com.imsl.stat

Class DiscriminantAnalysis

java.lang.Object

com.imsl.stat.DiscriminantAnalysis

public class DiscriminantAnalysis
extends Object

Performs a linear or a quadratic discriminant function analysis among several known groups.

DiscriminantAnalysis allows linear or a quadratic discrimination and the use of several classification rules, such as reclassification, split sample, or leave-out-one methods. One or more observations can be added to the rule during each invocation of the update method.

DiscriminantAnalysis results in the measure of distance between the groups,(see getMahalanobis method), a table summarizing the classification results, (see getClassTable), a matrix containing the posterior probabilities of group membership for each classified observation, (see getProbability), the within-sample means, (see getMeans) and covariance matrices computed from their LU factorizations, (see getCovariance). The linear discriminant function coefficients are also computed, (see getCoefficients method).

All observations can be input during one call to the update method; this has the advantage of simplicity. Alternatively, one or more rows of observations can be input during separate calls to update. This does not require all observations be memory resident, a significant advantage with large data sets. Note, however, to classify the same data set requires a second pass of the data to the classify method. During the first pass to the update method the discriminant functions are computed while in the second pass to the classify method the observations are classified. When known groups are available the method getClassTable is useful in comparing how well the algorithm classifies. Multiple calls to the classify method are also allowed. The class table, getClassTable, is an accumulation of all observations classified. The class membership and probabilities, returned in getClassMembership and getProbabilities, will contain the membership for each observation from the most recent invocation of the classify method.

Pooled only and pooled with group covariance computation cannot be mixed. By default, both pooled and group covariance matrices will be computed. An IllegalStateException will be thrown if an attempt is made to change the covariance computation after the first call to the update method. See the setCovarianceComputation method for more details on specifying the covariance computation.

The within-group means are updated for all valid observations in x. Observations with invalid group numbers are ignored, as are observations with missing values (Double.NaN). The LU factorization of the covariance matrices are updated by adding (or deleting) observations via Givens rotations. See the downdate method to delete observations.

During the algorithm's training process, or each invocation of the update method, each observation in x is added to the means and the factorizations of the covariance matrices. Statistics of interest are computed: the linear discriminant functions, the prior probabilities, the log of the determinant of each of the covariance matrices, and a test statistic for testing that all of the within-group covariance matrices are equal. The matrix of Mahalanobis distances, which consists of the distances between the groups, is computed via the pooled covariance matrix when linear discrimination is specified. The row covariance matrix is used when the discrimination is quadratic. Covariance matrices are defined as follows. Let \(N_i\) denote the sum of the frequencies of the observations in group i, and let \(M_i\) denote the number of observations in group i. Then, if \(S_i\) denotes the within-group i covariance matrix, $$S_i = \frac{1}{N_i - 1} \sum_{j=1}^{M_i} w_j f_j (x_j - \overline{x})(x_j - \overline{x})^T$$ where \(w_j\) is the weight of the j-th observation in group i, \(f_j\) is its frequency, \(x_j\) is the j-th observation column vector (in group i), and \(\overline{x}\) denotes the mean vector of the observations in group i. The mean vectors are computed as $$\overline{x} = \frac{1}{W_i} \sum_{j=1}^{M_i} w_j f_j x_j$$ where $$W_i = \sum_{j=1}^{M_i} w_j f_j$$ Given the means and the covariance matrices, the linear discriminant function for group i is computed as: $$z_i = \ln(p_i)-0.5\overline{x_i}^T S_{p}^{-1} \overline{x_i} + x^T S_{p}^{-1} \overline{x_i}$$ where \(\ln(p_i)\) is the natural log of the prior probability for the i-th group, x is the observation to be classified, and \(S_p\) denotes the pooled covariance matrix.

Let S denote either the pooled covariance matrix or one of the within-group covariance matrices \(S_i\). (S will be the pooled covariance matrix in linear discrimination, and \(S_i\) otherwise.) The Mahalanobis distance between group i and group j is computed as: $$D_{ij}^{2} = (\overline{x_i} - \overline{x_j})^T S^{-1} (\overline{x_i} - \overline{x_j})$$

Finally, the asymptotic chi-squared test for the equality of covariance matrices is computed as follows (Morrison 1976, page 252): $$\gamma = C^{-1} \sum_{i=1}^{k} n_i \{ ln( \left| S_p \right| ) - ln( \left| S_i \right| ) \}$$ where \(n_i\) is the number of degrees of freedom in the i-th sample covariance matrix, \(k\) is the number of groups, and $$C^{-1} = \frac{1-2p^2 + 3p - 1}{6(p + 1)(k - 1)} \left(\sum_{i=1}^{k} \frac{1}{n_i} - \frac{1}{\sum_{j}n_j} \right)$$ where \(p\) is the number of variables.

The estimated posterior probability of each observation x belonging to group i is computed using the prior probabilities and the sample mean vectors and estimated covariance matrices under a multivariate normal assumption. Under quadratic discrimination, the within-group covariance matrices are used to compute the estimated posterior probabilities. The estimated posterior probability of an observation x belonging to group i is $$\hat{q_i}(x) = \frac{e^{-\frac{1}{2}D_{i}^{2}(x)}}{\sum_{j=1}^{k} e^{-\frac{1}{2}D_{j}^{2}(x)}}$$ where $$D_{i}^{2}(x) = \left\{ \begin{array}{ll} (x - \overline{x_i})^T S_{i}^{-1} (x - \overline{x_i}) + ln \left|S_i \right| - 2 ln(p_i) & \mbox{Linear or Quadratic, pooled, group} \\ (x - \overline{x_i})^T S_{p}^{-1} (x - \overline{x_i}) - 2 ln(p_i) & \mbox{Linear, Pooled} \end{array} \right. $$

For the leave-out-one method of classification, the sample mean vector and sample covariance matrices in the formula for $$D_{i}^{2}(x)$$ are adjusted so as to remove the observation x from their computation. For linear discrimination, the linear discriminant function coefficients are actually used to compute the same posterior probabilities.

Using the posterior probabilities, each observation in x is classified into a group; the result is tabulated in the matrix returned by getClassTable and saved in the vector returned by getClassMembership. If a group variable is provided and the group number is out of range, the classification table is not altered at this stage. If the reclassification method is specified, then all observations with no missing values are classified. When the leaving-out-one method is used, observations with invalid group numbers, weights, frequencies or classification variables are not classified. Regardless of the frequency, a 1 is added (or subtracted) from the classification table for each row of x that is classified and contains a valid group number. When the leaving-out-one method is used, adjustment is made to the posterior probabilities to remove the effect of the observation in the classification rule. In this adjustment, each observation is presumed to have a weight of \(w_j\) and a frequency of 1.0. See Lachenbruch (1975, page 36) for the required adjustment.

See Also:

Discriminant Analysis Example

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	DiscriminantAnalysis.CovarianceSingularException The variance-covariance matrix is singular.
static class	DiscriminantAnalysis.EmptyGroupException There are no observations in a group.
static class	DiscriminantAnalysis.SumOfWeightsNegException The sum of the weights have become negative.

Field Summary

Fields

Modifier and Type	Field and Description
static int	LEAVE_OUT_ONE Indicates leave-out-one classification method.
static int	LINEAR Indicates a linear discrimination method.
static int	POOLED Indicates pooled covariances computation.
static int	POOLED_GROUP Indicates pooled, group covariances computation.
static int	PRIOR_EQUAL Indicates prior equal probabilities.
static int	PRIOR_PROPORTIONAL Indicates prior proportional probabilities.
static int	QUADRATIC Indicates a quadratic discrimination method.
static int	RECLASSIFICATION Indicates reclassification classification method.

Constructor Summary

Constructors

Constructor and Description
DiscriminantAnalysis(int nVariables, int nGroups) Constructs a DiscriminantAnalysis.

Method Summary

Modifier and Type	Method and Description
void	classify(double[][] x) Classify a set of observations using the linear or quadratic discriminant functions generated during the training process.
void	classify(double[][] x, int[] varIndex) Classify a set of observations using the linear or quadratic discriminant functions generated during the training process.
void	classify(double[][] x, int[] frequencies, double[] weights) Classify a set of observations and associated frequencies and weights using the linear or quadratic discriminant functions generated during the training process.
void	classify(double[][] x, int[] group, int[] varIndex) Classify a set of observations and compare against known groups using the linear or quadratic discriminant functions generated during the training process.
void	classify(double[][] x, int[] varIndex, int[] frequencies, double[] weights) Classify a set of observations and associated frequencies and weights using the linear or quadratic discriminant functions generated during the training process.
void	classify(double[][] x, int[] group, int[] varIndex, int[] frequencies, double[] weights) Classify a set of observations, associated frequencies and weights, and compare against known groups using the linear or quadratic discriminant functions generated during the training process.
void	downdate(double[][] x, int[] group) Removes a set of observations from the discriminant functions.
void	downdate(double[][] x, int[] group, int[] varIndex) Removes a set of observations from the discriminant functions.
void	downdate(double[][] x, int[] group, int[] frequencies, double[] weights) Removes a set of observations and associated frequencies and weights from the discriminant functions.
void	downdate(double[][] x, int[] group, int[] varIndex, int[] frequencies, double[] weights) Removes a set of observations and associated frequencies and weights from the discriminant functions.
int[]	getClassMembership() Returns the group number to which the observation was classified.
double[][]	getClassTable() Returns the classification table.
double[][]	getCoefficients() Returns the linear discriminant function coefficients.
double[][][]	getCovariance() Returns the array of covariances.
int[]	getGroupCounts() Returns the group counts.
double[][]	getMahalanobis() Returns the Mahalanobis distances between the group means.
double[][]	getMeans() Returns the variable means.
int	getNRowsMissing() Deprecated. Use DiscriminantAnalysis.getNumberOfRowsMissing() instead.
int	getNumberOfRowsMissing() Returns the number of rows of data encountered containing missing values (Double.NaN).
double[]	getPrior() Returns the prior probabilities.
double[][]	getProbability() Returns the posterior probabilities for each observation.
double[]	getStatistics() Returns statistics.
void	setClassificationMethod(int method) Specifies the classification method to be either reclassification or leave-out-one.
void	setCovarianceComputation(int type) Specifies the covariance matrix computation to be either pooled or pooled, group.
void	setDiscriminationMethod(int method) Specifies the discrimination method used to be either linear or quadratic discrimination.
void	setPrior(double[] prior) Specifies user supplied prior probabilities.
void	setPrior(int prior) Specifies the prior probabilities to be calculated as either equal or proportional priors.
void	update(double[][] x) Deprecated. Use DiscriminantAnalysis.update(double[][], int[]) instead.
void	update(double[][] x, double[] frequencies, double[] weights) Deprecated. Use DiscriminantAnalysis.update(double[][], int[], int[], double[]) instead.
void	update(double[][] x, int groupIndex) Deprecated. Use DiscriminantAnalysis.update(double[][], int[]) instead.
void	update(double[][] x, int[] group) Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among several known groups.
void	update(double[][] x, int[] varIndex, double[] frequencies, double[] weights) Deprecated. Use DiscriminantAnalysis.update(double[][], int[], int[], int[], double[]) instead.
void	update(double[][] x, int[] group, int[] varIndex) Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among several known groups.
void	update(double[][] x, int[] group, int[] frequencies, double[] weights) Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among several known groups.
void	update(double[][] x, int[] group, int[] varIndex, int[] frequencies, double[] weights) Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among several known groups.
void	update(double[][] x, int groupIndex, double[] frequencies, double[] weights) Deprecated. Use DiscriminantAnalysis.update(double[][], int[], int[], double[]) instead.
void	update(double[][] x, int groupIndex, int[] varIndex) Deprecated. Use DiscriminantAnalysis.update(double[][], int[], int[]) instead.
void	update(double[][] x, int groupIndex, int[] varIndex, double[] frequencies, double[] weights) Deprecated. Use DiscriminantAnalysis.update(double [][], int[], int[], int[], double[]) instead.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

LINEAR

public static final int LINEAR

Indicates a linear discrimination method.

See Also:

Constant Field Values

QUADRATIC

public static final int QUADRATIC

Indicates a quadratic discrimination method.

See Also:

Constant Field Values

POOLED

public static final int POOLED

Indicates pooled covariances computation.

See Also:

Constant Field Values

POOLED_GROUP

public static final int POOLED_GROUP

Indicates pooled, group covariances computation.

See Also:

Constant Field Values

RECLASSIFICATION

public static final int RECLASSIFICATION

Indicates reclassification classification method.

See Also:

Constant Field Values

LEAVE_OUT_ONE

public static final int LEAVE_OUT_ONE

Indicates leave-out-one classification method.

See Also:

Constant Field Values

PRIOR_PROPORTIONAL

public static final int PRIOR_PROPORTIONAL

Indicates prior proportional probabilities.

See Also:

Constant Field Values

PRIOR_EQUAL

public static final int PRIOR_EQUAL

Indicates prior equal probabilities.

See Also:

Constant Field Values

Constructor Detail

DiscriminantAnalysis

public DiscriminantAnalysis(int nVariables,
                            int nGroups)

Constructs a DiscriminantAnalysis.

Parameters:

nVariables - an int representing the number of variables to be used in the discrimination

nGroups - an int representing the number of groups in the data

Method Detail

update

public void update(double[][] x)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Deprecated. Use DiscriminantAnalysis.update(double[][], int[]) instead.

Trains a set of observations by performing a linear or quadratic discriminant function analysis among several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables + 1 columns. The column containing the group numbers must be in column nVariables of the input matrix. Specifically, the first nVariables columns correspond to the variables, and the last column contains the group numbers. The groups must be numbered 1,2, ..., nGroups. Any additional columns will be ignored.

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

update

public void update(double[][] x,
                   int groupIndex)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Deprecated. Use DiscriminantAnalysis.update(double[][], int[]) instead.

Trains a set of observations by performing a linear or quadratic discriminant function analysis among several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables + 1 columns. The first nVariables columns, excluding groupIndex column, correspond to the variables, The groupIndex column contains the group numbers. Any additional columns will be ignored.

groupIndex - an int containing the column index of x in which the group numbers are stored. The groups must be numbered 1,2, ..., nGroups. Any observations with a group number outside of this range will be skipped.

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

update

public void update(double[][] x,
                   int groupIndex,
                   int[] varIndex)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Deprecated. Use DiscriminantAnalysis.update(double[][], int[], int[]) instead.

Trains a set of observations by performing a linear or quadratic discriminant function analysis among the several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables + 1 columns. The columns indicated in varIndex correspond to the variables, and groupIndex column contains the group numbers. Any additional columns will be ignored.

groupIndex - an int containing the column index of x in which the group numbers are stored. The groups must be numbered 1,2, ..., nGroups.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis.

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

update

public void update(double[][] x,
                   double[] frequencies,
                   double[] weights)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Deprecated. Use DiscriminantAnalysis.update(double[][], int[], int[], double[]) instead.

Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among the several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables + 1 columns. The first nVariables columns correspond to the variables, and the last column (column nVariables) contains the group numbers. The groups must be numbered 1,2, ..., nGroups.

frequencies - a double array containing the associated frequencies for each observation.

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

update

public void update(double[][] x,
                   int groupIndex,
                   double[] frequencies,
                   double[] weights)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Deprecated. Use DiscriminantAnalysis.update(double[][], int[], int[], double[]) instead.

Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among the several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables + 1 columns. The first nVariables columns correspond to the variables, excluding the groupIndex column.

groupIndex - an int containing the column index of x in which the group numbers are stored. The groups must be numbered 1,2, ..., nGroups.

frequencies - a double array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

update

public void update(double[][] x,
                   int[] varIndex,
                   double[] frequencies,
                   double[] weights)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Deprecated. Use DiscriminantAnalysis.update(double[][], int[], int[], int[], double[]) instead.

Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among the several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables + 1 columns. The columns indicated in varIndex correspond to the variables, and the last column (column nVariables) contains the group numbers. The groups must be numbered 1,2, ..., nGroups.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis.

frequencies - a double array containing the associated frequencies for each observation.

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

update

public void update(double[][] x,
                   int groupIndex,
                   int[] varIndex,
                   double[] frequencies,
                   double[] weights)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Deprecated. Use DiscriminantAnalysis.update(double [][], int[], int[], int[], double[]) instead.

Trains a set of observations and associated frequencies and weights by performin a linear or quadratic discriminant function analysis among the several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables + 1 columns. The columns indicated in varIndex correspond to the variables, and groupIndex column contains the group numbers.

groupIndex - an int containing the column index of x in which the group numbers are stored. The groups must be numbered 1,2, ..., nGroups.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

frequencies - a double array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

update

public void update(double[][] x,
                   int[] group)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The first nVariables correspond to the variables. Any additional columns will be ignored.

group - an int array containing the group numbers. The groups must be numbered
1,2, ..., nGroups for each observation.

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

update

public void update(double[][] x,
                   int[] group,
                   int[] varIndex)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The columns indicated in varIndex correspond to the variables. Any additional columns will be ignored.

group - an int array containing the group numbers. The groups must be numbered
1,2, ..., nGroups for each observation.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

update

public void update(double[][] x,
                   int[] group,
                   int[] frequencies,
                   double[] weights)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The first nVariables correspond to the variables. Any additional columns will be ignored.

group - an int array containing the group numbers. The groups must be numbered
1,2, ..., nGroups for each observation.

frequencies - an int array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

update

public void update(double[][] x,
                   int[] group,
                   int[] varIndex,
                   int[] frequencies,
                   double[] weights)
            throws DiscriminantAnalysis.SumOfWeightsNegException

Trains a set of observations and associated frequencies and weights by performing a linear or quadratic discriminant function analysis among several known groups.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The columns indicated in varIndex correspond to the variables.

group - an int array containing the group numbers. The groups must be numbered
1,2, ..., nGroups for each observation.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

frequencies - an int array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

downdate

public void downdate(double[][] x,
                     int[] group)
              throws DiscriminantAnalysis.SumOfWeightsNegException

Removes a set of observations from the discriminant functions.

Parameters:

x - a double matrix containing the observations to be removed, with at least nVariables columns. The first nVariables columns correspond to the variables. Any additional columns will be ignored.

group - an int array containing the group numbers. The groups must be numbered 1,2, ..., nGroups for each observation.

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

downdate

public void downdate(double[][] x,
                     int[] group,
                     int[] varIndex)
              throws DiscriminantAnalysis.SumOfWeightsNegException

Removes a set of observations from the discriminant functions.

Parameters:

x - a double matrix containing the observations to be removed, with at least nVariables columns. The columns indicated in varIndex correspond to the variables.

group - an int array containing the group numbers. The groups must be numbered 1,2, ..., nGroups for each observation.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

downdate

public void downdate(double[][] x,
                     int[] group,
                     int[] frequencies,
                     double[] weights)
              throws DiscriminantAnalysis.SumOfWeightsNegException

Removes a set of observations and associated frequencies and weights from the discriminant functions.

Parameters:

x - a double matrix containing the observations to be removed, with at least nVariables columns. The columns indicated in varIndex correspond to the variables.

group - an int array containing the group numbers. The groups must be numbered 1,2, ..., nGroups for each observation.

frequencies - an int array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

downdate

public void downdate(double[][] x,
                     int[] group,
                     int[] varIndex,
                     int[] frequencies,
                     double[] weights)
              throws DiscriminantAnalysis.SumOfWeightsNegException

Removes a set of observations and associated frequencies and weights from the discriminant functions.

Parameters:

x - a double matrix containing the observations to be removed, with at least nVariables columns. The columns indicated in varIndex correspond to the variables.

group - an int array containing the group numbers. The groups must be numbered 1,2, ..., nGroups for each observation.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

frequencies - an int array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

classify

public void classify(double[][] x)
              throws DiscriminantAnalysis.SumOfWeightsNegException,
                     DiscriminantAnalysis.EmptyGroupException,
                     DiscriminantAnalysis.CovarianceSingularException

Classify a set of observations using the linear or quadratic discriminant functions generated during the training process.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The first nVariables columns correspond to the variables. Reclassification does not require group numbers be present. Any additional columns will be ignored.

Throws:

IllegalStateException - is thrown if the leave-out-one classification method is chosen.

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

classify

public void classify(double[][] x,
                     int[] varIndex)
              throws DiscriminantAnalysis.SumOfWeightsNegException,
                     DiscriminantAnalysis.EmptyGroupException,
                     DiscriminantAnalysis.CovarianceSingularException

Classify a set of observations using the linear or quadratic discriminant functions generated during the training process.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The columns indicated in varIndex correspond to the variables. Reclassification does not require group numbers be present. Additional columns will be ignored.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

Throws:

IllegalStateException - is thrown if the leave-out-one classification method is chosen.

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative.

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

classify

public void classify(double[][] x,
                     int[] frequencies,
                     double[] weights)
              throws DiscriminantAnalysis.SumOfWeightsNegException,
                     DiscriminantAnalysis.EmptyGroupException,
                     DiscriminantAnalysis.CovarianceSingularException

Classify a set of observations and associated frequencies and weights using the linear or quadratic discriminant functions generated during the training process.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The first nVariables columns correspond to the variables. Reclassification does not require group numbers be present. Any additional columns will be ignored.

frequencies - an int array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

IllegalStateException - is thrown if the leave-out-one classification method is chosen

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular

classify

public void classify(double[][] x,
                     int[] varIndex,
                     int[] frequencies,
                     double[] weights)
              throws DiscriminantAnalysis.SumOfWeightsNegException,
                     DiscriminantAnalysis.EmptyGroupException,
                     DiscriminantAnalysis.CovarianceSingularException

Classify a set of observations and associated frequencies and weights using the linear or quadratic discriminant functions generated during the training process.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The columns indicated in varIndex correspond to the variables. Reclassification does not require group numbers be present. Additional columns in x will be ignored.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

frequencies - an int array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

IllegalStateException - is thrown if the leave-out-one classification method is chosen

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular

classify

public void classify(double[][] x,
                     int[] group,
                     int[] varIndex)
              throws DiscriminantAnalysis.SumOfWeightsNegException,
                     DiscriminantAnalysis.EmptyGroupException,
                     DiscriminantAnalysis.CovarianceSingularException

Classify a set of observations and compare against known groups using the linear or quadratic discriminant functions generated during the training process.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The columns indicated in varIndex correspond to the variables. Any additional columns will be ignored.

group - an int array containing the group numbers. The groups must be numbered 1,2, ..., nGroups for each observation.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular

classify

public void classify(double[][] x,
                     int[] group,
                     int[] varIndex,
                     int[] frequencies,
                     double[] weights)
              throws DiscriminantAnalysis.SumOfWeightsNegException,
                     DiscriminantAnalysis.EmptyGroupException,
                     DiscriminantAnalysis.CovarianceSingularException

Classify a set of observations, associated frequencies and weights, and compare against known groups using the linear or quadratic discriminant functions generated during the training process.

Parameters:

x - a double matrix containing the observations with at least nVariables columns. The columns indicated in varIndex correspond to the variables. Additional columns are ignored.

group - an int array containing the group numbers. The groups must be numbered 1,2, ..., nGroups for each observation.

varIndex - an int array containing the column indices in x that correspond to the variables to be used in the analysis

frequencies - an int array containing the associated frequencies for each observation

weights - a double array containing the associated weights for each observation

Throws:

DiscriminantAnalysis.SumOfWeightsNegException - is thrown when the sum of the weights have become negative

DiscriminantAnalysis.EmptyGroupException

DiscriminantAnalysis.CovarianceSingularException

setDiscriminationMethod

public void setDiscriminationMethod(int method)

Specifies the discrimination method used to be either linear or quadratic discrimination.

Parameters:

method - an int scalar indicating the method of discrimination. Use class member LINEAR or QUADRATIC. By default, the LINEAR method is used.

setCovarianceComputation

public void setCovarianceComputation(int type)

Specifies the covariance matrix computation to be either pooled or pooled, group.

Parameters:

type - an int scalar indicating the type of covariance matrices to be computed. Use class member POOLED or POOLED_GROUP. By default, POOLED_GROUP is used.

setClassificationMethod

public void setClassificationMethod(int method)

Specifies the classification method to be either reclassification or leave-out-one.

Parameters:

method - an int indicating the method of classification. Use class member RECLASSIFICATION or LEAVE_OUT_ONE. By default, the RECLASSIFICATION method is used.

setPrior

public void setPrior(int prior)

Specifies the prior probabilities to be calculated as either equal or proportional priors.

Parameters:

prior - an int specifying how to calculate prior probabilities as either equal or proportional prior probabilities. Use class member PRIOR_EQUAL to set equal prior probabilities, calculated as 1.0/nGroups. Use class member PRIOR_PROPORTIONAL to calculate the priors to be proportional to the sample size in each group. The sum of all prior probabilities is equal to 1.0. If the values calculated for the priors are less than 1.0e-20, they will be converted to the Math.log(1.0e-20). Prior probabilities are used in calculating statistics, coefficients, Mahalanobis, and classification probabilities. By default, PRIOR_EQUAL is used.

setPrior

public void setPrior(double[] prior)

Specifies user supplied prior probabilities.

Parameters:

prior - a double vector of length nGroups containing the prior probabilities for each group. The elements of prior should sum to 1.0. If the values of prior are less than 1.0e-20, they will be converted to the Math.log(1.0e-20). By default, the prior probablities are calculated to be equal, see setPrior(int).

getPrior

public double[] getPrior()

Returns the prior probabilities.

Returns:

a double array of length nGroups containing the prior probabilities for each group.

getGroupCounts

public int[] getGroupCounts()

Returns the group counts.

Returns:

an int array of length nGroups containing the number of observations in each group. If an update has not preceeded the invocation of this method, an array of all zeros will be returned.

getMeans

public double[][] getMeans()
                    throws DiscriminantAnalysis.EmptyGroupException,
                           DiscriminantAnalysis.CovarianceSingularException

Returns the variable means.

Returns:

an nGroups by nVariables double matrix containing the variable means. The i-th row contains the variable means for group i.

If this method is invoked before classification, the unscaled means will be returned.

Throws:

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

getCovariance

public double[][][] getCovariance()
                           throws DiscriminantAnalysis.EmptyGroupException,
                                  DiscriminantAnalysis.CovarianceSingularException

Returns the array of covariances.

Returns:

a g by nVariables by nVariables double array containing the covariances. Where, g = nGroups+1 if pooled, group covariance computation is specified or g=1 if pooled covariance computation is specified. When pooled only covariance matrices are computed, the within-group covariance matrices are not computed. The pooled covariance matrix is always computed and is returned as the g-th covariance matrix.

If this method is invoked before classification, the unscaled covariance matrix will be returned.

Throws:

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

getCoefficients

public double[][] getCoefficients()
                           throws DiscriminantAnalysis.EmptyGroupException,
                                  DiscriminantAnalysis.CovarianceSingularException

Returns the linear discriminant function coefficients.

Returns:

an nGroups by nVariables double matrix containing the linear discriminant function coefficients. The first column of the matrix contains the constant term, and the remaining columns contain the variable coefficients. The i-th row of the returned matrix corresponds to group i. The coefficients are always computed as linear discriminant function coefficients even when quadratic discrimination is specified.

Throws:

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

getClassTable

public double[][] getClassTable()

Returns the classification table.

Returns:

an nGroups by nGroups double matrix containing the classification table. The accumulation of each observation that is classified and has a group number equal to 1, 2, ..., nGroups is entered into the table. If a known group is provided, the rows of the table correspond to the known group membership. The columns refer to the group to which the observation was classified. If a known group is not provided, the table will only contain the accumulated classified groups in the column coresponding to the group to which the observation was classified.

Throws:

IllegalStateException - is thrown if no data has been classified.

getMahalanobis

public double[][] getMahalanobis()
                          throws DiscriminantAnalysis.EmptyGroupException,
                                 DiscriminantAnalysis.CovarianceSingularException

Returns the Mahalanobis distances between the group means.

Returns:

an nGroups by nGroups double matrix containing the Mahalanobis distances between the group means. For linear discrimination, the Mahalanobis distance $$D_{ij}^2(x)$$ between group means i and j is computed using the within covariance matrix for group i in place of the pooled covariance matrix.

Throws:

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

getStatistics

public double[] getStatistics()
                       throws DiscriminantAnalysis.EmptyGroupException,
                              DiscriminantAnalysis.CovarianceSingularException

Returns statistics.

Returns:

a double array containing output statistics.

index	Description
0	Sum of the degrees of freedom for the within-covariance matrices.
1	Chi-squared statistic.
2	The degrees of freedom in the chi-squared statistic.
3	Probability of a greater chi-squared, respectively, of a test of the homogeneity of the within-covariance matrices. (Not computed when the pooled only covariance matrix is computed).
4 thru (4+nGroups)	Log of the determinant of each group's covariance matrix (not computed when the pooled only covariance matrix is computed) and of the pooled covariance matrix.
Last (nGroups + 1) elements	Sum of the weights within each group.
Last element	Sum of the weights in all groups.

Throws:

DiscriminantAnalysis.EmptyGroupException - is thrown when there are no observations in a group.

DiscriminantAnalysis.CovarianceSingularException - is thrown when the variance-covariance matrix is singular.

getClassMembership

public int[] getClassMembership()

Returns the group number to which the observation was classified.

Returns:

an int array containing the group to which the observation was classified. If an observation has an invalid group number, frequency, or weight when the leaving-out-one method has been specified, then the observation is not classified and the corresponding elements of the array are set to zero. Note this will return the classmembership of the last set of observations classified.

Throws:

IllegalStateException - is thrown if no data has been classified.

getProbability

public double[][] getProbability()

Returns the posterior probabilities for each observation.

Returns:

an x.length by nGroups double matrix containing the posterior probabilities for each observation. Note this will return the probabilities of the last set of observations classified.

Throws:

IllegalStateException - is thrown if no data has been classified.

getNRowsMissing

public int getNRowsMissing()

Deprecated. Use DiscriminantAnalysis.getNumberOfRowsMissing() instead.

Returns the number of rows of data encountered containing missing values (NaN).

Returns:

an int representing the number of rows of data encountered containing missing values (NaN) for the classification, group, weight, and/or frequency variables. If a row of data contains a missing value (NaN) for any of these variables, that row is excluded from the computations.

getNumberOfRowsMissing

public int getNumberOfRowsMissing()

Returns the number of rows of data encountered containing missing values (Double.NaN).

Returns:

an int representing the number of rows of data encountered containing missing values (Double.NaN) for the classification, group, weight, and/or frequency variables. If a row of data contains a missing value (Double.NaN) for any of these variables, that row is excluded from the computations.