Getting Started with NNS: Forecasting

Fred Viole

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

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 = NNS.ARMA(AirPassengers, 
               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

Nonlinear Regression

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

nns = NNS.ARMA(AirPassengers, 
               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

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, ncores = 1) - 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, 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

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 corrsponding periods, as well as the NNS.ARMA regression method used.

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

nns.optimal = NNS.ARMA.optim(AirPassengers, 
                             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 forecast when using the optimum parameters from the NNS.ARMA.optim call.

sqrt(mean((nns+nns.optimal$bias.shift - tail(AirPassengers, 44)) ^ 2))
## [1] 19.49762

Negative values and $bias.shift

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

nns <- pmax(0, nns+nns.optimal$bias.shift)
sqrt(mean((nns - tail(AirPassengers, 44)) ^ 2))
## [1] 19.49762

Extension of Estimates

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

Brief Notes on Other Parameters

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.

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".

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

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.

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.

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.

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.

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.

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:

References

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