# Correlation Matrix (Concurrency)

## Synopsis

This Operator determines correlation between all Attributes and it can produce a weights vector based on these correlations. Correlation is a statistical technique that can show whether and how strongly pairs of Attributes are related.## Description

A correlation is a number between -1 and +1 that measures the degree of association between two Attributes (call them X and Y). A positive value for the correlation implies a positive association. In this case large values of X tend to be associated with large values of Y and small values of X tend to be associated with small values of Y. A negative value for the correlation implies a negative or inverse association. In this case large values of X tend to be associated with small values of Y and vice versa.

Suppose we have two Attributes X and Y, with means X' and Y' respectively and standard deviations S(X) and S(Y) respectively. The correlation is computed as summation from 1 to n of the product *(X(i)-X').(Y(i)-Y') * and then dividing this summation by the product *(n-1).S(X).S(Y)* where *n* is total number of Examples and *i* is the increment variable of summation. There can be other formulas and definitions but let us stick to this one for simplicity.

As discussed earlier a positive value for the correlation implies a positive association. Suppose that an X value was above average, and that the associated Y value was also above average. Then the product *(X(i)-X').(Y(i)-Y')* would be the product of two positive numbers which would be positive. If the X value and the Y value were both below average, then the product above would be of two negative numbers, which would also be positive. Therefore, a positive correlation is evidence of a general tendency that large values of X are associated with large values of Y and small values of X are associated with small values of Y.

As discussed earlier a negative value for the correlation implies a negative or inverse association. Suppose that an X value was above average, and that the associated Y value was instead below average. Then the product *(X(i)-X').(Y(i)-Y')* would be the product of a positive and a negative number which would make the product negative. If the X value was below average and the Y value was above average, then the product above would also be negative. Therefore, a negative correlation is evidence of a general tendency that large values of X are associated with small values of Y and small values of X are associated with large values of Y.

This Operator can be used for creating a correlation matrix that shows correlations of all the Attributes of the input ExampleSet. The Attribute weights vector; based on the correlations can also be returned by this Operator. Using this weights vector, highly correlated Attributes can be removed from the ExampleSet with the help of the Select by Weights Operator. Highly correlated Attributes can be more easily removed by simply using the Remove Correlated Attributes Operator. Correlated Attributes are usually removed because they are similar in behavior and only have little influence when calculating predictions. They may also hamper run time and memory usage.

## Input

- example set (IOObject)
This input port expects an ExampleSet on which the correlation matrix will be calculated.

## Output

- example set (IOObject)
The ExampleSet, that was given as input is passed through without changes.

- matrix (IOObject)
The correlations of all Attributes of the input ExampleSet are calculated and the resultant correlation matrix is returned from this port. The correlation for nominal Attributes is not well defined and results in a missing value. When Attributes contain missing values, only pairwise complete tuples are used for calculating the correlation.

- weights (Average Vector)
The Attribute weights vector based on the correlations of the Attributes is delivered through this output port.

## Parameters

- attribute_filter_type
This parameter allows you to select the Attribute selection filter; the method you want to use for selecting Attributes. It has the following options:

- all: This option selects all the Attributes of the ExampleSet, no Attributes are removed. This is the default option.
- single: This option allows the selection of a single Attribute. The required Attribute is selected by the attribute parameter.
- subset: This option allows the selection of multiple Attributes through a list (see parameter attributes). If the meta data of the ExampleSet is known all Attributes are present in the list and the required ones can easily be selected.
- regular_expression: This option allows you to specify a regular expression for the Attribute selection. The regular expression filter is configured by the parameters regular expression, use except expression and except expression.
- value_type: This option allows selection of all the Attributes of a particular type. It should be noted that types are hierarchical. For example real and integer types both belong to the numeric type. The value type filter is configured by the parameters value type, use value type exception, except value type.
- block_type: This option allows the selection of all the Attributes of a particular block type. It should be noted that block types may be hierarchical. For example value_series_start and value_series_end block types both belong to the value_series block type. The block type filter is configured by the parameters block type, use block type exception, except block type.
- no_missing_values: This option selects all Attributes of the ExampleSet which do not contain a missing value in any Example. Attributes that have even a single missing value are removed.
- numeric_value_filter: All numeric Attributes whose Examples all match a given numeric condition are selected. The condition is specified by the numeric condition parameter. Please note that all nominal Attributes are also selected irrespective of the given numerical condition.

- attribute
The required Attribute can be selected from this option. The Attribute name can be selected from the drop down box of the parameter if the meta data is known.

Range: - attributes
The required Attributes can be selected from this option. This opens a new window with two lists. All Attributes are present in the left list. They can be shifted to the right list, which is the list of selected Attributes that will make it to the output port.

Range: - regular_expression
Attributes whose names match this expression will be selected. The expression can be specified through the

Range:*edit and preview regular expression*menu. This menu gives a good idea of regular expressions and it also allows you to try different expressions and preview the results simultaneously. - use_except_expression
If enabled, an exception to the first regular expression can be specified. This exception is specified by the

Range:*except regular expression*parameter. - except_regular_expression
This option allows you to specify a regular expression. Attributes matching this expression will be filtered out even if they match the first expression (expression that was specified in

Range:*regular expression*parameter). - value_type
This option allows to select a type of Attribute. One of the following types can be chosen: nominal, numeric, integer, real, text, binominal, polynominal, file_path, date_time, date, time.

Range: - use_value_type_exception
If enabled, an exception to the selected type can be specified. This exception is specified by the

Range:*except value type*parameter. - except_value_type
The Attributes matching this type will be removed from the final output even if they matched the before selected type, specified by the

Range:*value type*parameter. One of the following types can be selected here: nominal, numeric, integer, real, text, binominal, polynominal, file_path, date_time, date, time. - block_type
This option allows to select a block type of Attribute. One of the following types can be chosen: single_value, value_series, value_series_start, value_series_end, value_matrix, value_matrix_start, value_matrix_end, value_matrix_row_start.

Range: - use_block_type_exception
If enabled, an exception to the selected block type can be specified. This exception is specified by the

Range:*except block type*parameter. - except_block_type
The Attributes matching this block type will be removed from the final output even if they matched the before selected type by the

Range:*block type*parameter. One of the following block types can be selected here: single_value, value_series, value_series_start, value_series_end, value_matrix, value_matrix_start, value_matrix_end, value_matrix_row_start. - numeric_condition
The numeric condition used by the numeric condition filter type. A numeric Attribute is kept if all Examples match the specified condition for this Attribute. For example the numeric condition '> 6' will keep all numeric Attributes having a value of greater than 6 in every Example. A combination of conditions is possible: '> 6 && < 11' or '<= 5 || < 0'. But && and || cannot be used together in one numeric condition. Conditions like '(> 0 && < 2) || (>10 && < 12)' are not allowed because they use both && and ||. Nominal Attributes are always kept, regardless of the specified numeric condition.

Range: - include_special_attributes
Special Attributes are Attributes with special roles. These are: id, label, prediction, cluster, weight and batch. Also custom roles can be assigned to Attributes. By default all special Attributes are delivered to the output port irrespective of the conditions in the Select Attribute Operator. If this parameter is set to true, special Attributes are also tested against conditions specified in the Select Attribute Operator and only those Attributes are selected that match the conditions.

Range: - invert_selection
If this parameter is set to true the selection is reversed. In that case all Attributes matching the specified condition are removed and the other Attributes remain in the output ExampleSet. Special Attributes are kept independent of the

Range:*invert selection*parameter as along as the*include special attributes*parameter is not set to true. If so the condition is also applied to the special Attributes and the selection is reversed if this parameter is checked. - normalize_weightsThis parameter indicates if the weights of the resultant Attribute weights vector should be normalized. If set to true, all weights are normalized such that the minimum weight is 0 and the maximum weight is 1. Range: boolean
- squared_correlationThis parameter indicates if the squared correlation should be calculated. If set to true, the correlation matrix shows squares of correlations instead of simple correlations. Range: boolean

## Tutorial Processes

### Correlation matrix of the Golf data set

The 'Golf' data set is loaded using the Retrieve Operator. A breakpoint is inserted here so that you can view the ExampleSet. As you can see, the ExampleSet has 4 regular Attributes i.e. 'Outlook', 'Temperature', 'Humidity' and 'Wind' and the label Attribute 'Play'.

All Attributes with only two nominal values are converted to binominal Attributes using Nominal to Binominal. Then the Correlation Matrix Operator is applied on the result. The weights vector generated by this Operator is provided to the Select by Weights Operator along with the data set. The parameters of the Select by Weights Operator are adjusted such that the Attributes with weights greater than 0.5 are selected and all other Attributes are removed. This is why the resultant ExampleSet only has the 'Play' and the 'Temperature' Attribute.

The correlation matrix, weights vector and the resultant ExampleSet can be viewed in the Results Workspace. For the correlation matrix you can see that Outlook is a nominal Attribute, so no correlation can be calculated with it. The correlation of an Attribute to its self is always one, so the diagonal entries are all 1.