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I have been running different forecasting algorithms such as Facebook Prophet and Forecast (from R) and I note that despite all my time series values being positive, my seasonality values are negative.

For instance, here is the airline passengers dataset with no negative values-

     Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1949 112 118 132 129 121 135 148 148 136 119 104 118
1950 115 126 141 135 125 149 170 170 158 133 114 140
1951 145 150 178 163 172 178 199 199 184 162 146 166
1952 171 180 193 181 183 218 230 242 209 191 172 194
1953 196 196 236 235 229 243 264 272 237 211 180 201
1954 204 188 235 227 234 264 302 293 259 229 203 229
1955 242 233 267 269 270 315 364 347 312 274 237 278
1956 284 277 317 313 318 374 413 405 355 306 271 306
1957 315 301 356 348 355 422 465 467 404 347 305 336
1958 340 318 362 348 363 435 491 505 404 359 310 337
1959 360 342 406 396 420 472 548 559 463 407 362 405
1960 417 391 419 461 472 535 622 606 508 461 390 432

But the decomposed plot for it shows the following (taken from https://aneesha.shinyapps.io/ShinyTimeseriesForecasting/)-

enter image description here

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    $\begingroup$ observed = trend + seasonal + random where the last two are centred at zero. $\endgroup$
    – Nick Cox
    Commented Jul 15, 2020 at 17:21
  • $\begingroup$ @NickCox: do you want to post your comment(s) as an answer? Better to have a short answer than no answer at all. Anyone who has a better answer can post it. $\endgroup$ Commented Jul 15, 2020 at 18:44
  • $\begingroup$ @StephanKolassa Flattering that my comment is thought worth expanding and fair to be reminded that a better answer is certainly possible! $\endgroup$
    – Nick Cox
    Commented Jul 15, 2020 at 18:57

1 Answer 1

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The recipe followed is

observed $=$ trend $+$ seasonal $+$ random

where the last two are centred at zero. Commonly, trend is long-term change, seasonal is systematic departure given time of year and random is whatever is unpredicted otherwise.

That definition leaves scope for trend and seasonal terms to be estimated or fitted in various ways.

In this particular case, and many others, observed seasonality is itself variable over time: for example the peak month flips between July and August with occasional ties. But the graph shows that precisely the same seasonal terms are fitted additively for each month in every year. More complicated recipes allow seasonality to be more complicated, meaning variable.

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    $\begingroup$ Good answer. I'll add that additive seasonality is centered around 0, multiplicative seasonality is centered around 1. $\endgroup$
    – zbicyclist
    Commented Jul 16, 2020 at 13:38
  • $\begingroup$ Indeed; not to get side-tracked on the example but working on logarithmic scale (equivalently, a multiplicative model) looks a serious candidate for this dataset. The amplitude of seasonal fluctuation is certainly increasing. I think I've even used the example data myself in some paper or other. $\endgroup$
    – Nick Cox
    Commented Jul 16, 2020 at 15:52

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