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)
## [1] 35.39965Nonlinear Regression
Now we can try using a nonlinear regression on the relevant component
series using method = "nonlin".
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 75.16192Now 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.
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)
## [1] 20.19599Note: 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.
## $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 24Cross-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 = 20.1959877511828"
[1] "BEST method = 'nonlin' PATH MEMBER = c( 12 )"
[1] "BEST nonlin OBJECTIVE FUNCTION = 20.1959877511828"
[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 = 19.5082249052739"
[1] "BEST method = 'both' PATH MEMBER = c( 12 )"
[1] "BEST both OBJECTIVE FUNCTION = 19.5082249052739"
$periods
[1] 12
$weights
NULL
$obj.fn
[1] 19.50822
$method
[1] "both"
$shrink
[1] FALSE
$nns.regress
[1] FALSE
$bias.shift
[1] 11.34026
$errors
[1] -12.0495905 -19.5023885 -18.2981119 -30.4665605 -21.9967015 -16.3628298 -12.6732257 -4.2894621 -2.6001984
[10] 2.4174837 16.6574755 24.0964052 12.0029210 7.8864972 -0.7526824 -26.4198893 13.6743157 1.1898601
[19] 9.1072756 24.6494525 7.8148305 5.9940877 5.8100458 8.9687243 -11.4805859 12.7282091 -12.4809879
[28] -36.8363281 -8.2269378 -16.1171482 -15.1770286 -12.3754742 -19.5291985 -16.2952067 25.2265398 -24.0729594
[37] -30.8466826 -9.3810198 -42.3392122 -41.2587312 -17.0627050 -40.1705358 -16.0734602 -9.0786139
$results
[1] 354.2907 413.8379 458.0421 447.8737 393.3436 341.9774 303.6670 343.0508 348.7401 331.7577 389.9977 383.4367
[13] 380.9157 445.1133 493.9374 481.8833 422.2942 367.7482 326.1741 368.2520 374.0967 354.1451 416.6348 410.9915
[25] 410.2864 480.0025 531.6140 519.3302 454.3068 395.5812 350.6175 395.6273 399.2349 376.1436 443.3473 438.1093
[37] 438.3493 513.6632 569.8738 555.3985 485.0235 422.3001 374.1886 422.0488
$lower.pred.int
[1] 301.7733 361.3205 405.5248 395.3563 340.8262 289.4601 251.1497 290.5334 296.2227 279.2404 337.4804 330.9193
[13] 328.3984 392.5960 441.4200 429.3659 369.7768 315.2308 273.6567 315.7346 321.5793 301.6277 364.1174 358.4741
[25] 357.7690 427.4851 479.0966 466.8128 401.7894 343.0638 298.1001 343.1099 346.7175 323.6262 390.8299 385.5920
[37] 385.8320 461.1459 517.3564 502.8811 432.5061 369.7827 321.6712 369.5314
$upper.pred.int
[1] 390.2389 449.7861 493.9904 483.8219 429.2918 377.9257 339.6153 378.9990 384.6883 367.7060 425.9460 419.3849
[13] 416.8640 481.0616 529.8856 517.8315 458.2425 403.6964 362.1223 404.2002 410.0450 390.0933 452.5830 446.9397
[25] 446.2347 515.9507 567.5622 555.2784 490.2550 431.5294 386.5657 431.5755 435.1831 412.0918 479.2956 474.0576
[37] 474.2976 549.6115 605.8220 591.3467 520.9717 458.2483 410.1368 457.9970
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.