GLM¶

Introduction¶

Generalized Linear Models (GLM) estimate regression models for outcomes following exponential distributions. In addition to the Gaussian (i.e. normal) distribution, these include Poisson, binomial, and gamma distributions. Each serves a different purpose, and depending on distribution and link function choice, can be used either for prediction or classification.

The GLM suite includes:

• Gaussian regression
• Poisson regression
• Binomial regression (classification)
• Multinomial classification
• Gamma regression

Defining a GLM Model¶

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

• training_frame: (Required) Specify the dataset used to build the model. NOTE: In Flow, if you click the Build a model button from the Parse cell, the training frame is entered automatically.

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

• nfolds: Specify the number of folds for cross-validation.

• seed: Specify the random number generator (RNG) seed for algorithm components dependent on randomization. The seed is consistent for each H2O instance so that you can create models with the same starting conditions in alternative configurations.

• y: (Required) Specify the column to use as the independent variable.

  • For a regression model, this column must be numeric (Real or Int).
  • For a classification model, this column must be categorical (Enum or String). If the family is Binomial, the dataset cannot contain more than two levels.

• keep_cross_validation_predictions: Specify whether to keep the cross-validation predictions.

• keep_cross_validation_fold_assignment: Enable this option to preserve the cross-validation fold assignment.

• fold_assignment: (Applicable only if a value for nfolds is specified and fold_column is not specified) Specify the cross-validation fold assignment scheme. The available options are AUTO (which is Random), Random, Modulo, or 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) Specify the column or columns to be excluded from the model. In Flow, click the checkbox next to a column name to add it to the list of columns excluded from the model. To add all columns, click the All button. To remove a column from the list of ignored columns, click the X next to the column name. To remove all columns from the list of ignored columns, click the None button. To search for a specific column, type the column name in the Search field above the column list. To only show columns with a specific percentage of missing values, specify the percentage in the Only show columns with more than 0% missing values field. To change the selections for the hidden columns, use the Select Visible or Deselect Visible buttons.

• ignore_const_cols: Enable this option to ignore constant training columns, since no information can be gained from them. This option is enabled by default.

• score_each_iteration: (Optional) Enable this option to score during each iteration of the model training.

• offset_column: Specify a column to use as the offset; the value cannot be the same as the value for the weights_column.

  Note: Offsets are per-row “bias values” that are used during model training. For Gaussian distributions, they can be seen as simple corrections to the response (y) column. Instead of learning to predict the response (y-row), the model learns to predict the (row) offset of the response column. For other distributions, the offset corrections are applied in the linearized space before applying the inverse link function to get the actual response values. For more information, refer to the following link.

• 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.

• family: Specify the model type.

  • If the family is gaussian, the data must be numeric (Real or Int).
  • If the family is binomial, the data must be categorical 2 levels/classes or binary (Enum or Int).
  • If the family is multinomial, the data can be categorical with more than two levels/classes (Enum).
  • If the family is poisson, the data must be numeric and non-negative (Int).
  • If the family is gamma, the data must be numeric and continuous and positive (Real or Int).
  • If the family is tweedie, the data must be numeric and continuous (Real) and non-negative.
  • If the family is quasibinomial, the data must be numeric.

• tweedie_variance_power: (Only applicable if Tweedie is specified for Family) Specify the Tweedie variance power.

• tweedie_link_power: (Only applicable if Tweedie is specified for Family) Specify the Tweedie link power.

• solver: Specify the solver to use (AUTO, IRLSM, L_BFGS, COORDINATE_DESCENT_NAIVE, or COORDINATE_DESCENT). IRLSM is fast on problems with a small number of predictors and for lambda search with L1 penalty, while L_BFGS scales better for datasets with many columns. COORDINATE_DESCENT is IRLSM with the covariance updates version of cyclical coordinate descent in the innermost loop. COORDINATE_DESCENT_NAIVE is IRLSM with the naive updates version of cyclical coordinate descent in the innermost loop. COORDINATE_DESCENT_NAIVE and COORDINATE_DESCENT are currently experimental.

• alpha: Specify the regularization distribution between L1 and L2.

• lambda: Specify the regularization strength.

• lambda_search: Specify whether to enable lambda search, starting with lambda max. If you also specify a value for lambda_min_ratio, then this value is interpreted as lambda min. If you do not specify a value for lambda_min_ratio, then GLM will calculate the minimum lambda.

• early_stopping: Specify whether to stop early when there is no more relative improvement on the training or validation set.

• nlambdas: (Applicable only if lambda_search is enabled) Specify the number of lambdas to use in the search. The default is 100.

• standardize: Specify whether to standardize the numeric columns to have a mean of zero and unit variance. Standardization is highly recommended; if you do not use standardization, the results can include components that are dominated by variables that appear to have larger variances relative to other attributes as a matter of scale, rather than true contribution. This option is enabled by default.

• missing_values_handling: Specify how to handle missing values (Skip or MeanImputation).

• compute_p_values: Request computation of p-values. Only applicable with no penalty (lambda = 0 and no beta constraints). Setting remove_collinear_columns is recommended. H2O will return an error if p-values are requested and there are collinear columns and remove_collinear_columns flag is not enabled.

• remove_collinear_columns: Specify whether to automatically remove collinear columns during model-building. When enabled, collinear columns will be dropped from the model and will have 0 coefficient in the returned model. This can only be set if there is no regularization (lambda=0).

• intercept: Specify whether to include a constant term in the model. This option is enabled by default.

• non_negative: Specify whether to force coefficients to have non-negative values.

• max_iterations: Specify the number of training iterations.

• objective_epsilon: Specify a threshold for convergence. If the objective value is less than this threshold, the model is converged.

• beta_epsilon: Specify the beta epsilon value. If the L1 normalization of the current beta change is below this threshold, consider using convergence.

• gradient_epsilon: (For L-BFGS only) Specify a threshold for convergence. If the objective value (using the L-infinity norm) is less than this threshold, the model is converged.

• link: Specify a link function (Identity, Family_Default, Logit, Log, Inverse, or Tweedie).

  • If the family is Gaussian, then Identity, Log, and Inverse are supported.
  • If the family is Binomial, then Logit is supported.
  • If the family is Poisson, then Log and Identity are supported.
  • If the family is Gamma, then Inverse, Log, and Identity are supported.
  • If the family is Tweedie, then only Tweedie is supported.
  • If the family is Multinomial, then only Family_Default is supported. (This defaults to multinomial.)
  • If the family is Quasibinomial, then only Logit is supported.

• prior: Specify prior probability for p(y==1). Use this parameter for logistic regression if the data has been sampled and the mean of response does not reflect reality.

  Note: This is a simple method affecting only the intercept. You may want to use weights and offset for a better fit.

• lambda_min_ratio: Specify the minimum lambda to use for lambda search (specified as a ratio of lambda_max).

• beta_constraints: Specify a dataset to use beta constraints. The selected frame is used to constraint the coefficient vector to provide upper and lower bounds. The dataset must contain a names column with valid coefficient names.

• max_active_predictors: Specify the maximum number of active predictors during computation. This value is used as a stopping criterium to prevent expensive model building with many predictors.

• interactions: Specify a list of predictor column indices to interact. All pairwise combinations will be computed for this list.

Interpreting a GLM Model¶

By default, the following output displays:

• A graph of the normalized coefficient magnitudes
• Output (model category, model summary, scoring history, training metrics, validation metrics, best lambda, threshold, residual deviance, null deviance, residual degrees of freedom, null degrees of freedom, AIC, AUC, binomial, rank)
• Coefficients
• Coefficient magnitudes

Classification and Regression¶

GLM can produce two categories of models: classication (binary classication only) and regression. Logistic regression is the GLM performing binary classication.

The data type of the response column determines the model category. If the response is a categorical variable (also called a factor or an enum), then a classication model is created. If the response column data type is numeric (either integer or real), then a regression model is created.

Family and Link Functions¶

GLM problems consist of three main components:

• A random component \(f\) for the dependent variable \(y\): The density function \(f(y;\theta,\phi)\) has a probability distribution from the exponential family parametrized by \(\theta\) and \(\phi\). This removes the restriction on the distribution of the error and allows for non-homogeneity of the variance with respect to the mean vector.
• A systematic component (linear model) \(\eta\): \(\eta = X\beta\), where \(X\) is the matrix of all observation vectors \(x_i\).
• A link function \(g\): \(E(y) = \mu = {g^-1}(\eta)\) relates the expected value of the response \(\mu\) to the linear component \(\eta\). The link function can be any monotonic differentiable function. This relaxes the constraints on the additivity of the covariates, and it allows the response to belong to a restricted range of values depending on the chosen transformation \(g\).

Accordingly, in order to specify a GLM problem, you must choose a family function \(f\), link function \(g\), and any parameters needed to train the model.

Families¶

The family option specifies a probability distribution from an exponential family. You can specify one of the following, based on the response column type:

• gaussian: The data must be numeric (Real or Int). This is the default family.
• binomial: The data must be categorical 2 levels/classes or binary (Enum or Int).
• quasibinomial: The data must be numeric.
• multinomial: The data can be categorical with more than two levels/classes (Enum).
• poisson: The data must be numeric and non-negative (Int).
• gamma: The data must be numeric and continuous and positive (Real or Int).
• tweedie: The data must be numeric and continuous (Real) and non-negative.

Note: If your response column is binomial, then you must convert that column to a categorical (.asfactor() in Python and as.factor() in R) and set family = binomial. The following configurations can lead to unexpected results.

• If you DO convert the response column to categorical and DO NOT to set family=binomial, then you will receive an error message.
• If you DO NOT convert response column to categorical and DO NOT set the family, then GLM will assume the 0s and 1s are numbers and will provide a Gaussian solution to a regression problem.

Linear Regression (Gaussian Family)¶

Linear regression corresponds to the Gaussian family model. The link function \(g\) is the identity, and density \(f\) corresponds to a normal distribution. It is the simplest example of a GLM but has many uses and several advantages over other families. Specifically, it is faster and requires more stable computations. Gaussian models the dependency between a response \(y\) and a covariates vector \(x\) as a linear function:

\[\hat {y} = {x^T}\beta + {\beta_0}\]

The model is fitted by solving the least squares problem, which is equivalent to maximizing the liklihood for the Gaussian family.

\[^\text{max}_{\beta,\beta_0} - \dfrac {1} {2N} \sum_{i=1}^{N}(x_{i}^{T}\beta + \beta_0 - y_i)^2 - \lambda \Big( \alpha||\beta||_1 + \dfrac {1} {2}(1 - \alpha)||\beta||^2_2 \Big)\]

The deviance is the sum of the squared prediction errors:

\[D = \sum_{i=1}^{N}(y_i - \hat {y}_i)^2\]

Logistic Regression (Binomial Family)¶

Logistic regression is used for binary classification problems where the response is a categorical variable with two levels. It models the probability of an observation belonging to an output category given the data (for example, \(Pr(y=1|x)\)). The canonical link for the binomial family is the logit function (also known as log odds). Its inverse is the logistic function, which takes any real number and projects it onto the [0,1] range as desired to model the probability of belonging to a class. The corresponding s-curve is below:

The fitted model has the form:

\[\hat {y} = Pr(y=1|x) = \dfrac {e^{x{^T}\beta + {\beta_0}}} {1 + {e^{x{^T}\beta + {\beta_0}}}}\]

This can alternatively be written as:

\[\text{log} \Big( \dfrac {\hat {y}} {1-\hat {y}} \Big) = \text{log} \Big( \dfrac {Pr(y=1|x)} {Pr(y=0|x)} \Big) = x^T\beta + \beta_0\]

The model is fitted by maximizing the following penalized likelihood:

\[^\text{max}_{\beta,\beta_0} \dfrac {1} {N} \sum_{i=1}^{N} \Big( y_i(x_{i}^{T}\beta + \beta_0) - \text{log} (1 + e^{x{^T_i}\beta + {\beta_0}} ) \Big)- \lambda \Big( \alpha||\beta||_1 + \dfrac {1} {2}(1 - \alpha)||\beta||^2_2 \Big)\]

The corresponding deviance is equal to:

\[D = -2 \sum_{i=1}^{n} \big( y_i \text{log}(\hat {y}_i) + (1 - y_i) \text{log}(1 - \hat {y}_i) \big)\]

Pseudo-Logistic Regression (Quasibinomial Family)¶

The quasibinomial family option works in the same way as the aforementioned binomial family. The difference is that binomial models only support 0/1 for the values of the target. A quasibinomial model supports “pseudo” logistic regression and allows for two arbitrary integer values (for example -4, 7). Additional information about the quasibinomial option can be found in the “Estimating Effects on Rare Outcomes: Knowledge is Power” paper.

Multiclass Classification (Multinomial Family)¶

Multinomial family generalization of the binomial model is used for multi-class response variables. Similar to the binomail family, GLM models the conditional probability of observing class “c” given “x”. A vector of coefficients exists for each of the output classes. (\(\beta\) is a matrix.) The probabilities are defined as:

\[- \Big[ \dfrac {1} {N} \sum_{i=1}^N \sum_{k=1}^K \big( y_{i,k} (x^T_i \beta_k + \beta_{k0}) \big) - \text{log} \big( \sum_{k=1}^K e^{x{^T_i}\beta_k + {\beta_{k0}}} \big) \Big] + \lambda \Big[ \dfrac {(1-\alpha)} {2} ||\beta || ^2_F + \alpha \sum_{j=1}^P ||\beta_j ||_1 \Big]\]

where \(\beta_c\) is a vector of coefficients for class “c”, and \(y_{i,k}\) is the \(k\text{th}\) element of the binary vector produced by expanding the response variable using one-hot encoding (i.e., \(y_{i,k} == 1\) iff the response at the \(i\text{th}\) observation is “k”; otherwise it is 0.)

Poisson Models¶

Poisson regression is typically used for datasets where the response represents counts, and the errors are assumed to have a Poisson distribution. In general, it can be applied to any data where the response is non-negative. It models the dependency between the response and covariates as:

\[\hat {y} = e^{x{^T}\beta + {\beta_{0}}}\]

The model is fitted by maximizing the corresponding penalized likelihood:

\[^\text{max}_{\beta,\beta_0} \dfrac {1} {N} \sum_{i=1}^{N} \Big( y_i(x_{i}^{T}\beta + \beta_0) - e^{x{^T_i}\beta + {\beta_0}} \Big)- \lambda \Big( \alpha||\beta||_1 + \dfrac {1} {2}(1 - \alpha)||\beta||^2_2 \Big)\]

The corresponding deviance is equal to:

\[D = -2 \sum_{i=1}^{N} \big( y_i \text{log}(y_i / \hat {y}_i) - (y_i - \hat {y}_i) \big)\]

Gamma Models¶

The gamma distribution is useful for modeling a positive continuous response variable, where the conditional variance of the response grows with its mean, but the coefficientof variation of the response \(\sigma^2(y_i)/\mu_i\) is constant. It is usually used with the log link \(g(\mu_i) = \text{log}(\mu_i)\) or the inverse link \(g(\mu_i) = \dfrac {1} {\mu_i}\), which is equivalent to the canonical link.

The model is fitted by solving the following likelihood maximization:

\[^\text{max}_{\beta,\beta_0} - \dfrac {1} {N} \sum_{i=1}^{N} \dfrac {y_i} {x{^T_i}\beta + \beta_0} + \text{log} \big( x{^T_i}\beta + \beta_0 \big ) - \lambda \Big( \alpha||\beta||_1 + \dfrac {1} {2}(1 - \alpha)||\beta||^2_2 \Big)\]

The corresponding deviance is equal to:

\[D = 2 \sum_{i=1}^{N} - \text{log} \bigg (\dfrac {y_i} {\hat {y}_i} \bigg) + \dfrac {(y_i - \hat{y}_i)} {\hat {y}_i}\]

Tweedie Models¶

Tweedie distributions are a family of distributions that include gamma, normal, Poisson, and their combination. Tweedie distributions are especially useful for modeling positive continuous variables with exact zeros. The variance of the Tweedie distribution is proportional to the \(p\text{th}\) power of the mean \(var(y_i) = \phi\mu{^p_i}\).

The Tweedie distribution is parametrized by variance power \(p\). It is defined for all \(p\) values except in the (0,1) interval and has the following distributions as special cases:

• \(p = 0\): Normal
• \(p = 1\): Poisson
• \(p \in (1,2)\): Compound Poisson, non-negative with mass at zero
• \(p = 2\): Gamma
• \(p = 3\): Gaussian
• \(p > 2\): Stable, with support on the positive reals

For \(p > 1\), the model likelood to maximize has the form:

\[^\text{max}_{\beta,\beta_0} \sum_{i=1}^{N} \text{log} (a(y_i, \phi)) + \bigg( \dfrac {1} {\phi} \bigg(\dfrac {y_i\mu{^{1-p}_i}} {1-p} - \kappa(\mu_i, p) \bigg) \bigg) - \lambda \bigg ( \alpha || \beta ||_1 + \dfrac {1} {2} (1-\alpha) ||\beta||{^2_2} \bigg )\]

where \(\kappa(\mu,p) = \mu^{2-p} / (2-p)\) for \(p \neq 2\) and \(\kappa(\mu,p) = \text{log} (\mu)\) for \(p=2\), and where the function \(a(y_i,\phi)\) is evaluated using series expansion because it does not have an analytical solution. The link function in the GLM representation of the Tweedie distribution defaults to \(g(\mu) = \mu^q = \eta = X\beta\) with \(q=1-p\). The link power \(q\) can be set to other values, including \(q=0\), which is interpreated as \(\text{log}(\mu)=\eta\).

The corresponding deviance when \(p \neq 1\) and \(p \neq 2\) is equal to:

\[D = -2 \sum_{i=1}^{N} y_i(y_i^{1-p} - \hat{y}_1^{1-p}) - \dfrac {(y_i^{2-p} - \hat{y}_i^{2-p})} {(2-p)}\]

Regularization¶

Regularization is used to attempt to solve problems with overfitting that can occur in GLM. Penalties can be introduced to the model building process to avoid overfitting, to reduce variance of the prediction error, and to handle correlated predictors. The two most common penalized models are ridge regression and LASSO (least absolute shrinkage and selection operator). The elastic net combines both penalties using both the alpha and lambda options (i.e., values greater than 0 for both).

Solvers¶

This section provides general guidelines for best performance from the GLM implementation details. The optimal solver depends on the data properties and prior information regarding the variables (if available). In general, the data are considered sparse if the ratio of zeros to non-zeros in the input matrix is greater than 10. The solution is sparse when only a subset of the original set of variables is intended to be kept in the model. In a dense solution, all predictors have non-zero coefficients in the final model.

In GLM, you can specify one of the following solvers:

• IRLSM: Iteratively Reweighted Least Squares Method (default)
• L_BFGS: Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm
• AUTO: Sets the solver based on given data and parameters.
• COORDINATE_DESCENT: Coordinate Decent (experimental)
• COORDINATE_DESCENT_NAIVE: Coordinate Decent Naive (experimental)

Modifying or Creating Custom GLM Model¶

In R and python, the makeGLMModel call can be used to create an H2O model from given coefficients. It needs a source GLM model trained on the same dataset to extract the dataset information. To make a custom GLM model from R or python:

• R: call h2o.makeGLMModel. This takes a model, a vector of coefficients, and (optional) decision threshold as parameters.
• Pyton: H2OGeneralizedLinearEstimator.makeGLMModel (static method) takes a model, a dictionary containing coefficients, and (optional) decision threshold as parameters.

FAQ¶

• How does the algorithm handle missing values during training?

Depending on the selected missing value handling policy, they are either imputed mean or the whole row is skipped. The default behavior is mean imputation. Note that categorical variables are imputed by adding an extra “missing” level. Optionally, glm can skip all rows with any missing values.

• How does the algorithm handle missing values during testing?

Same as during training. If the missing value handling is set to skip and we are generating predictions, skipped rows will have Na (missing) prediction.

• What happens if the response has missing values?

The rows with missing response are ignored during model training and validation.

• What happens during prediction if the new sample has categorical levels not seen in training?

The value will be filled with either 0 or a special missing level (if trained with missing values, and missing_value_handling was set to MeanImputation).

• Does it matter if the data is sorted?

No.

• Should data be shuffled before training?

No.

• How does the algorithm handle highly imbalanced data in a response column?

GLM does not require special handling for imbalanced data.

• What if there are a large number of columns?

IRLS will get quadratically slower with the number of columns. Try L-BFGS for datasets with more than 5-10 thousand columns.

• What if there are a large number of categorical factor levels?

GLM internally one-hot encodes the categorical factor levels; the same limitations as with a high column count will apply.

• When building the model, does GLM use all features or a selection of the best features?

Typically, GLM picks the best predictors, especially if lasso is used (alpha = 1). By default, the GLM model includes an L1 penalty and will pick only the most predictive predictors.

• When running GLM, is it better to create a cluster that uses many smaller nodes or fewer larger nodes?

A rough heuristic would be:

\(nodes ~=M *N^2/(p * 1e8)\)

where \(M\) is the number of observations, \(N\) is the number of columns (categorical columns count as a single column in this case), and \(p\) is the number of CPU cores per node.

For example, a dataset with 250 columns and 1M rows would optimally use about 20 nodes with 32 cores each (following the formula \(250^2 *1000000/(32* 1e8) = 19.5 ~= 20)\).

• How is variable importance calculated for GLM?

For GLM, the variable importance represents the coefficient magnitudes.

• How does GLM define and check for convergence during logistic regression?

GLM includes three convergence criteria outside of max iterations:

• beta_epsilon: beta stops changing. This is used mostly with IRLSM.
• gradient_epsilon: gradient is too small. This is used mostly with L-BFGS.
• objective_epsilon: relative objective improvement is too small. This is used by all solvers.

The default values below are based on a heuristic:

• The default for beta_epsilon is 1e-4.
• The default for gradient_epsilon is 1e-6 if there is no regularization (lambda = 0) or you are running with lambda_search; 1e-4 otherwise.
• The default for objective_epsilon is 1e-6 if lambda = 0; 1e-4 otherwise.

The default for max_iterations depends on the solver type and whether you run with lambda search:

• for IRLSM, the default is 50 if no lambda search; 10* number of lambdas otherwise
• for LBFGS, the default is number of classes (1 if not classification) * max(20, number of predictors /4 ) if no lambda search; it is number of classes * 100 * n-lambdas with lambda search.

You will receive a warning if you reach the maximum number of iterations. In some cases, GLM can end prematurely if it can not progress forward via line search. This typically happens when running a lambda search with IRLSM solver. Note that using CoordinateDescent solver fixes the issue.

GLM Algorithm¶

Following the definitive text by P. McCullagh and J.A. Nelder (1989) on the generalization of linear models to non-linear distributions of the response variable Y, H2O fits GLM models based on the maximum likelihood estimation via iteratively reweighed least squares.

Let \(y_{1},…,y_{n}\) be n observations of the independent, random response variable \(Y_{i}\).

Assume that the observations are distributed according to a function from the exponential family and have a probability density function of the form:

\(f(y_{i})=exp[\frac{y_{i}\theta_{i} - b(\theta_{i})}{a_{i}(\phi)} + c(y_{i}; \phi)]\) where \(\theta\) and \(\phi\) are location and scale parameters, and \(a_{i}(\phi)\), \(b_{i}(\theta{i})\), and \(c_{i}(y_{i}; \phi)\) are known functions.

\(a_{i}\) is of the form \(a_{i}= \frac{\phi}{p_{i}}\) where \(p_{i}\) is a known prior weight.

When \(Y\) has a pdf from the exponential family:

\(E(Y_{i})=\mu_{i}=b^{\prime} var(Y_{i})=\sigma_{i}^2=b^{\prime\prime}(\theta_{i})a_{i}(\phi)\)

Let \(g(\mu_{i})=\eta_{i}\) be a monotonic, differentiable transformation of the expected value of \(y_{i}\). The function \(\eta_{i}\) is the link function and follows a linear model.

\(g(\mu_{i})=\eta_{i}=\mathbf{x_{i}^{\prime}}\beta\)

When inverted: \(\mu=g^{-1}(\mathbf{x_{i}^{\prime}}\beta)\)

Maximum Likelihood Estimation

For an initial rough estimate of the parameters \(\hat{\beta}\), use the estimate to generate fitted values: \(\mu_{i}=g^{-1}(\hat{\eta_{i}})\)

Let \(z\) be a working dependent variable such that \(z_{i}=\hat{\eta_{i}}+(y_{i}-\hat{\mu_{i}})\frac{d\eta_{i}}{d\mu_{i}}\),

where \(\frac{d\eta_{i}}{d\mu_{i}}\) is the derivative of the link function evaluated at the trial estimate.

Calculate the iterative weights: \(w_{i}=\frac{p_{i}}{[b^{\prime\prime}(\theta_{i})\frac{d\eta_{i}}{d\mu_{i}}^{2}]}\)

where \(b^{\prime\prime}\) is the second derivative of \(b(\theta_{i})\) evaluated at the trial estimate.

Assume \(a_{i}(\phi)\) is of the form \(\frac{\phi}{p_{i}}\). The weight \(w_{i}\) is inversely proportional to the variance of the working dependent variable \(z_{i}\) for current parameter estimates and proportionality factor \(\phi\).

Regress \(z_{i}\) on the predictors \(x_{i}\) using the weights \(w_{i}\) to obtain new estimates of \(\beta\).

\(\hat{\beta}=(\mathbf{X}^{\prime}\mathbf{W}\mathbf{X})^{-1}\mathbf{X}^{\prime}\mathbf{W}\mathbf{z}\)

where \(\mathbf{X}\) is the model matrix, \(\mathbf{W}\) is a diagonal matrix of \(w_{i}\), and \(\mathbf{z}\) is a vector of the working response variable \(z_{i}\).

This process is repeated until the estimates \(\hat{\beta}\) change by less than the specified amount.

Cost of computation

H2O can process large data sets because it relies on parallel processes. Large data sets are divided into smaller data sets and processed simultaneously and the results are communicated between computers as needed throughout the process.

In GLM, data are split by rows but not by columns, because the predicted Y values depend on information in each of the predictor variable vectors. If O is a complexity function, N is the number of observations (or rows), and P is the number of predictors (or columns) then

\(Runtime \propto p^3+\frac{(N*p^2)}{CPUs}\)

Distribution reduces the time it takes an algorithm to process because it decreases N.

Relative to P, the larger that (N/CPUs) becomes, the more trivial p becomes to the overall computational cost. However, when p is greater than (N/CPUs), O is dominated by p.

  \(Complexity = O(p^3 + N*p^2)\)

For more information about how GLM works, refer to the Generalized Linear Modeling booklet.

References¶

Breslow, N E. “Generalized Linear Models: Checking Assumptions and Strengthening Conclusions.” Statistica Applicata 8 (1996): 23-41.

Jerome Friedman, Trevor Hastie, and Rob Tibshirani. Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 2009.

Frome, E L. “The Analysis of Rates Using Poisson Regression Models.” Biometrics (1983): 665-674.

Goldberger, Arthur S. “Best Linear Unbiased Prediction in the Generalized Linear Regression Model.” Journal of the American Statistical Association 57.298 (1962): 369-375.

Guisan, Antoine, Thomas C Edwards Jr, and Trevor Hastie. “Generalized Linear and Generalized Additive Models in Studies of Species Distributions: Setting the Scene.” Ecological modeling 157.2 (2002): 89-100.

Nelder, John A, and Robert WM Wedderburn. “Generalized Linear Models.” Journal of the Royal Statistical Society. Series A (General) (1972): 370-384.

Pearce, Jennie, and Simon Ferrier. “Evaluating the Predictive Performance of Habitat Models Developed Using Logistic Regression.” Ecological modeling 133.3 (2000): 225-245.

Press, S James, and Sandra Wilson. “Choosing Between Logistic Regression and Discriminant Analysis.” Journal of the American Statistical Association 73.364 (April, 2012): 699–705.

Snee, Ronald D. “Validation of Regression Models: Methods and Examples.” Technometrics 19.4 (1977): 415-428.

Balzer, Laura B, and van der Laan, Mark J. “Estimating Effects on Rare Outcomes: Knowledge is Power.” U.C. Berkeley Division of Biostatistics Working Paper Series (2013).