EDIT: FIXED -- Computational instability in R Forecast package?

Question

Original Question:

I have the following time series data observed daily:

series <- c(10, 25, 8, 27, 18, 21, 12, 9, 31, 18, 8, 30, 14, 13, 10, 14, 
  14, 14, 6, 9, 22, 21, 22, 8, 7, 6, 22, 21, 36, 16, 2, 13, 23, 
  40, 12, 27, 18, 10, 11, 37, 44, 30, 40, 25, 13, 11, 58, 56, 46, 
  39, 28, 27, 19, 20, 97, 90, 70, 73, 30, 22, 97, 34)

and want to fit it using tbats from the R forecasts package. I also want to model it with weekly correlation:

 library(forecast)
 x.msts = msts(series,seasonal.periods = 7)
 model <- tbats(x.msts)
 # shows "--- loading profile ---"

Examing/plotting the model with str reveals a huge fitted variance of 4.9e+17.

And, plotting the forecast going forward, we observe massive swings:

> forecast(model)$mean

 Multi-Seasonal Time Series:
 Start: 9 7
 Seasonal Periods: 7
 Data:
 [1]  1.483789e+44 -1.399297e+42 -2.566455e+44 -1.374316e+43 -1.527758e+38
 [6]  2.036194e+42  5.639596e+42  8.231600e+40 -2.578859e+41 -1.355840e+43

Are these estimates the "correct" solution to the TBATS model fitting procedure, or is there a bug in the forecast package? If not a bug, can someone help me understanding mathematically why this normal-looking time series produces these estimates?

This is my first post on CV so apologies if this should be on SO!

Post-answer update:

I have filed a bug report on github

Also some people have noticed that I'm not using multiple seasonality factors, so I want to show here that the bug is still an issue:

x2.msts <- msts(series,seasonal.periods = c(7,30))
model_x2_1 <- tbats(x2.msts) # high variance
model_x2_2 <- tbats( series, seasonal.periods = c(7,30) ) # also high variance

Answer 1

This is perhaps the same problem as described here, so the reason is presumably a bug in the forecast package. I'm not sure if the following alternative will give you the desired result, but you can leave series as is and put seasonal.periods=7 in the call of tbats:

library(forecast)

series <- c(10, 25, 8, 27, 18, 21, 12, 9, 31, 18, 8, 30, 14, 13, 10, 14, 
            14, 14, 6, 9, 22, 21, 22, 8, 7, 6, 22, 21, 36, 16, 2, 13, 23, 
            40, 12, 27, 18, 10, 11, 37, 44, 30, 40, 25, 13, 11, 58, 56, 46, 
            39, 28, 27, 19, 20, 97, 90, 70, 73, 30, 22, 97, 34)

x.msts <- msts(series,seasonal.periods = 7)
model_1 <- tbats(x.msts)

model_2 <- tbats( series, seasonal.periods = 7 )

The variance of model_2 is much better than that of model_1:

> str(model_1)
List of 19
 $ lambda           : num 0.21
 $ alpha            : num 0.374
 $ beta             : NULL
 $ damping.parameter: NULL
 $ gamma.values     : NULL
 $ ar.coefficients  : num [1:2] 1.296 -0.911
 $ ma.coefficients  : num [1:2] -1.62 0.98
 $ likelihood       : num 549
 $ optim.return.code: int 0
 $ variance         : num 4.9e+17
 $ AIC              : num 571
 $ parameters       :List of 2
  ..$ vect   : num [1:6] 0.21 0.374 1.296 -0.911 -1.615 ...
  ..$ control:List of 6
  .. ..$ use.beta    : logi FALSE
  .. ..$ use.box.cox : logi TRUE
  .. ..$ use.damping : logi FALSE
  .. ..$ length.gamma: num 0
  .. ..$ p           : int 2
  .. ..$ q           : int 2
 $ seed.states      : num [1:5, 1] 4.16 0 0 0 0
 $ fitted.values    : Time-Series [1:62] from 1 to 9.71: 19.97 19.28 4.53 21.83 56.15 ...
  ..- attr(*, "msts")= num 7
 $ errors           : Time-Series [1:62] from 1 to 9.71: -1.206 0.496 0.828 0.415 -2.354 ...
  ..- attr(*, "msts")= num 7
 $ x                : num [1:5, 1:62] 3.71 -1.21 0 -1.21 0 ...
 $ seasonal.periods : NULL
 $ y                : Time-Series [1:62] from 1 to 9.71: 10 25 8 27 18 21 12 9 31 18 ...
  ..- attr(*, "msts")= num 7
 $ call             : language tbats(y = x.msts)
 - attr(*, "class")= chr "bats"
> 

.

> str(model_2)
List of 23
 $ lambda           : num 0.198
 $ alpha            : num 0.198
 $ beta             : NULL
 $ damping.parameter: NULL
 $ gamma.one.values : num -0.0157
 $ gamma.two.values : num 0.00991
 $ ar.coefficients  : NULL
 $ ma.coefficients  : NULL
 $ likelihood       : num 553
 $ optim.return.code: int 0
 $ variance         : num 0.969
 $ AIC              : num 571
 $ parameters       :List of 2
  ..$ vect   : num [1:4] 0.19842 0.19782 -0.0157 0.00991
  ..$ control:List of 6
  .. ..$ use.beta    : logi FALSE
  .. ..$ use.box.cox : logi TRUE
  .. ..$ use.damping : logi FALSE
  .. ..$ length.gamma: int 2
  .. ..$ p           : num 0
  .. ..$ q           : num 0
 $ seed.states      : num [1:5, 1] 4.1851 0.3176 0.0103 -0.5806 0.4447
 $ fitted.values    : Time-Series [1:62] from 1 to 62: 25.1 20 11.1 10.2 24.3 ...
 $ errors           : Time-Series [1:62] from 1 to 62: -1.594 0.41 -0.507 1.697 -0.552 ...
 $ x                : num [1:5, 1:62] 3.87 -0.231 0.456 -0.626 -0.125 ...
 $ seasonal.periods : num 7
 $ k.vector         : int 2
 $ y                : Time-Series [1:62] from 1 to 62: 10 25 8 27 18 21 12 9 31 18 ...
 $ p                : num 0
 $ q                : num 0
 $ call             : language tbats(y = series, seasonal.periods = 7)
 - attr(*, "class")= chr [1:2] "tbats" "bats"
>