This chapter describes Generalized Linear Models (GLM), a statistical technique for linear modeling. Oracle Data Mining supports GLM for both regression and classification mining functions.
This chapter includes the following topics:
Generalized Linear Models (GLM) include and extend the class of linear models described in "Linear Regression".
Linear models make a set of restrictive assumptions, most importantly, that the target (dependent variable y) is normally distributed conditioned on the value of predictors with a constant variance regardless of the predicted response value. The advantage of linear models and their restrictions include computational simplicity, an interpretable model form, and the ability to compute certain diagnostic information about the quality of the fit.
Generalized linear models relax these restrictions, which are often violated in practice. For example, binary (yes/no or 0/1) responses do not have same variance across classes. Furthermore, the sum of terms in a linear model typically can have very large ranges encompassing very negative and very positive values. For the binary response example, we would like the response to be a probability in the range [0,1].
Generalized linear models accommodate responses that violate the linear model assumptions through two mechanisms: a link function and a variance function. The link function transforms the target range to potentially infinity to +infinity so that the simple form of linear models can be maintained. The variance function expresses the variance as a function of the predicted response, thereby accommodating responses with nonconstant variances (such as the binary responses).
Oracle Data Mining includes two of the most popular members of the GLM family of models with their most popular link and variance functions:
Linear regression with the identity link and variance function equal to the constant 1 (constant variance over the range of response values). See "Linear Regression".
Logistic regression with the logit link and binomial variance functions. See "Logistic Regression".
GLM is a parametric modeling technique. Parametric models make assumptions about the distribution of the data. When the assumptions are met, parametric models can be more efficient than nonparametric models.
The challenge in developing models of this type involves assessing the extent to which the assumptions are met. For this reason, quality diagnostics are key to developing quality parametric models.
Oracle Data Mining GLM models are easy to interpret. Each model build generates many statistics and diagnostics. Transparency is also a key feature: model details describe key characteristics of the coefficients, and global details provide highlevel statistics.
Oracle Data Mining GLM is uniquely suited for handling wide data. The algorithm can build and score quality models that use a virtually limitless number of predictors (attributes). The only constraints are those imposed by system resources.
GLM has the ability to predict confidence bounds. In addition to predicting a best estimate and a probability (classification only) for each row, GLM identifies an interval wherein the prediction (regression) or probability (classification) will lie. The width of the interval depends upon the precision of the model and a userspecified confidence level.
The confidence level is a measure of how sure the model is that the true value will lie within a confidence interval computed by the model. A popular choice for confidence level is 95%. For example, a model might predict that an employee's income is $125K, and that you can be 95% sure that it lies between $90K and $160K. Oracle Data Mining supports 95% confidence by default, but that value is configurable.
Note:
Confidence bounds are returned with the coefficient statistics. You can also use thePREDICTION_BOUNDS
SQL function to obtain the confidence bounds of a model prediction. See Oracle Database SQL Language Reference.The best regression models are those in which the predictors correlate highly with the target, but there is very little correlation between the predictors themselves. Multicollinearity is the term used to describe multivariate regression with correlated predictors.
Ridge regression is a technique that compensates for multicollinearity. Oracle Data Mining supports ridge regression for both regression and classification mining functions. The algorithm automatically uses ridge if it detects singularity (exact multicollinearity) in the data.
Information about singularity is returned in the global model details. See "Global Model Statistics for Linear Regression" and "Global Model Statistics for Logistic Regression".
You can choose to explicitly enable ridge regression by specifying the GLMS_RIDGE_REGRESSION
setting. If you explicitly enable ridge, you can use the systemgenerated ridge parameter or you can supply your own. If ridge is used automatically, the ridge parameter is also calculated automatically.
The build settings for ridge are summarized as follows:
GLMS_RIDGE_REGRESSION
— Whether or not to override the automatic choice made by the algorithm regarding ridge regression
GLMS_RIDGE_VALUE
— The value of the ridge parameter, used only if you specifically enable ridge regression.
GLMS_VIF_FOR_RIDGE
— Whether or not to produce Variance Inflation Factor (VIF) statistics when ridge is being used for linear regression.
See Also:
Oracle Database PL/SQL Packages and Types ReferenceConfidence bounds are not supported by models built with ridge regression. See "Confidence Bounds".
GLM produces Variance Inflation Factor (VIF) statistics for linear regression models, unless they were built with ridge. You can explicitly request VIF with ridge by specifying the GLMS_VIF_FOR_RIDGE
setting. The algorithm will produce VIF with ridge only if enough system resources are available.
When ridge regression is enabled, different data preparation is likely to produce different results in terms of model coefficients and diagnostics. Oracle Corporation recommends that you enable Automatic Data Preparation for GLM models, especially when ridge regression is being used. See "Data Preparation for GLM".
The process of developing a GLM model typically involves a number of model builds. Each build generates many statistics that you can evaluate to determine the quality of your model. Depending on these diagnostics, you may want to try changing the model settings or making other modifications.
You can use build settings to specify:
Coefficient confidence — The GLMS_CONF_LEVEL
setting indicates the degree of certainty that the true coefficient lies within the confidence bounds computed by the model. The default confidence is.95.
Row weights — The ODMS_ROW_WEIGHT_COLUMN_NAME
setting identifies a column that contains a weighting factor for the rows.
Row diagnostics — The GLMS_DIAGNOSTICS_TABLE_NAME
setting identifies a table to contain rowlevel diagnostics.
Additional build settings are available to:
Control the use of ridge regression, as described in "Ridge Regression".
Specify the handling of missing values in the training data, as described in "Data Preparation for GLM".
Specify the target value to be used as a reference in a logistic regression model, as described in "Logistic Regression" .
See:
Oracle Database PL/SQL Packages and Types Reference for details about GLM settingsGLM models generate many metrics to help you evaluate the quality of the model.
The same set of statistics is returned for both linear and logistic regression, but statistics that do not apply to the mining function are returned as NULL. The coefficient statistics are described in "Coefficient Statistics for Linear Regression" and "Coefficient Statistics for Logistic Regression" .
Coefficient statistics are returned by the GET_MODEL_DETAILS_GLM
function in DBMS_DATA_MINING
.
Separate highlevel statistics describing the model as a whole, are returned for linear and logistic regression. When ridge regression is enabled, fewer global details are returned (See "Ridge Regression"). The global model statistics are described in "Global Model Statistics for Linear Regression" and "Global Model Statistics for Logistic Regression".
Global statistics are returned by the GET_MODEL_DETAILS_GLOBAL
function in DBMS_DATA_MINING
.
You can configure GLM models to generate perrow statistics by specifying the name of a diagnostics table in the build setting GLMS_DIAGNOSTICS_TABLE_NAME
. The row diagnostics are described in "Row Diagnostics for Linear Regression" and "Row Diagnostics for Logistic Regression".
GLM requires a case ID to generate row diagnostics. If you provide the name of a diagnostic table but the data does not include a case ID column, an exception is raised.
Automatic Data Preparation (ADP) implements suitable data transformations for both linear and logistic regression.
Note:
Oracle Corporation recommends that you use Automatic Data Preparation with GLM.When ADP is enabled, the build data are standardized using a widely used correlation transformation (Netter, et. al, 1990). The data are first centered by subtracting the attribute means from the attribute values for each observation. Then the data are scaled by dividing each attribute value in an observation by the square root of the sum of squares per attribute across all observations. This transformation is applied to both numeric and categorical attributes.
Prior to standardization, categorical attributes are exploded into N1 columns where N is the attribute cardinality. The most frequent value (mode) is omitted during the explosion transformation. In the case of highest frequency ties, the attribute values are sorted alphanumerically in ascending order, and the first value on the list is omitted during the explosion. This explosion transformation occurs whether or not ADP is enabled.
In the case of high cardinality categorical attributes, the described transformations (explosion followed by standardization) can increase the build data size because the resulting data representation is dense. To reduce memory, disk space, and processing requirements, an alternative approach needs to be used. For large datasets where the estimated internal dense representation would require more than 1Gb of disk space, categorical attributes are not standardized. Under these circumstances, the VIF statistic should be used with caution.
Reference:
Neter, J., Wasserman, W., and Kutner, M.H., "Applied Statistical Models", Richard D. Irwin, Inc., Burr Ridge, IL, 1990.Categorical attributes are exploded into N1 columns where N is the attribute cardinality. The most frequent value (mode) is omitted during the explosion transformation. In the case of highest frequency ties, the attribute values are sorted alphanumerically in ascending order and the first value on the list is omitted during the explosion. This explosion transformation occurs whether or not ADP is enabled.
When ADP is enabled, numerical attributes are standardized by scaling the attribute values by a measure of attribute variability. This measure of variability is computed as the standard deviation per attribute with respect to the origin (not the mean) (Marquardt, 1980).
Reference:
Marquardt, D.W., "A Critique of Some Ridge Regression Methods: Comment", Journal of the American Statistical Association, Vol. 75, No. 369 , 1980, pp. 8791.When building or applying a model, Oracle Data Mining automatically replaces missing values of numerical attributes with the mean and missing values of categorical attributes with the mode.
You can configure a GLM model to override the default treatment of missing values. With the ODMS_MISSING_VALUE_TREATMENT
setting, you can cause the algorithm to delete rows in the training data that have missing values instead of replacing them with the mean or the mode. However, when the model is applied, Oracle Data Mining will perform the usual mean/mode missing value replacement. As a result, statistics generated from scoring may not match the statistics generated from building the model.
If you want to delete rows with missing values in the scoring the model, you must perform the transformation explicitly. To make build and apply statistics match, you must remove the rows with NULLs from the scoring data before performing the apply operation. You can do this by creating a view.
CREATE VIEW viewname AS SELECT * from tablename WHERE column_name1 is NOT NULL AND column_name2 is NOT NULL AND column_name3 is NOT NULL .....
Note:
In Oracle Data Mining, missing values in nested data indicate sparsity, not values missing at random.The value ODMS_MISSING_VALUE_DELETE_ROW
is only valid for tables without nested columns. If this value is used with nested data, an exception is raised.
Linear regression is the GLM regression algorithm supported by Oracle Data Mining. The algorithm assumes no target transformation and constant variance over the range of target values.
GLM regression models generate the following coefficient statistics:
Linear coefficient estimate
Standard error of the coefficient estimate
tvalue of the coefficient estimate
Probability of the tvalue
Variance Inflation Factor (VIF)
Standardized estimate of the coefficient
Lower and upper confidence bounds of the coefficient
GLM regression models generate the following statistics that describe the model as a whole:
Model degrees of freedom
Model sum of squares
Model mean square
Model F statistic
Model F value probability
Error degrees of freedom
Error sum of squares
Error mean square
Corrected total degrees of freedom
Corrected total sum of squares
Root mean square error
Dependent mean
Coefficient of variation
RSquare
Adjusted RSquare
Akaike's information criterion
Schwarz's Baysian information criterion
Estimated mean square error of the prediction
Hocking Sp statistic
JP statistic (the final prediction error)
Number of parameters (the number of coefficients, including the intercept)
Number of rows
Whether or not the model converged
Whether or not a covariance matrix was computed
For linear regression, the diagnostics table has the columns described in Table 121. All the columns are NUMBER
, except the CASE_ID
column, which preserves the type from the training data.
Table 121 Diagnostics Table for GLM Regression Models
Column  Description 


Value of the case ID column 

Value of the target column 

Value predicted by the model for the target 

Value of the diagonal element of the hat matrix 

Measure of error 

Standard error of the residual 

Studentized residual 

Predicted residual 

Cook's D influence statistic 
Binary logistic regression is the GLM classification algorithm supported by Oracle Data Mining. The algorithm uses the logit link function and the binomial variance function.
You can use the build setting GLMS_REFERENCE_CLASS_NAME
to specify the target value to be used as a reference in a binary logistic regression model. Probabilities will be produced for the other (nonreference) class. By default, the algorithm chooses the value with the highest prevalence. If there are ties, the attributes are sorted alphanumerically in ascending order.
You can use the build setting CLAS_WEIGHTS_TABLE_NAME
to specify the name of a class weights table. Class weights influence the weighting of target classes during the model build.
GLM classification models generate the following coefficient statistics:
Name of the predictor
Coefficient estimate
Standard error of the coefficient estimate
Wald chisquare value of the coefficient estimate
Probability of the Wald chisquare value
Standardized estimate of the coefficient
Lower and upper confidence bounds of the coefficient
Exponentiated coefficient
Exponentiated coefficient for the upper and lower confidence bounds of the coefficient
GLM classification models generate the following statistics that describe the model as a whole:
Akaike's criterion for the fit of the intercept only model
Akaike's criterion for the fit of the intercept and the covariates (predictors) model
Schwarz's criterion for the fit of the intercept only model
Schwarz's criterion for the fit of the intercept and the covariates (predictors) model
2 log likelihood of the intercept only model
2 log likelihood of the model
Likelihood ratio degrees of freedom
Likelihood ratio chisquare probability value
Pseudo Rsquare Cox an Snell
Pseudo Rsquare Nagelkerke
Dependent mean
Percent of correct predictions
Percent of incorrect predictions
Percent of ties (probability for two cases is the same)
Number of parameters (the number of coefficients, including the intercept)
Number of rows
Whether or not the model converged
Whether or not a covariance matrix was computed.
For logistic regression, the diagnostics table has the columns described in Table 122. All the columns are NUMBER
, except the CASE_ID
and TARGET_VALUE
columns, which preserve the type from the training data.
Table 122 Row Diagnostics Table for Logistic Regression
Column  Description 


Value of the case ID column 

Value of the target value 

Probability associated with the target value 

Value of the diagonal element of the hat matrix 

Residual with respect to the adjusted dependent variable 

The raw residual scaled by the estimated standard deviation of the target 

Contribution to the overall goodness of fit of the model 

Confidence interval displacement diagnostic 

Confidence interval displacement diagnostic 

Change in the deviance due to deleting an individual observation 

Change in the Pearson chisquare 