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Isotonic Regression

Introduction​

An Isotonic Regression is a method of solving univariate regression problems by fitting a free-form line to an ordered sequence of observations such that the fitted line is non-decreasing while minimizing the distance of the fitted line from the observations.

H2O's Isotonic Regression implements a pool adjacent violators algorithm which uses an approach to parallelizing isotonic regression [1].

MOJO support​

Isotonic Regression models can be exported as MOJOs.

Defining an isotonic regression model​

All parameters are optional unless specified as required.

Algorithm-specific parameters​

  • out_of_bounds: Method for handling values of the X predictor that are outside of the bounds seen in training. Must be one of: clip or na (default). The default value prints "NA" for values outside of the training range.

Common parameters​

  • custom_metric_func: Specify a custom evaluation function.

  • fold_assignment: (Applicable only if a value for nfolds is specified and fold_column is not specified) Specify the cross-validation fold assignment scheme. One of:

    • AUTO (default; uses Random)
    • Random
    • Modulo (read more about Modulo)
    • Stratified (which will stratify the folds based on the response variable for classification problems)

  • fold_column: Specify the column that contains the cross-validation fold index assignment per observation.

  • ignored_columns: (Optional, Python only) Specify the column or columns to be excluded from the model.

  • keep_cross_validation_fold_assignment: Enable this option to preserve the cross-validation fold assignment. This option defaults to False (not enabled).

  • keep_cross_validation_models: Specify whether to keep the cross-validated models. Keeping cross-validation models may consume significantly more memory in the H2O cluster. This option defaults to True (enabled).

  • keep_cross_validation_predictions: Specify whether to keep the cross-validation predictions. This option defaults to False (not enabled).

  • model_id: Specify a custom name for the model to use as a reference. By default, H2O automatically generates a destination key.

  • nfolds: Specify the number of folds for cross-validation. This value defaults to 0 (no cross-validation).

  • training_frame: Required Specify the dataset used to build the model.

  • validation_frame: Specify the dataset used to evaluate the accuracy of the model.

  • weights_column: Specify a column to use for the observation weights, which are used for bias correction. The specified weights_column must be included in the specified training_frame.

    Python only: To use a weights column when passing an H2OFrame to x instead of a list of column names, the specified training_frame must contain the specified weights_column.

    Note: Weights are per-row observation weights and do not increase the size of the data frame. This is typically the number of times a row is repeated, but non-integer values are supported as well. During training, rows with higher weights matter more, due to the larger loss function pre-factor.

  • x: Specify the name of a single predictor variable to use when building the model. If x is missing, then all columns except y are used.

  • y: Required Specify the column to use as the dependent variable.

    • Isotonic Regression treats any input as a regression problem regardless of the type of dependent variable. For categorical variables, it trains the model on indices of the labels.

Examples​

Below are simple examples showing how to use Isotonic Regression in R and Python.

library(h2o)
h2o.init()

set.seed(1234)
N <- 100
x <- seq(N)
y <- sample(-50:50, N, replace=TRUE) + 50 * log1p(x)
train <- as.h2o(data.frame(x = x, y = y))

isotonic <- h2o.isotonicregression(x = "x", y = "y", training_frame = train)
print(isotonic)

References​

  1. Kearsley, A.J., Tapia, R.A., Trosset, M.W. (1996). An Approach to Parallelizing Isotonic Regression. In: Fischer, H., Riedmüller, B., Schäffler, S. (eds) Applied Mathematics and Parallel Computing. Physica-Verlag HD. https://doi.org/10.1007/978-3-642-99789-1_10
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