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## cern.colt.matrix.doublealgo
Class Statistic

    java.lang.Object
      cern.colt.matrix.doublealgo.Statistic

public class Statistic
extends Object

Basic statistics operations on matrices. Computation of covariance, correlation, distance matrix. Random sampling views. Conversion to histograms with and without OLAP cube operators. Conversion to bins with retrieval of statistical bin measures. Also see `cern.jet.stat` and `hep.aida.bin`, in particular `DynamicBin1D`.

Examples:

`A` | `covariance(A)` | `correlation(covariance(A))` | `distance(A,EUCLID)`
`4 x 3 matrix 1 2 3 2 4 6 3 6 9 4 -8 -10` | `3 x 3 matrix 1.25 -3.5 -4.5 -3.5 29 39 -4.5 39 52.5` | `3 x 3 matrix 1 -0.581318 -0.555492 -0.581318 1 0.999507 -0.555492 0.999507 1` | `3 x 3 matrix 0 12.569805 15.874508 12.569805 0 4.242641 15.874508 4.242641 0` ``

Version:
  1.0, 09/24/99

Nested Class Summary
`static interface` | `Statistic.VectorVectorFunction` Interface that represents a function object: a function that takes two argument vectors and returns a single value.

Field Summary
`static Statistic.VectorVectorFunction` | `BRAY_CURTIS` Bray-Curtis distance function; `Sum( abs(x[i]-y[i]) ) / Sum( x[i]+y[i] )`.
`static Statistic.VectorVectorFunction` | `CANBERRA` Canberra distance function; `Sum( abs(x[i]-y[i]) / abs(x[i]+y[i]) )`.
`static Statistic.VectorVectorFunction` | `EUCLID` Euclidean distance function; `Sqrt(Sum( (x[i]-y[i])^2 ))`.
`static Statistic.VectorVectorFunction` | `MANHATTAN` Manhattan distance function; `Sum( abs(x[i]-y[i]) )`.
`static Statistic.VectorVectorFunction` | `MAXIMUM` Maximum distance function; `Max( abs(x[i]-y[i]) )`.

Method Summary
`static DoubleMatrix2D` | `aggregate(DoubleMatrix2D matrix, BinFunction1D[] aggr, DoubleMatrix2D result)` Applies the given aggregation functions to each column and stores the results in a the result matrix.
`static DynamicBin1D` | `bin(DoubleMatrix1D vector)` Fills all cell values of the given vector into a bin from which statistics measures can be retrieved efficiently.
`static DoubleMatrix2D` | `correlation(DoubleMatrix2D covariance)` Modifies the given covariance matrix to be a correlation matrix (in-place).
`static DoubleMatrix2D` | `covariance(DoubleMatrix2D matrix)` Constructs and returns the covariance matrix of the given matrix.
`static IHistogram2D` | `cube(DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix1D weights)` 2-d OLAP cube operator; Fills all cells of the given vectors into the given histogram.
`static IHistogram3D` | `cube(DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix1D z, DoubleMatrix1D weights)` 3-d OLAP cube operator; Fills all cells of the given vectors into the given histogram.
`static void` | `demo1()` Demonstrates usage of this class.
`static void` | `demo2(int rows, int columns, boolean print)` Demonstrates usage of this class.
`static void` | `demo3(Statistic.VectorVectorFunction norm)` Demonstrates usage of this class.
`static DoubleMatrix2D` | `distance(DoubleMatrix2D matrix, Statistic.VectorVectorFunction distanceFunction)` Constructs and returns the distance matrix of the given matrix.
`static IHistogram1D` | `histogram(IHistogram1D histo, DoubleMatrix1D vector)` Fills all cells of the given vector into the given histogram.
`static IHistogram2D` | `histogram(IHistogram2D histo, DoubleMatrix1D x, DoubleMatrix1D y)` Fills all cells of the given vectors into the given histogram.
`static IHistogram2D` | `histogram(IHistogram2D histo, DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix1D weights)` Fills all cells of the given vectors into the given histogram.
`static IHistogram3D` | `histogram(IHistogram3D histo, DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix1D z, DoubleMatrix1D weights)` Fills all cells of the given vectors into the given histogram.
`static void` | `main(String[] args)` Benchmarks covariance computation.
`static DoubleMatrix1D` | `viewSample(DoubleMatrix1D matrix, double fraction, RandomEngine randomGenerator)` Constructs and returns a sampling view with a size of `round(matrix.size() * fraction)`.
`static DoubleMatrix2D` | `viewSample(DoubleMatrix2D matrix, double rowFraction, double columnFraction, RandomEngine randomGenerator)` Constructs and returns a sampling view with `round(matrix.rows() * rowFraction)` rows and `round(matrix.columns() * columnFraction)` columns.
`static DoubleMatrix3D` | `viewSample(DoubleMatrix3D matrix, double sliceFraction, double rowFraction, double columnFraction, RandomEngine randomGenerator)` Constructs and returns a sampling view with `round(matrix.slices() * sliceFraction)` slices and `round(matrix.rows() * rowFraction)` rows and `round(matrix.columns() * columnFraction)` columns.

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

Field Detail

### EUCLID

    public static final Statistic.VectorVectorFunction EUCLID

  Euclidean distance function; `Sqrt(Sum( (x[i]-y[i])^2 ))`.

### BRAY_CURTIS

    public static final Statistic.VectorVectorFunction BRAY_CURTIS

  Bray-Curtis distance function; `Sum( abs(x[i]-y[i]) ) / Sum( x[i]+y[i] )`.

### CANBERRA

    public static final Statistic.VectorVectorFunction CANBERRA

  Canberra distance function; `Sum( abs(x[i]-y[i]) / abs(x[i]+y[i]) )`.

### MAXIMUM

    public static final Statistic.VectorVectorFunction MAXIMUM

  Maximum distance function; `Max( abs(x[i]-y[i]) )`.

### MANHATTAN

    public static final Statistic.VectorVectorFunction MANHATTAN

  Manhattan distance function; `Sum( abs(x[i]-y[i]) )`.

Method Detail

### aggregate

    public static DoubleMatrix2D aggregate(DoubleMatrix2D matrix,
                                           BinFunction1D[] aggr,
                                           DoubleMatrix2D result)

  Applies the given aggregation functions to each column and stores the results in a the result matrix. If matrix has shape `m x n`, then result must have shape `aggr.length x n`. Tip: To do aggregations on rows use dice views (transpositions), as in `aggregate(matrix.viewDice(),aggr,result.viewDice())`.

  Parameters:
    `matrix` - any matrix; a column holds the values of a given variable.
    `aggr` - the aggregation functions to be applied to each column.
    `result` - the matrix to hold the aggregation results.
  Returns:
    `result` (for convenience only).
  See Also:
    `Formatter`, `BinFunction1D`, `BinFunctions1D`

### bin

    public static DynamicBin1D bin(DoubleMatrix1D vector)

  Fills all cell values of the given vector into a bin from which statistics measures can be retrieved efficiently. Cells values are copied.
  Tip: Use `System.out.println(bin(vector))` to print most measures computed by the bin. Example:

      Size: 20000
      Sum: 299858.02350278624
      SumOfSquares: 5399184.154095971
      Min: 0.8639113139711261
      Max: 59.75331890541892
      Mean: 14.992901175139313
      RMS: 16.43043540825375
      Variance: 45.17438077634358
      Standard deviation: 6.721188940681818
      Standard error: 0.04752598277592142
      Geometric mean: 13.516615397064466
      Product: Infinity
      Harmonic mean: 11.995174297952191
      Sum of inversions: 1667.337172700724
      Skew: 0.8922838940067878
      Kurtosis: 1.1915828121825598
      Sum of powers(3): 1.1345828465808412E8
      Sum of powers(4): 2.7251055344494686E9
      Sum of powers(5): 7.367125643433887E10
      Sum of powers(6): 2.215370909100143E12
      Moment(0,0): 1.0
      Moment(1,0): 14.992901175139313
      Moment(2,0): 269.95920770479853
      Moment(3,0): 5672.914232904206
      Moment(4,0): 136255.27672247344
      Moment(5,0): 3683562.8217169433
      Moment(6,0): 1.1076854545500715E8
      Moment(0,mean()): 1.0
      Moment(1,mean()): -2.0806734113421045E-14
      Moment(2,mean()): 45.172122057305664
      Moment(3,mean()): 270.92018671421
      Moment(4,mean()): 8553.8664869067
      Moment(5,mean()): 153357.41712233616
      Moment(6,mean()): 4273757.570142922
      25%, 50% and 75% Quantiles: 10.030074811938091, 13.977982089912224,
      18.86124362967137
      quantileInverse(mean): 0.559163335012079
      Distinct elements & frequencies not printed (too many).

  Parameters:
    `vector` - the vector to analyze.
  Returns:
    a bin holding the statistics measures of the vector.

### correlation

    public static DoubleMatrix2D correlation(DoubleMatrix2D covariance)

  Modifies the given covariance matrix to be a correlation matrix (in-place). The correlation matrix is a square, symmetric matrix consisting of nothing but correlation coefficients. The rows and the columns represent the variables, the cells represent correlation coefficients. The diagonal cells (i.e. the correlation between a variable and itself) will equal 1, for the simple reason that the correlation coefficient of a variable with itself equals 1. The correlation of two column vectors x and y is given by `corr(x,y) = cov(x,y) / (stdDev(x)*stdDev(y))` (Pearson's correlation coefficient). A correlation coefficient varies between -1 (for a perfect negative relationship) to +1 (for a perfect positive relationship). See the math definition and another def. Compares two column vectors at a time. Use dice views to compare two row vectors at a time.

  Parameters:
    `covariance` - a covariance matrix, as, for example, returned by method `covariance(DoubleMatrix2D)`.
  Returns:
    the modified covariance, now correlation matrix (for convenience only).

### covariance

    public static DoubleMatrix2D covariance(DoubleMatrix2D matrix)

  Constructs and returns the covariance matrix of the given matrix. The covariance matrix is a square, symmetric matrix consisting of nothing but covariance coefficients. The rows and the columns represent the variables, the cells represent covariance coefficients. The diagonal cells (i.e. the covariance between a variable and itself) will equal the variances. The covariance of two column vectors x and y is given by `cov(x,y) = (1/n) * Sum((x[i]-mean(x)) * (y[i]-mean(y)))`. See the math definition. Compares two column vectors at a time. Use dice views to compare two row vectors at a time.

  Parameters:
    `matrix` - any matrix; a column holds the values of a given variable.
  Returns:
    the covariance matrix (`n x n, n=matrix.columns`).

### cube

    public static IHistogram2D cube(DoubleMatrix1D x,
                                    DoubleMatrix1D y,
                                    DoubleMatrix1D weights)

  2-d OLAP cube operator; Fills all cells of the given vectors into the given histogram. If you use hep.aida.ref.Converter.toString(histo) on the result, the OLAP cube of x-"column" vs. y-"column" , summing the weights "column" will be printed. For example, aggregate sales by product by region.

  Computes the distinct values of x and y, yielding histogram axes that capture one distinct value per bin. Then fills the histogram.

  Example output:

      Cube:
         Entries=5000, ExtraEntries=0
         MeanX=4.9838, RmsX=NaN
         MeanY=2.5304, RmsY=NaN
         xAxis: Min=0, Max=10, Bins=11
         yAxis: Min=0, Max=5, Bins=6
      Heights:
            | X
            | 0   1   2   3   4   5   6   7   8   9   10  | Sum
      ----------------------------------------------------------
      Y 5   |  30  53  51  52  57  39  65  61  55  49  22 |  534
        4   |  43 106 112  96  92  94 107  98  98 110  47 | 1003
        3   |  39 134  87  93 102 103 110  90 114  98  51 | 1021
        2   |  44  81 113  96 101  86 109  83 111  93  42 |  959
        1   |  54  94 103  99 115  92  98  97 103  90  44 |  989
        0   |  24  54  52  44  42  56  46  47  56  53  20 |  494
      ----------------------------------------------------------
        Sum | 234 522 518 480 509 470 535 476 537 493 226 |

  Returns:
    the histogram containing the cube.
  Throws:
    `IllegalArgumentException` - if `x.size() != y.size() || y.size() != weights.size()`.

### cube

    public static IHistogram3D cube(DoubleMatrix1D x,
                                    DoubleMatrix1D y,
                                    DoubleMatrix1D z,
                                    DoubleMatrix1D weights)

  3-d OLAP cube operator; Fills all cells of the given vectors into the given histogram. If you use hep.aida.ref.Converter.toString(histo) on the result, the OLAP cube of x-"column" vs. y-"column" vs. z-"column", summing the weights "column" will be printed. For example, aggregate sales by product by region by time.

  Computes the distinct values of x and y and z, yielding histogram axes that capture one distinct value per bin. Then fills the histogram.

  Returns:
    the histogram containing the cube.
  Throws:
    `IllegalArgumentException` - if `x.size() != y.size() || x.size() != z.size() || x.size() != weights.size()`.

### demo1

    public static void demo1()

  Demonstrates usage of this class.

### demo2

    public static void demo2(int rows,
                             int columns,
                             boolean print)

  Demonstrates usage of this class.

### demo3

    public static void demo3(Statistic.VectorVectorFunction norm)

  Demonstrates usage of this class.

### distance

    public static DoubleMatrix2D distance(DoubleMatrix2D matrix,
                                          Statistic.VectorVectorFunction distanceFunction)

  Constructs and returns the distance matrix of the given matrix. The distance matrix is a square, symmetric matrix consisting of nothing but distance coefficients. The rows and the columns represent the variables, the cells represent distance coefficients. The diagonal cells (i.e. the distance between a variable and itself) will be zero. Compares two column vectors at a time. Use dice views to compare two row vectors at a time.

  Parameters:
    `matrix` - any matrix; a column holds the values of a given variable (vector).
    `distanceFunction` - (EUCLID, CANBERRA, ..., or any user defined distance function operating on two vectors).
  Returns:
    the distance matrix (`n x n, n=matrix.columns`).

### histogram

    public static IHistogram1D histogram(IHistogram1D histo,
                                         DoubleMatrix1D vector)

  Fills all cells of the given vector into the given histogram.

  Returns:
    `histo` (for convenience only).

### histogram

    public static IHistogram2D histogram(IHistogram2D histo,
                                         DoubleMatrix1D x,
                                         DoubleMatrix1D y)

  Fills all cells of the given vectors into the given histogram.

  Returns:
    `histo` (for convenience only).
  Throws:
    `IllegalArgumentException` - if `x.size() != y.size()`.

### histogram

    public static IHistogram2D histogram(IHistogram2D histo,
                                         DoubleMatrix1D x,
                                         DoubleMatrix1D y,
                                         DoubleMatrix1D weights)

  Fills all cells of the given vectors into the given histogram.

  Returns:
    `histo` (for convenience only).
  Throws:
    `IllegalArgumentException` - if `x.size() != y.size() || y.size() != weights.size()`.

### histogram

    public static IHistogram3D histogram(IHistogram3D histo,
                                         DoubleMatrix1D x,
                                         DoubleMatrix1D y,
                                         DoubleMatrix1D z,
                                         DoubleMatrix1D weights)

  Fills all cells of the given vectors into the given histogram.

  Returns:
    `histo` (for convenience only).
  Throws:
    `IllegalArgumentException` - if `x.size() != y.size() || x.size() != z.size() || x.size() != weights.size()`.

### main

    public static void main(String[] args)

  Benchmarks covariance computation.

### viewSample

    public static DoubleMatrix1D viewSample(DoubleMatrix1D matrix,
                                            double fraction,
                                            RandomEngine randomGenerator)

  Constructs and returns a sampling view with a size of `round(matrix.size() * fraction)`. Samples "without replacement" from the uniform distribution.

  Parameters:
    `matrix` - any matrix.
    `randomGenerator` - a uniform random number generator; set this parameter to `null` to use a default generator seeded with the current time.
  Returns:
    the sampling view.
  Throws:
    `IllegalArgumentException` - if `! (0 <= rowFraction <= 1 && 0 <= columnFraction <= 1)`.
  See Also:
    `RandomSampler`

### viewSample

    public static DoubleMatrix2D viewSample(DoubleMatrix2D matrix,
                                            double rowFraction,
                                            double columnFraction,
                                            RandomEngine randomGenerator)

  Constructs and returns a sampling view with `round(matrix.rows() * rowFraction)` rows and `round(matrix.columns() * columnFraction)` columns. Samples "without replacement". Rows and columns are randomly chosen from the uniform distribution. Examples:

  `matrix` | `rowFraction=0.2 columnFraction=0.2` | `rowFraction=0.2 columnFraction=1.0` | `rowFraction=1.0 columnFraction=0.2`
  `10 x 10 matrix 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100` | `2 x 2 matrix 43 50 53 60` | `2 x 10 matrix 41 42 43 44 45 46 47 48 49 50 91 92 93 94 95 96 97 98 99 100` | `10 x 2 matrix 4 8 14 18 24 28 34 38 44 48 54 58 64 68 74 78 84 88 94 98`

  Parameters:
    `matrix` - any matrix.
    `rowFraction` - the percentage of rows to be included in the view.
    `columnFraction` - the percentage of columns to be included in the view.
    `randomGenerator` - a uniform random number generator; set this parameter to `null` to use a default generator seeded with the current time.
  Returns:
    the sampling view.
  Throws:
    `IllegalArgumentException` - if `! (0 <= rowFraction <= 1 && 0 <= columnFraction <= 1)`.
  See Also:
    `RandomSampler`

### viewSample

    public static DoubleMatrix3D viewSample(DoubleMatrix3D matrix,
                                            double sliceFraction,
                                            double rowFraction,
                                            double columnFraction,
                                            RandomEngine randomGenerator)

  Constructs and returns a sampling view with `round(matrix.slices() * sliceFraction)` slices and `round(matrix.rows() * rowFraction)` rows and `round(matrix.columns() * columnFraction)` columns. Samples "without replacement". Slices, rows and columns are randomly chosen from the uniform distribution.

  Parameters:
    `matrix` - any matrix.
    `sliceFraction` - the percentage of slices to be included in the view.
    `rowFraction` - the percentage of rows to be included in the view.
    `columnFraction` - the percentage of columns to be included in the view.
    `randomGenerator` - a uniform random number generator; set this parameter to `null` to use a default generator seeded with the current time.
  Returns:
    the sampling view.
  Throws:
    `IllegalArgumentException` - if `! (0 <= sliceFraction <= 1 && 0 <= rowFraction <= 1 && 0 <= columnFraction <= 1)`.
  See Also:
    `RandomSampler`

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