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# math

#### package math

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### Type Members

12. #### class MatrixOrderOptimization extends Logging

Optimizes the order of matrix multiplication chain.

Optimizes the order of matrix multiplication chain. Matrix multiplication is associative. However, the complexity of matrix multiplication chain is not associative.

16. #### trait Operators extends AnyRef

High level feature selection operators.

29. #### sealed trait VectorExpression extends AnyRef

Vector Expression.

### Value Members

1. #### implicit def array2VectorExpression(x: Array[Double]): VectorLift

Definition Classes
Operators
2. #### def beta(x: Double, y: Double): Double

The beta function, also called the Euler integral of the first kind.

The beta function, also called the Euler integral of the first kind.

B(x, y) = 01 tx-1 (1-t)y-1dt

for x, y > 0 and the integration is over [0,1].The beta function is symmetric, i.e. B(x,y) = B(y,x).

Definition Classes
Operators
3. #### def chisqtest(table: Array[Array[Int]]): CorTest

Given a two-dimensional contingency table in the form of an array of integers, returns Chi-square test for independence.

Given a two-dimensional contingency table in the form of an array of integers, returns Chi-square test for independence. The rows of contingency table are labels by the values of one nominal variable, the columns are labels by the values of the other nominal variable, and whose entries are non-negative integers giving the number of observed events for each combination of row and column. Continuity correction will be applied when computing the test statistic for 2x2 tables: one half is subtracted from all |O-E| differences. The correlation coefficient is calculated as Cramer's V.

Definition Classes
Operators
4. #### def chisqtest(x: Array[Int], prob: Array[Double], constraints: Int = 1): ChiSqTest

One-sample chisq test.

One-sample chisq test. Given the array x containing the observed numbers of events, and an array prob containing the expected probabilities of events, and given the number of constraints (normally one), a small value of p-value indicates a significant difference between the distributions.

Definition Classes
Operators
5. #### def chisqtest2(x: Array[Int], y: Array[Int], constraints: Int = 1): ChiSqTest

Two-sample chisq test.

Two-sample chisq test. Given the arrays x and y, containing two sets of binned data, and given one constraint, a small value of p-value indicates a significant difference between two distributions.

Definition Classes
Operators
6. #### def cholesky(A: MatrixExpression): CholeskyDecomposition

Cholesky decomposition.

Cholesky decomposition.

Definition Classes
Operators
7. #### def cholesky(A: DenseMatrix): CholeskyDecomposition

Cholesky decomposition.

Cholesky decomposition.

Definition Classes
Operators
8. #### def cholesky(A: Array[Array[Double]]): CholeskyDecomposition

Cholesky decomposition.

Cholesky decomposition.

Definition Classes
Operators
9. #### def det(A: MatrixExpression): Double

Returns the determinant of matrix.

Returns the determinant of matrix.

Definition Classes
Operators
10. #### def det(A: DenseMatrix): Double

Returns the determinant of matrix.

Returns the determinant of matrix.

Definition Classes
Operators
11. #### def diag(A: Matrix): Array[Double]

Returns the diagonal elements of matrix.

Returns the diagonal elements of matrix.

Definition Classes
Operators
12. #### def digamma(x: Double): Double

The digamma function is defined as the logarithmic derivative of the gamma function.

The digamma function is defined as the logarithmic derivative of the gamma function.

Definition Classes
Operators
13. #### def eigen(A: DenseMatrix, k: Int): EigenValueDecomposition

Eigen decomposition.

Eigen decomposition.

Definition Classes
Operators
14. #### def eigen(A: MatrixExpression): EigenValueDecomposition

Eigen decomposition.

Eigen decomposition.

Definition Classes
Operators
15. #### def eigen(A: DenseMatrix): EigenValueDecomposition

Eigen decomposition.

Eigen decomposition.

Definition Classes
Operators
16. #### def eigen(A: Array[Array[Double]]): EigenValueDecomposition

Eigen decomposition.

Eigen decomposition.

Definition Classes
Operators
17. #### def erf(x: Double): Double

The error function (also called the Gauss error function) is a special function of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations.

The error function (also called the Gauss error function) is a special function of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations. It is defined as:

erf(x) = 0x e-t2dt

The complementary error function, denoted erfc, is defined as erfc(x) = 1 - erf(x). The error function and complementary error function are special cases of the incomplete gamma function.

Definition Classes
Operators
18. #### def erfc(x: Double): Double

The complementary error function.

The complementary error function.

Definition Classes
Operators
19. #### def erfcc(x: Double): Double

The complementary error function with fractional error everywhere less than 1.2 × 10-7.

The complementary error function with fractional error everywhere less than 1.2 × 10-7. This concise routine is faster than erfc.

Definition Classes
Operators
20. #### def eye(m: Int, n: Int): ColumnMajorMatrix

Returns an m-by-n identity matrix.

Returns an m-by-n identity matrix.

Definition Classes
Operators
21. #### def eye(n: Int): ColumnMajorMatrix

Returns an n-by-n identity matrix.

Returns an n-by-n identity matrix.

Definition Classes
Operators
22. #### def ftest(x: Array[Double], y: Array[Double]): FTest

Test if the arrays x and y have significantly different variances.

Test if the arrays x and y have significantly different variances. Small values of p-value indicate that the two arrays have significantly different variances.

Definition Classes
Operators
23. #### def gamma(x: Double): Double

Gamma function.

Gamma function. Lanczos approximation (6 terms).

Definition Classes
Operators
24. #### def inv(A: MatrixExpression): DenseMatrix

Returns the inverse of matrix.

Returns the inverse of matrix.

Definition Classes
Operators
25. #### def inv(A: DenseMatrix): DenseMatrix

Returns the inverse of matrix.

Returns the inverse of matrix.

Definition Classes
Operators
26. #### def inverf(p: Double): Double

The inverse error function.

The inverse error function.

Definition Classes
Operators
27. #### def inverfc(p: Double): Double

The inverse complementary error function.

The inverse complementary error function.

Definition Classes
Operators
28. #### def kendalltest(x: Array[Double], y: Array[Double]): CorTest

Kendall rank correlation test.

Kendall rank correlation test. The Kendall Tau Rank Correlation Coefficient is used to measure the degree of correspondence between sets of rankings where the measures are not equidistant. It is used with non-parametric data. The p-value is calculated by approximation, which is good for n > 10.

Definition Classes
Operators
29. #### def kstest(x: Array[Double], y: Array[Double]): KSTest

The two-sample KS test for the null hypothesis that the data sets are drawn from the same distribution.

The two-sample KS test for the null hypothesis that the data sets are drawn from the same distribution. Small values of p-value show that the cumulative distribution function of x is significantly different from that of y. The arrays x and y are modified by being sorted into ascending order.

Definition Classes
Operators
30. #### def kstest(x: Array[Double], y: Distribution): KSTest

The one-sample KS test for the null hypothesis that the data set x is drawn from the given distribution.

The one-sample KS test for the null hypothesis that the data set x is drawn from the given distribution. Small values of p-value show that the cumulative distribution function of x is significantly different from the given distribution. The array x is modified by being sorted into ascending order.

Definition Classes
Operators
31. #### def lgamma(x: Double): Double

log of the Gamma function.

log of the Gamma function. Lanczos approximation (6 terms)

Definition Classes
Operators
32. #### def lu(A: MatrixExpression): LUDecomposition

LU decomposition.

LU decomposition.

Definition Classes
Operators
33. #### def lu(A: DenseMatrix): LUDecomposition

LU decomposition.

LU decomposition.

Definition Classes
Operators
34. #### def lu(A: Array[Array[Double]]): LUDecomposition

LU decomposition.

LU decomposition.

Definition Classes
Operators
35. #### implicit def matrix2MatrixExpression(x: DenseMatrix): MatrixLift

Definition Classes
Operators
36. #### implicit def matrixExpression2Array(exp: MatrixExpression): DenseMatrix

Definition Classes
Operators
37. #### def ones(m: Int, n: Int): ColumnMajorMatrix

Returns an m-by-n matrix of all ones.

Returns an m-by-n matrix of all ones.

Definition Classes
Operators
38. #### def ones(n: Int): ColumnMajorMatrix

Returns an n-by-n matrix of all ones.

Returns an n-by-n matrix of all ones.

Definition Classes
Operators
39. #### def pearsontest(x: Array[Double], y: Array[Double]): CorTest

Pearson correlation coefficient test.

Pearson correlation coefficient test.

Definition Classes
Operators
40. #### implicit def pimpArray2D(data: Array[Array[Double]]): PimpedArray2D

Definition Classes
Operators
41. #### implicit def pimpDouble(x: Double): PimpedDouble

Definition Classes
Operators
42. #### implicit def pimpDoubleArray(data: Array[Double]): PimpedDoubleArray

Definition Classes
Operators
43. #### implicit def pimpIntArray(data: Array[Int]): PimpedArray[Int]

Definition Classes
Operators
44. #### implicit def pimpMatrix(matrix: DenseMatrix): PimpedMatrix

Definition Classes
Operators
45. #### def qr(A: MatrixExpression): QRDecomposition

QR decomposition.

QR decomposition.

Definition Classes
Operators
46. #### def qr(A: DenseMatrix): QRDecomposition

QR decomposition.

QR decomposition.

Definition Classes
Operators
47. #### def qr(A: Array[Array[Double]]): QRDecomposition

QR decomposition.

QR decomposition.

Definition Classes
Operators
48. #### def rank(A: MatrixExpression): Int

Returns the rank of matrix.

Returns the rank of matrix.

Definition Classes
Operators
49. #### def rank(A: DenseMatrix): Int

Returns the rank of matrix.

Returns the rank of matrix.

Definition Classes
Operators
50. #### def spearmantest(x: Array[Double], y: Array[Double]): CorTest

Spearman rank correlation coefficient test.

Spearman rank correlation coefficient test. The Spearman Rank Correlation Coefficient is a form of the Pearson coefficient with the data converted to rankings (ie. when variables are ordinal). It can be used when there is non-parametric data and hence Pearson cannot be used.

The raw scores are converted to ranks and the differences between the ranks of each observation on the two variables are calculated.

The p-value is calculated by approximation, which is good for n > 10.

Definition Classes
Operators
51. #### def svd(A: DenseMatrix, k: Int): SingularValueDecomposition

SVD decomposition.

SVD decomposition.

Definition Classes
Operators
52. #### def svd(A: MatrixExpression): SingularValueDecomposition

SVD decomposition.

SVD decomposition.

Definition Classes
Operators
53. #### def svd(A: DenseMatrix): SingularValueDecomposition

SVD decomposition.

SVD decomposition.

Definition Classes
Operators
54. #### def svd(A: Array[Array[Double]]): SingularValueDecomposition

SVD decomposition.

SVD decomposition.

Definition Classes
Operators
55. #### def trace(A: Matrix): Double

Returns the trace of matrix.

Returns the trace of matrix.

Definition Classes
Operators
56. #### def ttest(x: Array[Double], y: Array[Double]): TTest

Given the paired arrays x and y, test if they have significantly different means.

Given the paired arrays x and y, test if they have significantly different means. Small values of p-value indicate that the two arrays have significantly different means.

Definition Classes
Operators
57. #### def ttest(x: Array[Double], mean: Double): TTest

Independent one-sample t-test whether the mean of a normally distributed population has a value specified in a null hypothesis.

Independent one-sample t-test whether the mean of a normally distributed population has a value specified in a null hypothesis. Small values of p-value indicate that the array has significantly different mean.

Definition Classes
Operators
58. #### def ttest2(x: Array[Double], y: Array[Double], equalVariance: Boolean = false): TTest

Test if the arrays x and y have significantly different means.

Test if the arrays x and y have significantly different means. Small values of p-value indicate that the two arrays have significantly different means.

equalVariance

true if the data arrays are assumed to be drawn from populations with the same true variance. Otherwise, The data arrays are allowed to be drawn from populations with unequal variances.

Definition Classes
Operators
59. #### implicit def vectorExpression2Array(exp: VectorExpression): Array[Double]

Definition Classes
Operators
60. #### def zeros(m: Int, n: Int): ColumnMajorMatrix

Returns an m-by-n zero matrix.

Returns an m-by-n zero matrix.

Definition Classes
Operators
61. #### def zeros(n: Int): ColumnMajorMatrix

Returns an n-by-n zero matrix.

Returns an n-by-n zero matrix.

Definition Classes
Operators