<BaselineModel modelName="geo-cusum" functionName="baseline" >

    <TestDistributions field="congestion-score" 
      testStatistic="CUSUM" testType="threshold" threshold="21.0" resetValue="0.0" >
      <Baseline>
        <GaussianDistribution mean="550.2" variance="48.2" />
      </Baseline>
      <Alternate>
        <GaussianDistribution mean="460.4" variance="39.2" />
      </Alternate>
    </TestDistributions>

  </BaselineModel>

Scoring with this type of change detection model can be done as follows: given a sequence of events, the mining and derived fields are evaluated to compute the field for the TestDistributions. Then the test indicated (CUSUM in the example above) is performed with parameter values supplied. In the example above, if the TestField exceeds the threshold of 21.0, a TRUE is generated, otherwise a FALSE is generated. A reset value can be used to provide an alternative to 0 in the formula above defining Z[n].

Change detection model with generalized likelihood ratio (GLR) statistic. Often there is less knowledge in change detection applications about the alternate distribution. For example, its mean and standard deviation may be unknown or very difficult to estimate. In this case, the generalized likelihood ratio (GLR) test can be used.

This would be represented in PMML with the following fragment:

  <BaselineModel modelName="geo-glr" functionName="baseline" >

    <TestDistributions field="congestion-score" 
      testStatistic="GLR" testType="threshold" threshold="12.8" resetValue="0.0" >
      <Baseline>
        <GaussianDistribution mean="550.2" variance="48.2" />
      </Baseline>
    </TestDistributions>

  </BaselineModel>

Threshold break models using standard scores. For another very basic example, if the baseline model is a single distribution, then the value of mining field can be converted to standard values (which are also called z-values) and a simple threshold used to determine whether the field is closed enough to the mean of the distribution.

  <BaselineModel modelName="three-sigma-threshold" functionName="baseline" >

    <TestDistributions field="defects" 
      testStatistic="zValue" testType="threshold" threshold="3.0" >
      <Baseline>
        <GaussianDistribution mean="18.2" variance="4.2" />
      </Baseline>
    </TestDistributions>

  </BaselineModel>

Scoring with this type of model is very simple. Given an event with a mining or derived field called "defects," the value of defects is converted to a standard score or z-value using the formula

Contingency table models using chi-squared statistics and p-values. For categorical variables, contingency tables summarize how one field and its categorical (rows) related to another field and its categorical values (columns). The baseline or null hypothesis is that there is no association between the field defining the rows and the field defining the columns. The alternate hypothesis is that there is such association. A standard approach is to compute the chi-squared statistic for the table and then to use a p-value to determine whether the chi-squared is statistically significant.

 <BaselineModel modelName="merchant-name-populated-model" functionName="baseline" >

  <TestDistributions testStatistic="chiSquared" testType="twoSidedPValue" threshold="95.0" >

    <CountTable>
      <FieldValueCounts field="merchant_name_populated">
	<FieldValueCount field="payment_accepted" count="41"/>
	<FieldValueCount field="payment_declined" count="81"/>
      </FieldValueCounts>
      <FieldValueCounts field="merchant_name_missing">
	<FieldValueCount field="payment_accepted" count="12"/>
	<FieldValueCount field="payment_declined" count="30"/>
      </FieldValueCounts>
    </CountTable>

  </TestDistributions>

 </BaselineModel>

Scoring with this type of model is done as follows. Given such a table (with any number of rows or columns), one first computes row and column totals, as well as the total for the whole table. Given any cell in the table, the expected number for that cell is computed as follows

If the table has r rows and c columns, the degree of freedom of the chi-squared statistic is defined as (r-1)(c-1). The p-value is the probability that the chi-squared statistic with this degree of freedom exceeds the chi-squared value computed from the table. It is common to use thresholds of 95%, 97.5%, or 99%.

An alternative test could have simply specified:

  <TestDistributions testStatistic="chiSquared" testType="threshold" threshold="2.1" >

In this case, the chi-squared statistic for the contingency table would have been computed as described above, and a TRUE returned if the computed chi-squared is equal to or above the threshold; otherwise, a FALSE is returned.

Target and Output elements. The PMML Target element is used by the Baseline Model to return the result computed by the model, while the PMML Output element can be used to return auxilliary information, such as derived fields that are needed for subsequent computations.