<BaselineModel modelName="geo-cusum" functionName="regression">
  <MiningSchema>
    <MiningField name="congestion-score" optype="continuous"/>
    <MiningField name="cusum-score" optype="continuous" usageType="target"/>
  </MiningSchema>

  <TestDistributions field="congestion-score" testStatistic="CUSUM" 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/or 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.

Example 2: Baseline models using standard scores. For another very basic example, if the baseline model is a single distribution, then the value of a mining field can be converted to standard values (which are also called z-values) and a simple threshold is used to determine whether the field is close enough to the mean of the distribution.

<BaselineModel modelName="standard-score" functionName="regression">
  <MiningSchema>
    <MiningField name="defects" optype="continuous"/>
    <MiningField name="score" optype="continuous" usageType="target"/>
  </MiningSchema>

  <Output>
    <OutputField name="alert" optype="categorical" dataType="string" feature="decision">
      <Apply function="if">
        <Apply function="greaterThan">
          <FieldRef field="score"/>
          <Constant dataType="double">1</Constant>
        </Apply>
        <!-- Then case -->
        <Constant dataType="string">True</Constant>
        <!-- Else case -->
        <Constant dataType="string">False</Constant>
      </Apply>
    </OutputField>
  </Output>

  <TestDistributions field="defects" testStatistic="zValue">
    <Baseline>
      <GaussianDistribution mean="18.2" variance="17.64"/>
    </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

and if defects is 24 then score = (24 - 18.2) / sqrt(17.64) = (5.8) / (4.2) = ~1.38.

Example 3: Scalar product detects changes between distributions by measuring distance between vectors.

<BaselineModel modelName="website-model" functionName="regression">
  <MiningSchema>
    <MiningField name="bin" optype="categorical"/>
    <MiningField name="score" optype="continuous" usageType="target"/>
  </MiningSchema>

  <TestDistributions field="bin" testStatistic="scalarProduct" weightField="cnt" normalizationScheme="Independent">
    <Baseline>
      <CountTable sample="262">
        <FieldValueCount field="bin" count="100" value="bin1"/>
        <FieldValueCount field="bin" count="150" value="bin2"/>
        <FieldValueCount field="bin" count="10" value="bin3"/>
        <FieldValueCount field="bin" count="2" value="bin4"/>
      </CountTable>
    </Baseline>
  </TestDistributions>

</BaselineModel>

Scoring with this type of model is done as follows. Given an observed distribution associating count Ci with vector coordinate (or bin) Vi and an expected distribution with counts ci for the same respective bins, the scalar product is simply the sum over i of Ci x ci/N. N is an optional normalization factor implied by the normalizationScheme attribute of the TestDistributions element. If the attribute is not specified, no normalization factor is applied. If the normalizationScheme attribute is set to "Independent", the normalization is defined by requiring the scalar product of each vector with itself to be one. This is equivalent to setting

Example 4: Contingency tables and comparisons to distributions using chi-squared statistics and p-values. Chi-squared tests can be used within baseline models to test whether a sampled distribution is consistent with a known distribution or whether two distributions are independent of one another when the distributions either discrete or binned. Contingency tables summarize how one field and its categorical values (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. If the fields are independent of each other, the distributions represented by the columns and rows will be statistically similar to the expected distributions obtained by summing over the values of the contingent field. A standard approach is to compute the chi-squared statistic for the table. The chi-squared statistic could then be used to find the associated p-value and determine whether the differences are statistically significant, i.e. whether the fields are not independent.

A Chi-squared statistic may also be computed to compare a discrete distribution for an event sample with an expected distribution. The corresponding p-value would indicate the probability of observing a value of chi-square at least as large as that observed. A small p-value would indicate that the observed distribution differs from the expected distribution.

Scoring Baseline models with chi-squared tests is done as follows. For a contingency 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

where the sum is over all cells in the table.

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%.

For comparing a sampled distribution with an expected distribution, the value of chi-squared is calculated directly from the observed and expected counts for each bin or discrete value. The expected distribution should be normalized to the total count within the observed distribution and the degrees of freedom is the number of values or number of bins minus one.

These tests are available for Baseline models by specifying the testStatistic attribute to be either "chiSquareIndependence" or "chiSquareDistribution". For chiSquareIndependence, a contingency table is described to the model by providing references to the fields describing the rows in the table. For chiSquareDistribution, a reference to a field describing the sampled distribution and a discrete distribution describing the expectations is required. A field reference to a sampled distribution can be provided using the Aggregate element.

<BaselineModel modelName="chisquared" functionName="regression">
  <MiningSchema>
    <MiningField name="obs" optype="continuous"/>
    <MiningField name="bin" optype="categorical"/>
    <MiningField name="score" optype="continuous" usageType="target"/>
  </MiningSchema>

  <LocalTransformations>
    <DerivedField name="obsDist" optype="continuous" dataType="integer">
      <Aggregate field="obs" function="sum" groupField="bin"/>
    </DerivedField>
  </LocalTransformations>

  <TestDistributions field="obsDist" testStatistic="chiSquareDistribution">
    <Baseline>
      <CountTable sample="262">
        <FieldValueCount field="bin" count="100" value="bin1"/>
        <FieldValueCount field="bin" count="150" value="bin2"/>
        <FieldValueCount field="bin" count="10" value="bin3"/>
        <FieldValueCount field="bin" count="2" value="bin4"/>
      </CountTable>
    </Baseline>
  </TestDistributions>

</BaselineModel>

An example of specifying scoring through a contingency table is provided below. Here, the fields from which the contingency table should be formed and the chi-squared calculated are specified within the BaselineModel element.

<BaselineModel modelName="chisquared" functionName="regression">
  <MiningSchema>
    <MiningField name="Count" optype="continuous"/>
    <MiningField name="Animal" optype="categorical"/>
    <MiningField name="TimeOfDay" optype="categorical"/>
    <MiningField name="score" optype="continuous" usageType="target"/>
  </MiningSchema>

  <TestDistributions field="Count" testStatistic="chiSquareIndependence">
    <Baseline>
      <FieldRef field="Animal"/>
      <FieldRef field="TimeOfDay"/>
    </Baseline>
  </TestDistributions>

</BaselineModel>

Output elements. The PMML Output element can be used to return auxiliary information, such as derived fields that are needed for subsequent computations. For more information, please refer to the chapters on outputs.