All fable models with formula-based model specification support a highly flexible specification of transformations. Specified transformations are automatically back-transformed and bias adjusted to produce forecast means and fitted values on the original scale of the data.

The transformation used for the model is defined on the left of the
tilde (`~`

) in the formula. For example, when forecasting
Melbourne `Trips`

from the `tsibble::tourism`

dataset, a square root transformation can applied using
`sqrt(Trips)`

.

```
library(fable)
library(tsibble)
library(dplyr)
%>%
tourism filter(Region == "Melbourne") %>%
model(ETS(sqrt(Trips)))
#> # A mable: 4 x 4
#> # Key: Region, State, Purpose [4]
#> Region State Purpose `ETS(sqrt(Trips))`
#> <chr> <chr> <chr> <model>
#> 1 Melbourne Victoria Business <ETS(A,N,A)>
#> 2 Melbourne Victoria Holiday <ETS(A,A,A)>
#> 3 Melbourne Victoria Other <ETS(A,N,N)>
#> 4 Melbourne Victoria Visiting <ETS(M,N,A)>
```

Multiple transformations can be combined using this interface,
allowing more complicated transformations to be used. A simple example
of a combined transformation is \(f(x) =
log(x+1)\), as it involves both a `log`

transformation, and a `+1`

transformation. This
transformation is commonly used to overcome a limitation of using log
transformations to preserve non-negativity, on data which contains
zeroes.

Simple combined transformations and backtransformations can be constructed automatically.

```
library(tsibble)
%>%
tourism filter(Region == "Melbourne") %>%
model(ETS(log(Trips + 1)))
#> # A mable: 4 x 4
#> # Key: Region, State, Purpose [4]
#> Region State Purpose `ETS(log(Trips + 1))`
#> <chr> <chr> <chr> <model>
#> 1 Melbourne Victoria Business <ETS(A,N,A)>
#> 2 Melbourne Victoria Holiday <ETS(M,A,A)>
#> 3 Melbourne Victoria Other <ETS(A,N,N)>
#> 4 Melbourne Victoria Visiting <ETS(M,N,A)>
```

It is possible to extend the supported transformations by defining your own transformation with an appropriate back-transformation function. It is assumed that the first argument of your function is your data which is being transformed.

A useful transformation which is not readily supported by fable is
the scaled logit, which allows the forecasts to be bounded by a given
interval (*forecasting
within limits*). The appropriate transformation to ensure the
forecasted values are between \(a\) and
\(b\) (where \(a<b\)) is given by:

\[f(x) = \log\left(\dfrac{x-a}{b-x}\right)\]

Inverting this transformation gives the appropriate back-transformation of:

\[f^{-1}(x) = \dfrac{a + be^x}{1 + e^x} =
\dfrac{(b-a)e^x}{1 + e^x} + a\] To use this transformation for
modelling, we can pair the transformation with its back transformation
using the `new_transformation`

function from
`fabletools`

. This function which accepts two inputs: first
the transformation, and second the back-transformation.

```
<- function(x, lower=0, upper=1){
scaled_logit log((x-lower)/(upper-x))
}<- function(x, lower=0, upper=1){
inv_scaled_logit -lower)*exp(x)/(1+exp(x)) + lower
(upper
}<- new_transformation(scaled_logit, inv_scaled_logit) my_scaled_logit
```

Once you define your transformation as above, it is ready to use anywhere you would normally use a transformation.

```
cbind(mdeaths, fdeaths) %>%
as_tsibble(pivot_longer = FALSE) %>%
model(ETS(my_scaled_logit(mdeaths, 750, 3000) ~
error("A") + trend("N") + season("A"))) %>%
report()
#> Series: mdeaths
#> Model: ETS(A,N,A)
#> Transformation: my_scaled_logit(mdeaths, 750, 3000)
#> Smoothing parameters:
#> alpha = 0.1427297
#> gamma = 0.0001000037
#>
#> Initial states:
#> l[0] s[0] s[-1] s[-2] s[-3] s[-4] s[-5]
#> -0.576581 0.7181113 -0.1305675 -0.4690919 -1.197746 -1.114423 -0.7727465
#> s[-6] s[-7] s[-8] s[-9] s[-10] s[-11]
#> -0.6240044 -0.275529 0.4140919 0.9658113 1.272782 1.213311
#>
#> sigma^2: 0.1489
#>
#> AIC AICc BIC
#> 185.2488 193.8202 219.3988
```

When forecasting with transformations, the model is fitted and forecasted using the transformed data. To produce forecasts of the original data, the predicted values must be back-transformed. However this process of predicting transformed data and backtransforming predictions usually results in producing forecast medians. To convert the forecast medians into forecast means, a transformation bias adjustment is required:

\[\hat{y} = f^{-1}(\tilde{y}) + \dfrac{1}{2}\sigma^2\dfrac{\partial^2}{\partial \tilde{y}^2}f^{-1}(\tilde{y})\] Note that the forecast medians are given by \(f^{-1}(\tilde{y})\), and the adjustment needed to produce forecast means (\(\hat{y}\)) is \(\dfrac{1}{2}\sigma^2\dfrac{\partial^2}{\partial \tilde{y}^2}f^{-1}(\tilde{y})\).

The fable package automatically produces forecast means (by
back-transforming and adjusting the transformed forecasts). The forecast
medians can be obtained via the forecast intervals when
`level=0`

.

More information about adjusting forecasts to compute forecast means
can be found at *the
forecast mean after back-transformation*.