```
library(NNS)
library(data.table)
require(knitr)
require(rgl)
require(tdigest)
require(dtw)
require(meboot)
```

The underlying assumptions of traditional autoregressive models are well known. The resulting complexity with these models leads to observations such as,

*``We have found that choosing the wrong model or parameters can
often yield poor results, and it is unlikely that even experienced
analysts can choose the correct model and parameters efficiently given
this array of choices.’’*

`NNS`

simplifies the forecasting process. Below are some
examples demonstrating ** NNS.ARMA** and its

** NNS.ARMA** has the ability to fit a
linear regression to the relevant component series, yielding very fast
results. For our running example we will use the

`AirPassengers`

dataset loaded in base R.We will forecast 44 periods `h = 44`

of
`AirPassengers`

using the first 100 observations
`training.set = 100`

, returning estimates of the final 44
observations. We will then test this against our validation set of
`tail(AirPassengers,44)`

.

Since this is monthly data, we will try a
`seasonal.factor = 12`

.

Below is the linear fit and associated root mean squared error (RMSE)
using `method = "lin"`

.

```
= NNS.ARMA(AirPassengers,
nns h = 44,
training.set = 100,
method = "lin",
plot = TRUE,
seasonal.factor = 12,
seasonal.plot = FALSE, ncores = 1)
```

`sqrt(mean((nns - tail(AirPassengers, 44)) ^ 2))`

`## [1] 35.39965`

Now we can try using a nonlinear regression on the relevant component
series using `method = "nonlin"`

.

```
= NNS.ARMA(AirPassengers,
nns h = 44,
training.set = 100,
method = "nonlin",
plot = FALSE,
seasonal.factor = 12,
seasonal.plot = FALSE, ncores = 1)
sqrt(mean((nns - tail(AirPassengers, 44)) ^ 2))
```

`## [1] 19.49762`

We can test a series of `seasonal.factors`

and select the
best one to fit. The largest period to consider would be
`0.5 * length(variable)`

, since we need more than 2 points
for a regression! Remember, we are testing the first 100 observations of
`AirPassengers`

, not the full 144 observations.

```
= t(sapply(1 : 25, function(i) c(i, sqrt( mean( (NNS.ARMA(AirPassengers, h = 44, training.set = 100, method = "lin", seasonal.factor = i, plot=FALSE, ncores = 1) - tail(AirPassengers, 44)) ^ 2) ) ) ) )
seas
colnames(seas) = c("Period", "RMSE")
seas
```

```
## Period RMSE
## [1,] 1 75.67783
## [2,] 2 75.71250
## [3,] 3 75.87604
## [4,] 4 75.16563
## [5,] 5 76.07418
## [6,] 6 70.43185
## [7,] 7 77.98493
## [8,] 8 75.48997
## [9,] 9 79.16378
## [10,] 10 81.47260
## [11,] 11 106.56886
## [12,] 12 35.39965
## [13,] 13 90.98265
## [14,] 14 95.64979
## [15,] 15 82.05345
## [16,] 16 74.63052
## [17,] 17 87.54036
## [18,] 18 74.90881
## [19,] 19 96.96011
## [20,] 20 88.75015
## [21,] 21 100.21346
## [22,] 22 108.68674
## [23,] 23 85.06430
## [24,] 24 35.49018
## [25,] 25 75.16192
```

Now we know `seasonal.factor = 12`

is our best fit, we can
see if there’s any benefit from using a nonlinear regression.
Alternatively, we can define our best fit as the corresponding
`seas$Period`

entry of the minimum value in our
`seas$RMSE`

column.

`= seas[which.min(seas[ , 2]), 1] a `

Below you will notice the use of `seasonal.factor = a`

generates the same output.

```
= NNS.ARMA(AirPassengers,
nns h = 44,
training.set = 100,
method = "nonlin",
seasonal.factor = a,
plot = TRUE, seasonal.plot = FALSE, ncores = 1)
```

`sqrt(mean((nns - tail(AirPassengers, 44)) ^ 2))`

`## [1] 19.49762`

**Note:** You may experience instances with monthly data
that report `seasonal.factor`

close to multiples of 3, 4, 6
or 12. For instance, if the reported
`seasonal.factor = {37, 47, 71, 73}`

use
`(seasonal.factor = c(36, 48, 72))`

by setting the
`modulo`

parameter in
** NNS.seas(..., modulo = 12)**. The same
suggestion holds for daily data and multiples of 7, or any other time
series with logically inferred cyclical patterns. The nearest periods to
that

`modulo`

will be in the expanded output.`NNS.seas(AirPassengers, modulo = 12, plot = FALSE)`

```
## $all.periods
## Period Coefficient.of.Variation Variable.Coefficient.of.Variation
## 1: 48 0.4002249 0.4279947
## 2: 12 0.4059923 0.4279947
## 3: 60 0.4279947 0.4279947
## 4: 36 0.4279947 0.4279947
## 5: 24 0.4279947 0.4279947
##
## $best.period
## Period
## 48
##
## $periods
## [1] 48 12 60 36 24
```

`seasonal.factor`

NNS also offers a wrapper function
** NNS.ARMA.optim()** to test a given vector of

`seasonal.factor`

and returns the optimized objective
function (in this case RMSE written as
`obj.fn = expression( sqrt(mean((predicted - actual)^2)) )`

)
and the corrsponding periods, as well as the
`NNS.ARMA`

Given our monthly dataset, we will try multiple years by setting
`seasonal.factor = seq(12, 24, 6)`

every 6 months.

```
= NNS.ARMA.optim(AirPassengers,
nns.optimal training.set = 100,
seasonal.factor = seq(12, 24, 6),
obj.fn = expression( sqrt(mean((predicted - actual)^2)) ),
objective = "min",
ncores = 1)
```

```
## [1] "CURRNET METHOD: lin"
## [1] "COPY LATEST PARAMETERS DIRECTLY FOR NNS.ARMA() IF ERROR:"
## [1] "NNS.ARMA(... method = 'lin' , seasonal.factor = c( 12 ) ...)"
## [1] "CURRENT lin OBJECTIVE FUNCTION = 35.3996540135277"
## [1] "BEST method = 'lin', seasonal.factor = c( 12 )"
## [1] "BEST lin OBJECTIVE FUNCTION = 35.3996540135277"
## [1] "CURRNET METHOD: nonlin"
## [1] "COPY LATEST PARAMETERS DIRECTLY FOR NNS.ARMA() IF ERROR:"
## [1] "NNS.ARMA(... method = 'nonlin' , seasonal.factor = c( 12 ) ...)"
## [1] "CURRENT nonlin OBJECTIVE FUNCTION = 19.4976178189546"
## [1] "BEST method = 'nonlin' PATH MEMBER = c( 12 )"
## [1] "BEST nonlin OBJECTIVE FUNCTION = 19.4976178189546"
## [1] "CURRNET METHOD: both"
## [1] "COPY LATEST PARAMETERS DIRECTLY FOR NNS.ARMA() IF ERROR:"
## [1] "NNS.ARMA(... method = 'both' , seasonal.factor = c( 12 ) ...)"
## [1] "CURRENT both OBJECTIVE FUNCTION = 26.6112299452096"
## [1] "BEST method = 'both' PATH MEMBER = c( 12 )"
## [1] "BEST both OBJECTIVE FUNCTION = 26.6112299452096"
```

` nns.optimal`

```
## $periods
## [1] 12
##
## $weights
## NULL
##
## $obj.fn
## [1] 19.49762
##
## $method
## [1] "nonlin"
##
## $shrink
## [1] FALSE
##
## $bias.shift
## [1] 0
##
## $errors
## [1] -12.0495905 -19.5023885 -18.2981119 -30.4665605 -21.9967015 -16.3628298
## [7] -12.6732257 -5.7137170 -2.6001984 2.2792659 17.1994048 24.2420635
## [13] 6.6919485 -1.2269250 -8.4029057 -34.4569779 6.9539623 -2.5920976
## [19] 4.8338436 18.5863427 1.8098569 -0.3087157 -1.1892791 2.5325891
## [25] -22.4687006 -4.9819699 -27.7262972 -52.7041072 -21.5667488 -23.9122298
## [31] -23.6982624 -23.0856682 -29.9142644 -27.1628466 12.6507957 -35.1714729
## [37] -46.1877025 -34.0820674 -63.4664903 -63.3893474 -35.6270575 -51.0256013
## [43] -27.9853043 -23.5848310
##
## $results
## [1] 354.2580 421.2452 462.4395 453.0669 395.8280 338.4172 301.1178 338.6083
## [9] 347.7440 330.7530 393.0655 383.2619 390.9250 468.8563 511.8161 501.4936
## [17] 436.7415 370.9154 331.3098 371.0849 380.7716 361.0259 430.2580 418.6685
## [25] 427.7316 516.8815 561.5732 550.3086 478.0325 403.7194 361.7944 403.9807
## [33] 413.6136 390.9586 467.3674 453.9804 464.4469 564.6356 611.0813 598.8694
## [41] 519.0765 436.3233 392.0875 436.6022
```

Using our new parameters via the `nns.optimal$results`

yields the same results:

`sqrt(mean((nns.optimal$results - tail(AirPassengers, 44)) ^ 2))`

`## [1] 19.49762`

`$bias.shift`

** NNS.ARMA.optim** will return a

`$bias.shift`

, which is to be added to the ultimate
`NNS.ARMA`

`NNS.ARMA.optim`

`sqrt(mean((nns+nns.optimal$bias.shift - tail(AirPassengers, 44)) ^ 2))`

`## [1] 19.49762`

`$bias.shift`

If the variable cannot logically assume negative values, then simply
limit the `NNS`

estimates.

```
<- pmax(0, nns+nns.optimal$bias.shift)
nns sqrt(mean((nns - tail(AirPassengers, 44)) ^ 2))
```

`## [1] 19.49762`

Using our cross-validated parameters (`seasonal.factor`

and `method`

) we can forecast another 50 periods
out-of-sample (`h = 50`

), by dropping the
`training.set`

parameter while generating the 95% confidence
intervals.

```
NNS.ARMA(AirPassengers,
h = 50,
conf.intervals = .95,
seasonal.factor = nns.optimal$periods,
method = nns.optimal$method,
weights = nns.optimal$weights,
plot = TRUE, seasonal.plot = FALSE, ncores = 1) + nns.optimal$bias.shift
```

`seasonal.factor = c(1, 2, ...)`

We included the ability to use any number of specified seasonal periods simultaneously, weighted by their strength of seasonality. Computationally expensive when used with nonlinear regressions and large numbers of relevant periods.

`weights`

Instead of weighting by the `seasonal.factor`

strength of
seasonality, we offer the ability to weight each per any defined
compatible vector summing to 1.

Equal weighting would be `weights = "equal"`

.

`conf.intervals`

Provides the values for the specified confidence intervals within [0,1] for each forecasted point and plots the bootstrapped replicates for the forecasted points.

`seasonal.factor = FALSE`

We also included the ability to use all detected seasonal periods simultaneously, weighted by their strength of seasonality. Computationally expensive when used with nonlinear regressions and large numbers of relevant periods.

`best.periods`

This parameter restricts the number of detected seasonal periods to
use, again, weighted by their strength. To be used in conjunction with
`seasonal.factor = FALSE`

.

`modulo`

To be used in conjunction with `seasonal.factor = FALSE`

.
This parameter will ensure logical seasonal patterns (i.e.,
`modulo = 7`

for daily data) are included along with the
results.

`mod.only`

To be used in conjunction with
`seasonal.factor = FALSE & modulo != NULL`

. This
parameter will ensure empirical patterns are kept along with the logical
seasonal patterns.

`dynamic = TRUE`

This setting generates a new seasonal period(s) using the estimated
values as continuations of the variable, either with or without a
`training.set`

. Also computationally expensive due to the
recalculation of seasonal periods for each estimated value.

`plot`

,`seasonal.plot`

These are the plotting arguments, easily enabled or disabled with
`TRUE`

or `FALSE`

.
`seasonal.plot = TRUE`

will not plot without
`plot = TRUE`

. If a seasonal analysis is all that is desired,
`NNS.seas`

is the function specifically suited for that
task.

The extension to a generalized multivariate instance is provided in
the following documentation of the
** NNS.VAR()** function:

If the user is so motivated, detailed arguments and proofs are provided within the following: