Feed-forward neural networks with a single hidden layer and lagged inputs for forecasting univariate time series.

```
nnetar(
y,
p,
P = 1,
size,
repeats = 20,
xreg = NULL,
lambda = NULL,
model = NULL,
subset = NULL,
scale.inputs = TRUE,
x = y,
...
)
```

- y
A numeric vector or time series of class

`ts`

.- p
Embedding dimension for non-seasonal time series. Number of non-seasonal lags used as inputs. For non-seasonal time series, the default is the optimal number of lags (according to the AIC) for a linear AR(p) model. For seasonal time series, the same method is used but applied to seasonally adjusted data (from an stl decomposition).

- P
Number of seasonal lags used as inputs.

- size
Number of nodes in the hidden layer. Default is half of the number of input nodes (including external regressors, if given) plus 1.

- repeats
Number of networks to fit with different random starting weights. These are then averaged when producing forecasts.

- xreg
Optionally, a vector or matrix of external regressors, which must have the same number of rows as

`y`

. Must be numeric.- lambda
Box-Cox transformation parameter. If

`lambda="auto"`

, then a transformation is automatically selected using`BoxCox.lambda`

. The transformation is ignored if NULL. Otherwise, data transformed before model is estimated.- model
Output from a previous call to

`nnetar`

. If model is passed, this same model is fitted to`y`

without re-estimating any parameters.- subset
Optional vector specifying a subset of observations to be used in the fit. Can be an integer index vector or a logical vector the same length as

`y`

. All observations are used by default.- scale.inputs
If TRUE, inputs are scaled by subtracting the column means and dividing by their respective standard deviations. If

`lambda`

is not`NULL`

, scaling is applied after Box-Cox transformation.- x
Deprecated. Included for backwards compatibility.

- ...
Other arguments passed to

`nnet`

for`nnetar`

.

Returns an object of class "`nnetar`

".

The function `summary`

is used to obtain and print a summary of the
results.

The generic accessor functions `fitted.values`

and `residuals`

extract useful features of the value returned by `nnetar`

.

- model
A list containing information about the fitted model

- method
The name of the forecasting method as a character string

- x
The original time series.

- xreg
The external regressors used in fitting (if given).

- residuals
Residuals from the fitted model. That is x minus fitted values.

- fitted
Fitted values (one-step forecasts)

- ...
Other arguments

A feed-forward neural network is fitted with lagged values of `y`

as
inputs and a single hidden layer with `size`

nodes. The inputs are for
lags 1 to `p`

, and lags `m`

to `mP`

where
`m=frequency(y)`

. If `xreg`

is provided, its columns are also
used as inputs. If there are missing values in `y`

or
`xreg`

, the corresponding rows (and any others which depend on them as
lags) are omitted from the fit. A total of `repeats`

networks are
fitted, each with random starting weights. These are then averaged when
computing forecasts. The network is trained for one-step forecasting.
Multi-step forecasts are computed recursively.

For non-seasonal data, the fitted model is denoted as an NNAR(p,k) model, where k is the number of hidden nodes. This is analogous to an AR(p) model but with nonlinear functions. For seasonal data, the fitted model is called an NNAR(p,P,k)[m] model, which is analogous to an ARIMA(p,0,0)(P,0,0)[m] model but with nonlinear functions.

```
fit <- nnetar(lynx)
fcast <- forecast(fit)
plot(fcast)
## Arguments can be passed to nnet()
fit <- nnetar(lynx, decay=0.5, maxit=150)
plot(forecast(fit))
lines(lynx)
## Fit model to first 100 years of lynx data
fit <- nnetar(window(lynx,end=1920), decay=0.5, maxit=150)
plot(forecast(fit,h=14))
lines(lynx)
## Apply fitted model to later data, including all optional arguments
fit2 <- nnetar(window(lynx,start=1921), model=fit)
```