ModelSelection¶

To help users select the best predictor subsets from their datasets for model building, we have implemented the ModelSelection toolbox. There are currently two modes to select the predictor subsets:

mode = "allsubsets" where all possible combinations of predictor subsets are generated for a given subset size. A model is built for each subset and the one with the highest \(R^2\) is returned. The best subsets are also returned for subset size \(1, 2, ..., n\). This mode guarantees to return the predictor subset with the highest \(R^2\) value at the cost of computation complexity.
mode = "maxr" where a sequential replacement method is used to find the best subsets for subset size of \(1, 2, ..., n\). However, the predictor subsets are not guaranteed to have the highest \(R^2\) value.

This model only supports GLM regression families.

Defining a ModelSelection Model¶

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.
y: (Required) Specify the column to use as the dependent variable.
- For a regression model only, this column must be numeric (Real or Int).
mode: (Required) Specify the model selection algorithm to use. This can be set to either maxr (default) or allsubsets.
x: Specify a vector containing the names or indices of the predictor variables to use when building the model. If x is missing, then all columns except y are used.
validation_frame: Specify the dataset used to evaluate the accuracy of the model.
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.
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). This option defaults to AUTO.
fold_column: Specify the column that contains the cross-validation fold index assignment per observation.
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. This option defaults to -1 (time-based random number).
ignored_columns: (Python and Flow only) 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: Enable this option to score during each iteration of the model training. This option is disabled by default.
score_iteration_interval: Perform scoring for every score_iteration_interval iteration. Defaults to -1.
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.
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.
solver: Specify the solver to use (AUTO, IRLSM, L_BFGS, COORDINATE_DESCENT_NAIVE, COORDINATE_DESCENT, GRADIENT_DESCENT_LH, or GRADIENT_DESCENT_SQERR). 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. GRADIENT_DESCENT_LH and GRADIENT_DESCENT_SQERR can only be used with the Ordinal family. AUTO (default) will set the solver based on the given data and other parameters.
alpha: Specify the regularization distribution between L1 and L2. The default value of alpha is 0 when SOLVER = ‘L-BFGS’; otherwise it is 0.5.
lambda: Specify the regularization strength.
lambda_search: Specify whether to enable lambda search, starting with lambda max (the smallest \(\lambda\) that drives all coefficients to zero). 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. This option is disabled by default.
nlambdas: (Applicable only if lambda_search is enabled) Specify the number of lambdas to use in the search. When alpha > 0, the default value for lambda_min_ratio is \(1e^{-4}\), then the default value for nlambdas is 100. This gives a ratio of 0.912. (For best results when using strong rules, keep the ratio close to this default.) When alpha=0, the default value for nlamdas is set to 30 because fewer lambdas are needed for ridge regression. This value defaults to -1.
lambda_min_ratio: Specify the minimum lambda to use for lambda search (specified as a ratio of lambda_max, which is the smallest \(\lambda\) for which the solution is all zeros). This value defaults to -1.
early_stopping: Specify whether to stop early when there is no more relative improvement on the training or validation set. This option is enabled by default.
stopping_rounds: Stops training when the option selected for stopping_metric doesn’t improve for the specified number of training rounds, based on a simple moving average. To disable this feature, specify 0 (default).
Note: If cross-validation is enabled:
- All cross-validation models stop training when the validation metric doesn’t improve.
- The main model runs for the mean number of epochs.
- N+1 models may be off by the number specified for stopping_rounds from the best model, but the cross-validation metric estimates the performance of the main model for the resulting number of epochs (which may be fewer than the specified number of epochs).
stopping_metric: Specify the metric to use for early stopping. The available options are:
- AUTO: This defaults to logloss for classification, deviance for regression, and anomaly_score for Isolation Forest. Note that custom and custom_increasing can only be used in GBM and DRF with the Python Client. Must be one of: AUTO, anomaly_score. Defaults to AUTO.
- anomaly_score (Isolation Forest only)
- deviance
- logloss
- MSE
- RMSE
- MAE
- RMSLE
- AUC (area under the ROC curve)
- AUCPR (area under the Precision-Recall curve)
- lift_top_group
- misclassification
- mean_per_class_error
- custom (GBM/DRF Python client only)
- custom_increasing (GBM/DRF Python client only)
stopping_tolerance: Specify the relative tolerance for the metric-based stopping to stop training if the improvement is less than this value. Defaults to 0.001.
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, MeanImputation, or PlugValues). This value defaults to MeanImputation.
plug_values: When missing_values_handling="PlugValues", specify a single row frame containing values that will be used to impute missing values of the training/validation frame.
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. Note that this option is not available for family="multinomial" or family="ordinal". This option is disabled by default.
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). This option is disabled by default.
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 (defaults to false).
max_iterations: Specify the number of training iterations (defaults to -1).
objective_epsilon: If the objective value is less than this threshold, then the model is converged. If lambda_search=True, then this value defaults to .0001. If lambda_search=False and lambda is equal to zero, then this value defaults to .000001. For any other value of lambda, the default value of objective_epsilon is set to .0001. The default value is -1.
beta_epsilon: Converge if beta changes less than this value (using L-infinity norm). This only applies to IRLSM solver, and the value defaults to 0.0001.
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. If lambda_search=True, then this value defaults to .0001. If lambda_search=False and lambda is equal to zero, then this value defaults to .000001. For any other value of lambda, this value defaults to .0001. This value defaults to -1.
startval: The initial starting values for fixed and randomized coefficients in HGLM specified as a double array.
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. This value defaults to -1 and must be a value in the range (0,1).

Note: This is a simple method affecting only the intercept. You may want to use weights and offset for a better fit.
cold_start: Specify whether the model should be built from scratch. This parameter is only applicable when building a GLM model with multiple alpha/lambda values. If false and for a fixed alpha value, the next model with the next lambda value out of the lambda array will be built using the coefficients and the GLM state values of the current model. If true, the next GLM model will be built from scratch. The default value is false.

Note: If an alpha array is specified and for a brand new alpha, the model will be built from scratch regardless of the value of cold_start.
beta_constraints: Specify a dataset to use beta constraints. The selected frame is used to constrain 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. This value defaults to -1.
obj_reg: Specifies the likelihood divider in objective value computation. This defaults to 1/nobs.
max_runtime_secs: Maximum allowed runtime in seconds for model training. This defaults to 0 (unlimited).
custom_metric_func: Optionally specify a custom evaluation function.
nparallelism: Number of models to be built in parallel. Defaults to 0.0 (which is adaptive to the system’s capabilities).
max_predictor_number: Maximum number of predictors to be considered when building GLM models. Defaults to 1.

Understanding ModelSelection `mode = allsubsets`¶

Setting the H2O ModelSelection mode = allsubsets guarantees the return of the model with the best \(R^2\) value.

For each predictor subset size \(x\):

If there are \(n\) predictors and we are looking at using \(x\) predictors, all combinations of \(x\) from \(n\) predictors are first generated;
for each element in the combination of \(x\) predictors: generate the training frame, build the model, and look at the \(R^2\) value of the model;
the best \(R^2\) value, the predictor names, and the model_id of the best models are stored in arrays as well as H2OFrame;
access functions are written in Java/R/Python to extract coefficients associated with the models with the best \(R^2\) values.

The main disadvantage of this mode is the long computation time.

Understanding ModelSelection `mode = maxr`¶

The H2O ModelSelection mode = maxr is implemented using the sequential replacement method [1]. This consists of a forward step and a replacement step. The sequential replacement method goes like this (where the predictors are denoted by A, B, C, …, Z):

Start with the current subset = {} (empty)
Forward step for 1 predictor subset:
1. add each available predictor (from A to Z) to the current empty subset and build a GLM model with each predictor subset;
2. save the model with the highest \(R^2\) for all models built with predictor A, B, …, Z;
3. set the new current subset = {predictor with highest \(R^2\) } (for example, predictor A).
Forward step for 2 predictor subset (starting with current subset = {A} ):
1. add each available predictor (from B to Z) to the current subset and build a GLM model;
2. save the model with the highest \(R^2\) for all models with predictor subsets AB, AC, …, AZ;
3. set the new current subset = {model with highest \(R^2\) } and save the best subset (for example, {AB}).
Replacement for 2 predictor subset from best subset chosen from forward step for 2 predictor subsets (i.e. starting from best subset {AB} from previous step):

fixing the second predictor, choose a different predictor for the first predictor from the remaining predictors C, D, …, Z (skipping predictor A as it was chosen already by forward step; B is taken as the second predictor). Then, build a GLM model for each new subset of (CB, DB, EB, …, ZB). Save the model with the highest \(R^2\) (for example, {DB}) from all models built with predictor subsets (CB, DB, EB, …, ZB);

fixing the first predictor, choose a different second predictor from the remaining predictor subset. Then, build a GLM model for each new subset generated. Save the model with the highest \(R^2\) from all models built;

compare the \(R^2\) value from the models built with forward step, step 4(a), and step 4(b) and choose the subset with the highest \(R^2\). If the best model is built with {AB}, we are done because steps 4(a) and 4(b) generated no improvement. If the best model is built with {DB}, repeat steps 4(a), 4(b), and 4(c) until no improvement is found. For the two predictor case, the first 4(b) can be skipped since it is already done in the forward step.

Start with the best \(n\) predictor subset and forward step for \(n\) predictor subsets:

add each predictor available to the \(n\) predictor subset and build a GLM model;

save the model with the highest \(R^2\) for all models built with \(n+1\) predictor subsets;

Replacement for \(n+1\) predictor subsets:

Repeat for predictor in location 0,1,2,…,n:

keep all predictors fixed except in location k (k will be from 0,1,2,…,n) and switch out the predictor at location k with one predictor from the available predictors. If there are m predictors in the available predictor subset, m GLM models will be built and the model with the best \(R^2\) value will be saved;

from all the n best models found from step 6(a), if the best \(R^2\) value has improved from the forward step or the previous 6(a), return to 6(a). If no improvement is found, break and just take the best \(R^2\) model as the one to save.

Again, the best \(R^2\) value, the predictor names, and the model_id of the best models are stored in arrays as well as H2OFrame. Additionally, coefficients associated with the models built with all the predictor subset sizes are available and accessible as well.

Examples¶

library(h2o)
h2o.init()

# Import the prostate dataset:
prostate <- h2o.importFile("http://s3.amazonaws.com/h2o-public-test-data/smalldata/logreg/prostate.csv")
|======================================================================| 100%

# Set the predictors & response:
predictors <- c("AGE", "RACE", "CAPSULE", "DCAPS", "PSA", "VOL", "DPROS")
response <- "GLEASON"

# Build & train the model:
allsubsetsModel <- h2o.modelSelection(x = predictors,
                                      y = response,
                                      training_frame = prostate,
                                      seed = 12345,
                                      max_predictor_number = 7,
                                      mode = "allsubsets")
|======================================================================| 100%

# Retrieve the results (H2OFrame containing best model_ids, best_r2_value, & predictor subsets):
results <- h2o.result(allsubsetsModel)
print(results)
model_name                    model_id best_r2_value                   predictor_names
1 best 1 predictor(s) model  GLM_model_1637788524625_26     0.2058868  1 CAPSULE
2 best 2 predictor(s) model  GLM_model_1637788524625_37     0.2695678  2 CAPSULE, PSA
3 best 3 predictor(s) model  GLM_model_1637788524625_66     0.2862530  3 CAPSULE, DCAPS, PSA
4 best 4 predictor(s) model GLM_model_1637788524625_105     0.2904461  4 CAPSULE, DPROS, DCAPS, PSA
5 best 5 predictor(s) model GLM_model_1637788524625_130     0.2921695  5 CAPSULE, AGE, DPROS, DCAPS, PSA
6 best 6 predictor(s) model GLM_model_1637788524625_145     0.2924758  6 CAPSULE, AGE, RACE, DPROS, DCAPS, PSA
7 best 7 predictor(s) model GLM_model_1637788524625_152     0.2925563  7 CAPSULE, AGE, RACE, DPROS, DCAPS, PSA, VOL

# Retrieve the list of coefficients:
coeff <- h2o.coef(allsubsetsModel)
print(coeff)
[[1]]
Intercept   CAPSULE
5.978584  1.007438
[[2]]
Intercept    CAPSULE        PSA
5.83309940 0.81073054 0.01458179
[[3]]
Intercept    CAPSULE      DCAPS        PSA
5.34902149 0.75750144 0.47979555 0.01289096
[[4]]
Intercept    CAPSULE      DPROS      DCAPS        PSA
5.23924958 0.71845861 0.07616614 0.44257893 0.01248512
[[5]]
Intercept    CAPSULE        AGE      DPROS      DCAPS        PSA
4.78548229 0.72070240 0.00687360 0.07827698 0.43777710 0.01245014
[[6]]
Intercept      CAPSULE          AGE         RACE        DPROS        DCAPS          PSA
4.853286962  0.717393309  0.006790891 -0.060686926  0.079288081  0.438470913  0.012572276
[[7]]
Intercept       CAPSULE           AGE          RACE         DPROS         DCAPS           PSA           VOL
4.8526636043  0.7153633278  0.0069487980 -0.0584344031  0.0791810013  0.4353149856  0.0126060611  -0.0005196059

# Retrieve the list of coefficients for a subset size of 3:
coeff3 <- h2o.coeff(allsubsetsModel, 3)
print(coeff3)
[[3]]
Intercept    CAPSULE      DCAPS        PSA
5.34902149 0.75750144 0.47979555 0.01289096

# Retrieve the list of standardized coefficients:
coeff_norm <- h2o.coef_norm(allsubsetsModel)
print(coeff_norm)
[[1]]
Intercept   CAPSULE
6.3842105 0.4947269
[[2]]
Intercept   CAPSULE       PSA
6.3842105 0.3981290 0.2916004
[[3]]
Intercept   CAPSULE     DCAPS       PSA
6.3842105 0.3719895 0.1490516 0.2577879
[[4]]
Intercept    CAPSULE      DPROS      DCAPS        PSA
6.38421053 0.35281659 0.07617433 0.13749000 0.24967213
[[5]]
Intercept    CAPSULE        AGE      DPROS      DCAPS        PSA
6.38421053 0.35391845 0.04486448 0.07828541 0.13599828 0.24897265
[[6]]
Intercept     CAPSULE         AGE        RACE       DPROS       DCAPS         PSA
6.38421053  0.35229345  0.04432463 -0.01873850  0.07929661  0.13621382  0.25141500
[[7]]
Intercept      CAPSULE          AGE         RACE        DPROS        DCAPS          PSA          VOL
6.384210526  0.351296573  0.045355300 -0.018042981  0.079189523  0.135233408  0.252090622 -0.009533532

# Retrieve the list of standardized coefficients for a subset size of 3:
coeff_norm3 <- h2o.coef_norm(allsubsetsModel)
print(coeff_norm3)
[[3]]
Intercept   CAPSULE     DCAPS       PSA
6.3842105 0.3719895 0.1490516 0.2577879

import h2o
from h2o.estimators import H2OModelSelectionEstimator
h2o.init()

# Import the prostate dataset:
prostate = h2o.import_file("http://s3.amazonaws.com/h2o-public-test-data/smalldata/logreg/prostate.csv")
Parse progress: =======================================  (done)| 100%

# Set the predictors & response:
predictors = ["AGE","RACE","CAPSULE","DCAPS","PSA","VOL","DPROS"]
response = "GLEASON"

# Build & train the model:
maxrModel = H2OModelSelectionEstimator(max_predictor_number=7,
                                       seed=12345,
                                       mode="maxr")
maxrModel.train(x=predictors, y=response, training_frame=prostate)
maxr Model Build progress: ======================================= (done)| 100%

# Retrieve the results (H2OFrame containing best model_ids, best_r2_value, & predictor subsets):
results = maxrModel.result()
print(results)
model_name                 model_id                       best_r2_value  predictor_names
-------------------------  ---------------------------  ---------------  ------------------------------------------
best 1 predictor(s) model  GLM_model_1638380984255_2           0.205887  CAPSULE
best 2 predictor(s) model  GLM_model_1638380984255_13          0.269568  CAPSULE, PSA
best 3 predictor(s) model  GLM_model_1638380984255_42          0.286253  CAPSULE, DCAPS, PSA
best 4 predictor(s) model  GLM_model_1638380984255_81          0.290446  CAPSULE, DPROS, DCAPS, PSA
best 5 predictor(s) model  GLM_model_1638380984255_106         0.29217   CAPSULE, AGE, DPROS, DCAPS, PSA
best 6 predictor(s) model  GLM_model_1638380984255_121         0.292476  CAPSULE, AGE, RACE, DPROS, DCAPS, PSA
best 7 predictor(s) model  GLM_model_1638380984255_128         0.292556  CAPSULE, AGE, RACE, DPROS, DCAPS, PSA, VOL

[7 rows x 4 columns]

# Retrieve the list of coefficients:
coeff = maxrModel.coef()
print(coeff)
# [{‘Intercept’: 5.978584176203302, ‘CAPSULE’: 1.0074379937434323},
# {‘Intercept’: 5.83309940166519, ‘CAPSULE’: 0.8107305373380133, ‘PSA’: 0.01458178860012023},
# {‘Intercept’: 5.349021488372978, ‘CAPSULE’: 0.757501440465183, ‘DCAPS’: 0.47979554935185015, ‘PSA’: 0.012890961277678725},
# {‘Intercept’: 5.239249580225221, ‘CAPSULE’: 0.7184586144005665, ‘DPROS’: 0.07616613714619831, ‘DCAPS’: 0.4425789341205361, ‘PSA’: 0.012485121785672872},
# {‘Intercept’: 4.785482292681689, ‘CAPSULE’: 0.7207023955198935, ‘AGE’: 0.006873599969264931, ‘DPROS’: 0.07827698214607832, ‘DCAPS’: 0.4377770966619996, ‘PSA’: 0.012450143759298283},
# {‘Intercept’: 4.853286962151182, ‘CAPSULE’: 0.7173933092205801, ‘AGE’: 0.00679089119920351, ‘RACE’: -0.06068692599374028, ‘DPROS’: 0.07928808123744804, ‘DCAPS’: 0.4384709133624667, ‘PSA’: 0.012572275831333262},
# {‘Intercept’: 4.852663604264297, ‘CAPSULE’: 0.7153633277776693, ‘AGE’: 0.006948797960002643, ‘RACE’: -0.05843440305164041, ‘DPROS’: 0.07918100130777159, ‘DCAPS’: 0.43531498557623927, ‘PSA’: 0.012606061059188276, ‘VOL’: -0.0005196059470357373}]

# Retrieve the list of coefficients for a subset size of 3:
coeff3 = maxrModel.coef(3)
print(coeff3)
# {'Intercept': 5.349021488372978, 'CAPSULE': 0.757501440465183, 'DCAPS': 0.47979554935185015, 'PSA': 0.012890961277678725}

# Retrieve the list of standardized coefficients:
coeff_norm = maxrModel.coef_norm()
print(coeff_norm)
# [{‘Intercept’: 6.38421052631579, ‘CAPSULE’: 0.49472694682382257},
# {‘Intercept’: 6.38421052631579, ‘CAPSULE’: 0.39812896270042736, ‘PSA’: 0.29160037716849074},
# {‘Intercept’: 6.38421052631579, ‘CAPSULE’: 0.37198951914000183, ‘DCAPS’: 0.1490515817762952, ‘PSA’: 0.25778793491797924},
# {‘Intercept’: 6.38421052631579, ‘CAPSULE’: 0.3528165891390707, ‘DPROS’: 0.07617433400499243, ‘DCAPS’: 0.13749000023165447, ‘PSA’: 0.24967213018482057},
# {‘Intercept’: 6.38421052631579, ‘CAPSULE’: 0.353918452469022, ‘AGE’: 0.04486447687517968, ‘DPROS’: 0.07828540617010687, ‘DCAPS’: 0.1359982784564225, ‘PSA’: 0.2489726545605919},
# {‘Intercept’: 6.38421052631579, ‘CAPSULE’: 0.352293445102015, ‘AGE’: 0.044324630838403115, ‘RACE’: -0.018738499858626197, ‘DPROS’: 0.07929661407409055, ‘DCAPS’: 0.1362138170890904, ‘PSA’: 0.2514149995462732},
# {‘Intercept’: 6.38421052631579, ‘CAPSULE’: 0.35129657330683034, ‘AGE’: 0.04535529952002336, ‘RACE’: -0.018042981011017332, ‘DPROS’: 0.07918952262067014, ‘DCAPS’: 0.13523340776861126, ‘PSA’: 0.25209062209542776, ‘VOL’: -0.009533532448945743}]

# Retrieve the list of standardized coefficients for a subset size of 3:
coeff_norm3 = maxrModel.coef_norm(3)
print(coeff_norm3)
# {‘Intercept’: 6.38421052631579, ‘CAPSULE’: 0.37198951914000183, ‘DCAPS’: 0.1490515817762952, ‘PSA’: 0.25778793491797924}

References¶

1. Alan Miller, Subset Selection in Regression, section 3.5, Second Edition, 2002 Chapman & Hall/CRC.