General Interface for "Boosted" ARIMA Regression Models

arima_boost() is a way to generate a specification of a time series model that uses boosting to improve modeling errors (residuals) on Exogenous Regressors. It works with both "automated" ARIMA (auto.arima) and standard ARIMA (arima). The main algorithms are:

Auto ARIMA + XGBoost Errors (engine = auto_arima_xgboost, default)
ARIMA + XGBoost Errors (engine = arima_xgboost)

Usage

arima_boost(
  mode = "regression",
  seasonal_period = NULL,
  non_seasonal_ar = NULL,
  non_seasonal_differences = NULL,
  non_seasonal_ma = NULL,
  seasonal_ar = NULL,
  seasonal_differences = NULL,
  seasonal_ma = NULL,
  mtry = NULL,
  trees = NULL,
  min_n = NULL,
  tree_depth = NULL,
  learn_rate = NULL,
  loss_reduction = NULL,
  sample_size = NULL,
  stop_iter = NULL
)

Arguments

mode: A single character string for the type of model. The only possible value for this model is "regression".
seasonal_period: A seasonal frequency. Uses "auto" by default. A character phrase of "auto" or time-based phrase of "2 weeks" can be used if a date or date-time variable is provided. See Fit Details below.
non_seasonal_ar: The order of the non-seasonal auto-regressive (AR) terms. Often denoted "p" in pdq-notation.
non_seasonal_differences: The order of integration for non-seasonal differencing. Often denoted "d" in pdq-notation.
non_seasonal_ma: The order of the non-seasonal moving average (MA) terms. Often denoted "q" in pdq-notation.
seasonal_ar: The order of the seasonal auto-regressive (SAR) terms. Often denoted "P" in PDQ-notation.
seasonal_differences: The order of integration for seasonal differencing. Often denoted "D" in PDQ-notation.
seasonal_ma: The order of the seasonal moving average (SMA) terms. Often denoted "Q" in PDQ-notation.
mtry: A number for the number (or proportion) of predictors that will be randomly sampled at each split when creating the tree models (specific engines only)
trees: An integer for the number of trees contained in the ensemble.
min_n: An integer for the minimum number of data points in a node that is required for the node to be split further.
tree_depth: An integer for the maximum depth of the tree (i.e. number of splits) (specific engines only).
learn_rate: A number for the rate at which the boosting algorithm adapts from iteration-to-iteration (specific engines only). This is sometimes referred to as the shrinkage parameter.
loss_reduction: A number for the reduction in the loss function required to split further (specific engines only).
sample_size: number for the number (or proportion) of data that is exposed to the fitting routine.
stop_iter: The number of iterations without improvement before stopping (xgboost only).

Details

The data given to the function are not saved and are only used to determine the mode of the model. For arima_boost(), the mode will always be "regression".

The model can be created using the fit() function using the following engines:

"auto_arima_xgboost" (default) - Connects to forecast::auto.arima() and xgboost::xgb.train
"arima_xgboost" - Connects to forecast::Arima() and xgboost::xgb.train

Main Arguments

The main arguments (tuning parameters) for the ARIMA model are:

seasonal_period: The periodic nature of the seasonality. Uses "auto" by default.
non_seasonal_ar: The order of the non-seasonal auto-regressive (AR) terms.
non_seasonal_differences: The order of integration for non-seasonal differencing.
non_seasonal_ma: The order of the non-seasonal moving average (MA) terms.
seasonal_ar: The order of the seasonal auto-regressive (SAR) terms.
seasonal_differences: The order of integration for seasonal differencing.
seasonal_ma: The order of the seasonal moving average (SMA) terms.

The main arguments (tuning parameters) for the model XGBoost model are:

mtry: The number of predictors that will be randomly sampled at each split when creating the tree models.
trees: The number of trees contained in the ensemble.
min_n: The minimum number of data points in a node that are required for the node to be split further.
tree_depth: The maximum depth of the tree (i.e. number of splits).
learn_rate: The rate at which the boosting algorithm adapts from iteration-to-iteration.
loss_reduction: The reduction in the loss function required to split further.
sample_size: The amount of data exposed to the fitting routine.
stop_iter: The number of iterations without improvement before stopping.

These arguments are converted to their specific names at the time that the model is fit.

Other options and argument can be set using set_engine() (See Engine Details below).

If parameters need to be modified, update() can be used in lieu of recreating the object from scratch.

Engine Details

The standardized parameter names in modeltime can be mapped to their original names in each engine:

Model 1: ARIMA:

modeltime	forecast::auto.arima	forecast::Arima
seasonal_period	ts(frequency)	ts(frequency)
non_seasonal_ar, non_seasonal_differences, non_seasonal_ma	max.p(5), max.d(2), max.q(5)	order = c(p(0), d(0), q(0))
seasonal_ar, seasonal_differences, seasonal_ma	max.P(2), max.D(1), max.Q(2)	seasonal = c(P(0), D(0), Q(0))

Model 2: XGBoost:

modeltime	xgboost::xgb.train
tree_depth	max_depth (6)
trees	nrounds (15)
learn_rate	eta (0.3)
mtry	colsample_bynode (1)
min_n	min_child_weight (1)
loss_reduction	gamma (0)
sample_size	subsample (1)
stop_iter	early_stop

Other options can be set using set_engine().

auto_arima_xgboost (default engine)

Model 1: Auto ARIMA (forecast::auto.arima):

#> function (y, d = NA, D = NA, max.p = 5, max.q = 5, max.P = 2, max.Q = 2, 
#>     max.order = 5, max.d = 2, max.D = 1, start.p = 2, start.q = 2, start.P = 1, 
#>     start.Q = 1, stationary = FALSE, seasonal = TRUE, ic = c("aicc", "aic", 
#>         "bic"), stepwise = TRUE, nmodels = 94, trace = FALSE, approximation = (length(x) > 
#>         150 | frequency(x) > 12), method = NULL, truncate = NULL, xreg = NULL, 
#>     test = c("kpss", "adf", "pp"), test.args = list(), seasonal.test = c("seas", 
#>         "ocsb", "hegy", "ch"), seasonal.test.args = list(), allowdrift = TRUE, 
#>     allowmean = TRUE, lambda = NULL, biasadj = FALSE, parallel = FALSE, 
#>     num.cores = 2, x = y, ...)

Parameter Notes:

All values of nonseasonal pdq and seasonal PDQ are maximums. The auto.arima will select a value using these as an upper limit.
xreg - This should not be used since XGBoost will be doing the regression

Model 2: XGBoost (xgboost::xgb.train):

#> function (params = list(), data, nrounds, watchlist = list(), obj = NULL, 
#>     feval = NULL, verbose = 1, print_every_n = 1L, early_stopping_rounds = NULL, 
#>     maximize = NULL, save_period = NULL, save_name = "xgboost.model", xgb_model = NULL, 
#>     callbacks = list(), ...)

Parameter Notes:

XGBoost uses a params = list() to capture. Parsnip / Modeltime automatically sends any args provided as ... inside of set_engine() to the params = list(...).

Fit Details

Date and Date-Time Variable

It's a requirement to have a date or date-time variable as a predictor. The fit() interface accepts date and date-time features and handles them internally.

fit(y ~ date)

Seasonal Period Specification

The period can be non-seasonal (seasonal_period = 1) or seasonal (e.g. seasonal_period = 12 or seasonal_period = "12 months"). There are 3 ways to specify:

seasonal_period = "auto": A period is selected based on the periodicity of the data (e.g. 12 if monthly)
seasonal_period = 12: A numeric frequency. For example, 12 is common for monthly data
seasonal_period = "1 year": A time-based phrase. For example, "1 year" would convert to 12 for monthly data.

Univariate (No xregs, Exogenous Regressors):

For univariate analysis, you must include a date or date-time feature. Simply use:

Formula Interface (recommended): fit(y ~ date) will ignore xreg's.
XY Interface: fit_xy(x = data[,"date"], y = data$y) will ignore xreg's.

Multivariate (xregs, Exogenous Regressors)

The xreg parameter is populated using the fit() or fit_xy() function:

Only factor, ordered factor, and numeric data will be used as xregs.
Date and Date-time variables are not used as xregs
character data should be converted to factor.

Xreg Example: Suppose you have 3 features:

y (target)
date (time stamp),
month.lbl (labeled month as a ordered factor).

The month.lbl is an exogenous regressor that can be passed to the arima_boost() using fit():

fit(y ~ date + month.lbl) will pass month.lbl on as an exogenous regressor.
fit_xy(data[,c("date", "month.lbl")], y = data$y) will pass x, where x is a data frame containing month.lbl and the date feature. Only month.lbl will be used as an exogenous regressor.

Note that date or date-time class values are excluded from xreg.

Examples

library(dplyr)
library(lubridate)
#> 
#> Attaching package: ‘lubridate’
#> The following object is masked from ‘package:greybox’:
#> 
#>     hm
#> The following objects are masked from ‘package:base’:
#> 
#>     date, intersect, setdiff, union
library(parsnip)
library(rsample)
library(timetk)


# Data
m750 <- m4_monthly %>% filter(id == "M750")

# Split Data 80/20
splits <- initial_time_split(m750, prop = 0.9)

# MODEL SPEC ----

# Set engine and boosting parameters
model_spec <- arima_boost(

    # ARIMA args
    seasonal_period = 12,
    non_seasonal_ar = 0,
    non_seasonal_differences = 1,
    non_seasonal_ma = 1,
    seasonal_ar     = 0,
    seasonal_differences = 1,
    seasonal_ma     = 1,

    # XGBoost Args
    tree_depth = 6,
    learn_rate = 0.1
) %>%
    set_engine(engine = "arima_xgboost")

# FIT ----

# \donttest{
# Boosting - Happens by adding numeric date and month features
model_fit_boosted <- model_spec %>%
    fit(value ~ date + as.numeric(date) + month(date, label = TRUE),
        data = training(splits))

model_fit_boosted
#> parsnip model object
#> 
#> ARIMA(0,1,1)(0,1,1)[12] w/ XGBoost Errors
#> ---
#> Model 1: Standard ARIMA
#> Series: outcome 
#> ARIMA(0,1,1)(0,1,1)[12] 
#> 
#> Coefficients:
#>           ma1     sma1
#>       -0.3405  -0.4781
#> s.e.   0.0652   0.0628
#> 
#> sigma^2 = 25114:  log likelihood = -1699.55
#> AIC=3405.1   AICc=3405.19   BIC=3415.8
#> 
#> ---
#> Model 2: XGBoost Errors
#> 
#> xgboost::xgb.train(params = list(eta = 0.1, max_depth = 6, gamma = 0, 
#>     colsample_bytree = 1, colsample_bynode = 1, min_child_weight = 1, 
#>     subsample = 1), data = x$data, nrounds = 15, watchlist = x$watchlist, 
#>     verbose = 0, objective = "reg:squarederror", nthread = 1)
# }