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If an original predictor of a recommended predictor exists, then we also collect bivariate statistics between this original predictor and the target; if an original predictor has a recast version, then we use the recast version. Computing predictive power Predictive power is used to measure the usefulness of a predictor and is computed with respect to the transformed target. If an original predictor of a recommended predictor exists, then we also compute predictive power for this original predictor; if an original predictor has a recast version, then we use the recast version.

When the target is continuous, we fit a linear regression model and predictive power is computed as follows. Categorical target. An analysis of transformations. Goodman, L. Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74, — A Bayesian network model consists of the graph G together with a conditional probability table for each node given values of its parent nodes.

Given the value of its parents, each node is assumed to be independent of all the nodes that are not its descendents. Given set of variables V and a corresponding sample dataset, we are presented with the task of fitting an appropriate Bayesian network model. The task of determining the appropriate edges in the graph G is called structure learning, while the task of estimating the conditional probability tables given parents for each node is called parameter learning.

This algorithm is used mainly for classification. It efficiently creates a simple Bayesian network model. Its main advantages are its classification accuracy and favorable performance compared with general Bayesian network models. Its disadvantage is also due to its simplicity; it imposes much restriction on the dependency structure uncovered among its nodes.

Markov blanket identifies all the variables in the network that are needed to predict the target variable. This can produce more complex networks, but also takes longer to produce. Using feature selection preprocessing can significantly improve performance of this algorithm. Denote the number of records in for which take its jth value and for which for which takes its jth value.

The number of non-redundant parameters of TAN The Markov blanket boundary about target A subset of A subset of with respect to , such that variables and are conditionally independent in G. A directed arc from to in G. A variable set which represents all the adjacent variables of variable in G, ignoring the edge directions.

The conditional independence CI test function which returns the p-value of the test. The significance level for CI tests between two variables. If the p-value of the test is larger than then they are independent, and vice-versa. The cardinality of , The cardinality of the parent set of.

Target variables must be discrete flag or set type. Numeric predictors are discretized into 5 equal-width bins before the BN model is built. Feature Selection via Breadth-First Search Feature selection preprocessing works as follows: E It begins by searching for the direct neighbors of a given target Y, based on statistical tests of independence.

These variables are known as the parents or children of Y, denoted by E For each , we look for E For each , we add it to The explicit algorithm is given below. One important method is to relax independence assumption. An example of this structure is shown below. The algorithm for the TAN classifier first learns a tree structure over using mutual information conditioned on. Then it adds a link or arc from the target node to each predictor node.

The TAN learning procedure is: 1. Take the training data D, and as input. Learn a tree-like network structure over below. Add as a parent of every where 4. Learning the parameters of TAN network. This method associates a weight to each edge corresponding to the mutual information between the two variables.

When the weight matrix is created, the MWST algorithm Prim, gives an undirected tree that can be oriented with the choice of a root. Compute between each pair of variables. Then it finds an unmarked variable whose weight with one of the marked variables is maximal, then marks this variable and adds the edge to the tree. This process is repeated until all variables are marked. Transform the resulting undirected tree to directed one by choosing the direction of all edges to be outward from it. Let denote the cardinality of the parent set of , that is, the number of different values to which the parent of can be instantiated.

So it can be calculated as. Note implies. We use to denote the number of takes its jth value. We use to denote the number of records in records in D for which take its jth value and for which takes its kth value. D for which Maximum Likelihood Estimation The closed form solution for the parameters and that maximize the log likelihood score is 43 Bayesian Networks Algorithms where denotes the number of cases with Note that if , then in the training data.

Let well as for each of the sets and denote corresponding Dirichlet distribution parameters such that and. Upon observing the dataset D, we obtain Dirichlet posterior distributions with the following sets of parameters: The posterior estimation is always used for model updating.

Adjustment for small cell counts To overcome problems caused by zero or very small cell counts, parameters can be estimated and using as posterior parameters and. Using statistical tests such as chi-squared test or G test , this algorithm finds the conditional independence relationships among the nodes and uses these relationships as constraints to construct a BN structure.

This algorithm is referred to as a dependency-analysis-based or constraint-based algorithm. Markov Blanket Conditional Independence Test The conditional independence CI test tests whether two variables are conditionally independent with respect to a conditional variable set. There are two familiar methods to compute the CI test: Pearson chi-square test and log likelihood ratio test. The test statistic for this Suppose that N is the total number of cases in D, is the number of cases in D where takes its ith category, and and are the corresponding numbers for Y and S.

So is the number of cases in D where takes its ith category and takes its jth category. We have: Because where distribution, we get the p-value for is the degrees of freedom for the as follows: As we know, the larger p-value, the less likely we are to reject the null hypothesis. For a given are significance level , if the p-value is greater than we can not reject the hypothesis that independent. We can easily generalize this independence test into a conditional independence test: The degree of freedom for is: Likelihood Ratio Test We assume the null hypothesis is that hypothesis is are independent.

The test statistic for this 45 Bayesian Networks Algorithms or equivalently, The conditional version of the independence test is The test is asymptotically distributed as a same as in the test. So the p-value for the distribution, where degrees of freedom are the test is In the following parts of this document, we use to uniformly represent the p-value of whichever test is applied.

If , we say variable X and Y are independent, and if , we say variable X and Y are conditionally independent given variable set S. Markov Blanket Structure Learning This algorithm aims at learning a Bayesian networks structure from a dataset. It starts with a , and compute for each variable pair in G. If complete graph G. Let , remove the arc between. Then for each arc perform an exhaustive to find the smallest conditional variable set S such that. After this, orientation rules are applied to orient the arcs in G.

If such S exist, delete arc Markov Blanket Arc Orientation Rules Arcs in the derived structure are oriented based on the following rules: 1. All patterns of the of the form 2. Patterns of the form 3. Patterns of the form 4. Patterns of the form or are updated so that are updated to are updated to if 46 Bayesian Networks Algorithms are updated so that After the last step, if there are still undirected arcs in the graph, return to step 2 and repeat until all arcs are oriented.

Given a Bayesian Network G and a target variable Y, to derive the Markov Blanket of Y, we should select all the directed and all the parents of Y in G denoted as , all the directed children of Y in G denoted as in G denoted as.

Markov Blanket Parameter Learning Maximum Likelihood Estimation The closed form solution for the parameters the log likelihood score is Note that if , then that maximize. The number of parameters K is Posterior Estimation Assume that Dirichlet prior distributions are specified for each of the sets Heckerman et al. Let denote corresponding Dirichlet distributed parameters such that. Upon observing the dataset D, we obtain Dirichlet posterior distributions with the following sets of parameters: The posterior estimate is always used for model updating.

If the Use only complete records option is deselected, then for each pairwise comparison between fields, all records containing valid values for the two fields in question are used. The target category with the highest posterior probability is the predicted category for this case, , is predicted by Markov Blanket Models The scoring function uses the estimated model to compute the probabilities of Y belongs to each category for a new case.

Suppose is the parent set of Y, and denotes the given case , denotes the direct children set of Y, configuration of denotes the parent set excluding Y of the ith variable in. The score for each category of Y is computed by: , , where the joint probability that , and is: 48 Bayesian Networks Algorithms where Note that c is never actually computed during scoring because its value cancels from the numerator and denominator of the scoring equation given above.

For details on how each model type is built, see the appropriate algorithm documentation for the model type. The node also reports several comparison metrics for each model, to help you select the optimal model for your application. The following metrics are available: Maximum Profit This gives the maximum amount of profit, based on the model and the profit and cost settings. It is calculated as Profit where is defined as if is a hit otherwise r is the user-specified revenue amount per hit, and c is the user-specified cost per record.

The default value of q is 30, but this value can be modified in the binary classifier node options. The ROC curve plots the true positive rate where the model predicts the target response and the response is observed against the false positive rate where the model predicts the target response but a nonresponse is observed.

For a good model, the curve will rise sharply near the left axis and cut across near the top, so that nearly all the area in the unit square falls below the curve. For an uninformative model, the curve will approximate a diagonal line from the lower left to the upper right corner of the graph. Thus, the closer the AUC is to 1. Figure ROC curves for a good model left and an uninformative model right The AUC is computed by identifying segments as unique combinations of predictor values that determine subsets of records which all have the same predicted probability of the target value.

Note: Modeler 13 upgraded the C5. See the RuleQuest website for more information. Scoring A record is scored with the class and confidence of the rule that fires for that record. If a rule set is directly generated from the C5. For each record, all rules are examined and each rule that applies to the record is used to generate a prediction and an associated confidence.

The sum of confidence figures for each output value is computed, and the value with the greatest confidence sum is chosen as the final prediction. The confidence for the final prediction is the confidence sum for that value divided by the number of rules that fired for that record.

Scores with boosted C5. The voting for boosted C5 classifiers is as follows. For each record, each composite classifier rule set or decision tree assigns a prediction and a confidence. The confidence for the final prediction by the boosted classifier is the confidence sum for that value divided by confidence sum for all values. It uses only two data passes and delivers results for much lower support levels than Apriori. In addition, it allows changes in the support level during execution.

Carma deals with items and itemsets that make up transactions. Deriving Rules Carma proceeds in two stages. First it identifies frequent itemsets in the data, and then it generates rules from the lattice of frequent itemsets. Frequent Itemsets Carma uses a two-phase method of identifying frequent itemsets. Phase I: Estimation In the estimation phase, Carma uses a single data pass to identify frequent itemset candidates.

A lattice is used to store information on itemsets. An itemset Y is an ancestor of itemset X if X contains every item in Y. More specifically, Y is a parent of X if X contains every item in Y plus one additional item. Initially the lattice contains no itemsets.

As each transaction is read, the lattice is updated in three steps: E Increment statistics. For each itemset in the lattice that exists in the current transaction, increment the count value. For each itemset v in the transaction that is not already in the lattice, check all subsets of the itemset in the lattice. E Prune the lattice. Every k transactions where k is the pruning value, set to by default , the lattice is examined and small itemsets are removed.

Phase II: Validation After the frequent itemset candidates have been identified, a second data pass is made to compute exact frequencies for the candidates, and the final list of frequent itemsets is determined based on these frequencies. The first step in Phase II is to remove infrequent itemsets from the lattice.

When all nodes in the lattice are marked as exact, phase II terminates. Generating Rules Carma uses a common rule-generating algorithm for extracting rules from the lattice of itemsets that tends to eliminate redundant rules Aggarwal and Yu, An itemset Y is a maximal ancestor of itemset X if , where c is the specified confidence threshold for rules.

Maximum rule size. Sets the limit on the number of items that will be considered as an itemset. Exclude rules with multiple consequents. This option restricts rules in the final rule list to those with a single item as consequent. Set pruning value. Sets the number of transactions to process between pruning passes.

Vary support. Allows support to vary in order to enhance training during the early transactions in the training data. Allow rules without antecedents. Allows rules that are consequent only, which are simple statements of co-occuring items, along with traditional if-then rules. Varying support If the vary support option is selected, the target support value changes as transactions are processed to provide more efficient training.

The support value starts large and decreases in four steps as transactions are processed. The first support value s1 applies to the first 9 transactions, the second value s2 applies to the next 90 transactions, the third value s3 applies to transactions , and the fourth value s4 applies to all remaining transactions. If we call the final support value s, and the estimated number of transactions t, then the following constraints are used to determine the support values: E If E If E If or , set , set , set.

There is an exception to this: when a numeric field is examined based on a split point, user-defined missing values are included in the comparison. It is a recursive process—each of those two subsets is then split again, and the process repeats until the homogeneity criterion is reached or until some other stopping criterion is satisfied as do all of the tree-growing methods. The same predictor field may be used several times at different levels in the tree.

It uses surrogate splitting to make the best use of data with missing values. It allows unequal misclassification costs to be considered in the tree growing process. It also allows you to specify the prior probability distribution in a classification problem. Primary Calculations The calculations directly involved in building the model are described below.

Frequency and Case Weight Fields Frequency and case weight fields are useful for reducing the size of your dataset. Each has a distinct function, though. If a case weight field is mistakenly specified to be a frequency field, or vice versa, the resulting analysis will be incorrect.

For the calculations described below, if no frequency or case weight fields are specified, assume that frequency and case weights for all records are equal to 1. Frequency Fields A frequency field represents the total number of observations represented by each record. It is useful for analyzing aggregate data, in which a record represents more than one individual. The sum of the values for a frequency field should always be equal to the total number of observations in the sample.

Note that output and statistics are the same whether you use a frequency field or case-by-case data. The table below shows a hypothetical example, with the predictor fields sex and employment and the target field response. The frequency field tells us, for example, that 10 employed men responded yes to the target question, and 19 unemployed women responded no.

Case weights The use of a case weight field gives unequal treatment to the records in a dataset. When a case weight field is used, the contribution of a record in the analysis is weighted in proportion to the population units that the record represents in the sample. For example, suppose that in a direct marketing promotion, 10, households respond and 1,, households do not respond.

You can do this if you define a case weight equal to 1 for responders and for nonresponders. Here purity refers to similarity of values of the target field. In a completely pure node, all of the records have the same value for the target field. Sort the field values for records in the node from smallest to largest. Choose each point in turn as a split point, and compute the impurity statistic for the resulting child nodes of the split.

Select the best split point for the field as the one that yields the largest decrease in impurity relative to the impurity of the node being split. Examine each possible combination of values as two subsets. For each combination, calculate the impurity of the child nodes for the split based on that combination. Find the best split for the node. Check stopping rules, and recurse.

If no stopping rules are triggered by the split or by the parent node, apply the split to create two child nodes. Apply the algorithm again to each child node. Surrogate splitting is used to handle blanks for predictor fields. If the best predictor field to be used for a split has a blank or missing value at a particular node, another field that yields a split similar to the predictor field in the context of that node is used as a surrogate for the predictor field, and its value is used to assign the record to one of the child nodes.

Unless, of course, this record also has a missing value on X. In such a situation, the next best surrogate is used, and so on, up to the limit of number of surrogates specified. In the interest of speed and memory conservation, only a limited number of surrogates is identified for each split in the tree. If a record has missing values for the split field and all surrogate fields, it is assigned to the child node with the higher weighted probability, calculated as where Nf,j t is the sum of frequency weights for records in category j for node t, and Nf t is the sum of frequency weights for all records in node t.

Predictive measure of association Let resp. Let be the probability of sending a case in to the same child by both and , and be the split with maximized probability. For symbolic target fields, you can choose Gini or twoing. For continuous targets, the least-squared deviation LSD method is automatically selected. Note that when the Gini index is used to find the improvement for a split during tree growth, only those records in node t and the root node with valid values for the split-predictor are used to compute Nj t and Nj, respectively.

When all records in the node belong to the same category, the Gini index equals 0. Twoing The twoing index is based on splitting the target categories into two superclasses, and then finding the best split on the predictor field based on those two superclasses. The twoing criterion function for split s at node t is defined as where tL and tR are the nodes created by the split s. The split s is chosen as the split that maximizes this criterion. The LSD measure R t is simply the weighted within-node variance for node t, and it is equal to the resubstitution estimate of risk for the node.

It is defined as where NW t is the weighted number of records in node t, wi is the value of the weighting field for record i if any , fi is the value of the frequency field if any , yi is the value of the target field, and y t is the weighted mean for node t.

Stopping Rules Stopping rules control how the algorithm decides when to stop splitting nodes in the tree. Tree growth proceeds until every leaf node in the tree triggers at least one stopping rule. Profits Profits are numeric values associated with categories of a symbolic target field that can be used to estimate the gain or loss associated with a segment. They define the relative value of each value of the target field. Values are used in computing gains but not in tree growing.

Profit for each node in the tree is calculated as where j is the target field category, fj t is the sum of frequency field values for all records in node t with category j for the target field, and Pj is the user-defined profit value for category j. Priors Prior probabilities are numeric values that influence the misclassification rates for categories of the target field. They specify the proportion of records expected to belong to each category of the target field prior to the analysis.

The values are involved both in tree growing and risk estimation. There are three ways to derive prior probabilities. Empirical Priors By default, priors are calculated based on the training data. The prior probability assigned to each target category is the weighted proportion of records in the training data belonging to that category, In tree-growing and class assignment, the Ns take both case weights and frequency weights into account if defined ; in risk estimation, only frequency weights are included in calculating empirical priors.

The values specified for the priors must conform to the probability constraint: the sum of priors for all categories must equal 1. Costs Gini. If costs are specified, the Gini index is computed as where C i j specifies the cost of misclassifying a category j record as category i. Costs, if specified, are not taken into account in splitting nodes using the twoing criterion. However, costs will be incorporated into node assignment and risk estimation, as described in Predicted Values and Risk Estimates, below.

Costs do not apply to regression trees. Pruning Pruning refers to the process of examining a fully grown tree and removing bottom-level splits that do not contribute significantly to the accuracy of the tree. In pruning the tree, the software tries to create the smallest tree whose misclassification risk is not too much greater than that of the largest tree possible.

It removes a tree branch if the cost associated with having a more complex tree exceeds the gain associated with having another level of nodes branch. It uses an index that measures both the misclassification risk and the complexity of the tree, since we want to minimize both of these things.

This cost-complexity measure is defined as follows: R T is the misclassification risk of tree T, and is the number of terminal nodes for tree T. Cost-complexity pruning works by removing the weakest split. Determining the threshold is a simple computation. Prune the branch from the tree, and calculate the risk estimate of the pruned tree. E Repeat the previous step until only the root node is left, yielding a series of trees, T1, T2, E If the standard error rule option is selected, choose the smallest tree Topt for which E If the standard error rule option is not selected, then the tree with the smallest risk estimate R T is selected.

Risk Estimates Risk estimates describe the risk of error in predicted values for specific nodes of the tree and for the tree as a whole. If the model uses user-specified priors, the risk estimate is calculated as Note that case weights are not considered in calculating risk estimates.

Risk Estimates for numeric target field For regression trees with a numeric target field , the risk estimate r t of a node t is computed as where fi is the frequency weight for record i a record assigned to node t , yi is the value of the is the weighted mean of the target field for all records in node t.

Gain Summary The gain summary provides descriptive statistics for the terminal nodes of a tree. If profits are defined for the tree, the gain is the average profit value for each terminal node, where P xi is the profit value assigned to the target value observed in record xi. This weighted mean is calculated as where Nw t is defined as Confidence For classification trees, confidence values for records passed through the generated model are calculated as follows. For regression trees, no confidence value is assigned.

It is a highly efficient statistical technique for segmentation, or tree growing, developed by Kass, Using the significance of a statistical test as a criterion, CHAID evaluates all of the values of a potential predictor field. It merges values that are judged to be statistically homogeneous similar with respect to the target variable and maintains all other values that are heterogeneous dissimilar. It then selects the best predictor to form the first branch in the decision tree, such that each child node is made of a group of homogeneous values of the selected field.

This process continues recursively until the tree is fully grown. The statistical test used depends upon the measurement level of the target field. If the target field is continuous, an F test is used. If the target field is categorical, a chi-squared test is used.

CHAID is not a binary tree method; that is, it can produce more than two categories at any particular level in the tree. Therefore, it tends to create a wider tree than do the binary growing methods. It works for all types of variables, and it accepts both case weights and frequency variables.

It handles missing values by treating them all as a single valid category. In particular, sometimes CHAID may not find the optimal split for a variable, since it stops merging categories as soon as it finds that all remaining categories are statistically different. Exhaustive CHAID remedies this by continuing to merge categories of the predictor variable until only two supercategories are left.

It then examines the series of merges for the predictor and finds the set of categories that gives the strongest association with the target variable, and computes an adjusted p-value for that association. Thus, Exhaustive CHAID can find the best split for each predictor, and then choose which predictor to split on by comparing the adjusted p-values.

Because its method of combining categories of variables is more thorough than that of CHAID, it takes longer to compute. Binning of Scale-Level Predictors Scale level continuous predictor fields are automatically discretized or binned into a set of ordinal categories. The binned categories are determined as follows: 1. The data values yi are sorted.

For each unique value, starting with the smallest, calculate the relative weighted frequency of values less than or equal to the current value yi: where wk is the weight for record k or 1. Determine the bin to which the value belongs by comparing the relative frequency with the ideal bin percentile cutpoints of 0.

However, when the number of records having a single value is large or a set of records with the same value has a large combined weighted frequency , the binning may result in fewer bins. This will also happen if there are fewer than k distinct values for the binned field for records in the training data. However, continuous predictor fields are automatically categorized for the purpose of the analysis. Each final category of a predictor field X will represent a child node if X is used to split the node.

The following steps are applied to each predictor field X: 1. If X has one or two categories, no more categories are merged, so proceed to node splitting below. Find the eligible pair of categories of X that is least significantly different most similar as determined by the p-value of the appropriate statistical test of association with the target field.

For ordinal fields, only adjacent categories are eligible for merging; for nominal fields, all pairs are eligible. Otherwise, skip to step 6. If the user has selected the Allow splitting of merged categories option, and the newly formed compound category contains three or more original categories, then find the best binary split within the compound category that for which the p-value of the statistical test is smallest.

Continue merging categories from step 1 for this predictor field. Any category with fewer than the user-specified minimum segment size records is merged with the most similar other category that which gives the largest p-value when compared with the small category.

For each predictor variable X, find the pair of categories of X that is least significantly different that is, has the largest p-value with respect to the target variable Y. The method used to calculate the p-value depends on the measurement level of Y. Merge into a compound category the pair that gives the largest p-value. Calculate the p-value based on the new set of categories of X. This represents one set of categories for X. Remember the p-value and its corresponding set of categories.

Repeat steps 1, 2, and 3 until only two categories remain. Then, compare the sets of categories of X generated during each step of the merge sequence, and find the one for which the p-value in step 3 is the smallest. That set is the set of merged categories for X to be used in determining the split at the current node. Splitting Nodes When categories have been merged for all predictor fields, each field is evaluated for its association with the target field, based on the adjusted p-value of the statistical test of association, as described below.

Each of the merged categories of the split field defines a child node of the split. Processing proceeds recursively until one or more stopping rules are triggered for every unsplit node, and no further splits can be made. Statistical Tests Used Calculations of the unadjusted p-values depend on the type of the target field. During the merge step, categories are compared pairwise, that is, one possibly compound category is compared against another possibly compound category.

For such comparisons, only records belonging to one of the comparison categories in the current node are considered. During the split step, all categories are considered in calculating the p-value, thus all records in the current node are used. Scale Target Field F Test. For models with a scale-level target field, the p-value is calculated based on a standard ANOVA F-test comparing the target field means across categories of the predictor field under consideration.

To do the test, a contingency count table is formed using classes of Y as columns and categories of the predictor X as rows. The expected cell frequencies under the null hypothesis of independence are estimated. The observed cell frequencies and the expected cell frequencies are used to calculate the chi-squared statistic, and the p-value is based on the calculated statistic.

Expected Frequencies for Chi-Square Test Likelihood-ratio Chi-squared test The likelihood-ratio chi-square is calculated based on the expected and observed frequencies, as described above. If , stop and output , , and , , and as the final estimates of. Otherwise, increment k and repeat from step 2. Ordinal Target Field Row Effects Model If the target field Y is ordinal, the null hypothesis of independence of X and Y is tested against the row effects model, with the rows being the categories of X and the columns the categories under the hypothesis of of Y Goodman, Two sets of expected cell frequencies, independence and under the hypothesis that the data follow the row effects model , are both estimated.

By default, the order of each category is used as the category score. Users can specify their own set of scores. Parameter estimates procedure: , , , and hence are calculated using the following iterative 1. If estimates of , stop and set , , , and , , , and as the final. Bonferroni Adjustment The adjusted p-value is calculated as the p-value times a Bonferroni multiplier. The Bonferroni multiplier controls the overall p-value across multiple statistical tests. Suppose that a predictor field originally has I categories, and it is reduced to r categories after the merging step.

The Bonferroni multiplier B is the number of possible ways that I categories can be merged into r categories. If case weights are specified and the case weight for a record is blank, zero, or negative, the record is ignored, and likewise for frequency weights. For other records, blanks in predictor fields are treated as an additional category for the field.

Ordinal Predictors The algorithm first generates the best set of categories using all non-blank information. Then the algorithm identifies the category that is most similar to the blank category. Finally, two p-values are calculated: one for the set of categories formed by merging the blank category with its most similar category, and the other for the set of categories formed by adding the blank category as a separate category.

The set of categories with the smallest p-value is used. Nominal Predictors The missing category is treated the same as other categories in the analysis. Effect of Options Stopping Rules Stopping rules control how the algorithm decides when to stop splitting nodes in the tree. They define the order and distance between categories of an ordinal categorical target field. Values of scores are involved in tree growing. If user-specified scores are provided, they are used in calculation of expected cell frequencies, as described above.

Secondary Calculations Secondary calculations are not directly related to building the model, but give you information about the model and its performance. Note that case weights are not considered in calculating risk estimates. If profits are defined for the tree, the gain is the average profit value for each terminal node, 84 CHAID Algorithms where P xi is the profit value assigned to the target value observed in record xi.

This weighted mean is calculated as where Nw t is defined as 85 CHAID Algorithms Confidence For classification trees, confidence values for records passed through the generated model are calculated as follows. For nodes where there were no blanks in the training data, a blank category will not exist for the split of that node.

In that case, records with a blank value for the split field are assigned a null value. Cluster Evaluation Algorithms This document describes measures used for evaluating clustering models. It can be used to evaluate individual objects, clusters, and models. For both range numeric and discrete variables, the higher the importance measure, the less likely the variation for a variable between clusters is due to chance and more likely due to some underlying difference.

Notation The following notation is used throughout this chapter unless otherwise stated: Continuous variable k in case i standardized. The sth category of variable k in case i one-of-c coding. N Total number of valid cases. The number of cases in cluster j. Y Variable with J cluster labels. The centroid of cluster j for variable k. The distance between case i and the centroid of cluster j. The distance between the overall mean and the centroid of cluster j.

Goodness Measures The average Silhouette coefficient is simply the average over all cases of the following calculation for each individual case: where A is the average distance from the case to every other case assigned to the same cluster and B is the minimal average distance from the case to cases of a different cluster across all clusters.

Unfortunately, this coefficient is computationally expensive. As found by Kaufman and Rousseeuw , an average silhouette greater than 0. Data Preparation Before calculating Silhouette coefficient, we need to transform cases as follows: 1. Recode categorical variables using one-of-c coding.

If a variable has c categories, then it is stored as c vectors, with the first category denoted 1,0, The order of the categories is based on the ascending sort or lexical order of the data values. Rescale continuous variables. This normalization tries to equalize the contributions of continuous and categorical features to the distance computations.

Basic Statistics The following statistics are collected in order to compute the goodness measures: the centroid of variable k for cluster j, the distance between a case and the centroid, and the overall mean u. For with an ordinal or continuous variable k, we average all standardized values of variable k within cluster j. For nominal variables, is a vector of probabilities of occurrence for each state s of variable k for cluster j. Note that in counting , we do not consider cases with missing values in variable k.

If the value of variable k is missing for all cases within cluster j, is marked as missing. At this point, we do not consider differential weights, thus equals 1 if the variable k in case i is valid, 0 if not. If all equal 0, set. The distance component is calculated as follows for ordinal and continuous variables For binary or nominal variables, it is 89 Cluster Evaluation Algorithms where variable k uses one-of-c coding, and is the number of its states. The calculation of is the same as that of is used in place of.

If equals 0, the Silhouette of case i is not used in the average operations. In order to compare between models, we will use the averaged form, defined as: Average SSB Predictor Importance The importance of field i is defined as 90 Cluster Evaluation Algorithms where denotes the set of predictor and evaluation fields, is the significance or equals zero, set p-value computed from applying a certain test, as described below.

If , where MinDouble is the minimal double value. The degrees of freedom are The p-value for continuous fields is based on an F test. The chi-square statistic for cluster j is computed as follows If , the importance is set to be undefined or unknown; If , subtract one from I for each such category to obtain If , the importance is set to be undefined or unknown.

The degrees of freedom are The null hypothesis for continuous fields is that the mean in cluster j is the same as the overall mean. References Kaufman, L. Finding groups in data: An introduction to cluster analysis. New York: John Wiley and Sons. Tan, P. Steinbach, and V. Introduction to Data Mining. These models are called proportional hazards models. Under the proportional hazards assumption, the hazard function h of t given X is of the form where x is a known vector of regressor variables associated with the individual, is a vector of unknown parameters, and is the baseline hazard function for an individual with.

Hence, for any two covariates sets and , the log hazard functions and should be parallel across time. When a factor does not affect the hazard function multiplicatively, stratification may be useful in model building. Suppose that individuals can be assigned to one of m different strata, defined by the levels of one or more factors. The hazard function for an individual in the jth stratum is defined as There are two unknown components in the model: the regression parameter and the baseline.

The estimation for the parameters is described below. Let denote the probability density function pdf of T given a regressor be the survivor function the probability of an individual surviving until time x and let t. The approach we use here is to estimate from the partial likelihood function and then to maximize the full likelihood for.

Let stratum and defined by be the observed uncensored failure time of the individuals in the jth be the corresponding covariates. Then the partial likelihood function is where is the sum of case weights of individuals whose lifetime is equal to and is individuals, is the case weight of the weighted sum of the regression vector x for those individual l, and is the set of individuals alive and uncensored just prior to in the jth stratum.

Thus the log-likelihood arising from the partial likelihood function is 95 COXREG Algorithms and the first derivatives of l are is the rth component of. The maximum partial likelihood estimate MPLE of is obtained by setting equal to zero for , where p is the number of can usually be independent variables in the model.

The equations solved by using the Newton-Raphson method. All the covariates are centered by their corresponding overall mean. The overall mean of a covariate is defined as the sum of the product of weight and covariate for all the censored and uncensored cases in each stratum.

For notational simplicity, used in the Estimation Section denotes centered covariates. The r, s -th is the information matrix containing minus the second partial derivatives of element of I is defined by We can also write I in a matrix form as 96 COXREG Algorithms where is a matrix which represents the p covariate variables in the model evaluated is the number of distinct individuals in , and is a matrix with at time , the lth diagonal element defined by and the l, k element defined by Estimation of the Baseline Function is estimated separately for After the MPLE of is found, the baseline survivor function each stratum.

Assume that, for a stratum, are observed lifetimes in the sample. There are at risk and deaths at , and in the interval there are censored times. Further, it follows that and , the observed likelihood function is of the form Writing where is the set of individuals dying at and is the set of individuals with censored times in.

Note that if the last observation is uncensored, is empty and If we let Differentiating , with respect to We then plug the MPLE of can be written as and setting the equations equal to zero, we get into this equation and solve these k equations separately. A good initial value for is where Once the is the weight sum for set , are found,. See Lawless, , p. The asymptotic variance of is estimated by Selection Statistics for Stepwise Methods The same methods for variable selection are offered as in binary logistic regression.

Here we will only define the three removal statistics—Wald, LR, and Conditional—and the Score entry statistic. First we compute the information matrix I for all eligible variables based on the parameter estimates for the variables in the model and zero parameter estimates for the variables not in the model. Then we partition the resulting I into four submatrices as follows: where and are square matrices for variables in the model and variables not in the model, respectively, and is the cross-product matrix for variables in and out.

The score statistic for variable is defined by where is the first derivative of the log-likelihood with respect to all the parameters associated is equal to , and and are the submatrices with and in and associated with variable. Wald Statistic The Wald statistic is calculated for the variables in the model to select variables for removal. The Wald statistic for variable is defined by where is the parameter estimate associated with with. Assume that r variables are in the current model and let us call the current model the full model.

For each of r variables deleted from the full model, MPLES are found and the reduced log-likelihood function, l reduced , is calculated. Then LR statistic is defined as —2 l reduced — l full Conditional Statistic The conditional statistic is also computed for every variable in the model. The formula for conditional statistic is the same as LR statistic except that the parameter estimates for each reduced model are conditional estimates, not MPLES.

Let be the MPLES for the r variables blocks and C be the asymptotic covariance for the parameters left in the model given is where is the MPLE for the parameter s associated with and is without the covariance between the parameter estimates left in the model and , and covariance of. Then the conditional statistic for variable is defined by , is is the b where is the log-likelihood function evaluated at. Note that all these four statistics have a chi-square distribution with degrees of freedom equal to the number of parameters the corresponding model has.

Statistics The following output statistics are available. Initial Model Information The initial model for the first method is for a model that does not include covariates. The log-likelihood function l is equal to where is the sum of weights of individuals in set. The previous model is the model from the last step. The degrees of freedom are equal to the absolute value of the difference between the number of parameters estimated in these two models. Model Chi-Square —2 log-likelihood function for initial model — —2 log-likelihood function for current model.

The initial model is the final model from the previous method. The degrees of freedom are equal to the absolute value of the difference between the number of parameters estimated in these two model. Note: The values of the model chi-square and improvement chi-square can be less than or equal to zero. If the degrees of freedom are equal to zero, the chi-square is not printed.

Overall Chi-Square The overall chi-square statistic tests the hypothesis that all regression coefficients for the variables in the model are identically zero. This statistic is defined as where represents the vector of first derivatives of the partial log-likelihood function evaluated. Otherwise R is set to zero. For a multiple category variable, only the Wald statistic, df, significance, and partial R are printed, where R is defined by Wald df 2 log-likelihood for the intial model if Wald df.

The partial R for variables not in the equation is defined similarly to the R for the variables in the equation by changing the Wald statistic to the Score statistic. There is one overall statistic called the residual chi-square. This statistic tests if all regression coefficients for the variables not in the equation are zero. Finally, and survival function are estimated by and, for a given x, The asymptotic variances are and Plots For a specified pattern, the covariate values plots available for Cox regression.

If the plot shows a covariate. For stratum j, parallelism among strata, then the stratum variable should be a covariate. Blank Handling All records with missing values for any input or output field are excluded from the estimation of the model.

References Breslow, N. Covariance analysis of censored survival data. Biometrics, 30, 89— Approximate case influence for the proportional hazards regression model with censored data. Biometrics, 40, — Cox, D. Regression models and life tables with discussion. Kalbfleisch, J.

The statistical analysis of failure time data, 2 ed. Lawless, R. Statistical models and methods for lifetime data. Storer, B. A diagnostic for Cox regression and general conditional likelihoods. Journal of the American Statistical Association, 80, — Decision List Algorithms The objective of decision lists is to find a group of individuals with a distinct behavior pattern; for example, a high probability of buying a product. A decision list model consists of a set of decision rules.

A decision rule is an if-then rule, which has two parts: antecedent and consequent. The antecedent is a Boolean expression of predictors, and the consequent is the predicted value of the target field when the antecedent is true. The simplest construct of a decision rule is a segment. If a case is covered by one of the rules in a decision list, then it is considered to be covered by the list. In a decision list, order of rules is significant; if a case is covered by a rule, it will be ignored by subsequent rules.

Algorithm Overview The decision list algorithm can be summarized as follows: E Candidate rules are found from the original dataset. E The best rules are appended to the decision list. E Records covered by the decision list are removed from the dataset. E New rules are found based on the reduced dataset. The process repeats until one or more of the stopping criteria are met. Terminology of Decision List Algorithm The following terms are used in describing the decision list algorithm: Model.

A decision list model. In every rule discovery cycle, a set of candidate rules will be found. They will then be added to the model under construction. The resulting models will be inputs to the next cycle. Another name for a variable or field in the dataset. Source attribute.

Another name for predictor field. Extending the model. Adding decision rules to a decision list or adding segments to a decision rule. A subset of records in the dataset. Another name for group. Columns are fields attributes , and rows are records cases. A collection of list models. The ith list model of L. A list model that contains no rules. The estimated response probability of list Li. N Total population size The value of the mth field column for the nth record row of X.

The subset of records in X that are covered by list model Li. Y The target field in X. The value of the target field for the nth record. A Collection of all attributes fields of X. The jth attribute of X. R A collection of rules to extend a preceding rule list. The kth rule in rule collection R. T ResultSet A set of candidate list models. A collection of decision list models. Primary Algorithm The primary algorithm for creating a decision list model is as follows: 1.

Initialize the model. Loop over all elements E Select the records of L. For more information, Decision List Algorithms E Construct a set of new candidate models by appending each rule in R to. E Save extended list s to T. Select list models from T. E Calculate the estimated response probability of each list model in T as E Select the w lists in T with the highest.

Add as to ResultSet. With decision rules, groups are searched for significantly increased occurrence of the target value. Decision rules will search for groups with a higher or lower probability as required. Notation The following notations are used in describing the decision list algorithm: Data matrix. The ith rule in rule collection R. A special rule that covers all the cases in X.

The estimated response probability of Ri. N Total population size. The value of the mth field column for the nth record row of X. The subset of records in X that are covered by rule Ri. If Allow attribute re-use is false, A excludes attributes existing in the preceding rule. The rule split algorithm for deriving rules about Aj and records in X. A set of candidate list models. Initialize the rule set. Loop over all rules E Select records covered by rule. E Create an empty set S of new segments.

E Construct a set of new candidate rules by extending E Save extended rules to T. Select rules from T. E Calculate the estimated response probability E Select the w rules with the highest Add as for each extended rule in T as. The records and the attribute from which to generate segments should be given.

This algorithm is applicable to all ordinal attributes, and the ordinal attribute should have values that are unambiguously ordered. This decision rule split algorithm is sometimes referred to as the sea-level method. C A sorted list of attribute values categories to split. Values are sorted in ascending order. The ith category in the list of categories C. The value of the split field attribute for the nth record row of X. N M Total population size.

I am running a decision tree classification using SPSS on a data set with around 20 predictors categorical with few categories. What are the implications of using one method over the other? So depending on what you need it for I'd suggest to use CHAID if the sample is of some size and the aspects of interpretation are more important. All single-tree methods involve a staggering number of multiple comparisons that bring great instability to the result. That is why to achieve satisfactory predictive discrimination some form of tree averaging bagging, boosting, random forests is necessary except that you lose the advantage of trees - interpretability.

The simplicity of single trees is largely an illusion. They are simple because they are wrong in the sense that training the tree to multiple large subsets of the data will reveal great disagreement between tree structures.

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Highest score default Date modified newest first Date created oldest first. This may or may not be desired it can lead to better segments or easier interpretation. What it definitely does, though, is thin out the sample size in the nodes and thus lead to less deep trees. When used for segmentation purposes this can backfire soon as CHAID needs a large sample sizes to work well. CART does binary splits each node is split into two daughter nodes by default. CART can definitely do regression and classification.

CHAID uses a pre-pruning idea. A node is only split if a significance criterion is fulfilled. This ties in with the above problem of needing large sample sizes as the Chi-Square test has only little power in small samples which is effectively reduced even further by a Bonferroni correction for multiple testing.

CART on the other hand grows a large tree and then post-prunes the tree back to a smaller version. Thus CHAID tries to prevent overfitting right from the start only split is there is significant association , whereas CART may easily overfit unless the tree is pruned back. This is largely irrelevant when the trees are used for prediction but is an important issue when trees are used for interpretation: A tree that has those two parts of the algorithm highly confounded is said to be "biased in variable selection" an unfortunate name.

Compute the mean and standard deviation from the raw data. Split the continuous variable into , where non-intersecting intervals: , and. Calculate univariate statistics in each interval: , , 3. Between two tail intervals 5. If , then is 0. If it does, then Else and , find one interval with the least number of cases. Check if and is less than a threshold default , go to step 4; otherwise, go to step 6.

Check if is less than a threshold, and , go to step 4; otherwise, go to step 6. Compute the robust mean 7. See below for details. If it is, then within the range satisfies the conditions: or where cutoff is positive number default is 3 , then is detected as an outlier. Robust Mean and Standard Deviation Robust mean and standard deviation within the range as follows: are calculated and where and. Missing Value Handling Continuous variables.

Z-score Transformation Suppose a continuous variable has mean and standard deviation sd. The z-score transformation is where is the transformed value of continuous variable X for case i. Since we do not take into account the analysis weight in the rescaling formula, the rescaled values follow a normal distribution. If there is a tie on counts, then ties will be broken by ascending sort or lexical order of the data values. Continuous Target The transformation proposed by Box and Cox transforms a continuous variable into one that is more normally distributed.

We apply the Box-Cox transformation followed by the z score transformation so that the rescaled target has the user-specified mean and standard deviation. Box-Cox transformation. This transforms a non-normal variable Y to a more normally distributed variable: where are observations of variable Y, and c is a constant such that all values are positive. Here, we choose. We perform a grid search over a user-specified finite set [a,b] with increment s.

The algorithm can be described as follows: 1. Compute where j is an integer such that. For each , compute the following statistics: Mean: Standard deviation: Skewness: Sum of logarithm transformation: 3. For each , compute the log-likelihood function. Find the value of j with the largest log-likelihood function, breaking ties by selecting the smallest value of. Also find the , and. Update univariate statistics.

Continuous target or no target and all continuous predictors If there is a continuous target and some continuous predictors, then we need to calculate the covariance and correlations between all pairs of continuous variables. If there is no continuous target, then we only calculate the covariance and correlations between all pairs of continuous predictors.

We suppose there are there are m continuous variables, and denote the covariance , with element , and the correlation matrix as , with element. Start with. Reordering Categories For a nominal predictor, we rearrange categories from lowest to highest counts. The new field values start with 0 as the least frequent category. Note that the new field will be numeric even if the original field is a string.

Since we use pairwise deletion to handle missing values when we collect bivariate statistics, for a category i of a categorical we may have some categories with zero cases; that is, predictor. When we calculate p-values, these categories will be excluded. If there is only one category or no category after excluding categories with zero cases, we set the p-value to be 1 and this predictor will not be selected. Exclude all categories with zero case count.

If X has 0 categories, merge all excluded categories into one category, then stop. If X has 1 category, go to step 7. Else, find the allowable pair of categories of X that is most similar. This is the pair whose test statistic gives the largest p-value with respect to the target. An allowable pair of categories for an ordinal predictor is two adjacent categories; for a nominal predictor it is any two categories.

Note that for an ordinal predictor, if categories between the ith category and jth categories are excluded because of zero cases, then the ith category and jth categories are two adjacent categories. For the pair having the largest p-value, check if its p-value is larger than a specified alpha-level default is 0. If it does, this pair is merged into a single compound category and at the same time we calculate the bivariate statistics of this new category.

Then a new set of categories of X is formed. If it does not, then go to step 6. Go to step 3. For an ordinal predictor, find the maximum value in each new category. Sort these maximum values in ascending order. Suppose we have r new categories, and the maximum values are: , then we get the merge rule as: the first new category will contain all original , the second new category will contain all original categories such that categories such that ,…, and the last new category will contain all original categories such that.

For a nominal predictor, all categories excluded at step 1 will be merged into the new category with the lowest count. If there are ties on categories with the lowest counts, then ties are broken by selecting the category with the smallest value by ascending sort or lexical order of the original category values which formed the new categories with the lowest counts. Scale target. If the categories i and can be merged based on p-value, then the bivariate statistics should be calculated as: Categorical target.

If the categories i and can be merged based on p-value, then the bivariate statistics should be calculated as: Update univariate and bivariate statistics At the end of the supervised merge step, we calculate the bivariate statistics for each new category. For univariate statistics, the counts for each new category will be sum of the counts of each original categories which formed the new category. P-value Calculations Each p-value calculation is based on the appropriate statistical test of association between the predictor and target.

Based on F statistics, the p-value can be derived as where is a random variable following a F distribution with and degrees of freedom. At the merge step we calculate the F statistic and p-value between two categories i and where is the mean of Y for a new category and of X as merged by i and : is a random variable following a F distribution with 1 and degrees of freedom.

Nominal target The null hypothesis of independence of X and Y is tested. First a contingency or count table is formed using classes of Y as columns and categories of the predictor X as rows. Then the expected cell frequencies under the null hypothesis are estimated. The observed cell frequencies and the expected cell frequencies are used to calculate the Pearson chi-squared statistic and the p-value: where expected cell frequency for cell.

How to estimate then is the observed cell frequency and is the estimated following the independence model. If , is described below. When we investigate whether two categories i and statistic is revised as follows a chi-squared of X can be merged, the Pearson chi-squared 29 Automated Data Preparation Algorithms and the p-value is given by. Then the null hypothesis of the independence of X and Y is tested against the row effects model with the rows being the categories of X and columns the classes of Y proposed by Goodman Two sets of expected under the hypothesis of independence and under the hypothesis that cell frequencies, the data follow a row effects model , are both estimated.

The likelihood ratio statistic is where The p-value is given by. Estimated expected cell frequencies independence assumption If analysis weights are specified, the expected cell frequency under the null hypothesis of independence is of the form where and are parameters to be estimated, and Parameter estimates 1. If default is 0. Otherwise, and go to step 2. By default, the order of a class of Y is used as the class score. These orders will be standardized via the following linear transformation such that the largest score is and the lowest score is 0.

Where and are the smallest and largest order, respectively. The expected cell frequency under the row effects model is given by where parameters to be estimated. Parameter estimates 1. Otherwise, and as the final estimates and go to step 2. Unsupervised Merge If there is no target, we merge categories based on counts. Suppose that X has I categories which are sorted in ascending order. For an ordinal predictor, we sort it according to its values, while for nominal predictor we rearrange categories from lowest to highest count, with ties broken 31 Automated Data Preparation Algorithms by ascending sort or lexical order of the data values.

Let be the number of cases for the ith be the total number of cases for X. Then we use the equal frequency method category, and to merge sparse categories. If , then ; otherwise the original categories will be merged into the new category g and let , and , then go to step 2. If , then merge categories using one of the following rules: i If , then categories unmerged.

Output the merge rule and merged predictor. When original categories are merged into one new category, then the number of cases in this new category will be. At the end of the merge step, we get new categories and the number of cases in each category. Continuous Predictor Handling Continuous predictor handling includes supervised binning when the target is categorical, predictor selection when the target is continuous and predictor construction when the target is continuous or there is no target in the dataset.

Any derived predictors that are constant, or have all missing values, are excluded from further analysis. Suppose that we have already collected the bivariate statistics between the categorical target and a continuous predictor. The supervised algorithm follows: 1. Sort the means.

If , then can be considered a homogeneous subset. At the and same time we compute the mean and standard deviation of this subset: , where and then set 4. If and ; Otherwise ,. Else compute the cut point of bins. Output the binning rules. Category 1: :. The selected predictors are grouped if they are highly correlated. In each group, we will derive a new predictor using principal component analysis.

However, if there is no target, we will do not implement predictor selection. To identify highly correlated predictors, we compute the correlation between a scale and a group as follows: suppose that X is a continuous predictor and continuous predictors form a group G. Then the correlation between X and group G is defined as: where is correlation between X and. The predictor selection and predictor construction algorithm is as follows: 1. Target is continuous and predictor selection is in effect If the p-value between a continuous predictor and target is larger than a threshold default is 0.

If of each source predictor. In addition, output the remaining predictors in the correlation matrix. Find the two most correlated predictors such that their correlation in absolute value is larger than , and put them in group i. If there are no predictors to be chosen, then go to step 9.

Add one predictor to group i such that the predictor is most correlated with group i and the. Repeat this step until the number of predictors in group i is correlation is larger than greater than a threshold default is 5 or there is no predictor to be chosen. Derive a new predictor from the group i using principal component analysis. Both predictor selection and predictor construction are in effect Compute partial correlations between the other continuous predictors and the target, controlling for values of the new predictor.

Also compute the p-values based on partial correlation. If the p-value based on partial correlation between a continuous predictor and continuous target is larger than a threshold default is 0. Remove predictors that are in the group from the correlation matrix. If only predictor construction is needed, then we implement all steps except step 1 and step 7.

If both predictor selection and predictor construction are needed, then all steps are implemented. Principal Component Analysis Let as follows: 1. Input be m continuous predictors. Principal component analysis can be described , the covariance matrix of. Calculate the eigenvectors and eigenvalues of the covariance matrix. Sort the eigenvalues and corresponding eigenvectors in descending order,. Derive new predictors. Suppose the elements of the first component.

Partial correlation and P-value For two continuous variables, X and Y, we can calculate the partial correlation between them controlling for the values of a new continuous variable Z: Since the new variable Z is always a linear combination of several continuous variables, we compute the correlation of Z and a continuous variable using a property of the covariance rather than the original dataset. Suppose the new derived predictor Z is a linear combination of original : predictors Then for any a continuous variable X continuous predictor or continuous target , the correlation between X and Z is where , and.

This may occur with pairwise deletion. Based on partial correlation, the p-value is derived from the t test where and is a random variable with a t distribution with degrees of freedom,. Discretization for calculating predictive power If the transformed target is categorical, we use the equal width bins method to discretize a continuous predictor into a number of bins equal to the number of categories of the target. Discretization for creating histograms We use the equal width bins method to discretize a continuous predictor into a maximum of bins.

If their original variables are also continuous, then the original variables will be discretized. After discretization, the number of cases and mean in each bin are collected to create histograms. Predictive Power Collect bivariate statistics for predictive power We collect bivariate statistics between recommended predictors and the transformed target. If an original predictor of a recommended predictor exists, then we also collect bivariate statistics between this original predictor and the target; if an original predictor has a recast version, then we use the recast version.

Computing predictive power Predictive power is used to measure the usefulness of a predictor and is computed with respect to the transformed target. If an original predictor of a recommended predictor exists, then we also compute predictive power for this original predictor; if an original predictor has a recast version, then we use the recast version.

When the target is continuous, we fit a linear regression model and predictive power is computed as follows. Categorical target. An analysis of transformations. Goodman, L. Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74, — A Bayesian network model consists of the graph G together with a conditional probability table for each node given values of its parent nodes. Given the value of its parents, each node is assumed to be independent of all the nodes that are not its descendents.

Given set of variables V and a corresponding sample dataset, we are presented with the task of fitting an appropriate Bayesian network model. The task of determining the appropriate edges in the graph G is called structure learning, while the task of estimating the conditional probability tables given parents for each node is called parameter learning. This algorithm is used mainly for classification. It efficiently creates a simple Bayesian network model.

Its main advantages are its classification accuracy and favorable performance compared with general Bayesian network models. Its disadvantage is also due to its simplicity; it imposes much restriction on the dependency structure uncovered among its nodes. Markov blanket identifies all the variables in the network that are needed to predict the target variable.

This can produce more complex networks, but also takes longer to produce. Using feature selection preprocessing can significantly improve performance of this algorithm. Denote the number of records in for which take its jth value and for which for which takes its jth value. The number of non-redundant parameters of TAN The Markov blanket boundary about target A subset of A subset of with respect to , such that variables and are conditionally independent in G. A directed arc from to in G.

A variable set which represents all the adjacent variables of variable in G, ignoring the edge directions. The conditional independence CI test function which returns the p-value of the test. The significance level for CI tests between two variables. If the p-value of the test is larger than then they are independent, and vice-versa. The cardinality of , The cardinality of the parent set of. Target variables must be discrete flag or set type.

Numeric predictors are discretized into 5 equal-width bins before the BN model is built. Feature Selection via Breadth-First Search Feature selection preprocessing works as follows: E It begins by searching for the direct neighbors of a given target Y, based on statistical tests of independence.

These variables are known as the parents or children of Y, denoted by E For each , we look for E For each , we add it to The explicit algorithm is given below. One important method is to relax independence assumption. An example of this structure is shown below. The algorithm for the TAN classifier first learns a tree structure over using mutual information conditioned on.

Then it adds a link or arc from the target node to each predictor node. The TAN learning procedure is: 1. Take the training data D, and as input. Learn a tree-like network structure over below. Add as a parent of every where 4. Learning the parameters of TAN network.

This method associates a weight to each edge corresponding to the mutual information between the two variables. When the weight matrix is created, the MWST algorithm Prim, gives an undirected tree that can be oriented with the choice of a root. Compute between each pair of variables. Then it finds an unmarked variable whose weight with one of the marked variables is maximal, then marks this variable and adds the edge to the tree.

This process is repeated until all variables are marked. Transform the resulting undirected tree to directed one by choosing the direction of all edges to be outward from it. Let denote the cardinality of the parent set of , that is, the number of different values to which the parent of can be instantiated.

So it can be calculated as. Note implies. We use to denote the number of takes its jth value. We use to denote the number of records in records in D for which take its jth value and for which takes its kth value. D for which Maximum Likelihood Estimation The closed form solution for the parameters and that maximize the log likelihood score is 43 Bayesian Networks Algorithms where denotes the number of cases with Note that if , then in the training data.

Let well as for each of the sets and denote corresponding Dirichlet distribution parameters such that and. Upon observing the dataset D, we obtain Dirichlet posterior distributions with the following sets of parameters: The posterior estimation is always used for model updating.

Adjustment for small cell counts To overcome problems caused by zero or very small cell counts, parameters can be estimated and using as posterior parameters and. Using statistical tests such as chi-squared test or G test , this algorithm finds the conditional independence relationships among the nodes and uses these relationships as constraints to construct a BN structure. This algorithm is referred to as a dependency-analysis-based or constraint-based algorithm. Markov Blanket Conditional Independence Test The conditional independence CI test tests whether two variables are conditionally independent with respect to a conditional variable set.

There are two familiar methods to compute the CI test: Pearson chi-square test and log likelihood ratio test. The test statistic for this Suppose that N is the total number of cases in D, is the number of cases in D where takes its ith category, and and are the corresponding numbers for Y and S.

So is the number of cases in D where takes its ith category and takes its jth category. We have: Because where distribution, we get the p-value for is the degrees of freedom for the as follows: As we know, the larger p-value, the less likely we are to reject the null hypothesis. For a given are significance level , if the p-value is greater than we can not reject the hypothesis that independent. We can easily generalize this independence test into a conditional independence test: The degree of freedom for is: Likelihood Ratio Test We assume the null hypothesis is that hypothesis is are independent.

The test statistic for this 45 Bayesian Networks Algorithms or equivalently, The conditional version of the independence test is The test is asymptotically distributed as a same as in the test. So the p-value for the distribution, where degrees of freedom are the test is In the following parts of this document, we use to uniformly represent the p-value of whichever test is applied.

If , we say variable X and Y are independent, and if , we say variable X and Y are conditionally independent given variable set S. Markov Blanket Structure Learning This algorithm aims at learning a Bayesian networks structure from a dataset. It starts with a , and compute for each variable pair in G. If complete graph G. Let , remove the arc between. Then for each arc perform an exhaustive to find the smallest conditional variable set S such that.

After this, orientation rules are applied to orient the arcs in G. If such S exist, delete arc Markov Blanket Arc Orientation Rules Arcs in the derived structure are oriented based on the following rules: 1.

All patterns of the of the form 2. Patterns of the form 3. Patterns of the form 4. Patterns of the form or are updated so that are updated to are updated to if 46 Bayesian Networks Algorithms are updated so that After the last step, if there are still undirected arcs in the graph, return to step 2 and repeat until all arcs are oriented.

Given a Bayesian Network G and a target variable Y, to derive the Markov Blanket of Y, we should select all the directed and all the parents of Y in G denoted as , all the directed children of Y in G denoted as in G denoted as. Markov Blanket Parameter Learning Maximum Likelihood Estimation The closed form solution for the parameters the log likelihood score is Note that if , then that maximize.

The number of parameters K is Posterior Estimation Assume that Dirichlet prior distributions are specified for each of the sets Heckerman et al. Let denote corresponding Dirichlet distributed parameters such that. Upon observing the dataset D, we obtain Dirichlet posterior distributions with the following sets of parameters: The posterior estimate is always used for model updating. If the Use only complete records option is deselected, then for each pairwise comparison between fields, all records containing valid values for the two fields in question are used.

The target category with the highest posterior probability is the predicted category for this case, , is predicted by Markov Blanket Models The scoring function uses the estimated model to compute the probabilities of Y belongs to each category for a new case. Suppose is the parent set of Y, and denotes the given case , denotes the direct children set of Y, configuration of denotes the parent set excluding Y of the ith variable in.

The score for each category of Y is computed by: , , where the joint probability that , and is: 48 Bayesian Networks Algorithms where Note that c is never actually computed during scoring because its value cancels from the numerator and denominator of the scoring equation given above. For details on how each model type is built, see the appropriate algorithm documentation for the model type.

The node also reports several comparison metrics for each model, to help you select the optimal model for your application. The following metrics are available: Maximum Profit This gives the maximum amount of profit, based on the model and the profit and cost settings. It is calculated as Profit where is defined as if is a hit otherwise r is the user-specified revenue amount per hit, and c is the user-specified cost per record.

The default value of q is 30, but this value can be modified in the binary classifier node options. The ROC curve plots the true positive rate where the model predicts the target response and the response is observed against the false positive rate where the model predicts the target response but a nonresponse is observed.

For a good model, the curve will rise sharply near the left axis and cut across near the top, so that nearly all the area in the unit square falls below the curve. For an uninformative model, the curve will approximate a diagonal line from the lower left to the upper right corner of the graph. Thus, the closer the AUC is to 1. Figure ROC curves for a good model left and an uninformative model right The AUC is computed by identifying segments as unique combinations of predictor values that determine subsets of records which all have the same predicted probability of the target value.

Note: Modeler 13 upgraded the C5. See the RuleQuest website for more information. Scoring A record is scored with the class and confidence of the rule that fires for that record. If a rule set is directly generated from the C5. For each record, all rules are examined and each rule that applies to the record is used to generate a prediction and an associated confidence. The sum of confidence figures for each output value is computed, and the value with the greatest confidence sum is chosen as the final prediction.

The confidence for the final prediction is the confidence sum for that value divided by the number of rules that fired for that record. Scores with boosted C5. The voting for boosted C5 classifiers is as follows. For each record, each composite classifier rule set or decision tree assigns a prediction and a confidence.

The confidence for the final prediction by the boosted classifier is the confidence sum for that value divided by confidence sum for all values. It uses only two data passes and delivers results for much lower support levels than Apriori. In addition, it allows changes in the support level during execution.

Carma deals with items and itemsets that make up transactions. Deriving Rules Carma proceeds in two stages. First it identifies frequent itemsets in the data, and then it generates rules from the lattice of frequent itemsets. Frequent Itemsets Carma uses a two-phase method of identifying frequent itemsets. Phase I: Estimation In the estimation phase, Carma uses a single data pass to identify frequent itemset candidates.

A lattice is used to store information on itemsets. An itemset Y is an ancestor of itemset X if X contains every item in Y. More specifically, Y is a parent of X if X contains every item in Y plus one additional item. Initially the lattice contains no itemsets. As each transaction is read, the lattice is updated in three steps: E Increment statistics. For each itemset in the lattice that exists in the current transaction, increment the count value.

For each itemset v in the transaction that is not already in the lattice, check all subsets of the itemset in the lattice. E Prune the lattice. Every k transactions where k is the pruning value, set to by default , the lattice is examined and small itemsets are removed. Phase II: Validation After the frequent itemset candidates have been identified, a second data pass is made to compute exact frequencies for the candidates, and the final list of frequent itemsets is determined based on these frequencies.

The first step in Phase II is to remove infrequent itemsets from the lattice. When all nodes in the lattice are marked as exact, phase II terminates. Generating Rules Carma uses a common rule-generating algorithm for extracting rules from the lattice of itemsets that tends to eliminate redundant rules Aggarwal and Yu, An itemset Y is a maximal ancestor of itemset X if , where c is the specified confidence threshold for rules.

Maximum rule size. Sets the limit on the number of items that will be considered as an itemset. Exclude rules with multiple consequents. This option restricts rules in the final rule list to those with a single item as consequent. Set pruning value. Sets the number of transactions to process between pruning passes. Vary support. Allows support to vary in order to enhance training during the early transactions in the training data.

Allow rules without antecedents. Allows rules that are consequent only, which are simple statements of co-occuring items, along with traditional if-then rules. Varying support If the vary support option is selected, the target support value changes as transactions are processed to provide more efficient training.

The support value starts large and decreases in four steps as transactions are processed. The first support value s1 applies to the first 9 transactions, the second value s2 applies to the next 90 transactions, the third value s3 applies to transactions , and the fourth value s4 applies to all remaining transactions.

If we call the final support value s, and the estimated number of transactions t, then the following constraints are used to determine the support values: E If E If E If or , set , set , set. There is an exception to this: when a numeric field is examined based on a split point, user-defined missing values are included in the comparison. It is a recursive process—each of those two subsets is then split again, and the process repeats until the homogeneity criterion is reached or until some other stopping criterion is satisfied as do all of the tree-growing methods.

The same predictor field may be used several times at different levels in the tree. It uses surrogate splitting to make the best use of data with missing values. It allows unequal misclassification costs to be considered in the tree growing process. It also allows you to specify the prior probability distribution in a classification problem. Primary Calculations The calculations directly involved in building the model are described below.

Frequency and Case Weight Fields Frequency and case weight fields are useful for reducing the size of your dataset. Each has a distinct function, though. If a case weight field is mistakenly specified to be a frequency field, or vice versa, the resulting analysis will be incorrect.

For the calculations described below, if no frequency or case weight fields are specified, assume that frequency and case weights for all records are equal to 1. Frequency Fields A frequency field represents the total number of observations represented by each record. It is useful for analyzing aggregate data, in which a record represents more than one individual. The sum of the values for a frequency field should always be equal to the total number of observations in the sample.

Note that output and statistics are the same whether you use a frequency field or case-by-case data. The table below shows a hypothetical example, with the predictor fields sex and employment and the target field response. The frequency field tells us, for example, that 10 employed men responded yes to the target question, and 19 unemployed women responded no.

Case weights The use of a case weight field gives unequal treatment to the records in a dataset. When a case weight field is used, the contribution of a record in the analysis is weighted in proportion to the population units that the record represents in the sample. For example, suppose that in a direct marketing promotion, 10, households respond and 1,, households do not respond. You can do this if you define a case weight equal to 1 for responders and for nonresponders.

Here purity refers to similarity of values of the target field. In a completely pure node, all of the records have the same value for the target field. Sort the field values for records in the node from smallest to largest.

Choose each point in turn as a split point, and compute the impurity statistic for the resulting child nodes of the split. Select the best split point for the field as the one that yields the largest decrease in impurity relative to the impurity of the node being split. Examine each possible combination of values as two subsets. For each combination, calculate the impurity of the child nodes for the split based on that combination.

Find the best split for the node. Check stopping rules, and recurse. If no stopping rules are triggered by the split or by the parent node, apply the split to create two child nodes. Apply the algorithm again to each child node. Surrogate splitting is used to handle blanks for predictor fields. If the best predictor field to be used for a split has a blank or missing value at a particular node, another field that yields a split similar to the predictor field in the context of that node is used as a surrogate for the predictor field, and its value is used to assign the record to one of the child nodes.

Unless, of course, this record also has a missing value on X. In such a situation, the next best surrogate is used, and so on, up to the limit of number of surrogates specified. In the interest of speed and memory conservation, only a limited number of surrogates is identified for each split in the tree.

If a record has missing values for the split field and all surrogate fields, it is assigned to the child node with the higher weighted probability, calculated as where Nf,j t is the sum of frequency weights for records in category j for node t, and Nf t is the sum of frequency weights for all records in node t. Predictive measure of association Let resp. Let be the probability of sending a case in to the same child by both and , and be the split with maximized probability. For symbolic target fields, you can choose Gini or twoing.

For continuous targets, the least-squared deviation LSD method is automatically selected. Note that when the Gini index is used to find the improvement for a split during tree growth, only those records in node t and the root node with valid values for the split-predictor are used to compute Nj t and Nj, respectively. When all records in the node belong to the same category, the Gini index equals 0. Twoing The twoing index is based on splitting the target categories into two superclasses, and then finding the best split on the predictor field based on those two superclasses.

The twoing criterion function for split s at node t is defined as where tL and tR are the nodes created by the split s. The split s is chosen as the split that maximizes this criterion. The LSD measure R t is simply the weighted within-node variance for node t, and it is equal to the resubstitution estimate of risk for the node. It is defined as where NW t is the weighted number of records in node t, wi is the value of the weighting field for record i if any , fi is the value of the frequency field if any , yi is the value of the target field, and y t is the weighted mean for node t.

Stopping Rules Stopping rules control how the algorithm decides when to stop splitting nodes in the tree. Tree growth proceeds until every leaf node in the tree triggers at least one stopping rule. Profits Profits are numeric values associated with categories of a symbolic target field that can be used to estimate the gain or loss associated with a segment.

They define the relative value of each value of the target field. Values are used in computing gains but not in tree growing. Profit for each node in the tree is calculated as where j is the target field category, fj t is the sum of frequency field values for all records in node t with category j for the target field, and Pj is the user-defined profit value for category j.

Priors Prior probabilities are numeric values that influence the misclassification rates for categories of the target field. They specify the proportion of records expected to belong to each category of the target field prior to the analysis. The values are involved both in tree growing and risk estimation. There are three ways to derive prior probabilities. Empirical Priors By default, priors are calculated based on the training data. The prior probability assigned to each target category is the weighted proportion of records in the training data belonging to that category, In tree-growing and class assignment, the Ns take both case weights and frequency weights into account if defined ; in risk estimation, only frequency weights are included in calculating empirical priors.

The values specified for the priors must conform to the probability constraint: the sum of priors for all categories must equal 1. Costs Gini. If costs are specified, the Gini index is computed as where C i j specifies the cost of misclassifying a category j record as category i. Costs, if specified, are not taken into account in splitting nodes using the twoing criterion. However, costs will be incorporated into node assignment and risk estimation, as described in Predicted Values and Risk Estimates, below.

Costs do not apply to regression trees. Pruning Pruning refers to the process of examining a fully grown tree and removing bottom-level splits that do not contribute significantly to the accuracy of the tree. In pruning the tree, the software tries to create the smallest tree whose misclassification risk is not too much greater than that of the largest tree possible.

It removes a tree branch if the cost associated with having a more complex tree exceeds the gain associated with having another level of nodes branch. It uses an index that measures both the misclassification risk and the complexity of the tree, since we want to minimize both of these things.

This cost-complexity measure is defined as follows: R T is the misclassification risk of tree T, and is the number of terminal nodes for tree T. Cost-complexity pruning works by removing the weakest split. Determining the threshold is a simple computation. Prune the branch from the tree, and calculate the risk estimate of the pruned tree. E Repeat the previous step until only the root node is left, yielding a series of trees, T1, T2, E If the standard error rule option is selected, choose the smallest tree Topt for which E If the standard error rule option is not selected, then the tree with the smallest risk estimate R T is selected.

Risk Estimates Risk estimates describe the risk of error in predicted values for specific nodes of the tree and for the tree as a whole. If the model uses user-specified priors, the risk estimate is calculated as Note that case weights are not considered in calculating risk estimates.

Risk Estimates for numeric target field For regression trees with a numeric target field , the risk estimate r t of a node t is computed as where fi is the frequency weight for record i a record assigned to node t , yi is the value of the is the weighted mean of the target field for all records in node t. Gain Summary The gain summary provides descriptive statistics for the terminal nodes of a tree.

If profits are defined for the tree, the gain is the average profit value for each terminal node, where P xi is the profit value assigned to the target value observed in record xi. This weighted mean is calculated as where Nw t is defined as Confidence For classification trees, confidence values for records passed through the generated model are calculated as follows. For regression trees, no confidence value is assigned.

It is a highly efficient statistical technique for segmentation, or tree growing, developed by Kass, Using the significance of a statistical test as a criterion, CHAID evaluates all of the values of a potential predictor field. It merges values that are judged to be statistically homogeneous similar with respect to the target variable and maintains all other values that are heterogeneous dissimilar. It then selects the best predictor to form the first branch in the decision tree, such that each child node is made of a group of homogeneous values of the selected field.

This process continues recursively until the tree is fully grown. The statistical test used depends upon the measurement level of the target field. If the target field is continuous, an F test is used. If the target field is categorical, a chi-squared test is used. CHAID is not a binary tree method; that is, it can produce more than two categories at any particular level in the tree.

Therefore, it tends to create a wider tree than do the binary growing methods. It works for all types of variables, and it accepts both case weights and frequency variables. It handles missing values by treating them all as a single valid category. In particular, sometimes CHAID may not find the optimal split for a variable, since it stops merging categories as soon as it finds that all remaining categories are statistically different.

Exhaustive CHAID remedies this by continuing to merge categories of the predictor variable until only two supercategories are left. It then examines the series of merges for the predictor and finds the set of categories that gives the strongest association with the target variable, and computes an adjusted p-value for that association. Thus, Exhaustive CHAID can find the best split for each predictor, and then choose which predictor to split on by comparing the adjusted p-values.

Because its method of combining categories of variables is more thorough than that of CHAID, it takes longer to compute. Binning of Scale-Level Predictors Scale level continuous predictor fields are automatically discretized or binned into a set of ordinal categories. The binned categories are determined as follows: 1. The data values yi are sorted. For each unique value, starting with the smallest, calculate the relative weighted frequency of values less than or equal to the current value yi: where wk is the weight for record k or 1.

Determine the bin to which the value belongs by comparing the relative frequency with the ideal bin percentile cutpoints of 0. However, when the number of records having a single value is large or a set of records with the same value has a large combined weighted frequency , the binning may result in fewer bins. This will also happen if there are fewer than k distinct values for the binned field for records in the training data.

However, continuous predictor fields are automatically categorized for the purpose of the analysis. Each final category of a predictor field X will represent a child node if X is used to split the node. The following steps are applied to each predictor field X: 1. If X has one or two categories, no more categories are merged, so proceed to node splitting below. Find the eligible pair of categories of X that is least significantly different most similar as determined by the p-value of the appropriate statistical test of association with the target field.

For ordinal fields, only adjacent categories are eligible for merging; for nominal fields, all pairs are eligible. Otherwise, skip to step 6. If the user has selected the Allow splitting of merged categories option, and the newly formed compound category contains three or more original categories, then find the best binary split within the compound category that for which the p-value of the statistical test is smallest.

Continue merging categories from step 1 for this predictor field. Any category with fewer than the user-specified minimum segment size records is merged with the most similar other category that which gives the largest p-value when compared with the small category.

For each predictor variable X, find the pair of categories of X that is least significantly different that is, has the largest p-value with respect to the target variable Y. The method used to calculate the p-value depends on the measurement level of Y. Merge into a compound category the pair that gives the largest p-value.

Calculate the p-value based on the new set of categories of X. This represents one set of categories for X. Remember the p-value and its corresponding set of categories. Repeat steps 1, 2, and 3 until only two categories remain. Then, compare the sets of categories of X generated during each step of the merge sequence, and find the one for which the p-value in step 3 is the smallest. That set is the set of merged categories for X to be used in determining the split at the current node.

Splitting Nodes When categories have been merged for all predictor fields, each field is evaluated for its association with the target field, based on the adjusted p-value of the statistical test of association, as described below. Each of the merged categories of the split field defines a child node of the split. Processing proceeds recursively until one or more stopping rules are triggered for every unsplit node, and no further splits can be made.

Statistical Tests Used Calculations of the unadjusted p-values depend on the type of the target field. During the merge step, categories are compared pairwise, that is, one possibly compound category is compared against another possibly compound category.

For such comparisons, only records belonging to one of the comparison categories in the current node are considered. During the split step, all categories are considered in calculating the p-value, thus all records in the current node are used. Scale Target Field F Test. For models with a scale-level target field, the p-value is calculated based on a standard ANOVA F-test comparing the target field means across categories of the predictor field under consideration. To do the test, a contingency count table is formed using classes of Y as columns and categories of the predictor X as rows.

The expected cell frequencies under the null hypothesis of independence are estimated. The observed cell frequencies and the expected cell frequencies are used to calculate the chi-squared statistic, and the p-value is based on the calculated statistic. Expected Frequencies for Chi-Square Test Likelihood-ratio Chi-squared test The likelihood-ratio chi-square is calculated based on the expected and observed frequencies, as described above.

If , stop and output , , and , , and as the final estimates of. Otherwise, increment k and repeat from step 2. Ordinal Target Field Row Effects Model If the target field Y is ordinal, the null hypothesis of independence of X and Y is tested against the row effects model, with the rows being the categories of X and the columns the categories under the hypothesis of of Y Goodman, Two sets of expected cell frequencies, independence and under the hypothesis that the data follow the row effects model , are both estimated.

By default, the order of each category is used as the category score. Users can specify their own set of scores. Parameter estimates procedure: , , , and hence are calculated using the following iterative 1. If estimates of , stop and set , , , and , , , and as the final.

Bonferroni Adjustment The adjusted p-value is calculated as the p-value times a Bonferroni multiplier. The Bonferroni multiplier controls the overall p-value across multiple statistical tests. Suppose that a predictor field originally has I categories, and it is reduced to r categories after the merging step. The Bonferroni multiplier B is the number of possible ways that I categories can be merged into r categories.

If case weights are specified and the case weight for a record is blank, zero, or negative, the record is ignored, and likewise for frequency weights. For other records, blanks in predictor fields are treated as an additional category for the field.

Ordinal Predictors The algorithm first generates the best set of categories using all non-blank information. Then the algorithm identifies the category that is most similar to the blank category. Finally, two p-values are calculated: one for the set of categories formed by merging the blank category with its most similar category, and the other for the set of categories formed by adding the blank category as a separate category. The set of categories with the smallest p-value is used.

Nominal Predictors The missing category is treated the same as other categories in the analysis. Effect of Options Stopping Rules Stopping rules control how the algorithm decides when to stop splitting nodes in the tree.

They define the order and distance between categories of an ordinal categorical target field. Values of scores are involved in tree growing. If user-specified scores are provided, they are used in calculation of expected cell frequencies, as described above. Secondary Calculations Secondary calculations are not directly related to building the model, but give you information about the model and its performance. Note that case weights are not considered in calculating risk estimates.

If profits are defined for the tree, the gain is the average profit value for each terminal node, 84 CHAID Algorithms where P xi is the profit value assigned to the target value observed in record xi. This weighted mean is calculated as where Nw t is defined as 85 CHAID Algorithms Confidence For classification trees, confidence values for records passed through the generated model are calculated as follows. For nodes where there were no blanks in the training data, a blank category will not exist for the split of that node.

In that case, records with a blank value for the split field are assigned a null value. Cluster Evaluation Algorithms This document describes measures used for evaluating clustering models. It can be used to evaluate individual objects, clusters, and models. For both range numeric and discrete variables, the higher the importance measure, the less likely the variation for a variable between clusters is due to chance and more likely due to some underlying difference.

Notation The following notation is used throughout this chapter unless otherwise stated: Continuous variable k in case i standardized. The sth category of variable k in case i one-of-c coding. N Total number of valid cases. The number of cases in cluster j. Y Variable with J cluster labels. The centroid of cluster j for variable k. The distance between case i and the centroid of cluster j. The distance between the overall mean and the centroid of cluster j.

Goodness Measures The average Silhouette coefficient is simply the average over all cases of the following calculation for each individual case: where A is the average distance from the case to every other case assigned to the same cluster and B is the minimal average distance from the case to cases of a different cluster across all clusters.

Unfortunately, this coefficient is computationally expensive. As found by Kaufman and Rousseeuw , an average silhouette greater than 0. Data Preparation Before calculating Silhouette coefficient, we need to transform cases as follows: 1. Recode categorical variables using one-of-c coding. If a variable has c categories, then it is stored as c vectors, with the first category denoted 1,0, The order of the categories is based on the ascending sort or lexical order of the data values.

Rescale continuous variables. This normalization tries to equalize the contributions of continuous and categorical features to the distance computations. Basic Statistics The following statistics are collected in order to compute the goodness measures: the centroid of variable k for cluster j, the distance between a case and the centroid, and the overall mean u.

For with an ordinal or continuous variable k, we average all standardized values of variable k within cluster j. For nominal variables, is a vector of probabilities of occurrence for each state s of variable k for cluster j.

Note that in counting , we do not consider cases with missing values in variable k. If the value of variable k is missing for all cases within cluster j, is marked as missing. At this point, we do not consider differential weights, thus equals 1 if the variable k in case i is valid, 0 if not. If all equal 0, set. The distance component is calculated as follows for ordinal and continuous variables For binary or nominal variables, it is 89 Cluster Evaluation Algorithms where variable k uses one-of-c coding, and is the number of its states.

The calculation of is the same as that of is used in place of. If equals 0, the Silhouette of case i is not used in the average operations. In order to compare between models, we will use the averaged form, defined as: Average SSB Predictor Importance The importance of field i is defined as 90 Cluster Evaluation Algorithms where denotes the set of predictor and evaluation fields, is the significance or equals zero, set p-value computed from applying a certain test, as described below.

If , where MinDouble is the minimal double value. The degrees of freedom are The p-value for continuous fields is based on an F test. The chi-square statistic for cluster j is computed as follows If , the importance is set to be undefined or unknown; If , subtract one from I for each such category to obtain If , the importance is set to be undefined or unknown. The degrees of freedom are The null hypothesis for continuous fields is that the mean in cluster j is the same as the overall mean.

References Kaufman, L. Finding groups in data: An introduction to cluster analysis. New York: John Wiley and Sons. Tan, P. Steinbach, and V. Introduction to Data Mining. These models are called proportional hazards models. Under the proportional hazards assumption, the hazard function h of t given X is of the form where x is a known vector of regressor variables associated with the individual, is a vector of unknown parameters, and is the baseline hazard function for an individual with.

Hence, for any two covariates sets and , the log hazard functions and should be parallel across time. When a factor does not affect the hazard function multiplicatively, stratification may be useful in model building. Suppose that individuals can be assigned to one of m different strata, defined by the levels of one or more factors. The hazard function for an individual in the jth stratum is defined as There are two unknown components in the model: the regression parameter and the baseline.

Create a free Team Why Teams? Learn more. Asked 9 years ago. Modified 3 years, 9 months ago. Viewed 25k times. Improve this question. COOLSerdash Placidia Placidia Add a comment. Sorted by: Reset to default. Highest score default Date modified newest first Date created oldest first.

This may or may not be desired it can lead to better segments or easier interpretation. What it definitely does, though, is thin out the sample size in the nodes and thus lead to less deep trees. When used for segmentation purposes this can backfire soon as CHAID needs a large sample sizes to work well.

CART does binary splits each node is split into two daughter nodes by default. CART can definitely do regression and classification. CHAID uses a pre-pruning idea. A node is only split if a significance criterion is fulfilled. This ties in with the above problem of needing large sample sizes as the Chi-Square test has only little power in small samples which is effectively reduced even further by a Bonferroni correction for multiple testing. CART on the other hand grows a large tree and then post-prunes the tree back to a smaller version.

Thus CHAID tries to prevent overfitting right from the start only split is there is significant association , whereas CART may easily overfit unless the tree is pruned back. This is largely irrelevant when the trees are used for prediction but is an important issue when trees are used for interpretation: A tree that has those two parts of the algorithm highly confounded is said to be "biased in variable selection" an unfortunate name.

This means that split variable selection prefers variables with many possible splits say metric predictors. With surrogate splits CART knows how to handle missing values surrogate splits means that with missing values NAs for predictor variables the algorithm uses other predictor variables that are not as "good" as the primary split variable but mimic the splits produced by the primary splitter.

CHAID has no such thing afaik. Improve this answer. Momo Momo 8, 3 3 gold badges 46 46 silver badges 59 59 bronze badges. Nice overview. Could you explain what "multiway splits" and "surrogate splits" are? Are multiway splits if the splits are not dichotomous? Regarding multiway splits, I've found the following interesting statement from Hastie et al. They can also be different.

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