# Generalized Linear Model¶

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

The GLM suite includes

Gaussian regression

Poisson regression

binomial regression

gamma regression

Tweedie regression

## Defining a GLM Model¶

Response:

Response is the model dependent variable, often noted as Y. The specific features of a dependent variable should be considered when choosing the appropriate distribution for estimating a model.

Gaussian Y variables must be continuous and real valued.

Binomial Y variables are discrete and valued only at 0 or 1.

Poisson Y variables are discrete and valued strictly greater than 0. Poisson models are used to model count data.

Gamma Y variables are discrete and valued strictly greater than 0.

Tweedie Y variables follow a Poisson-Gamma mixed compound distribution. This is often also called a zero-inflated Poisson, and is used when Y variables follow a distribution with a large mass at 0, and integer valued counts for all non-zero observations.

Ignored Columns:

This field will auto populate a list of the columns from the data set in use. The user selected set of columns will be omitted from the modeling process. H2O omits the dependent variable specified in Y, as well as any columns with a constant value. Constant columns are omitted because the variances of such columns are 0. In this case Y is independent of X, and X is not an explanatory variable.

H2O factors (also called categorical variables or enumerators) as if they are collapsed columns of binomial variables at each factor level. When a factor is encountered, H2O determines the cardinality of the variable, and generates a unique regression coefficient for all but one of the factor levels. The omitted factor level becomes the reference level. H2O omits the first level in the ordered set. For instance, if factor levels are A, B, and C, level A will be omitted.

Please note that H2O does not currently return a warning when users predict on data outside of the range on which the model was originally specified. For example, H2O allows a model to be trained on data with X between (-1, 10), and then applied to predicting on data where the range of X is (-10, 10) without warning. This is also true in the analogous case for predicting and training on factors. It is the user’s responsibility to ensure that out of data prediction is undertaken with caution, as the veracity of the original results are often constrained to the data range used in the original model.

Max Iter:

The maximum number of iterations to be performed for training the model via gradient descent. . If Max Iter is set to 100, the algorithm will repeat the gradient descent 100 times, or until the model converges, whichever comes first. If the model will not converge after 100 cycles, modeling will stop.

Standardize:

An option that transforms variables into standardized variables, each with mean 0 and unit variance. Variables and coefficients are now expressed in terms of their relative position to 0, and in standard units.

N Folds:

N folds specifies the number of cross validation models to be generated simultaneously to training a model on the full data set. If N folds is sent to 10, additional models will be generated with 1/10 of the data used to train each. The purpose of N folds is to evaluate the stability of the parameter estimates produced.

Each of the given options differs in the assumptions made about the Y variable - the target of prediction. Each family is associated with a default link function, which defines the specialized transformation on the set of X variables chosen to predict Y.

Gaussian (identity):

Y are quantitative, continuous (or discrete predicted values can be meaningfully interpreted as approximately continuos).

Binomial (logit):

Dependent variables take on two values, coded as 0 and 1, and follow a binomial distribution. Binomial dependent variables can be understood as a categorical Y with two possible outcomes

Poisson (log):

Dependent variable is a count - a quantitative, discrete value that expresses the number of times some event occurred.

Gamma (inverse):

Dependent variable is a survival measure, or is distributed as Poisson where variance is greater than the mean of the distribution.

Tweedie Variance Power:

Tweedie distributions are distributions of the dependent variable Y where $$var(Y)=a[E(Y)]^{p}$$

where a and p are constants, and p is determined on the basis of the distribution of Y. Guidelines for selecting Tweedie power are given below.

Tweedie power is chosen based on the distribution of the dependent variable.

p Response distribution

0 Normal 1 Poisson (1, 2) Compound Poisson, non-negative with mass at zero 2 Gamma 3 Inverse-Gaussian > 2 Stable, with support on the positive reals

Alpha:

A user defined tuning regularization parameter. H2O sets Alpha to 0.5 by default, but the parameter can take any value between 0 and 1, inclusive. It functions such that there is an added penalty taken against the estimated fit of the model as the number of parameters increases. An Alpha of 1 is the lasso penalty, and an alpha of 0 is the ridge penalty.

Lambda:

H2O provides a default value, but this can also be user defined. Lambda is a regularization parameter that is designed to prevent overfitting. The best value(s) of lambda depends on the desired level of agreement.

Beta Epsilon:

Precision of the vector of coefficients. Computation stops when the maximum difference between two beta vectors is below than beta epsilon.

Higher Accuracy:

The higher accuracy option implements line search optimization. Line search is an optimization approach that calculates an adaptive step size at each iteration of the gradient descent. Because line search is a direct search algorithm it can improve model convergence without specification of additional regularization. Line search can slow model training.

Lambda Search:

The lambda search option allows users to start at 0.90*Lambda max, where lambda max is the value for lambda at which the model returned estimates all coefficients as zero. An additional 50 values of lambda are estimated. These values are successively smaller, and are log scaled. Models for each are returned, along with the ratio of the explained deviance to nonzero parameter estimates.

## GLMgrid Models¶

GLMgrid models can be generated for sets of regularization parameters by entering the parameters either as a list of comma separated values, or ranges in steps. For example, if users wish to evaluate a model for alpha=(0, .5, 1), entering 0, .5, 1 or 0:1:.5 will achieve the desired outcome.

## Interpreting a Model¶

Degrees of Freedom:

Null (total)
Defined as (n-1), where n is the number of observations or rows in the data set. Quantity (n-1) is used rather than n to account for the condition that the residuals must sum to zero, which calls for a loss of one degree of freedom.
Residual
Defined as (n-1)-p. This is the null degrees of freedom less the number of parameters being estimated in the model.

Residual Deviance:

The difference between the predicted value and the observed value for each example or observation in the data. Deviance is a function of the specific model in question. Even when the same data set is used between two models, deviance statistics will change, because the predicted values of Y are model dependent.

Null Deviance:

The deviance associated with the full model (also known as the saturated model). Heuristically, this can be thought of as the disturbance representing stochastic processes when all of determinants of Y are known and accounted for.

Residual Deviance:

The deviance associated with the reduced model, a model defined by some subset of explanatory variables.

AIC:

A model selection criterion that penalizes models having large numbers of predictors. AIC stands for Akiaike Information Criterion. It is defined as $$AIC = 2k + n Log(\frac{RSS}{n}$$

Where $$k$$ is the number of model parameters, $$n$$ is the number of observations, and $$RSS$$ is the residual sum of squares.

AUC:

Area Under Curve. The curve in question is the receiver operating characteristic curve. The criteria is a commonly used metric for evaluating the performance of classifier models. It gives the probability that a randomly chosen positive observation is correctly ranked greater than a randomly chosen negative observation. In machine learning, AUC is usually seen as the preferred evaluative criteria for a model (over accuracy) for classification models. AUC is not an output for Gaussian regression, but is output for classification models like binomial.

Confusion Matrix:

The accuracy of the classifier can be evaluated from the confusion matrix, which reports actual versus predicted classifications, and the error rates of both.

## Validate GLM¶

After running the GLM Model, a .hex key associated with the model is generated.
1. Select the “Validate on Another Dataset” option in the horizontal menu at the top of your results page. You can also access this at a later time by going to the drop down menu Score and selecting GLM.
2. In the validation generation page enter the .hex key for the model you wish to validate in the Model Key field.
3. In the key field enter the .hex for a testing data set matching the structure of your training data set.
4. Push the Submit button.

## Cross Validation¶

The model resulting from a GLM analysis in H2O can be presented with cross validated models at the user’s request. The coefficients presented in the result model are independent of those in any of the cross validated models, and are generated via least squares on the full data set. Cross validated models are generated by taking a 90% random subsample of the data, training a model, and testing that model on the remaining 10%. This process is repeated as many times as the user specifies in the Nfolds field during model specification.

## Cost of Computation¶

H2O is able to process large data sets because it relies on paralleled processes. Large data sets are divided into smaller data sets and processed simultaneously, with results being 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 we let O be a complexity function, N be the number of observations (or rows), and P be 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.

$Complexity = O(p^3 + N*p^2)$

## 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:

$$f(y_{i})=exp[\frac{y_{i}\theta_{i} - b(\theta_{i})}{a_{i}(\phi)} + c(y_{i}; \phi)]$$

$$where\: \theta \:and \: \phi \:are \: location \: and \: scale\: parameters,$$ $$and \: a_{i}(\phi), \:b_{i}(\theta_{i}),\: c_{i}(y_{i}; \phi)\:are\:known\:functions.$$

$$a_{i}\:is\:of\:the\: form: \:a_{i}=\frac{\phi}{p_{i}}; p_{i}\: is\: a\: known\: prior\: weight.$$

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

Suppose some 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 a specified amount.

## References¶

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

Frome, E L. “The Analysis of Rates Using Poisson Regression Models.” Biometrics (1983): 665-674. http://www.csm.ornl.gov/~frome/BE/FP/FromeBiometrics83.pdf

Goldberger, Arthur S. “Best Linear Unbiased Prediction in the Generalized Linear Regression Model.” Journal of the American Statistical Association 57.298 (1962): 369-375. http://people.umass.edu/~bioep740/yr2009/topics/goldberger-jasa1962-369.pdf

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. http://www.stanford.edu/~hastie/Papers/GuisanEtAl_EcolModel-2003.pdf

Nelder, John A, and Robert WM Wedderburn. “Generalized Linear Models.” Journal of the Royal Statistical Society. Series A (General) (1972): 370-384. http://biecek.pl/MIMUW/uploads/Nelder_GLM.pdf

Niu, Feng, et al. “Hogwild!: A lock-free approach to parallelizing stochastic gradient descent.” Advances in Neural Information Processing Systems 24 (2011): 693-701.*implemented algorithm on p.5 http://www.eecs.berkeley.edu/~brecht/papers/hogwildTR.pdf

Pearce, Jennie, and Simon Ferrier. “Evaluating the Predictive Performance of Habitat Models Developed Using Logistic Regression.” Ecological modeling 133.3 (2000): 225-245. http://www.whoi.edu/cms/files/Ecological_Modelling_2000_Pearce_53557.pdf

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. http://www.statpt.com/logistic/press_1978.pdf

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