MathUtil (ELKI: Environment for DeveLoping KDD-Applications Supported by Index-Structures)

java.lang.Object
- de.lmu.ifi.dbs.elki.math.MathUtil

```
public final class MathUtil
extends Object
```
A collection of math related utility functions.

Field Summary

Fields
Modifier and Type	Field and Description
`(package private) static double[]`	`ERFAPP_A` Coefficients for erf approximation.
`(package private) static double[]`	`ERFAPP_B` Coefficients for erf approximation.
`(package private) static double[]`	`ERFAPP_C` Coefficients for erf approximation.
`(package private) static double[]`	`ERFAPP_D` Coefficients for erf approximation.
`(package private) static double[]`	`ERFAPP_P` Coefficients for erf approximation.
`(package private) static double[]`	`ERFAPP_Q` Coefficients for erf approximation.
`(package private) static double[]`	`ERFINV_A` Coefficients for erfinv approximation, rational version
`(package private) static double[]`	`ERFINV_B` Coefficients for erfinv approximation, rational version
`(package private) static double[]`	`ERFINV_C` Coefficients for erfinv approximation, rational version
`(package private) static double[]`	`ERFINV_D` Coefficients for erfinv approximation, rational version
`(package private) static double[]`	`LANCZOS` LANCZOS-Coefficients for Gamma approximation.
`(package private) static double`	`NUM_PRECISION` Numerical precision to use
`static double`	`ONE_BY_SQRTPI` Precomputed value of 1 / sqrt(pi)
`(package private) static double`	`P_HIGH` Treshold for switching nethods for erfinv approximation
`(package private) static double`	`P_LOW` Treshold for switching nethods for erfinv approximation
`static double`	`SQRT2` Square root of 2.
`static double`	`SQRTHALF` Square root of 0.5 == 1 / Sqrt(2)
`static double`	`SQRTTWOPI` Squre root of two times Pi.
`static double`	`TWOPI` Two times Pi.

Constructor Summary

Constructors
Modifier Constructor and Description

private MathUtil()
Fake constructor for static class.

Constructors
Modifier	Constructor and Description
`private`	`MathUtil()` Fake constructor for static class.

Method Summary

Methods
Modifier and Type	Method and Description
`static double`	`approximateBinomialCoefficient(int n, int k)` Binomial coefficent, also known as "n choose k")
`static double`	`approximateFactorial(int n)` Compute the Factorial of n, often written as `c!`
`static long`	`binomialCoefficient(long n, long k)` Binomial coefficient, also known as "n choose k".
`static double`	`cosineSimilarity(Vector v1, Vector v2)` Compute the cosine similarity for two vectors.
`static double`	`deg2rad(double deg)` Convert Degree to Radians
`static double`	`erf(double x)` Error function for Gaussian distributions = Normal distributions.
`static double`	`erfc(double x)` Complementary error function for Gaussian distributions = Normal distributions.
`static double`	`erfinv(double x)` Inverse error function.
`static BigInteger`	`factorial(BigInteger n)` Compute the Factorial of n, often written as `c!`
`static long`	`factorial(int n)` Compute the Factorial of n, often written as `c!`
`static double`	`hypotenuse(double a, double b)` Computes the square root of the sum of the squared arguments without under or overflow.
`static double`	`latlngDistance(double lat1, double lon1, double lat2, double lon2)` Compute the approximate on-earth-surface distance of two points.
`static double`	`logGamma(double x)` Compute logGamma.
`static double`	`mahalanobisDistance(Matrix weightMatrix, Vector o1_minus_o2)` Compute the Mahalanobis distance using the given weight matrix
`static double`	`normalCDF(double x, double mu, double sigma)` Cumulative probability density function (CDF) of a normal distribution.
`static double`	`normalPDF(double x, double mu, double sigma)` Probability density function of the normal distribution.
`static double`	`normalProbit(double x, double mu, double sigma)` Inverse cumulative probability density function (probit) of a normal distribution.
`static double`	`pearsonCorrelationCoefficient(double[] x, double[] y)` Provides the Pearson product-moment correlation coefficient for two FeatureVectors.
`static double`	`pearsonCorrelationCoefficient(NumberVector<?,?> x, NumberVector<?,?> y)` Provides the Pearson product-moment correlation coefficient for two FeatureVectors.
`static double`	`rad2deg(double rad)` Radians to Degree
`static double[]`	`randomDoubleArray(int len)` Produce an array of random numbers in [0:1]
`static double[]`	`randomDoubleArray(int len, Random r)` Produce an array of random numbers in [0:1]
`static double`	`regularizedGammaP(double a, double x)` Returns the regularized gamma function P(a, x).
`static double`	`regularizedGammaQ(double a, double x)` Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
`static double`	`standardNormalProbit(double d)` Approximate the inverse error function for normal distributions.
`static long`	`sumFirstIntegers(long i)` Compute the sum of the i first integers.
`static double`	`weightedPearsonCorrelationCoefficient(double[] x, double[] y, double[] weights)` Provides the Pearson product-moment correlation coefficient for two FeatureVectors.
`static double`	`weightedPearsonCorrelationCoefficient(NumberVector<?,?> x, NumberVector<?,?> y, double[] weights)` Provides the Pearson product-moment correlation coefficient for two FeatureVectors.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Field Detail
  - TWOPI
```
public static final double TWOPI
```
    Two times Pi.
    
    See Also:
    Constant Field Values
  - SQRTTWOPI
```
public static final double SQRTTWOPI
```
    Squre root of two times Pi.
  - SQRT2
```
public static final double SQRT2
```
    Square root of 2.
  - SQRTHALF
```
public static final double SQRTHALF
```
    Square root of 0.5 == 1 / Sqrt(2)
  - ONE_BY_SQRTPI
```
public static final double ONE_BY_SQRTPI
```
    Precomputed value of 1 / sqrt(pi)
  - ERFAPP_A
```
static final double[] ERFAPP_A
```
    Coefficients for erf approximation. Loosely based on http://www.netlib.org/specfun/erf
  - ERFAPP_B
```
static final double[] ERFAPP_B
```
    Coefficients for erf approximation. Loosely based on http://www.netlib.org/specfun/erf
  - ERFAPP_C
```
static final double[] ERFAPP_C
```
    Coefficients for erf approximation. Loosely based on http://www.netlib.org/specfun/erf
  - ERFAPP_D
```
static final double[] ERFAPP_D
```
    Coefficients for erf approximation. Loosely based on http://www.netlib.org/specfun/erf
  - ERFAPP_P
```
static final double[] ERFAPP_P
```
    Coefficients for erf approximation. Loosely based on http://www.netlib.org/specfun/erf
  - ERFAPP_Q
```
static final double[] ERFAPP_Q
```
    Coefficients for erf approximation. Loosely based on http://www.netlib.org/specfun/erf
  - P_LOW
```
static final double P_LOW
```
    Treshold for switching nethods for erfinv approximation
    
    See Also:
    Constant Field Values
  - P_HIGH
```
static final double P_HIGH
```
    Treshold for switching nethods for erfinv approximation
    
    See Also:
    Constant Field Values
  - ERFINV_A
```
static final double[] ERFINV_A
```
    Coefficients for erfinv approximation, rational version
  - ERFINV_B
```
static final double[] ERFINV_B
```
    Coefficients for erfinv approximation, rational version
  - ERFINV_C
```
static final double[] ERFINV_C
```
    Coefficients for erfinv approximation, rational version
  - ERFINV_D
```
static final double[] ERFINV_D
```
    Coefficients for erfinv approximation, rational version
  - LANCZOS
```
static final double[] LANCZOS
```
    LANCZOS-Coefficients for Gamma approximation. These are said to have higher precision than those in "Numerical Recipes". They probably come from Paul Godfrey: http://my.fit.edu/~gabdo/gamma.txt
  - NUM_PRECISION
```
static final double NUM_PRECISION
```
    Numerical precision to use
    
    See Also:
    Constant Field Values
- Constructor Detail
  - MathUtil
```
private MathUtil()
```
    Fake constructor for static class.
- Method Detail
  - hypotenuse
```
public static double hypotenuse(double a,
                double b)
```
    Computes the square root of the sum of the squared arguments without under or overflow.
    
    Parameters:
    a - first cathetus
    b - second cathetus
    
    Returns:
    sqrt(a<sup>2</sup> + b<sup>2</sup>)
  - mahalanobisDistance
```
public static double mahalanobisDistance(Matrix weightMatrix,
                         Vector o1_minus_o2)
```
    Compute the Mahalanobis distance using the given weight matrix
    
    Parameters:
    weightMatrix - Weight Matrix
    o1_minus_o2 - Delta vector
    
    Returns:
    Mahalanobis distance
  - pearsonCorrelationCoefficient
```
public static double pearsonCorrelationCoefficient(NumberVector<?,?> x,
                                   NumberVector<?,?> y)
```
    Provides the Pearson product-moment correlation coefficient for two FeatureVectors.
    
    Parameters:
    x - first FeatureVector
    y - second FeatureVector
    
    Returns:
    the Pearson product-moment correlation coefficient for x and y
  - weightedPearsonCorrelationCoefficient
```
public static double weightedPearsonCorrelationCoefficient(NumberVector<?,?> x,
                                           NumberVector<?,?> y,
                                           double[] weights)
```
    Provides the Pearson product-moment correlation coefficient for two FeatureVectors.
    
    Parameters:
    x - first FeatureVector
    y - second FeatureVector
    
    Returns:
    the Pearson product-moment correlation coefficient for x and y
  - pearsonCorrelationCoefficient
```
public static double pearsonCorrelationCoefficient(double[] x,
                                   double[] y)
```
    Provides the Pearson product-moment correlation coefficient for two FeatureVectors.
    
    Parameters:
    x - first FeatureVector
    y - second FeatureVector
    
    Returns:
    the Pearson product-moment correlation coefficient for x and y
  - weightedPearsonCorrelationCoefficient
```
public static double weightedPearsonCorrelationCoefficient(double[] x,
                                           double[] y,
                                           double[] weights)
```
    Provides the Pearson product-moment correlation coefficient for two FeatureVectors.
    
    Parameters:
    x - first FeatureVector
    y - second FeatureVector
    
    Returns:
    the Pearson product-moment correlation coefficient for x and y
  - factorial
```
public static BigInteger factorial(BigInteger n)
```
    Compute the Factorial of n, often written as c! in mathematics.
    
    Use this method if for large values of n.
    
    Parameters:
    n - Note: n >= 0. This BigInteger n will be 0 after this method finishes.
    
    Returns:
    n * (n-1) * (n-2) * ... * 1
  - factorial
```
public static long factorial(int n)
```
    Compute the Factorial of n, often written as c! in mathematics.
    
    Parameters:
    n - Note: n >= 0
    
    Returns:
    n * (n-1) * (n-2) * ... * 1
  - binomialCoefficient
```
public static long binomialCoefficient(long n,
                       long k)
```
    Binomial coefficient, also known as "n choose k".
    
    Parameters:
    n - Total number of samples. n > 0
    k - Number of elements to choose. n >= k, k >= 0
    
    Returns:
    n! / (k! * (n-k)!)
  - approximateFactorial
```
public static double approximateFactorial(int n)
```
    Compute the Factorial of n, often written as c! in mathematics.
    
    Parameters:
    n - Note: n >= 0
    
    Returns:
    n * (n-1) * (n-2) * ... * 1
  - approximateBinomialCoefficient
```
public static double approximateBinomialCoefficient(int n,
                                    int k)
```
    Binomial coefficent, also known as "n choose k")
    
    Parameters:
    n - Total number of samples. n > 0
    k - Number of elements to choose. n >= k, k >= 0
    
    Returns:
    n! / (k! * (n-k)!)
  - erfc
```
public static double erfc(double x)
```
    Complementary error function for Gaussian distributions = Normal distributions. Numerical approximation using taylor series. Implementation loosely based on http://www.netlib.org/specfun/erf
    
    Parameters:
    x - parameter value
    
    Returns:
    erfc(x)
  - erf
```
public static double erf(double x)
```
    Error function for Gaussian distributions = Normal distributions. Numerical approximation using taylor series. Implementation loosely based on http://www.netlib.org/specfun/erf
    
    Parameters:
    x - parameter value
    
    Returns:
    erf(x)
  - erfinv
```
public static double erfinv(double x)
```
    Inverse error function.
    
    Parameters:
    x - parameter value
    
    Returns:
    erfinv(x)
  - standardNormalProbit
```
public static double standardNormalProbit(double d)
```
    Approximate the inverse error function for normal distributions. Largely based on:
    http://www.math.uio.no/~jacklam/notes/invnorm/index.html
    by Peter John Acklam
    
    Parameters:
    d - Quantile. Must be in [0:1], obviously.
    
    Returns:
    Inverse erf.
  - normalPDF
```
public static double normalPDF(double x,
               double mu,
               double sigma)
```
    Probability density function of the normal distribution.
```
 1/(SQRT(2*pi*sigma^2)) * e^(-(x-mu)^2/2sigma^2)
 
```
    Parameters:
    x - The value.
    mu - The mean.
    sigma - The standard deviation.
    
    Returns:
    PDF of the given normal distribution at x.
  - normalCDF
```
public static double normalCDF(double x,
               double mu,
               double sigma)
```
    Cumulative probability density function (CDF) of a normal distribution.
    
    Parameters:
    x - value to evaluate CDF at
    mu - Mean value
    sigma - Standard deviation.
    
    Returns:
    The CDF of the normal given distribution at x.
  - normalProbit
```
public static double normalProbit(double x,
                  double mu,
                  double sigma)
```
    Inverse cumulative probability density function (probit) of a normal distribution.
    
    Parameters:
    x - value to evaluate probit function at
    mu - Mean value
    sigma - Standard deviation.
    
    Returns:
    The probit of the normal given distribution at x.
  - logGamma
```
public static double logGamma(double x)
```
    Compute logGamma. Based loosely on "Numerical Recpies" and the work of Paul Godfrey at http://my.fit.edu/~gabdo/gamma.txt TODO: find out which approximation really is the best...
    
    Parameters:
    x - Parameter x
    
    Returns:
    @return log(Γ(x))
  - regularizedGammaP
```
public static double regularizedGammaP(double a,
                       double x)
```
    Returns the regularized gamma function P(a, x). Includes the quadrature way of computing. TODO: find "the" most accurate version of this. We seem to agree with others for the first 10+ digits, but diverge a bit later than that.
    
    Parameters:
    a - Parameter a
    x - Parameter x
    
    Returns:
    Gamma value
  - regularizedGammaQ
```
public static double regularizedGammaQ(double a,
                       double x)
```
    Returns the regularized gamma function Q(a, x) = 1 - P(a, x). Includes the continued fraction way of computing, based loosely on the book "Numerical Recipes"; but probably not with the exactly same precision, since we reimplemented this in our coding style, not literally. TODO: find "the" most accurate version of this. We seem to agree with others for the first 10+ digits, but diverge a bit later than that.
    
    Parameters:
    a - parameter a
    x - parameter x
    
    Returns:
    Result
  - sumFirstIntegers
```
public static long sumFirstIntegers(long i)
```
    Compute the sum of the i first integers.
    
    Parameters:
    i - maximum summand
    
    Returns:
    Sum
  - randomDoubleArray
```
public static double[] randomDoubleArray(int len)
```
    Produce an array of random numbers in [0:1]
    
    Parameters:
    len - Length
    
    Returns:
    Array
  - randomDoubleArray
```
public static double[] randomDoubleArray(int len,
                         Random r)
```
    Produce an array of random numbers in [0:1]
    
    Parameters:
    len - Length
    r - Random generator
    
    Returns:
    Array
  - deg2rad
```
public static double deg2rad(double deg)
```
    Convert Degree to Radians
    
    Parameters:
    deg - Degree value
    
    Returns:
    Radian value
  - rad2deg
```
public static double rad2deg(double rad)
```
    Radians to Degree
    
    Parameters:
    rad - Radians value
    
    Returns:
    Degree value
  - latlngDistance
```
public static double latlngDistance(double lat1,
                    double lon1,
                    double lat2,
                    double lon2)
```
    Compute the approximate on-earth-surface distance of two points.
    
    Parameters:
    lat1 - Latitude of first point in degree
    lon1 - Longitude of first point in degree
    lat2 - Latitude of second point in degree
    lon2 - Longitude of second point in degree
    
    Returns:
    Distance in km (approximately)
  - cosineSimilarity
```
public static double cosineSimilarity(Vector v1,
                      Vector v2)
```
    Compute the cosine similarity for two vectors.
    
    Parameters:
    v1 - First vector
    v2 - Second vector
    
    Returns:
    Cosine similarity

Class MathUtil

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

TWOPI

SQRTTWOPI

SQRT2

SQRTHALF

ONE_BY_SQRTPI

ERFAPP_A

ERFAPP_B

ERFAPP_C

ERFAPP_D

ERFAPP_P

ERFAPP_Q

P_LOW

P_HIGH

ERFINV_A

ERFINV_B

ERFINV_C

ERFINV_D

LANCZOS

NUM_PRECISION

Constructor Detail

MathUtil

Method Detail

hypotenuse

mahalanobisDistance

pearsonCorrelationCoefficient

weightedPearsonCorrelationCoefficient

pearsonCorrelationCoefficient

weightedPearsonCorrelationCoefficient

factorial

factorial

binomialCoefficient

approximateFactorial

approximateBinomialCoefficient

erfc

erf

erfinv

standardNormalProbit

normalPDF

normalCDF

normalProbit

logGamma

regularizedGammaP

regularizedGammaQ

sumFirstIntegers

randomDoubleArray

randomDoubleArray

deg2rad

rad2deg

latlngDistance

cosineSimilarity