrix consists of strings. gn <- rbind(rep(1:2, each = 8), rep(1:8, 2)) rownames(gc) <- rownames(gn) <- c("sex", "State") x <- gts(abc, groups = gc) y <- gts(abc, groups = gn) plot(x,level=3) plot(x,level=2) plot(x,level=1) plot(x,level=0) Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 15

32 Example using R Example 2 with two simple hierarchies (geography and product) to show the argument "characters" bnames1 <- c("vicmelba1", "VICMelbA2", "VICGeelA1", "VICGeelA2", "VICMelbB1", "VICMelbB2", "VICGeelB1", "VICGeelB2", "NSWSyndA1", "NSWSyndA2", "NSWWollA1", "NSWWollA2", "NSWSyndB1", "NSWSyndB2", "NSWWollB1", "NSWWollB2") bts1 <- matrix(ts(rnorm(160)), ncol = 16) colnames(bts1) <- bnames1 x1 <- gts(bts1, characters = list(c(3, 4), c(1, 1)), gnames = c("state","state.city","product","product.size", "State.Product","State.Product.Size","State.City.Product")) plot(x1, level=1) plot(x1, level=7) Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 16

33 Example using R Example 3 with a non-hierarchical grouped time series of 3 grouping variables (state, age and sex) bnames2 <- c("vic1f","vic1m","vic2f","vic2m","vic3f","vic3m", "NSW1F","NSW1M","NSW2F","NSW2M","NSW3F","NSW3M") bts2 <- matrix(ts(rnorm(120)), ncol = 12) colnames(bts2) <- bnames2 x2 <- gts(bts2, characters = c(3, 1, 1), gnames=c("state","age","sex", "State.Age","State.Sex","Age.Sex")) plot(x2, level=3) Optimal forecast reconciliation for big time series data Hierarchical and grouped time series 17

34 Outline 1 Hierarchical and grouped time series 2 BLUF: Best Linear Unbiased Forecasts 3 Application: Australian tourism 4 Temporal hierarchies 5 References Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 18

35 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. (They may 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 Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 19

36 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. (They may 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 Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 19

37 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. (They may 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 Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 19

38 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. (They may 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 Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 19

39 Forecasting notation Let ŷ n (h) be vector of initial h-step forecasts, made at time n, stacked in same order as y t. (They may 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 Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 19

40 General properties: bias ỹ n (h) = SPŷ n (h) Assume: base forecasts ŷ n (h) are unbiased: E[ŷ n (h) y 1,..., y n ] = E[y n+h y 1,..., y n ] Let ˆbn (h) be bottom level base forecasts with β n (h) = E[ˆbn (h) y 1,..., y n ]. Then E[ŷ n (h)] = Sβ n (h). We want the revised forecasts to be unbiased: E[ỹ n (h)] = SPSβ n (h) = Sβ n (h). Revised forecasts are unbiased iff SPS = S. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 20

41 General properties: bias ỹ n (h) = SPŷ n (h) Assume: base forecasts ŷ n (h) are unbiased: E[ŷ n (h) y 1,..., y n ] = E[y n+h y 1,..., y n ] Let ˆbn (h) be bottom level base forecasts with β n (h) = E[ˆbn (h) y 1,..., y n ]. Then E[ŷ n (h)] = Sβ n (h). We want the revised forecasts to be unbiased: E[ỹ n (h)] = SPSβ n (h) = Sβ n (h). Revised forecasts are unbiased iff SPS = S. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 20

42 General properties: bias ỹ n (h) = SPŷ n (h) Assume: base forecasts ŷ n (h) are unbiased: E[ŷ n (h) y 1,..., y n ] = E[y n+h y 1,..., y n ] Let ˆbn (h) be bottom level base forecasts with β n (h) = E[ˆbn (h) y 1,..., y n ]. Then E[ŷ n (h)] = Sβ n (h). We want the revised forecasts to be unbiased: E[ỹ n (h)] = SPSβ n (h) = Sβ n (h). Revised forecasts are unbiased iff SPS = S. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 20

43 General properties: bias ỹ n (h) = SPŷ n (h) Assume: base forecasts ŷ n (h) are unbiased: E[ŷ n (h) y 1,..., y n ] = E[y n+h y 1,..., y n ] Let ˆbn (h) be bottom level base forecasts with β n (h) = E[ˆbn (h) y 1,..., y n ]. Then E[ŷ n (h)] = Sβ n (h). We want the revised forecasts to be unbiased: E[ỹ n (h)] = SPSβ n (h) = Sβ n (h). Revised forecasts are unbiased iff SPS = S. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 20

44 General properties: bias ỹ n (h) = SPŷ n (h) Assume: base forecasts ŷ n (h) are unbiased: E[ŷ n (h) y 1,..., y n ] = E[y n+h y 1,..., y n ] Let ˆbn (h) be bottom level base forecasts with β n (h) = E[ˆbn (h) y 1,..., y n ]. Then E[ŷ n (h)] = Sβ n (h). We want the revised forecasts to be unbiased: E[ỹ n (h)] = SPSβ n (h) = Sβ n (h). Revised forecasts are unbiased iff SPS = S. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 20

45 General properties: bias ỹ n (h) = SPŷ n (h) Assume: base forecasts ŷ n (h) are unbiased: E[ŷ n (h) y 1,..., y n ] = E[y n+h y 1,..., y n ] Let ˆbn (h) be bottom level base forecasts with β n (h) = E[ˆbn (h) y 1,..., y n ]. Then E[ŷ n (h)] = Sβ n (h). We want the revised forecasts to be unbiased: E[ỹ n (h)] = SPSβ n (h) = Sβ n (h). Revised forecasts are unbiased iff SPS = S. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 20

46 General properties: bias ỹ n (h) = SPŷ n (h) Assume: base forecasts ŷ n (h) are unbiased: E[ŷ n (h) y 1,..., y n ] = E[y n+h y 1,..., y n ] Let ˆbn (h) be bottom level base forecasts with β n (h) = E[ˆbn (h) y 1,..., y n ]. Then E[ŷ n (h)] = Sβ n (h). We want the revised forecasts to be unbiased: E[ỹ n (h)] = SPSβ n (h) = Sβ n (h). Revised forecasts are unbiased iff SPS = S. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 20

47 General properties: variance ỹ n (h) = SPŷ n (h) Let error variance of h-step base forecasts ŷ n (h) be W h = Var[y n+h ŷ n (h) y 1,..., y n ] Then the error variance of the corresponding revised forecasts is Var[y n+h ỹ n (h) y 1,..., y n ] = SPW h P S Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 21

48 General properties: variance ỹ n (h) = SPŷ n (h) Let error variance of h-step base forecasts ŷ n (h) be W h = Var[y n+h ŷ n (h) y 1,..., y n ] Then the error variance of the corresponding revised forecasts is Var[y n+h ỹ n (h) y 1,..., y n ] = SPW h P S Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 21

49 BLUF via trace minimization Theorem For any P satisfying SPS = S, then min P = trace[spw h P S ] has solution P = (S W h S) 1 S W h. W h is generalized inverse of W h. Problem: W h hard to estimate, especially for h > 1. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 22

50 BLUF via trace minimization Theorem For any P satisfying SPS = S, then min P = trace[spw h P S ] has solution P = (S W h S) 1 S W h. W h is generalized inverse of W h. Problem: W h hard to estimate, especially for h > 1. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 22

51 BLUF via trace minimization Theorem For any P satisfying SPS = S, then min P = trace[spw h P S ] has solution P = (S W h S) 1 S W h. W h is generalized inverse of W h. Problem: W h hard to estimate, especially for h > 1. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 22

52 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 1: OLS Assume forecast errors have the same aggregation constraints as the data. Then W h = SΩ h S where Ω h is covariance matrix of bottom level errors. If Moore-Penrose generalized inverse used, then (S W h S) 1 S W h = (S S) 1 S. ỹ n (h) = S(S S) 1 S ŷ n (h) Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 23

53 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 1: OLS Assume forecast errors have the same aggregation constraints as the data. Then W h = SΩ h S where Ω h is covariance matrix of bottom level errors. If Moore-Penrose generalized inverse used, then (S W h S) 1 S W h = (S S) 1 S. ỹ n (h) = S(S S) 1 S ŷ n (h) Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 23

54 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 1: OLS Assume forecast errors have the same aggregation constraints as the data. Then W h = SΩ h S where Ω h is covariance matrix of bottom level errors. If Moore-Penrose generalized inverse used, then (S W h S) 1 S W h = (S S) 1 S. ỹ n (h) = S(S S) 1 S ŷ n (h) Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 23

55 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 1: OLS Assume forecast errors have the same aggregation constraints as the data. Then W h = SΩ h S where Ω h is covariance matrix of bottom level errors. If Moore-Penrose generalized inverse used, then (S W h S) 1 S W h = (S S) 1 S. ỹ n (h) = S(S S) 1 S ŷ n (h) Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 23

56 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 1: OLS Assume forecast errors have the same aggregation constraints as the data. Then W h = SΩ h S where Ω h is covariance matrix of bottom level errors. If Moore-Penrose generalized inverse used, then (S W h S) 1 S W h = (S S) 1 S. ỹ n (h) = S(S S) 1 S ŷ n (h) Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 23

57 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 1: OLS Assume forecast errors have the same aggregation constraints as the data. Then W h = SΩ h S where Ω h is covariance matrix of bottom level errors. If Moore-Penrose generalized inverse used, then (S W h S) 1 S W h = (S S) 1 S. ỹ n (h) = S(S S) 1 S ŷ n (h) Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 23

58 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 2: WLS Suppose we approximate W 1 by its diagonal and assume that W h W 1. Easy to estimate, and places weight where we have best forecasts. Empirically, WLS gives better forecasts than OLS. Still need to estimate covariance matrix to produce prediction intervals. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 24

59 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 2: WLS Suppose we approximate W 1 by its diagonal and assume that W h W 1. Easy to estimate, and places weight where we have best forecasts. Empirically, WLS gives better forecasts than OLS. Still need to estimate covariance matrix to produce prediction intervals. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 24

60 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 2: WLS Suppose we approximate W 1 by its diagonal and assume that W h W 1. Easy to estimate, and places weight where we have best forecasts. Empirically, WLS gives better forecasts than OLS. Still need to estimate covariance matrix to produce prediction intervals. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 24

61 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 2: WLS Suppose we approximate W 1 by its diagonal and assume that W h W 1. Easy to estimate, and places weight where we have best forecasts. Empirically, WLS gives better forecasts than OLS. Still need to estimate covariance matrix to produce prediction intervals. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 24

62 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 2: WLS Suppose we approximate W 1 by its diagonal and assume that W h W 1. Easy to estimate, and places weight where we have best forecasts. Empirically, WLS gives better forecasts than OLS. Still need to estimate covariance matrix to produce prediction intervals. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 24

63 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 3: GLS Estimate W 1 using shrinkage to the diagonal and assume that W h W 1. Allows for covariances. Empirically, GLS gives better forecasts than WLS or OLS. Difficult to compute for large numbers of time series. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 25

64 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 3: GLS Estimate W 1 using shrinkage to the diagonal and assume that W h W 1. Allows for covariances. Empirically, GLS gives better forecasts than WLS or OLS. Difficult to compute for large numbers of time series. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 25

65 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 3: GLS Estimate W 1 using shrinkage to the diagonal and assume that W h W 1. Allows for covariances. Empirically, GLS gives better forecasts than WLS or OLS. Difficult to compute for large numbers of time series. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 25

66 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 3: GLS Estimate W 1 using shrinkage to the diagonal and assume that W h W 1. Allows for covariances. Empirically, GLS gives better forecasts than WLS or OLS. Difficult to compute for large numbers of time series. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 25

67 Optimal combination forecasts ỹ n (h) = S(S W h S) 1 S W hŷn(h) Revised forecasts Base forecasts Solution 3: GLS Estimate W 1 using shrinkage to the diagonal and assume that W h W 1. Allows for covariances. Empirically, GLS gives better forecasts than WLS or OLS. Difficult to compute for large numbers of time series. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 25

68 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))) # Forecast 10-step-ahead using WLS combination method # ETS used for each series by default fc <- forecast(y, h=10) # Select your own methods ally <- aggts(y) allf <- matrix(, nrow=10, ncol=ncol(ally)) for(i in 1:ncol(ally)) allf[,i] <- mymethod(ally[,i], h=10) allf <- ts(allf, start=2004) # Reconcile forecasts so they add up fc2 <- combinef(allf, nodes=y$nodes) Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 26

69 forecast.gts function Usage forecast(object, h, method = c("comb", "bu", "mo", "tdgsf", "tdgsa", "tdfp"), fmethod = c("ets", "rw", "arima"), weights = c("sd", "none", "nseries"), positive = FALSE, 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 positive If TRUE, forecasts are forced to be strictly positive weights Weights used for "optimal combination" method. When weights = "sd", it takes account of the standard deviation of forecasts. parallel If TRUE, allow parallel processing num.cores If parallel = TRUE, specify how many cores are going to be used Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 27

70 Exercise Use the infantgts data to: 1 Plot various levels of aggregation. 2 Forecast the series using auto.arima with the default WLS reconciliation method. 3 Plot the reconciled forecasts at various levels of aggregation. Optimal forecast reconciliation for big time series data BLUF: Best Linear Unbiased Forecasts 28

71 Outline 1 Hierarchical and grouped time series 2 BLUF: Best Linear Unbiased Forecasts 3 Application: Australian tourism 4 Temporal hierarchies 5 References Optimal forecast reconciliation for big time series data Application: Australian tourism 29

72 Australian tourism Optimal forecast reconciliation for big time series data Application: Australian tourism 30

73 Australian tourism Hierarchy: States (7) Zones (27) Regions (82) Optimal forecast reconciliation for big time series data Application: Australian tourism 30

74 Australian tourism Hierarchy: States (7) Zones (27) Regions (82) Base forecasts ETS (exponential smoothing) models Optimal forecast reconciliation for big time series data Application: Australian tourism 30

75 Base forecasts Domestic tourism forecasts: Total Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

76 Base forecasts Domestic tourism forecasts: NSW Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

77 Base forecasts Domestic tourism forecasts: VIC Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

78 Base forecasts Domestic tourism forecasts: Nth.Coast.NSW Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

79 Base forecasts Domestic tourism forecasts: Metro.QLD Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

80 Base forecasts Domestic tourism forecasts: Sth.WA Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

81 Base forecasts Domestic tourism forecasts: X201.Melbourne Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

82 Base forecasts Domestic tourism forecasts: X402.Murraylands Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

83 Base forecasts Domestic tourism forecasts: X809.Daly Visitor nights Year Optimal forecast reconciliation for big time series data Application: Australian tourism 31

84 Forecast evaluation Training sets time Optimal forecast reconciliation for big time series data Application: Australian tourism 32

85 Forecast evaluation Training sets Test set h = 1 time Optimal forecast reconciliation for big time series data Application: Australian tourism 32

86 Forecast evaluation Training sets Test set h = 2 time Optimal forecast reconciliation for big time series data Application: Australian tourism 32

87 Forecast evaluation Training sets Test set h = 3 time Optimal forecast reconciliation for big time series data Application: Australian tourism 32

88 Forecast evaluation Training sets Test set h = 4 time Optimal forecast reconciliation for big time series data Application: Australian tourism 32

89 Forecast evaluation Training sets Test set h = 5 time Optimal forecast reconciliation for big time series data Application: Australian tourism 32

90 Forecast evaluation Training sets Test set h = 6 time Optimal forecast reconciliation for big time series data Application: Australian tourism 32

91 Hierarchy: states, zones, regions Forecast horizon RMSE h = 1 h = 2 h = 3 h = 4 h = 5 h = 6 Ave Australia Base Bottom OLS WLS GLS States Base Bottom OLS WLS GLS Regions Base Bottom OLS WLS GLS Optimal forecast reconciliation for big time series data Application: Australian tourism 33

92 Outline 1 Hierarchical and grouped time series 2 BLUF: Best Linear Unbiased Forecasts 3 Application: Australian tourism 4 Temporal hierarchies 5 References Optimal forecast reconciliation for big time series data Temporal hierarchies 34

93 Temporal hierarchies Annual Semi-Annual 1 Semi-Annual 2 Q 1 Q 2 Q 3 Q 4 Basic idea: Forecast series at each available frequency. Optimally combine forecasts within the same year. Optimal forecast reconciliation for big time series data Temporal hierarchies 35

94 Temporal hierarchies Annual Semi-Annual 1 Semi-Annual 2 Q 1 Q 2 Q 3 Q 4 Basic idea: Forecast series at each available frequency. Optimally combine forecasts within the same year. Optimal forecast reconciliation for big time series data Temporal hierarchies 35

95 Monthly series Annual Semi-Annual 1 Semi-Annual 2 Q 1 Q 2 Q 3 Q 4 M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11 M 12 k = 2, 4, 12 nodes k = 3, 6, 12 nodes Why not k = 2, 3, 4, 6, 12 nodes? Optimal forecast reconciliation for big time series data Temporal hierarchies 36

96 Monthly series Annual FourM 1 FourM 2 FourM 3 BiM 1 BiM 2 BiM 3 BiM 4 BiM 5 BiM 6 M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11 M 12 k = 2, 4, 12 nodes k = 3, 6, 12 nodes Why not k = 2, 3, 4, 6, 12 nodes? Optimal forecast reconciliation for big time series data Temporal hierarchies 36

97 Monthly series Annual FourM 1 FourM 2 FourM 3 BiM 1 BiM 2 BiM 3 BiM 4 BiM 5 BiM 6 M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11 M 12 k = 2, 4, 12 nodes k = 3, 6, 12 nodes Why not k = 2, 3, 4, 6, 12 nodes? Optimal forecast reconciliation for big time series data Temporal hierarchies 36

98 Monthly data A SemiA 1 SemiA 2 FourM 1 FourM 2 FourM 3 Q 1. Q 4 BiM 1. BiM 6 M 1. M 12 }{{} (28 1) = I 12 } {{ } S M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11 M 12 }{{} Optimal forecast reconciliation for big time series data Temporal hierarchies 37 B t

99 In general For a time series y 1,..., y T, observed at frequency m, we generate aggregate series y [k] j = jk t=1+(j 1)k y t, for j = 1,..., T/k k F(m) = {factors of m}. A single unique hierarchy is only possible when there are no coprime pairs in F(m). M k = m/k is seasonal period of aggregated series. Optimal forecast reconciliation for big time series data Temporal hierarchies 38

100 In general For a time series y 1,..., y T, observed at frequency m, we generate aggregate series y [k] j = jk t=1+(j 1)k y t, for j = 1,..., T/k k F(m) = {factors of m}. A single unique hierarchy is only possible when there are no coprime pairs in F(m). M k = m/k is seasonal period of aggregated series. Optimal forecast reconciliation for big time series data Temporal hierarchies 38

101 In general For a time series y 1,..., y T, observed at frequency m, we generate aggregate series y [k] j = jk t=1+(j 1)k y t, for j = 1,..., T/k k F(m) = {factors of m}. A single unique hierarchy is only possible when there are no coprime pairs in F(m). M k = m/k is seasonal period of aggregated series. Optimal forecast reconciliation for big time series data Temporal hierarchies 38

102 WLS weights Hierarchy variance scaling Λ H : diagonal. Series variance scaling Λ V : elements equal within aggregation level. Structural scaling Λ S = diag(s1): elements equal to # nodes at each level. Depends only on seasonal period m. Independent of data and model. Allows forecasts where no errors available. Quarterly example Λ H = (ˆσ ) diag A, 2 ˆσ 2 S 1, ˆσ 2 S 2, ˆσ 2 Q 1, ˆσ 2 Q 2, ˆσ 2 Q 3, ˆσ 2 Q 4 Λ V = (ˆσ ) diag A, 2 ˆσ S, 2 ˆσ S, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ 2 Q Λ S = ( diag 4, 2, 2, 1, 1, 1, ) 1 Optimal forecast reconciliation for big time series data Temporal hierarchies 39

103 WLS weights Hierarchy variance scaling Λ H : diagonal. Series variance scaling Λ V : elements equal within aggregation level. Structural scaling Λ S = diag(s1): elements equal to # nodes at each level. Depends only on seasonal period m. Independent of data and model. Allows forecasts where no errors available. Quarterly example Λ H = (ˆσ ) diag A, 2 ˆσ 2 S 1, ˆσ 2 S 2, ˆσ 2 Q 1, ˆσ 2 Q 2, ˆσ 2 Q 3, ˆσ 2 Q 4 Λ V = (ˆσ ) diag A, 2 ˆσ S, 2 ˆσ S, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ 2 Q Λ S = ( diag 4, 2, 2, 1, 1, 1, ) 1 Optimal forecast reconciliation for big time series data Temporal hierarchies 39

104 WLS weights Hierarchy variance scaling Λ H : diagonal. Series variance scaling Λ V : elements equal within aggregation level. Structural scaling Λ S = diag(s1): elements equal to # nodes at each level. Depends only on seasonal period m. Independent of data and model. Allows forecasts where no errors available. Quarterly example Λ H = (ˆσ ) diag A, 2 ˆσ 2 S 1, ˆσ 2 S 2, ˆσ 2 Q 1, ˆσ 2 Q 2, ˆσ 2 Q 3, ˆσ 2 Q 4 Λ V = (ˆσ ) diag A, 2 ˆσ S, 2 ˆσ S, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ 2 Q Λ S = ( diag 4, 2, 2, 1, 1, 1, ) 1 Optimal forecast reconciliation for big time series data Temporal hierarchies 39

105 WLS weights Hierarchy variance scaling Λ H : diagonal. Series variance scaling Λ V : elements equal within aggregation level. Structural scaling Λ S = diag(s1): elements equal to # nodes at each level. Depends only on seasonal period m. Independent of data and model. Allows forecasts where no errors available. Quarterly example Λ H = (ˆσ ) diag A, 2 ˆσ 2 S 1, ˆσ 2 S 2, ˆσ 2 Q 1, ˆσ 2 Q 2, ˆσ 2 Q 3, ˆσ 2 Q 4 Λ V = (ˆσ ) diag A, 2 ˆσ S, 2 ˆσ S, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ 2 Q Λ S = ( diag 4, 2, 2, 1, 1, 1, ) 1 Optimal forecast reconciliation for big time series data Temporal hierarchies 39

106 WLS weights Hierarchy variance scaling Λ H : diagonal. Series variance scaling Λ V : elements equal within aggregation level. Structural scaling Λ S = diag(s1): elements equal to # nodes at each level. Depends only on seasonal period m. Independent of data and model. Allows forecasts where no errors available. Quarterly example Λ H = (ˆσ ) diag A, 2 ˆσ 2 S 1, ˆσ 2 S 2, ˆσ 2 Q 1, ˆσ 2 Q 2, ˆσ 2 Q 3, ˆσ 2 Q 4 Λ V = (ˆσ ) diag A, 2 ˆσ S, 2 ˆσ S, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ Q, 2 ˆσ 2 Q Λ S = ( diag 4, 2, 2, 1, 1, 1, ) 1 Optimal forecast reconciliation for big time series data Temporal hierarchies 39

107 Experimental setup: M3 forecasting competition (Makridakis and Hibon, 2000, IJF). In total 3003 series. 1,428 monthly series with a test sample of 12 observations each. 756 quarterly series with a test sample of 8 observations each. Forecast each series with ETS models. Optimal forecast reconciliation for big time series data Temporal hierarchies 40

108 Experimental setup: M3 forecasting competition (Makridakis and Hibon, 2000, IJF). In total 3003 series. 1,428 monthly series with a test sample of 12 observations each. 756 quarterly series with a test sample of 8 observations each. Forecast each series with ETS models. Optimal forecast reconciliation for big time series data Temporal hierarchies 40

109 Experimental setup: M3 forecasting competition (Makridakis and Hibon, 2000, IJF). In total 3003 series. 1,428 monthly series with a test sample of 12 observations each. 756 quarterly series with a test sample of 8 observations each. Forecast each series with ETS models. Optimal forecast reconciliation for big time series data Temporal hierarchies 40

110 Experimental setup: M3 forecasting competition (Makridakis and Hibon, 2000, IJF). In total 3003 series. 1,428 monthly series with a test sample of 12 observations each. 756 quarterly series with a test sample of 8 observations each. Forecast each series with ETS models. Optimal forecast reconciliation for big time series data Temporal hierarchies 40

111 Results: Monthly MAE percent difference relative to base max h BU WLS H WLS V WLS S Annual Semi-annual Four-monthly Quarterly Bi-monthly Monthly Optimal forecast reconciliation for big time series data Temporal hierarchies 41

112 Results: Quarterly MAE percent difference relative to base max h BU WLS H WLS V WLS S Annual Semi-annual Quarterly Optimal forecast reconciliation for big time series data Temporal hierarchies 42

113 UK Accidents and Emergency Demand Annual (k=52) Forecast Semi annual (k=26) Forecast Quarterly (k=13) Forecast Monthly (k=4) Bi weekly (k=2) Weekly (k=1) Forecast Forecast base reconciled Forecast Optimal forecast reconciliation for big time series data Temporal hierarchies 43

114 UK Accidents and Emergency Demand 1 Type 1 Departments Major A&E 2 Type 2 Departments Single Specialty 3 Type 3 Departments Other A&E/Minor Injury 4 Total Attendances 5 Type 1 Departments Major A&E > 4 hrs 6 Type 2 Departments Single Specialty > 4 hrs 7 Type 3 Departments Other A&E/Minor Injury > 4 hrs 8 Total Attendances > 4 hrs 9 Emergency Admissions via Type 1 A&E 10 Total Emergency Admissions via A&E 11 Other Emergency Admissions (i.e., not via A&E) 12 Total Emergency Admissions 13 Number of patients spending > 4 hrs from decision to admission Optimal forecast reconciliation for big time series data Temporal hierarchies 44

115 UK Accidents and Emergency Demand Minimum training set: all data except the last year Base forecasts using auto.arima(). Reconciled using WLS V. Mean Absolute Scaled Errors for 1, 4 and 13 weeks ahead using a rolling origin. Aggr. Level h Base Reconciled Change Weekly % Weekly % Weekly % Weekly % Annual % Optimal forecast reconciliation for big time series data Temporal hierarchies 45

116 UK Accidents and Emergency Demand Minimum training set: all data except the last year Base forecasts using auto.arima(). Reconciled using WLS V. Mean Absolute Scaled Errors for 1, 4 and 13 weeks ahead using a rolling origin. Aggr. Level h Base Reconciled Change Weekly % Weekly % Weekly % Weekly % Annual % Optimal forecast reconciliation for big time series data Temporal hierarchies 45

117 UK Accidents and Emergency Demand Minimum training set: all data except the last year Base forecasts using auto.arima(). Reconciled using WLS V. Mean Absolute Scaled Errors for 1, 4 and 13 weeks ahead using a rolling origin. Aggr. Level h Base Reconciled Change Weekly % Weekly % Weekly % Weekly % Annual % Optimal forecast reconciliation for big time series data Temporal hierarchies 45

118 UK Accidents and Emergency Demand Minimum training set: all data except the last year Base forecasts using auto.arima(). Reconciled using WLS V. Mean Absolute Scaled Errors for 1, 4 and 13 weeks ahead using a rolling origin. Aggr. Level h Base Reconciled Change Weekly % Weekly % Weekly % Weekly % Annual % Optimal forecast reconciliation for big time series data Temporal hierarchies 45

119 UK Accidents and Emergency Demand Minimum training set: all data except the last year Base forecasts using auto.arima(). Reconciled using WLS V. Mean Absolute Scaled Errors for 1, 4 and 13 weeks ahead using a rolling origin. Aggr. Level h Base Reconciled Change Weekly % Weekly % Weekly % Weekly % Annual % Optimal forecast reconciliation for big time series data Temporal hierarchies 45

120 UK Accidents and Emergency Demand Minimum training set: all data except the last year Base forecasts using auto.arima(). Reconciled using WLS V. Mean Absolute Scaled Errors for 1, 4 and 13 weeks ahead using a rolling origin. Aggr. Level h Base Reconciled Change Weekly % Weekly % Weekly % Weekly % Annual % Optimal forecast reconciliation for big time series data Temporal hierarchies 45

121 Outline 1 Hierarchical and grouped time series 2 BLUF: Best Linear Unbiased Forecasts 3 Application: Australian tourism 4 Temporal hierarchies 5 References Optimal forecast reconciliation for big time series data References 46

122 References RJ Hyndman, RA Ahmed, G Athanasopoulos, and HL Shang (2011). Optimal combination forecasts for hierarchical time series. Computational statistics & data analysis 55(9), RJ Hyndman, AJ Lee, and E Wang (2014). Fast computation of reconciled forecasts for hierarchical and grouped time series. Working paper 17/14. Monash University SL Wickramasuriya, G Athanasopoulos, and RJ Hyndman (2015). Forecasting hierarchical and grouped time series through trace minimization. Working paper 15/15. Monash University G Athanasopoulos, RJ Hyndman, N Kourentzes, and F Petropoulos (2015). Forecasting with temporal hierarchies. Working paper. Monash University RJ Hyndman, AJ Lee, and E Wang (2015). hts: Hierarchical and grouped time series. R package v4.5 on CRAN. Optimal forecast reconciliation for big time series data References 47