0
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I have a daily seasonal time series of 1461 observations, which I seasonally differenced and I am running multiple different models. My trained data is of 1023 obs and the test data is of 438 obs.

Here's train and test data (without any seasonal differencing).

clean_train
Time Series:
Start = c(2015, 4, 01) 
Frequency = 365 
   [1] 49.00000 58.00000 36.00000 44.00000 64.00000 46.00000 46.00000 39.00000 58.00000 61.00000 51.00000
  [12] 52.00000 48.00000 24.00000 43.00000 56.00000 44.00000 47.00000 44.00000 48.00000 31.00000 32.00000
  [23] 45.00000 43.00000 40.00000 45.00000 50.00000 38.00000 29.00000 48.00000 51.00000 44.00000 40.00000
  [34] 33.00000 34.00000 33.00000 49.00000 46.00000 49.00000 44.00000 41.00000 22.00000 40.00000 43.00000
  [45] 46.00000 45.00000 49.00000 47.00000 28.00000 47.00000 44.00000 51.00000 48.00000 37.00000 49.00000
  [56] 50.00000 40.00000 44.00000 48.00000 37.00000 47.00000 48.00000 35.00000 41.00000 46.00000 43.00000
  [67] 66.00000 51.00000 46.00000 26.00000 32.00000 46.00000 40.00000 30.00000 49.00000 42.00000 27.00000
  [78] 38.00000 43.00000 36.00000 44.00000 44.00000 50.00000 37.00000 30.00000 46.00000 44.00000 41.00000
  [89] 52.00000 49.00000 23.00000 30.00000 44.00000 30.00000 35.00000 46.00000 39.00000 37.00000 35.00000
 [100] 40.00000 31.00000 35.00000 40.00000 34.00000 27.00000 31.00000 43.00000 39.00000 38.00000 39.00000
 [111] 36.00000 31.00000 32.00000 45.00000 40.00000 44.00000 35.00000 45.00000 37.00000 25.00000 50.00000
 [122] 34.00000 27.00000 47.00000 36.00000 22.00000 20.00000 50.00000 49.00000 46.00000 26.00000 38.00000
 [133] 24.00000 40.00000 39.00000 49.00000 42.00000 36.00000 35.00000 24.00000 27.00000 42.00000 33.00000
 [144] 34.00000 32.00000 52.00000 25.00000 37.00000 29.00000 34.00000 25.00000 36.00000 26.00000 30.00000
 [155] 35.00000 35.00000 38.00000 34.00000 33.00000 39.00000 25.00000 33.00000 46.00000 40.00000 32.00000
 [166] 33.00000 34.00000 26.00000 27.00000 39.00000 40.00000 28.00000 47.00000 43.00000 34.00000 36.00000
 [177] 44.00000 46.00000 47.00000 45.00000 51.00000 31.00000 50.00000 41.00000 45.00000 43.00000 42.00000
 [188] 42.00000 41.00000 36.00000 44.00000 41.00000 43.00000 40.00000 42.00000 45.00000 45.00000 53.00000
 [199] 33.00000 43.00000 53.00000 45.00000 41.00000 32.00000 44.00000 46.00000 46.00000 44.00000 39.00000
 [210] 32.00000 33.00000 48.00000 43.00000 51.00000 47.00000 41.00000 42.00000 35.00000 52.00000 43.00000
 [221] 34.00000 36.00000 49.00000 27.00000 30.00000 49.00000 35.00000 51.00000 45.00000 50.00000 30.00000
 [232] 28.00000 46.00000 30.00000 44.00000 35.00000 47.00000 34.00000 34.00000 58.00000 38.00000 41.00000
 [243] 48.00000 44.00000 30.00000 32.00000 47.00000 37.00000 64.00000 41.00000 52.00000 33.00000 47.00000
 [254] 58.00000 58.00000 54.00000 61.00000 73.00000 45.00000 46.00000 55.00000 47.00000 53.00000 51.00000
 [265] 45.00000 52.00000 48.00000 59.00000 42.00000 59.00000 81.00000 79.00000 57.00000 60.00000 89.00000
 [276] 54.00000 66.00000 69.00000 61.00000 57.00000 44.00000 55.00000 58.00000 68.00000 53.00000 69.00000
 [287] 40.00000 41.00000 72.00000 55.00000 49.00000 44.00000 51.00000 37.00000 43.00000 85.00000 44.00000
 [298] 51.00000 46.00000 45.00000 46.00000 40.00000 56.00000 58.00000 53.00000 55.00000 56.00000 37.00000
 [309] 30.00000 66.00000 49.00000 47.00000 47.00000 54.00000 41.00000 44.00000 62.00000 51.00000 55.00000
 [320] 55.00000 60.00000 45.00000 57.00000 68.00000 48.00000 44.00000 49.00000 61.00000 47.00000 48.00000
 [331] 54.00000 47.00000 45.00000 71.00000 61.00000 38.00000 57.00000 62.00000 59.00000 82.00000 66.00000
 [342] 62.00000 47.00000 52.00000 50.00000 45.00000 68.00000 55.00000 79.00000 50.00000 54.00000 54.00000
 [353] 59.00000 50.00000 54.00000 74.00000 44.00000 58.00000 56.00000 61.00000 70.00000 77.00000 62.00000
 [364] 62.00000 50.00000 64.00000 58.00000 71.00000 53.00000 64.00000 49.00000 47.00000 69.00000 54.00000
 [375] 58.00000 64.00000 71.00000 43.00000 56.00000 66.00000 48.00000 56.00000 55.00000 43.00000 29.00000
 [386] 44.00000 46.00000 38.00000 45.00000 37.00000 41.00000 30.00000 43.00000 53.00000 46.00000 45.00000
 [397] 37.00000 43.00000 29.00000 34.00000 47.00000 39.00000 47.00000 52.00000 48.00000 46.00000 32.00000
 [408] 41.00000 40.00000 45.00000 28.00000 53.00000 23.00000 45.00000 46.00000 37.00000 47.00000 45.00000
 [419] 41.00000 34.00000 42.00000 37.00000 58.00000 47.00000 41.00000 38.00000 43.00000 28.00000 50.00000
 [430] 42.00000 49.00000 43.00000 39.00000 28.00000 30.00000 41.00000 46.00000 45.00000 32.00000 35.00000
 [441] 28.00000 35.00000 49.00000 35.00000 38.00000 45.00000 34.00000 36.00000 25.00000 45.00000 41.00000
 [452] 38.00000 40.00000 48.00000 33.00000 36.00000 37.00000 47.00000 34.00000 43.00000 40.00000 22.00000
 [463] 31.00000 48.00000 40.00000 33.00000 30.00000 37.00000 36.00000 31.00000 44.00000 39.00000 39.00000
 [474] 50.00000 34.00000 25.00000 39.00000 48.00000 47.00000 34.00000 34.00000 29.00000 42.00000 30.00000
 [485] 45.00000 32.00000 34.00000 31.00000 35.00000 34.73238 30.00000 45.00000 38.00000 39.00000 33.00000
 [496] 50.00000 27.00000 24.00000 46.00000 27.00000 36.00000 52.00000 42.00000 23.00000 27.00000 41.00000
 [507] 33.00000 32.00000 46.00000 45.00000 26.00000 23.00000 32.00000 43.00000 39.00000 29.00000 39.00000
 [518] 30.00000 36.00000 40.00000 30.00000 24.00000 49.00000 50.00000 24.00000 29.00000 32.00000 25.00000
 [529] 33.00000 36.00000 44.00000 28.00000 26.00000 30.00000 35.00000 40.00000 29.00000 39.00000 35.00000
 [540] 43.00000 51.00000 33.00000 33.00000 39.00000 47.00000 32.00000 32.00000 44.00000 46.00000 46.00000
 [551] 41.00000 50.00000 29.00000 33.00000 41.00000 38.00000 39.00000 40.00000 43.00000 32.00000 31.00000
 [562] 49.00000 64.00000 36.00000 38.00000 50.00000 32.00000 35.00000 48.00000 37.00000 43.00000 31.00000
 [573] 51.00000 35.00000 34.00000 47.00000 35.00000 40.00000 42.00000 35.00000 37.00000 42.00000 41.00000
 [584] 41.00000 31.00000 43.00000 36.00000 27.00000 35.00000 41.00000 51.00000 51.00000 32.00000 39.00000
 [595] 35.00000 39.00000 35.00000 32.00000 45.00000 43.00000 42.00000 28.00000 39.00000 46.00000 44.00000
 [606] 39.00000 50.00000 53.00000 50.00000 37.00000 68.00000 37.00000 49.00000 49.00000 36.00000 48.00000
 [617] 45.00000 63.00000 52.00000 51.00000 66.00000 64.00000 34.00000 44.00000 66.00000 40.00000 49.00000
 [628] 57.00000 55.00000 40.00000 46.00000 56.00000 54.00000 39.00000 59.00000 82.00000 59.00000 45.00000
 [639] 83.00000 61.00000 43.00000 64.00000 82.00000 59.00000 60.00000 77.00000 57.00000 53.00000 60.00000
 [650] 61.00000 29.00000 39.00000 59.00000 46.00000 42.00000 45.00000 39.00000 37.00000 41.00000 48.00000
 [661] 55.00000 48.00000 46.00000 52.00000 41.00000 39.00000 48.00000 47.00000 58.00000 55.00000 59.00000
 [672] 37.00000 42.00000 62.00000 28.00000 58.00000 35.00000 60.00000 52.00000 42.00000 63.00000 50.00000
 [683] 45.00000 52.00000 56.00000 42.00000 39.00000 67.00000 52.00000 35.00000 50.00000 51.00000 39.00000
 [694] 41.00000 55.00000 44.00000 47.00000 37.00000 51.00000 45.00000 42.00000 60.00000 63.00000 42.00000
 [705] 52.00000 48.00000 39.00000 33.00000 57.00000 43.00000 43.00000 47.00000 49.00000 37.00000 48.00000
 [716] 55.00000 58.00000 44.00000 76.00000 53.00000 52.00000 63.00000 66.00000 70.00000 54.00000 47.00000
 [727] 70.00000 56.00000 60.00000 67.00000 48.00000 55.00000 47.00000 67.00000 44.00000 49.00000 58.00000
 [738] 45.00000 54.00000 61.00000 53.00000 37.00000 44.00000 56.00000 42.00000 60.00000 44.00000 34.00000
 [749] 39.00000 44.00000 50.00000 55.00000 42.00000 55.00000 56.00000 39.00000 36.00000 49.00000 44.00000
 [760] 44.00000 57.00000 39.00000 34.00000 37.00000 55.00000 40.00000 34.00000 51.00000 37.00000 31.00000
 [771] 42.00000 39.00000 57.00000 32.00000 49.00000 43.00000 40.00000 47.00000 54.00000 45.00000 41.00000
 [782] 48.00000 55.00000 41.00000 46.00000 35.00000 56.00000 55.00000 56.00000 45.00000 36.00000 31.00000
 [793] 41.00000 38.00000 39.00000 51.00000 49.00000 38.00000 41.00000 44.00000 43.00000 39.00000 41.00000
 [804] 41.00000 23.00000 37.00000 52.00000 31.00000 43.00000 44.00000 34.00000 35.00000 30.00000 50.00000
 [815] 49.00000 43.00000 50.00000 43.00000 40.00000 27.00000 49.00000 46.00000 44.00000 36.00000 39.00000
 [826] 30.00000 45.00000 55.00000 39.00000 45.00000 44.00000 34.00000 40.00000 30.00000 42.00000 43.00000
 [837] 43.00000 39.00000 46.00000 22.00000 39.00000 45.00000 49.00000 26.00000 34.00000 47.00000 32.00000
 [848] 25.00000 50.00000 44.00000 38.00000 42.00000 34.00000 33.00000 33.00000 31.00000 38.00000 33.00000
 [859] 35.00000 38.00000 33.00000 30.00000 46.00000 41.00000 43.00000 30.00000 42.00000 33.00000 33.00000
 [870] 44.00000 37.00000 41.00000 46.00000 45.00000 35.00000 42.00000 45.00000 35.00000 55.00000 37.00000
 [881] 46.00000 34.00000 26.00000 48.00000 45.00000 42.00000 37.00000 40.00000 27.00000 32.00000 35.00000
 [892] 40.00000 36.00000 36.00000 43.00000 38.00000 23.00000 40.00000 34.00000 43.00000 49.00000 47.00000
 [903] 31.00000 43.00000 44.00000 44.00000 43.00000 39.00000 35.00000 33.00000 39.00000 47.00000 50.00000
 [914] 29.00000 40.00000 37.00000 30.00000 26.00000 41.00000 40.00000 40.00000 38.00000 55.00000 33.00000
 [925] 29.00000 58.00000 55.00000 52.00000 48.00000 58.00000 40.00000 33.00000 35.00000 48.00000 39.00000
 [936] 43.00000 40.00000 29.00000 28.00000 36.00000 51.00000 38.00000 43.00000 56.00000 38.00000 36.00000
 [947] 53.00000 39.00000 34.00000 35.00000 56.00000 45.00000 35.00000 51.00000 39.00000 46.00000 40.00000
 [958] 49.00000 38.00000 30.00000 55.00000 60.00000 43.00000 52.00000 42.00000 37.00000 43.00000 39.00000
 [969] 46.00000 38.00000 57.00000 50.00000 42.00000 37.00000 51.00000 37.00000 41.00000 54.00000 64.00000
 [980] 39.00000 53.00000 36.00000 45.00000 61.00000 39.00000 66.00000 53.00000 56.00000 68.00000 49.00000
 [991] 63.00000 64.00000 78.00000 59.00000 65.00000 79.00000 77.00000 62.00000 56.00000 77.00000 97 92 86 82 96 77 97 87 81 90 74 80 82 90 55 53 76 84 51 76 65 50 55

> clean_test
Time Series:
Start = c(2018, 01, 18) 
End = c(2019, 4) 
Frequency = 365 
  [1] 72.00000 64.00000 73.00000 65.00000 50.00000 52.00000 49.00000 70.00000 73.00000 43.00000 70.00000
 [12] 69.00000 61.00000 55.00000 71.00000 62.00000 56.00000 66.00000 70.00000 48.00000 56.00000 77.00000
 [23] 57.00000 72.00000 64.00000 69.00000 74.00000 63.00000 82.00000 76.00000 65.00000 68.00000 49.00000
 [34] 72.00000 58.00000 73.00000 60.00000 57.00000 50.00000 58.00000 44.00000 55.00000 68.00000 60.00000
 [45] 56.00000 44.00000 58.00000 50.00000 47.00000 62.00000 55.00000 49.00000 59.00000 55.00000 47.00000
 [56] 33.00000 63.00000 47.00000 53.00000 45.00000 59.00000 41.00000 53.00000 65.00000 62.00000 63.00000
 [67] 43.00000 66.00000 49.00000 57.00000 62.00000 47.00000 56.00000 64.00000 43.00000 52.00000 53.00000
 [78] 58.00000 65.00000 47.00000 54.00000 58.00000 43.00000 43.00000 61.00000 51.00000 42.00000 42.00000
 [89] 68.00000 28.00000 32.00000 51.00000 36.00000 50.00000 55.00000 48.00000 41.00000 38.00000 54.00000
[100] 42.00000 49.00000 54.00000 52.00000 40.00000 46.00000 47.00000 51.00000 54.00000 50.00000 46.00000
[111] 39.00000 42.00000 37.00000 46.00000 43.00000 42.00000 53.00000 33.00000 31.00000 45.00000 51.00000
[122] 45.00000 56.00000 46.00000 45.00000 34.00000 46.00000 35.00000 46.00000 57.00000 55.00000 42.00000
[133] 33.00000 44.00000 53.00000 47.00000 52.00000 51.00000 29.00000 30.00000 43.00000 36.00000 36.00000
[144] 50.00000 35.00000 35.00000 37.00000 56.00000 47.00000 48.00000 41.00000 48.00000 31.00000 26.00000
[155] 47.00000 49.00000 49.00000 33.00000 46.00000 30.00000 32.00000 51.00000 52.00000 40.00000 49.00000
[166] 51.00000 35.00000 42.00000 41.00000 33.00000 43.00000 46.00000 47.00000 30.00000 34.00000 38.00000
[177] 36.00000 35.00000 38.00000 43.00000 30.00000 31.00000 43.00000 43.00000 42.00000 38.00000 39.00000
[188] 20.00000 27.00000 53.00000 36.00000 43.00000 45.00000 30.00000 32.00000 32.00000 44.00000 40.00000
[199] 36.00000 33.00000 34.00000 24.00000 37.00000 42.00000 45.00000 33.00000 41.00000 44.00000 36.00000
[210] 39.00000 42.00000 41.00000 43.00000 39.00000 41.00000 28.00000 34.00000 39.00000 36.00000 42.00000
[221] 38.00000 33.00000 21.00000 25.00000 33.00000 43.00000 46.00000 31.00000 42.00000 31.00000 45.00000
[232] 40.00000 43.00000 39.00000 40.00000 50.00000 35.00000 37.00000 49.00000 52.00000 44.00000 39.00000
[243] 43.00000 39.00000 38.00000 49.00000 50.00000 50.00000 47.00000 43.00000 44.00000 36.00000 64.00000
[254] 42.00000 46.00000 54.00000 70.00000 43.00000 50.00000 47.00000 64.00000 43.00000 42.00000 48.00000
[265] 23.00000 46.00000 54.00000 43.00000 43.00000 53.00000 53.00000 42.00000 41.00000 55.00000 35.00000
[276] 46.00000 53.00000 51.00000 40.00000 45.00000 37.00000 41.00000 37.00000 40.00000 59.00000 43.00000
[287] 35.00000 55.00000 52.00000 57.00000 64.00000 56.00000 46.00000 42.00000 55.00000 65.00000 51.00000
[298] 62.00000 55.00000 36.00000 41.00000 59.00000 45.00000 39.00000 53.00000 49.00000 44.00000 43.00000
[309] 43.00000 45.00000 51.00000 60.00000 55.00000 50.00000 39.00000 60.00000 52.00000 47.00000 50.00000
[320] 44.00000 50.00000 48.00000 54.00000 58.00000 49.00000 63.00000 48.00000 47.00000 46.00000 65.00000
[331] 72.00000 67.00000 55.00000 59.00000 46.00000 51.00000 61.00000 60.00000 67.00000 52.00000 48.00000
[342] 70.00000 79.00000 82.00000 95.00000 78.00000 69.00000 63.00000 78.00000 66.00000 79.00000 76.00000
[353] 55.00000 89.00000 84.00000 64.00000 43.00000 76.00000 68.00000 55.00000 64.00000 64.00000 51.00000
[364] 57.00000 60.00000 61.00000 68.00000 54.00000 90.00000 40.00000 53.00000 76.00000 68.00000 54.00000
[375] 78.00000 82.00000 55.00000 61.00000 65.00000 65.00000 78.00000 59.00000 78.00000 61.00000 54.00000
[386] 64.00000 77.00000 79.00000 48.00000 74.00000 54.00000 65.00000 62.00000 64.00000 57.00000 61.00000
[397] 52.00000 50.00000 69.00000 83.00000 83.00000 68.00000 60.00000 73.00000 53.00000 63.00000 58.00000
[408] 92.00000 57.00000 56.00000 62.00000 60.00000 74.00000 69.00000 66.00000 66.00000 60.00000 63.00000
[419] 56.00000 49.00000 72.00000 73.00000 64.00000 61.00000 54.00000 43.00000 51.00000 56.00000 52.00000
[430] 67.00000 64.00000 41.00000 42.00000 45.00000 38.00000 39.00000 47.00000 50.77037

Amongst them I am trying to run the meanf() model with the differenced data, then undifferencing it to plot and to check accuracy and such.

This is my initial code:

train_seasdiff <- diff(clean_train, lag=7)
test_seasdiff <- diff(clean_test, lag = 7)
mean_diff <- meanf(train_seasdiff, h=14, level= 95)

Now, I am in doubt regarding the composition of the diffinv() I should use, so I have two options:

# undoes the seasonal diff on the forecast
# Option 1
inv_f <- diffinv(mean_diff$mean, lag=7, xi=clean_train[1017:1023])
# Option 2
inv_f <- diffinv(mean_diff$mean, lag=1, xi=clean_train[1023])

The plots that come out of these two commands are the following:

Option 1 Option 1

Opion 2 Option 2

And the code is:

plot(clean_train,
     type="l",
     xaxt="n",
     main = 'Mean Forecast Level',
     xlab = 'Date',
     ylab = 'Number of patients', 
     xlim = c(2017.78,2017.85),
     ylim =c(20, 100))
lines(clean_test, col = 'red', lty = 2)
lines(inv_f, col="blue") 

Which option should I use? I am a bit confused with the model, the lags, the seasonality. I am aware the meanf model just gets the mean of the series as forecast, so this would be a lineal forecast in the case of a stationary series, but since I had to un-difference my data to see the actual forecast, I am not sure which shape it should take.

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  • $\begingroup$ Post your data and I will try and help you $\endgroup$
    – IrishStat
    Mar 30, 2020 at 23:48
  • $\begingroup$ @IrishStat I added the last 100 obs from the train set and the firs 100 from the test set, let me know if you need something else $\endgroup$
    – confucius
    Mar 31, 2020 at 0:25
  • $\begingroup$ all you did was post the train set (438 values ) post all 1023 values as all the history should be used to get the best model . By withholding 585 values you are ignoring their model building identification information. and assuming that the best model with parameters didn't change . $\endgroup$
    – IrishStat
    Mar 31, 2020 at 16:59
  • $\begingroup$ I put the test set too which completes the 1461 obs. $\endgroup$
    – confucius
    Mar 31, 2020 at 17:28
  • $\begingroup$ Anyways, I dont need help with deciding the model that would best forecast this. I'm just trying to find the correct way to un-difference the forecasted values. The mean forecast is just an example of a forecasting technique that would require my data to be stationary to work. $\endgroup$
    – confucius
    Mar 31, 2020 at 17:32

1 Answer 1

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the title of your question confuses me along with the amount of observed data you are trying to use . are you eying to identify with 823 .. re-estimate parameters of the model developed on the 823 using 823 + 100 and them with this model and re-tuned parameters make a forecast for observations starting at 924 ?

Since you didn't proved the data for the 823 values... I could not develop a model for that time range.

Since time series models are based upon capturing the signal/equation from history and measuring it's true out-of-sample accuracy in order to place reasonable expectations going into the true future … Based upon a specified horizon length ( say 7 periods) I would take the 200 observations and go back to period 193, identify a model and parameters and forecast the next 7 periods (observations 194-200 ) and obtain a weighted mape of 18.67 .

I would then go back to period 186 develop a model and parameters and predict observations 187-193 and obtain 34.36. Doing this 12 more times yields the following table showing the actual summed and forecast summed the 14th iteration predicts observations 103-109

Note that a new model is developed using all of the data up to and including the values observed before the forecast.

I am unfamiliar with what you are trying to do and can't answer your question as I don't precisely know what you want. One additional option is to identify a model say with 102 values and fix the model and parameters ( God knows why ! ) and use that model and parameters to generate the 14 sets of 7 period out forecasts . Here are the results for the 14 origins using a 7 period out-of-sample .

enter image description here

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  • $\begingroup$ I have 1461 daily observations from which 1023 are Train set and 438 are Test set. I need to forecast the next 7 days after the test set. The method I am applying is fitting many models to the train set, forecasting 7 days from there and comparing that to the first 7 days of the test set, to get an idea of accuracy. After the model's parameters are chosen that way, I will fit the test set with those parameters and forecast 7 more days, for which I obviously don't have data to compare them to. $\endgroup$
    – confucius
    Mar 31, 2020 at 14:52
  • $\begingroup$ @IrisihStat Since my data is non stationary, to apply methods like naive forecast, mean forecast, MA, SES, I need to make the data stationary. To do that I applied a seasonal difference with lag 7 (I tested it and that makes it stationary). I can run the models with no problem using the differenced/stationary data, but those forecasts are equivalent to the differenced forecasts and not the "level" forecasts. SO what I wanted to know was how to properly "undifference" the forecasts to get the level forecasts $\endgroup$
    – confucius
    Mar 31, 2020 at 14:53
  • $\begingroup$ I tried pasting all 1461 values but thought it was a bit much to be honest... that's why I thought just 100 observations was enough to run a mean model... should I just paste all the observations on the initial post and that's it? $\endgroup$
    – confucius
    Mar 31, 2020 at 14:55
  • $\begingroup$ OK so with 1461 daily values (post them with starting date and country of origin ) what precisely do you want to do .? do you want to withhold observations …and identify a model and then make a projection ,,for how long and from what origins ? if you have differenced up front hen simply reverse that process to get forecasts in the original metric. $\endgroup$
    – IrishStat
    Mar 31, 2020 at 15:46
  • $\begingroup$ " if you have differenced up front hen simply reverse that process to get forecasts in the original metric. " that's exactly what I want to do but I'm not sure how to do it, that's why I did it two different ways in my original post but I am not sure which is the correct one. $\endgroup$
    – confucius
    Mar 31, 2020 at 16:39

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