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The ETS taxonomy: what the three letters mean

Today let's understand the ETS taxonomy: the naming system behind every exponential smoothing model this course has fitted so far.

Every model fit in this course so far already had a three-letter name like ETS(A,A,M), even though nobody spelled it out yet. Each one is a point on the same three-letter scale, and by the end of this lesson you can read that scale directly, for any model in the family.

Here is Western Australia's domestic holiday trips again, the 80 quarters of real tourism numbers every one of those models was fit on.

Every rise and fall in that line is the same series each model in this course has tried to explain.

Every model you have fit already has a three-letter name

Fit that same three-part idea properly in fable and it hands the name straight back to you. The formula error("A") + trend("A") + season("M") says additive error, additive trend, multiplicative season, in that order. Error always comes first, then trend, then season. That order never changes.

Fit it on the WA series and read the model line report() prints.

RInteractive R
# Build the WA holiday trips series and fit ETS(A,A,M) on it library(fable) library(tsibble) wa_trips <- c(772.5305, 719.6978, 754.9296, 813.8203, 943.0276, 845.0692, 823.1308, 697.4556, 905.4482, 904.5783, 671.5013, 715.9972, 709.2501, 738.4847, 761.6353, 812.5854, 871.0173, 772.7951, 788.5008, 643.1905, 846.2764, 714.0563, 722.1808, 765.7262, 941.6153, 685.453, 772.0889, 726.8811, 817.3691, 645.5258, 649.5991, 624.3395, 863.3794, 761.078, 679.3309, 799.3583, 851.7432, 786.3122, 701.9789, 762.1728, 899.8878, 653.8436, 735.9842, 655.3446, 747.5275, 642.4111, 570.141, 583.5386, 636.0903, 627.149, 602.5672, 567.0174, 732.6574, 647.6459, 612.5326, 665.9596, 778.8399, 679.9151, 628.522, 691.6633, 917.7258, 745.1224, 679.8861, 690.6982, 1265.7146, 1065.8001, 854.9872, 897.896, 1167.7514, 990.3225, 928.0221, 960.3365, 1165.718, 1053.7725, 803.5157, 982.7526, 1134.352, 997.8522, 879.9885, 1026.286) wa <- tsibble(Quarter = yearquarter(seq(as.Date("1998-01-01"), by = "quarter", length.out = 80)), Trips = wa_trips, index = Quarter) fit_aam <- wa |> model(ets = ETS(Trips ~ error("A") + trend("A") + season("M"))) report(fit_aam) #> Series: Trips #> Model: ETS(A,A,M) #> Smoothing parameters: #> alpha = 0.4063169 #> beta = 0.0001000025 #> gamma = 0.1403905 #> #> Initial states: #> l[0] b[0] s[0] s[-1] s[-2] s[-3] #> 798.6236 7.225898 0.9595522 0.9689424 0.9623976 1.109108 #> #> sigma^2: 7867.922 #> #> AIC AICc BIC #> 1077.777 1080.349 1099.215 

  

  





report() printed Model: ETS(A,A,M). That is fable reading back the exact three arguments you gave error(), trend() and season(), and stringing their letters together in that fixed order: error first, then trend, then season. The formula you wrote and the label report() printed are the same statement, just written two different ways.