StatSkewness Method (DoubleMatrix, Boolean, DataOperation)

public static DoubleMatrix Skewness(
	DoubleMatrix data,
	bool adjustForBias,
	DataOperation dataOperation
)

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

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

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

Parameters

data

Type: Novacta.AnalyticsDoubleMatrix
The data.

adjustForBias

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

dataOperation

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

Return Value

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



where and are the cumulants of order 2 and 3, 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 skewness 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 skewness 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 correction for bias is undefined, 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 skewness 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 2, it can be corrected for bias and the corresponding skewness 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 3, i.e. the correction for bias is undefined, 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 skewness estimates in a data matrix are computed.

using System;

namespace Novacta.Analytics.CodeExamples
{
    public class SkewnessExample0  
    {
        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);

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

            // Compute the skewness on columns.
            var skewnessOnColumns = Stat.Skewness(matrix, adjustForBias, DataOperation.OnColumns);

            Console.WriteLine();
            Console.WriteLine("Skewness on columns:");
            Console.WriteLine(skewnessOnColumns);

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

            Console.WriteLine();
            Console.WriteLine("Skewness on rows:");
            Console.WriteLine(skewnessOnRows);
        }
    }
}

// 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               
// 
// 
// 
// Skewness on columns:
// -0.436662085     NaN              -0.355715685     1.13762437       0                
// 
// 
// 
// Skewness on rows:
// 0.603194073      
// 1.57340612       
// 0.458582855      
// -0.208147907     
// 

Reference

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