Author: Fred Viole

# Getting Started with NNS: Forecasting

#### Fred Viole

    library(NNS)
    library(data.table)
    require(knitr)
    require(rgl)

# Forecasting

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 assumption free, minimal parameter forecasting method.

## Linear Regression

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

    sqrt(mean((nns_lin - tail(AirPassengers, 44)) ^ 2))

    ## [1] 35.39965

## Nonlinear Regression

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

    nns_nonlin = NNS.ARMA(AirPassengers,
                   h = 44,
                   training.set = 100,
                   method = "nonlin",
                   plot = FALSE,
                   seasonal.factor = 12,
                   seasonal.plot = FALSE)

    sqrt(mean((nns_nonlin - tail(AirPassengers, 44)) ^ 2))

    [1] 18.1809

## Cross-Validation

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.

    seas = 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) - tail(AirPassengers, 44)) ^ 2) ) ) ) )

    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.

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

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

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

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

    ## [1] 18.1809

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     24                0.4279947                         0.4279947
    ## 4     36                0.4279947                         0.4279947
    ## 5     60                0.4279947                         0.4279947
    ##
    ## $best.period
    ## [1] 48
    ##
    ## $periods
    ## [1] 48 12 24 36 60

## Cross-Validating All Combinations of `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 corresponding periods, as well as the `NNS.ARMA` regression method used. Alternatively, using external package objective functions work as well such as `obj.fn = expression(Metrics::rmse(actual, predicted))`.

`NNS.ARMA.optim()` will also test whether to regress the underlying data first, `shrink` the estimates to their subset mean values, include a `bias.shift` based on its internal validation errors, and compare different `weights` of both linear and nonlinear estimates.

Given our monthly dataset, we will try multiple years by setting `seasonal.factor = seq(12, 60, 6)` every 6 months based on our NNS.seas() insights above.

    nns.optimal = NNS.ARMA.optim(AirPassengers,
                                 training.set = 100,
                                 seasonal.factor = seq(12, 60, 6),
                                 obj.fn = expression( sqrt(mean((predicted - actual)^2)) ),
                                 objective = "min",
                                 pred.int = .95, plot = TRUE)

    nns.optimal

    [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 = 18.1809033101955"
    [1] "BEST method = 'nonlin' PATH MEMBER = c( 12 )"
    [1] "BEST nonlin OBJECTIVE FUNCTION = 18.1809033101955"
    [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 = 22.7363330823967"
    [1] "BEST method = 'both' PATH MEMBER = c( 12 )"
    [1] "BEST both OBJECTIVE FUNCTION = 22.7363330823967"
    >
    > nns.optimal
    $periods
    [1] 12

    $weights
    NULL

    $obj.fn
    [1] 18.1809

    $method
    [1] "nonlin"

    $shrink
    [1] FALSE

    $nns.regress
    [1] FALSE

    $bias.shift
    [1] 0

    $errors
     [1]  -6.0626221 -10.8434613 -10.7646998 -22.7134790 -15.3519569 -12.9673866  -9.1626428   3.9393939   7.4882812  12.3750000  29.1132812  34.3281250  19.7002739
    [14]  20.0656989  11.8833952 -15.1389735  24.1108241   7.4289721  15.2385271  38.3826941  19.2903993  17.4644272  19.3331767  19.8155057  -4.0856291  26.3260739
    [27]   2.6153110 -24.3491085   3.9057436  -8.8271346  -7.9236143   5.9867956  -3.9068174  -0.7986170  42.1995863 -10.1324609 -20.0852820   8.6573328 -21.3067790
    [40] -24.3403514  -0.6332912 -29.8418247  -5.8572216  14.8998761

    $results
     [1] 348.9374 411.1565 454.2353 444.2865 388.6480 334.0326 295.8374 339.9394 347.4883 330.3750 391.1133 382.3281 382.7003 455.0657 502.8834 489.8610 428.1108
    [18] 366.4290 325.2385 375.3827 379.2904 359.4644 425.3332 415.8155 415.9144 498.3261 550.6153 534.6509 466.9057 398.1729 354.0764 410.9868 413.0932 390.2014
    [35] 461.1996 450.8675 451.9147 543.6573 600.6932 581.6596 507.3667 431.1582 384.1428 446.8999

    $lower.pred.int
     [1] 310.8588 373.0779 416.1567 406.2079 350.5694 295.9540 257.7588 301.8608 309.4097 292.2964 353.0347 344.2495 344.6217 416.9871 464.8048 451.7824 390.0322
    [18] 328.3504 287.1599 337.3041 341.2118 321.3858 387.2546 377.7369 377.8358 460.2475 512.5367 496.5723 428.8271 360.0943 315.9978 372.9082 375.0146 352.1228
    [35] 423.1210 412.7889 413.8361 505.5787 562.6146 543.5810 469.2881 393.0796 346.0642 408.8213

    $upper.pred.int
     [1] 387.0160 449.2351 492.3139 482.3651 426.7266 372.1112 333.9160 378.0180 385.5669 368.4536 429.1919 420.4067 420.7789 493.1443 540.9620 527.9396 466.1894
    [18] 404.5076 363.3171 413.4613 417.3690 397.5430 463.4118 453.8941 453.9930 536.4047 588.6939 572.7295 504.9843 436.2515 392.1550 449.0654 451.1718 428.2800
    [35] 499.2782 488.9461 489.9933 581.7359 638.7718 619.7382 545.4453 469.2368 422.2214 484.9785

## Extension of Estimates

We can forecast another 50 periods out-of-sample (`h = 50`), by dropping the `training.set` parameter while generating the 95% prediction intervals.

    NNS.ARMA.optim(AirPassengers,
                    seasonal.factor = seq(12, 60, 6),
                    obj.fn = expression( sqrt(mean((predicted - actual)^2)) ),
                    objective = "min",
                    pred.int = .95, h = 50, plot = TRUE)

## Brief Notes on Other Parameters

- `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"`.

- `pred.int`

Provides the values for the specified prediction 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.

# Multivariate Time Series Forecasting

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

- Multivariate Time Series Forecasting: Nonparametric Vector Autoregression Using NNS

# References

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

- Nonlinear Nonparametric Statistics: Using Partial Moments

- Forecasting Using NNS