StatKurtosis Method (ReadOnlyDoubleMatrix, Boolean, DataOperation)

public static DoubleMatrix Kurtosis(
	ReadOnlyDoubleMatrix data,
	bool adjustForBias,
	DataOperation dataOperation
)

Public Shared Function Kurtosis ( 
	data As ReadOnlyDoubleMatrix,
	adjustForBias As Boolean,
	dataOperation As DataOperation
) As DoubleMatrix

public:
static DoubleMatrix^ Kurtosis(
	ReadOnlyDoubleMatrix^ data, 
	bool adjustForBias, 
	DataOperation dataOperation
)

static member Kurtosis : 
        data : ReadOnlyDoubleMatrix * 
        adjustForBias : bool * 
        dataOperation : DataOperation -> DoubleMatrix 

Parameters

data

Type: Novacta.AnalyticsReadOnlyDoubleMatrix
The data.

adjustForBias

Type: SystemBoolean
If set to true signals that the kurtosis is adjusted for bias.

dataOperation

Type: Novacta.AnalyticsDataOperation
A constant to specify if the kurtosis is to be computed for rows or columns.

Return Value

The kurtosis of a random variable can be defined as follows:



where and are the cumulants of order 2 and 4, respectively, and



is the central moment of order .

By interpreting the rows or the columns of data as samples drawn from random variables, this method returns the kurtosis estimates of such variables. Let and be the data number of rows and columns, respectively, and define

Operating on rows

If dataOperation is OnRows, then the method returns a column vector whose length equals the number of rows of data. The i-th entry of such column returns the kurtosis of the i-th data row, say

The parameter can be estimated through the coefficient



where



is the sample -th central moment of the i-th row.

Note that is undefined if the standard deviation of the i-th row of data is zero.

The statistic is a biased estimator of However, provided that the number of columns in data is greater than 3, it can be corrected for bias and the corresponding kurtosis evaluated through the coefficient

If adjustForBias is set to false, then is estimated through if it is defined, otherwise the i-th position in the returned value evaluates to NaN.

If adjustForBias is set to true, then this method operates as follows.

• If is less than 4, i.e. the cumulant estimators cannot be corrected for bias, then an exception is thrown.
• Differently, if is defined, then is stored in the i-th position of the returned value, otherwise such position stores NaN.

Operating on columns

If dataOperation is OnColumns, then the method returns a row vector whose length is the data number of columns. The j-th entry of the returned row exposes the kurtosis of the j-th data column, say

The parameter can be estimated through the coefficient



where



is the sample -th central moment of the j-th column.

Note that is undefined if the standard deviation of the j-th column of data is zero.

The statistic is a biased estimator of . However, provided that the number of rows in data is greater than 3, it can be corrected for bias and the corresponding kurtosis evaluated through the coefficient

If adjustForBias is set to false, then is estimated through if it is defined, otherwise the j-th position in the returned value evaluates to NaN.

If adjustForBias is set to true, then this method operates as follows.

• If is less than 4, i.e. the cumulant estimators cannot be corrected for bias, then an exception is thrown.
• Differently, if is defined, then is stored in the j-th position of the returned value, otherwise such position stores NaN.

In the following example, row and column kurtosis estimates in a data matrix are computed.

using System;

namespace Novacta.Analytics.CodeExamples
{
    public class KurtosisExample0  
    {
        public void Main()
        {
            // Create a matrix.
            var data = new double[20] {
                1, 2, -3,  6, -2,
                2, 2,  2,  0,  7,
               -3, 2,  3,  2,  9,
                5, 2,  7, -1, -4
            };
            var matrix = DoubleMatrix.Dense(4, 5, data, StorageOrder.RowMajor);
            Console.WriteLine("The data matrix:");
            Console.WriteLine(matrix);

            // Kurtosis can be adjusted for bias.
            bool adjustForBias = true;

            // Compute the kurtosis on columns.
            var kurtosisOnColumns = Stat.Kurtosis(matrix, adjustForBias, DataOperation.OnColumns);

            Console.WriteLine();
            Console.WriteLine("Kurtosis on columns:");
            Console.WriteLine(kurtosisOnColumns);

            // Kurtosis is overloaded to accept data as a read-only matrix:
            // compute the kurtosis on rows using a read-only wrapper of the data matrix.
            ReadOnlyDoubleMatrix readOnlyMatrix = matrix.AsReadOnly();
            var kurtosisOnRows = Stat.Kurtosis(readOnlyMatrix, adjustForBias, DataOperation.OnRows);

            Console.WriteLine();
            Console.WriteLine("Kurtosis on rows:");
            Console.WriteLine(kurtosisOnRows);
        }
    }
}

// Executing method Main() produces the following output:
// 
// The data matrix:
// 1                2                -3               6                -2               
// 2                2                2                0                7                
// -3               2                3                2                9                
// 5                2                7                -1               -4               
// 
// 
// 
// Kurtosis on columns:
// 1.16566634       NaN              1.2821471        0.757655955      -5.07072         
// 
// 
// 
// Kurtosis on rows:
// -0.22506045      
// 3.37802768       
// 1.9340679        
// -1.46435105      
// 

Reference

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