SMI model estimation through a greedy search for penalty parameters

Performs a greedy search over a given grid of penalty parameter combinations (lambda0, lambda2), and fits SMI model(s) with best (lowest validation set MSE) penalty parameter combination(s). If the optimal combination lies on the edge of the grid, the penalty parameters are adjusted by ±10%, and a second round of grid search is performed. If a grouping variable is used, penalty parameters are tuned separately for each individual model.

Usage

greedy_smimodel(
  data,
  val.data,
  yvar,
  neighbour = 0,
  family = gaussian(),
  index.vars,
  initialise = c("ppr", "additive", "linear", "multiple", "userInput"),
  num_ind = 5,
  num_models = 5,
  seed = 123,
  index.ind = NULL,
  index.coefs = NULL,
  s.vars = NULL,
  linear.vars = NULL,
  nlambda = 100,
  lambda.min.ratio = 1e-04,
  refit = TRUE,
  M = 10,
  max.iter = 50,
  tol = 0.001,
  tolCoefs = 0.001,
  TimeLimit = Inf,
  MIPGap = 1e-04,
  NonConvex = -1,
  verbose = FALSE,
  parallel = FALSE,
  workers = NULL,
  exclude.trunc = NULL,
  recursive = FALSE,
  recursive_colRange = NULL
)

Arguments

data: Training data set on which models will be trained. Must be a data set of class tsibble.(Make sure there are no additional date or time related variables except for the index of the tsibble). If multiple models are fitted, the grouping variable should be the key of the tsibble. If a key is not specified, a dummy key with only one level will be created.
val.data: Validation data set. (The data set on which the penalty parameter selection will be performed.) Must be a data set of class tsibble. (Once the penalty parameter selection is completed, the best model will be re-fitted for the combined data set data + val.data.)
yvar: Name of the response variable as a character string.
neighbour: If multiple models are fitted: Number of neighbours of each key (i.e. grouping variable) to be considered in model fitting to handle smoothing over the key. Should be an integer. If neighbour = x, x number of keys before the key of interest and x number of keys after the key of interest are grouped together for model fitting. The default is neighbour = 0 (i.e. no neighbours are considered for model fitting).
family: A description of the error distribution and link function to be used in the model (see glm and family).
index.vars: A character vector of names of the predictor variables for which indices should be estimated.
initialise: The model structure with which the estimation process should be initialised. The default is "ppr", where the initial model is derived from projection pursuit regression. The other options are "additive" - nonparametric additive model, "linear" - linear regression model (i.e. a special case single-index model, where the initial values of the index coefficients are obtained through a linear regression), "multiple" - multiple models are fitted starting with different initial models (number of indices = num_ind; num_models random instances of the model (i.e. the predictor assignment to indices and initial index coefficients are generated randomly) are considered), and the final optimal model with the lowest loss is returned, and "userInput" - user specifies the initial model structure (i.e. the number of indices and the placement of index variables among indices) and the initial index coefficients through index.ind and index.coefs arguments respectively.
num_ind: If initialise = "ppr" or "multiple": an integer that specifies the number of indices to be used in the model(s). The default is num_ind = 5.
num_models: If initialise = "multiple": an integer that specifies the number of starting models to be checked. The default is num_models = 5.
seed: If initialise = "multiple": the seed to be set when generating random starting points.
index.ind: If initialise = "userInput": an integer vector that assigns group index for each predictor in index.vars.
index.coefs: If initialise = "userInput": a numeric vector of index coefficients.
s.vars: A character vector of names of the predictor variables for which splines should be fitted individually (rather than considering as part of an index).
linear.vars: A character vector of names of the predictor variables that should be included linearly into the model.
nlambda: The number of values for lambda0 (penalty parameter for L0 penalty) - default is 100.
lambda.min.ratio: Smallest value for lambda0, as a fraction of lambda0.max (data derived).
refit: Whether to refit the model combining training and validation sets after parameter tuning. If FALSE, the final model will be estimated only on the training set.
M: Big-M value used in MIP.
max.iter: Maximum number of MIP iterations performed to update index coefficients for a given model.
tol: Tolerance for the objective function value (loss) of MIP.
tolCoefs: Tolerance for coefficients.
TimeLimit: A limit for the total time (in seconds) expended in a single MIP iteration.
MIPGap: Relative MIP optimality gap.
NonConvex: The strategy for handling non-convex quadratic objectives or non-convex quadratic constraints in Gurobi solver.
verbose: The option to print detailed solver output.
parallel: The option to use parallel processing in fitting SMI models for different penalty parameter combinations.
workers: If parallel = TRUE: Number of cores to use.
exclude.trunc: The names of the predictor variables that should not be truncated for stable predictions as a character string. (Since the nonlinear functions are estimated using splines, extrapolation is not desirable. Hence, if any predictor variable in val.data that is treated non-linearly in the estimated model, will be truncated to be in the in-sample range before obtaining predictions. If any variables are listed here will be excluded from such truncation.)
recursive: Whether to obtain recursive forecasts or not (default - FALSE).
recursive_colRange: If recursive = TRUE, the range of column numbers in val.data to be filled with forecasts. Recursive/autoregressive forecasting is required when the lags of the response variable itself are used as predictor variables into the model. Make sure such lagged variables are positioned together in increasing lag order (i.e. lag_1, lag_2, ..., lag_m, lag_m = maximum lag used) in val.data, with no break in the lagged variable sequence even if some of the intermediate lags are not used as predictors.

Value

An object of class smimodel. This is a tibble with two columns:

key: The level of the grouping variable (i.e. key of the training data set).
fit: Information of the fitted model corresponding to the key.

Each row of the column fit contains a list with six elements:

initial: A list of information of the model initialisation. (For descriptions of the list elements see make_smimodelFit).
best: A list of information of the final optimised model. (For descriptions of the list elements see make_smimodelFit).
best_lambdas: Selected penalty parameter combination.
lambda0_seq: Sequence of values for lambda0 used to construct the initial grid.
lambda2_seq: Sequence of values for lambda2 used to construct the initial grid.
searched: A tibble containing the penalty parameter combinations searched during the two-step greedy search and the corresponding validation set MSEs.

The number of rows of the tibble equals to the number of levels in the grouping variable.

References

Palihawadana, N.K., Hyndman, R.J. & Wang, X. (2024). Sparse Multiple Index Models for High-Dimensional Nonparametric Forecasting. https://www.monash.edu/business/ebs/research/publications/ebs/2024/wp16-2024.pdf.

Examples

if (FALSE) { # \dontrun{
library(dplyr)
library(ROI)
library(tibble)
library(tidyr)
library(tsibble)

# Simulate data
n = 1205
set.seed(123)
sim_data <- tibble(x_lag_000 = runif(n)) |>
  mutate(
    # Add x_lags
    x_lag = lag_matrix(x_lag_000, 5)) |>
  unpack(x_lag, names_sep = "_") |>
  mutate(
    # Response variable
    y1 = (0.9*x_lag_000 + 0.6*x_lag_001 + 0.45*x_lag_003)^3 + rnorm(n, sd = 0.1),
    # Add an index to the data set
    inddd = seq(1, n)) |>
  drop_na() |>
  select(inddd, y1, starts_with("x_lag")) |>
  # Make the data set a `tsibble`
  as_tsibble(index = inddd)

# Training set
sim_train <- sim_data[1:1000, ]
# Validation set
sim_val <- sim_data[1001:1200, ]

# Index variables
index.vars <- colnames(sim_data)[3:8]

# Model fitting
smi_greedy <- greedy_smimodel(data = sim_train,
                              val.data = sim_val,
                              yvar = "y1",
                              index.vars = index.vars,
                              initialise = "ppr",
                              lambda.min.ratio = 0.1)

# Best (optimised) fitted model
smi_greedy$fit[[1]]$best

# Selected penalty parameter combination
smi_greedy$fit[[1]]$best_lambdas
} # }