cern.jet.stat

## Class Descriptive

- java.lang.Object
- - cern.jet.stat.Descriptive

-     public class Descriptive
      extends java.lang.Object

  Basic descriptive statistics.
  Version:
    0.91, 08-Dec-99

- - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `static double` | `autoCorrelation(DoubleArrayList data, int lag, double mean, double variance)` Returns the auto-correlation of a data sequence.
    `static double` | `correlation(DoubleArrayList data1, double standardDev1, DoubleArrayList data2, double standardDev2)` Returns the correlation of two data sequences.
    `static double` | `covariance(DoubleArrayList data1, DoubleArrayList data2)` Returns the covariance of two data sequences, which is `cov(x,y) = (1/(size()-1)) * Sum((x[i]-mean(x)) * (y[i]-mean(y)))`.
    `static double` | `durbinWatson(DoubleArrayList data)` Durbin-Watson computation.
    `static void` | `frequencies(DoubleArrayList sortedData, DoubleArrayList distinctValues, IntArrayList frequencies)` Computes the frequency (number of occurances, count) of each distinct value in the given sorted data.
    `static double` | `geometricMean(DoubleArrayList data)` Returns the geometric mean of a data sequence.
    `static double` | `geometricMean(int size, double sumOfLogarithms)` Returns the geometric mean of a data sequence.
    `static double` | `harmonicMean(int size, double sumOfInversions)` Returns the harmonic mean of a data sequence.
    `static void` | `incrementalUpdate(DoubleArrayList data, int from, int to, double[] inOut)` Incrementally maintains and updates minimum, maximum, sum and sum of squares of a data sequence.
    `static void` | `incrementalUpdateSumsOfPowers(DoubleArrayList data, int from, int to, int fromSumIndex, int toSumIndex, double[] sumOfPowers)` Incrementally maintains and updates various sums of powers of the form `Sum(data[i]k)`.
    `static void` | `incrementalWeightedUpdate(DoubleArrayList data, DoubleArrayList weights, int from, int to, double[] inOut)` Incrementally maintains and updates sum and sum of squares of a weighted data sequence.
    `static double` | `kurtosis(DoubleArrayList data, double mean, double standardDeviation)` Returns the kurtosis (aka excess) of a data sequence, which is `-3 + moment(data,4,mean) / standardDeviation4`.
    `static double` | `kurtosis(double moment4, double standardDeviation)` Returns the kurtosis (aka excess) of a data sequence.
    `static double` | `lag1(DoubleArrayList data, double mean)` Returns the lag-1 autocorrelation of a dataset; Note that this method has semantics different from `autoCorrelation(..., 1)`;
    `static double` | `max(DoubleArrayList data)` Returns the largest member of a data sequence.
    `static double` | `mean(DoubleArrayList data)` Returns the arithmetic mean of a data sequence; That is `Sum( data[i] ) / data.size()`.
    `static double` | `meanDeviation(DoubleArrayList data, double mean)` Returns the mean deviation of a dataset.
    `static double` | `median(DoubleArrayList sortedData)` Returns the median of a sorted data sequence.
    `static double` | `min(DoubleArrayList data)` Returns the smallest member of a data sequence.
    `static double` | `moment(DoubleArrayList data, int k, double c)` Returns the moment of `k`-th order with constant `c` of a data sequence, which is `Sum( (data[i]-c)k ) / data.size()`.
    `static double` | `moment(int k, double c, int size, double[] sumOfPowers)` Returns the moment of `k`-th order with constant `c` of a data sequence, which is `Sum( (data[i]-c)k ) / data.size()`.
    `static double` | `pooledMean(int size1, double mean1, int size2, double mean2)` Returns the pooled mean of two data sequences.
    `static double` | `pooledVariance(int size1, double variance1, int size2, double variance2)` Returns the pooled variance of two data sequences.
    `static double` | `product(DoubleArrayList data)` Returns the product of a data sequence, which is `Prod( data[i] )`.
    `static double` | `product(int size, double sumOfLogarithms)` Returns the product, which is `Prod( data[i] )`.
    `static double` | `quantile(DoubleArrayList sortedData, double phi)` Returns the `phi-`quantile; that is, an element `elem` for which holds that `phi` percent of data elements are less than `elem`.
    `static double` | `quantileInverse(DoubleArrayList sortedList, double element)` Returns how many percent of the elements contained in the receiver are `<= element`.
    `static DoubleArrayList` | `quantiles(DoubleArrayList sortedData, DoubleArrayList percentages)` Returns the quantiles of the specified percentages.
    `static double` | `rankInterpolated(DoubleArrayList sortedList, double element)` Returns the linearly interpolated number of elements in a list less or equal to a given element.
    `static double` | `rms(int size, double sumOfSquares)` Returns the RMS (Root-Mean-Square) of a data sequence.
    `static double` | `sampleKurtosis(DoubleArrayList data, double mean, double sampleVariance)` Returns the sample kurtosis (aka excess) of a data sequence.
    `static double` | `sampleKurtosis(int size, double moment4, double sampleVariance)` Returns the sample kurtosis (aka excess) of a data sequence.
    `static double` | `sampleKurtosisStandardError(int size)` Return the standard error of the sample kurtosis.
    `static double` | `sampleSkew(DoubleArrayList data, double mean, double sampleVariance)` Returns the sample skew of a data sequence.
    `static double` | `sampleSkew(int size, double moment3, double sampleVariance)` Returns the sample skew of a data sequence.
    `static double` | `sampleSkewStandardError(int size)` Return the standard error of the sample skew.
    `static double` | `sampleStandardDeviation(int size, double sampleVariance)` Returns the sample standard deviation.
    `static double` | `sampleVariance(DoubleArrayList data, double mean)` Returns the sample variance of a data sequence.
    `static double` | `sampleVariance(int size, double sum, double sumOfSquares)` Returns the sample variance of a data sequence.
    `static double` | `sampleWeightedVariance(double sumOfWeights, double sumOfProducts, double sumOfSquaredProducts)` Returns the sample weighted variance of a data sequence.
    `static double` | `skew(DoubleArrayList data, double mean, double standardDeviation)` Returns the skew of a data sequence, which is `moment(data,3,mean) / standardDeviation3`.
    `static double` | `skew(double moment3, double standardDeviation)` Returns the skew of a data sequence.
    `static DoubleArrayList[]` | `split(DoubleArrayList sortedList, DoubleArrayList splitters)` Splits (partitions) a list into sublists such that each sublist contains the elements with a given range.
    `static double` | `standardDeviation(double variance)` Returns the standard deviation from a variance.
    `static double` | `standardError(int size, double variance)` Returns the standard error of a data sequence.
    `static void` | `standardize(DoubleArrayList data, double mean, double standardDeviation)` Modifies a data sequence to be standardized.
    `static double` | `sum(DoubleArrayList data)` Returns the sum of a data sequence.
    `static double` | `sumOfInversions(DoubleArrayList data, int from, int to)` Returns the sum of inversions of a data sequence, which is `Sum( 1.0 / data[i])`.
    `static double` | `sumOfLogarithms(DoubleArrayList data, int from, int to)` Returns the sum of logarithms of a data sequence, which is `Sum( Log(data[i])`.
    `static double` | `sumOfPowerDeviations(DoubleArrayList data, int k, double c)` Returns `Sum( (data[i]-c)k )`; optimized for common parameters like `c == 0.0` and/or `k == -2 ..`
    `static double` | `sumOfPowerDeviations(DoubleArrayList data, int k, double c, int from, int to)` Returns `Sum( (data[i]-c)k )` for all `i = from ..`
    `static double` | `sumOfPowers(DoubleArrayList data, int k)` Returns the sum of powers of a data sequence, which is `Sum ( data[i]k )`.
    `static double` | `sumOfSquaredDeviations(int size, double variance)` Returns the sum of squared mean deviation of of a data sequence.
    `static double` | `sumOfSquares(DoubleArrayList data)` Returns the sum of squares of a data sequence.
    `static double` | `trimmedMean(DoubleArrayList sortedData, double mean, int left, int right)` Returns the trimmed mean of a sorted data sequence.
    `static double` | `variance(double standardDeviation)` Returns the variance from a standard deviation.
    `static double` | `variance(int size, double sum, double sumOfSquares)` Returns the variance of a data sequence.
    `static double` | `weightedMean(DoubleArrayList data, DoubleArrayList weights)` Returns the weighted mean of a data sequence.
    `static double` | `weightedRMS(double sumOfProducts, double sumOfSquaredProducts)` Returns the weighted RMS (Root-Mean-Square) of a data sequence.
    `static double` | `winsorizedMean(DoubleArrayList sortedData, double mean, int left, int right)` Returns the winsorized mean of a sorted data sequence.

    - ### Methods inherited from class java.lang.Object

      `equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

- - ### Method Detail

    - #### autoCorrelation

          public static double autoCorrelation(DoubleArrayList data,
                               int lag,
                               double mean,
                               double variance)

      Returns the auto-correlation of a data sequence.

    - #### correlation

          public static double correlation(DoubleArrayList data1,
                           double standardDev1,
                           DoubleArrayList data2,
                           double standardDev2)

      Returns the correlation of two data sequences. That is `covariance(data1,data2)/(standardDev1*standardDev2)`.

    - #### covariance

          public static double covariance(DoubleArrayList data1,
                          DoubleArrayList data2)

      Returns the covariance of two data sequences, which is `cov(x,y) = (1/(size()-1)) * Sum((x[i]-mean(x)) * (y[i]-mean(y)))`. See the math definition.

    - #### durbinWatson

          public static double durbinWatson(DoubleArrayList data)

      Durbin-Watson computation.

    - #### frequencies

          public static void frequencies(DoubleArrayList sortedData,
                         DoubleArrayList distinctValues,
                         IntArrayList frequencies)

      Computes the frequency (number of occurances, count) of each distinct value in the given sorted data. After this call returns both `distinctValues` and `frequencies` have a new size (which is equal for both), which is the number of distinct values in the sorted data.

      Distinct values are filled into `distinctValues`, starting at index 0. The frequency of each distinct value is filled into `frequencies`, starting at index 0. As a result, the smallest distinct value (and its frequency) can be found at index 0, the second smallest distinct value (and its frequency) at index 1, ..., the largest distinct value (and its frequency) at index `distinctValues.size()-1`. Example:
      `elements = (5,6,6,7,8,8) --> distinctValues = (5,6,7,8), frequencies = (1,2,1,2)`

      Parameters:
        `sortedData` - the data; must be sorted ascending.
        `distinctValues` - a list to be filled with the distinct values; can have any size.
        `frequencies` - a list to be filled with the frequencies; can have any size; set this parameter to `null` to ignore it.

    - #### geometricMean

          public static double geometricMean(int size,
                             double sumOfLogarithms)

      Returns the geometric mean of a data sequence. Note that for a geometric mean to be meaningful, the minimum of the data sequence must not be less or equal to zero.
      The geometric mean is given by `pow( Product( data[i] ), 1/size)` which is equivalent to `Math.exp( Sum( Log(data[i]) ) / size)`.

    - #### geometricMean

          public static double geometricMean(DoubleArrayList data)

      Returns the geometric mean of a data sequence. Note that for a geometric mean to be meaningful, the minimum of the data sequence must not be less or equal to zero.
      The geometric mean is given by `pow( Product( data[i] ), 1/data.size())`. This method tries to avoid overflows at the expense of an equivalent but somewhat slow definition: `geo = Math.exp( Sum( Log(data[i]) ) / data.size())`.

    - #### harmonicMean

          public static double harmonicMean(int size,
                            double sumOfInversions)

      Returns the harmonic mean of a data sequence.
      Parameters:
        `size` - the number of elements in the data sequence.
        `sumOfInversions` - `Sum( 1.0 / data[i])`.

    - #### incrementalUpdate

          public static void incrementalUpdate(DoubleArrayList data,
                               int from,
                               int to,
                               double[] inOut)

      Incrementally maintains and updates minimum, maximum, sum and sum of squares of a data sequence. Assume we have already recorded some data sequence elements and know their minimum, maximum, sum and sum of squares. Assume further, we are to record some more elements and to derive updated values of minimum, maximum, sum and sum of squares.

      This method computes those updated values without needing to know the already recorded elements. This is interesting for interactive online monitoring and/or applications that cannot keep the entire huge data sequence in memory.

      Definition of sumOfSquares: `sumOfSquares(n) = Sum ( data[i] * data[i] )`.

      Parameters:
        `data` - the additional elements to be incorporated into min, max, etc.
        `from` - the index of the first element within `data` to consider.
        `to` - the index of the last element within `data` to consider. The method incorporates elements `data[from], ..., data[to]`.
        `inOut` - the old values in the following format:
        - `inOut[0]` is the old minimum.
        - `inOut[1]` is the old maximum.
        - `inOut[2]` is the old sum.
        - `inOut[3]` is the old sum of squares.
        If no data sequence elements have so far been recorded set the values as follows
        - `inOut[0] = Double.POSITIVE_INFINITY` as the old minimum.
        - `inOut[1] = Double.NEGATIVE_INFINITY` as the old maximum.
        - `inOut[2] = 0.0` as the old sum.
        - `inOut[3] = 0.0` as the old sum of squares.

    - #### incrementalUpdateSumsOfPowers

          public static void incrementalUpdateSumsOfPowers(DoubleArrayList data,
                                           int from,
                                           int to,
                                           int fromSumIndex,
                                           int toSumIndex,
                                           double[] sumOfPowers)

      Incrementally maintains and updates various sums of powers of the form `Sum(data[i]k)`. Assume we have already recorded some data sequence elements `data[i]` and know the values of `Sum(data[i]from), Sum(data[i]from+1), ..., Sum(data[i]to)`. Assume further, we are to record some more elements and to derive updated values of these sums.

      This method computes those updated values without needing to know the already recorded elements. This is interesting for interactive online monitoring and/or applications that cannot keep the entire huge data sequence in memory. For example, the incremental computation of moments is based upon such sums of powers:

      The moment of `k`-th order with constant `c` of a data sequence, is given by `Sum( (data[i]-c)k ) / data.size()`. It can incrementally be computed by using the equivalent formula

      `moment(k,c) = m(k,c) / data.size()` where
      `m(k,c) = Sum( -1i * b(k,i) * ci * sumOfPowers(k-i))` for `i = 0 .. k` and
      `b(k,i) =` `binomial(k,i)` and
      `sumOfPowers(k) = Sum( data[i]k )`.

      Parameters:
        `data` - the additional elements to be incorporated into min, max, etc.
        `from` - the index of the first element within `data` to consider.
        `to` - the index of the last element within `data` to consider. The method incorporates elements `data[from], ..., data[to]`.
        `inOut` - the old values of the sums in the following format:
        - `sumOfPowers[0]` is the old `Sum(data[i]fromSumIndex)`.
        - `sumOfPowers[1]` is the old `Sum(data[i]fromSumIndex+1)`.
        - ...
        - `sumOfPowers[toSumIndex-fromSumIndex]` is the old `Sum(data[i]toSumIndex)`.
        If no data sequence elements have so far been recorded set all old values of the sums to `0.0`.

    - #### incrementalWeightedUpdate

          public static void incrementalWeightedUpdate(DoubleArrayList data,
                                       DoubleArrayList weights,
                                       int from,
                                       int to,
                                       double[] inOut)

      Incrementally maintains and updates sum and sum of squares of a weighted data sequence. Assume we have already recorded some data sequence elements and know their sum and sum of squares. Assume further, we are to record some more elements and to derive updated values of sum and sum of squares.

      This method computes those updated values without needing to know the already recorded elements. This is interesting for interactive online monitoring and/or applications that cannot keep the entire huge data sequence in memory.

      Definition of sum: `sum = Sum ( data[i] * weights[i] )`.
      Definition of sumOfSquares: `sumOfSquares = Sum ( data[i] * data[i] * weights[i])`.

      Parameters:
        `data` - the additional elements to be incorporated into min, max, etc.
        `weights` - the weight of each element within `data`.
        `from` - the index of the first element within `data` (and `weights`) to consider.
        `to` - the index of the last element within `data` (and `weights`) to consider. The method incorporates elements `data[from], ..., data[to]`.
        `inOut` - the old values in the following format:
        - `inOut[0]` is the old sum.
        - `inOut[1]` is the old sum of squares.
        If no data sequence elements have so far been recorded set the values as follows
        - `inOut[0] = 0.0` as the old sum.
        - `inOut[1] = 0.0` as the old sum of squares.

    - #### kurtosis

          public static double kurtosis(double moment4,
                        double standardDeviation)

      Returns the kurtosis (aka excess) of a data sequence.
      Parameters:
        `moment4` - the fourth central moment, which is `moment(data,4,mean)`.
        `standardDeviation` - the standardDeviation.

    - #### kurtosis

          public static double kurtosis(DoubleArrayList data,
                        double mean,
                        double standardDeviation)

      Returns the kurtosis (aka excess) of a data sequence, which is `-3 + moment(data,4,mean) / standardDeviation4`.

    - #### lag1

          public static double lag1(DoubleArrayList data,
                    double mean)

      Returns the lag-1 autocorrelation of a dataset; Note that this method has semantics different from `autoCorrelation(..., 1)`;

    - #### max

          public static double max(DoubleArrayList data)

      Returns the largest member of a data sequence.

    - #### mean

          public static double mean(DoubleArrayList data)

      Returns the arithmetic mean of a data sequence; That is `Sum( data[i] ) / data.size()`.

    - #### meanDeviation

          public static double meanDeviation(DoubleArrayList data,
                             double mean)

      Returns the mean deviation of a dataset. That is `Sum (Math.abs(data[i]-mean)) / data.size())`.

    - #### median

          public static double median(DoubleArrayList sortedData)

      Returns the median of a sorted data sequence.
      Parameters:
        `sortedData` - the data sequence; must be sorted ascending.

    - #### min

          public static double min(DoubleArrayList data)

      Returns the smallest member of a data sequence.

    - #### moment

          public static double moment(int k,
                      double c,
                      int size,
                      double[] sumOfPowers)

      Returns the moment of `k`-th order with constant `c` of a data sequence, which is `Sum( (data[i]-c)k ) / data.size()`.
      Parameters:
        `sumOfPowers` - `sumOfPowers[m] == Sum( data[i]m) )` for `m = 0,1,..,k` as returned by method `incrementalUpdateSumsOfPowers(DoubleArrayList,int,int,int,int,double[])`. In particular there must hold `sumOfPowers.length == k+1`.
        `size` - the number of elements of the data sequence.

    - #### moment

          public static double moment(DoubleArrayList data,
                      int k,
                      double c)

      Returns the moment of `k`-th order with constant `c` of a data sequence, which is `Sum( (data[i]-c)k ) / data.size()`.

    - #### pooledMean

          public static double pooledMean(int size1,
                          double mean1,
                          int size2,
                          double mean2)

      Returns the pooled mean of two data sequences. That is `(size1 * mean1 + size2 * mean2) / (size1 + size2)`.
      Parameters:
        `size1` - the number of elements in data sequence 1.
        `mean1` - the mean of data sequence 1.
        `size2` - the number of elements in data sequence 2.
        `mean2` - the mean of data sequence 2.

    - #### pooledVariance

          public static double pooledVariance(int size1,
                              double variance1,
                              int size2,
                              double variance2)

      Returns the pooled variance of two data sequences. That is `(size1 * variance1 + size2 * variance2) / (size1 + size2)`;
      Parameters:
        `size1` - the number of elements in data sequence 1.
        `variance1` - the variance of data sequence 1.
        `size2` - the number of elements in data sequence 2.
        `variance2` - the variance of data sequence 2.

    - #### product

          public static double product(int size,
                       double sumOfLogarithms)

      Returns the product, which is `Prod( data[i] )`. In other words: `data[0]*data[1]*...*data[data.size()-1]`. This method uses the equivalent definition: `prod = pow( exp( Sum( Log(x[i]) ) / size(), size())`.

    - #### product

          public static double product(DoubleArrayList data)

      Returns the product of a data sequence, which is `Prod( data[i] )`. In other words: `data[0]*data[1]*...*data[data.size()-1]`. Note that you may easily get numeric overflows.

    - #### quantile

          public static double quantile(DoubleArrayList sortedData,
                        double phi)

      Returns the `phi-`quantile; that is, an element `elem` for which holds that `phi` percent of data elements are less than `elem`. The quantile need not necessarily be contained in the data sequence, it can be a linear interpolation.
      Parameters:
        `sortedData` - the data sequence; must be sorted ascending.
        `phi` - the percentage; must satisfy `0 <= phi <= 1`.

    - #### quantileInverse

          public static double quantileInverse(DoubleArrayList sortedList,
                               double element)

      Returns how many percent of the elements contained in the receiver are `<= element`. Does linear interpolation if the element is not contained but lies in between two contained elements.
      Parameters:
        `sortedList` - the list to be searched (must be sorted ascending).
        `element` - the element to search for.
      Returns:
        the percentage `phi` of elements `<= element` (`0.0 <= phi <= 1.0)`.

    - #### quantiles

          public static DoubleArrayList quantiles(DoubleArrayList sortedData,
                                  DoubleArrayList percentages)

      Returns the quantiles of the specified percentages. The quantiles need not necessarily be contained in the data sequence, it can be a linear interpolation.
      Parameters:
        `sortedData` - the data sequence; must be sorted ascending.
        `percentages` - the percentages for which quantiles are to be computed. Each percentage must be in the interval `[0.0,1.0]`.
      Returns:
        the quantiles.

    - #### rankInterpolated

          public static double rankInterpolated(DoubleArrayList sortedList,
                                double element)

      Returns the linearly interpolated number of elements in a list less or equal to a given element. The rank is the number of elements <= element. Ranks are of the form `{0, 1, 2,..., sortedList.size()}`. If no element is <= element, then the rank is zero. If the element lies in between two contained elements, then linear interpolation is used and a non integer value is returned.
      Parameters:
        `sortedList` - the list to be searched (must be sorted ascending).
        `element` - the element to search for.
      Returns:
        the rank of the element.

    - #### rms

          public static double rms(int size,
                   double sumOfSquares)

      Returns the RMS (Root-Mean-Square) of a data sequence. That is `Math.sqrt(Sum( data[i]*data[i] ) / data.size())`. The RMS of data sequence is the square-root of the mean of the squares of the elements in the data sequence. It is a measure of the average "size" of the elements of a data sequence.
      Parameters:
        `sumOfSquares` - `sumOfSquares(data) == Sum( data[i]*data[i] )` of the data sequence.
        `size` - the number of elements in the data sequence.

    - #### sampleKurtosis

          public static double sampleKurtosis(int size,
                              double moment4,
                              double sampleVariance)

      Returns the sample kurtosis (aka excess) of a data sequence. Ref: R.R. Sokal, F.J. Rohlf, Biometry: the principles and practice of statistics in biological research (W.H. Freeman and Company, New York, 1998, 3rd edition) p. 114-115.
      Parameters:
        `size` - the number of elements of the data sequence.
        `moment4` - the fourth central moment, which is `moment(data,4,mean)`.
        `sampleVariance` - the sample variance.

    - #### sampleKurtosis

          public static double sampleKurtosis(DoubleArrayList data,
                              double mean,
                              double sampleVariance)

      Returns the sample kurtosis (aka excess) of a data sequence.

    - #### sampleKurtosisStandardError

          public static double sampleKurtosisStandardError(int size)

      Return the standard error of the sample kurtosis. Ref: R.R. Sokal, F.J. Rohlf, Biometry: the principles and practice of statistics in biological research (W.H. Freeman and Company, New York, 1998, 3rd edition) p. 138.
      Parameters:
        `size` - the number of elements of the data sequence.

    - #### sampleSkew

          public static double sampleSkew(int size,
                          double moment3,
                          double sampleVariance)

      Returns the sample skew of a data sequence. Ref: R.R. Sokal, F.J. Rohlf, Biometry: the principles and practice of statistics in biological research (W.H. Freeman and Company, New York, 1998, 3rd edition) p. 114-115.
      Parameters:
        `size` - the number of elements of the data sequence.
        `moment3` - the third central moment, which is `moment(data,3,mean)`.
        `sampleVariance` - the sample variance.

    - #### sampleSkew

          public static double sampleSkew(DoubleArrayList data,
                          double mean,
                          double sampleVariance)

      Returns the sample skew of a data sequence.

    - #### sampleSkewStandardError

          public static double sampleSkewStandardError(int size)

      Return the standard error of the sample skew. Ref: R.R. Sokal, F.J. Rohlf, Biometry: the principles and practice of statistics in biological research (W.H. Freeman and Company, New York, 1998, 3rd edition) p. 138.
      Parameters:
        `size` - the number of elements of the data sequence.

    - #### sampleStandardDeviation

          public static double sampleStandardDeviation(int size,
                                       double sampleVariance)

      Returns the sample standard deviation. Ref: R.R. Sokal, F.J. Rohlf, Biometry: the principles and practice of statistics in biological research (W.H. Freeman and Company, New York, 1998, 3rd edition) p. 53.
      Parameters:
        `size` - the number of elements of the data sequence.
        `sampleVariance` - the sample variance.

    - #### sampleVariance

          public static double sampleVariance(int size,
                              double sum,
                              double sumOfSquares)

      Returns the sample variance of a data sequence. That is `(sumOfSquares - mean*sum) / (size - 1)` with `mean = sum/size`.
      Parameters:
        `size` - the number of elements of the data sequence.
        `sum` - `== Sum( data[i] )`.
        `sumOfSquares` - `== Sum( data[i]*data[i] )`.

    - #### sampleVariance

          public static double sampleVariance(DoubleArrayList data,
                              double mean)

      Returns the sample variance of a data sequence. That is `Sum ( (data[i]-mean)^2 ) / (data.size()-1)`.

    - #### sampleWeightedVariance

          public static double sampleWeightedVariance(double sumOfWeights,
                                      double sumOfProducts,
                                      double sumOfSquaredProducts)

      Returns the sample weighted variance of a data sequence. That is `(sumOfSquaredProducts - sumOfProducts * sumOfProducts / sumOfWeights) / (sumOfWeights - 1)`.
      Parameters:
        `sumOfWeights` - `== Sum( weights[i] )`.
        `sumOfProducts` - `== Sum( data[i] * weights[i] )`.
        `sumOfSquaredProducts` - `== Sum( data[i] * data[i] * weights[i] )`.

    - #### skew

          public static double skew(double moment3,
                    double standardDeviation)

      Returns the skew of a data sequence.
      Parameters:
        `moment3` - the third central moment, which is `moment(data,3,mean)`.
        `standardDeviation` - the standardDeviation.

    - #### skew

          public static double skew(DoubleArrayList data,
                    double mean,
                    double standardDeviation)

      Returns the skew of a data sequence, which is `moment(data,3,mean) / standardDeviation3`.

    - #### split

          public static DoubleArrayList[] split(DoubleArrayList sortedList,
                                DoubleArrayList splitters)

      Splits (partitions) a list into sublists such that each sublist contains the elements with a given range. `splitters=(a,b,c,...,y,z)` defines the ranges `[-inf,a), [a,b), [b,c), ..., [y,z), [z,inf]`.

      Examples:

      `data = (1,2,3,4,5,8,8,8,10,11)`.
      `splitters=(2,8)` yields 3 bins: `(1), (2,3,4,5) (8,8,8,10,11)`.
      `splitters=()` yields 1 bin: `(1,2,3,4,5,8,8,8,10,11)`.
      `splitters=(-5)` yields 2 bins: `(), (1,2,3,4,5,8,8,8,10,11)`.
      `splitters=(100)` yields 2 bins: `(1,2,3,4,5,8,8,8,10,11), ()`.
      Parameters:
        `sortedList` - the list to be partitioned (must be sorted ascending).
        `splitters` - the points at which the list shall be partitioned (must be sorted ascending).
      Returns:
        the sublists (an array with `length == splitters.size() + 1`. Each sublist is returned sorted ascending.

    - #### standardDeviation

          public static double standardDeviation(double variance)

      Returns the standard deviation from a variance.

    - #### standardError

          public static double standardError(int size,
                             double variance)

      Returns the standard error of a data sequence. That is `Math.sqrt(variance/size)`.
      Parameters:
        `size` - the number of elements in the data sequence.
        `variance` - the variance of the data sequence.

    - #### standardize

          public static void standardize(DoubleArrayList data,
                         double mean,
                         double standardDeviation)

      Modifies a data sequence to be standardized. Changes each element `data[i]` as follows: `data[i] = (data[i]-mean)/standardDeviation`.

    - #### sum

          public static double sum(DoubleArrayList data)

      Returns the sum of a data sequence. That is `Sum( data[i] )`.

    - #### sumOfInversions

          public static double sumOfInversions(DoubleArrayList data,
                               int from,
                               int to)

      Returns the sum of inversions of a data sequence, which is `Sum( 1.0 / data[i])`.
      Parameters:
        `data` - the data sequence.
        `from` - the index of the first data element (inclusive).
        `to` - the index of the last data element (inclusive).

    - #### sumOfLogarithms

          public static double sumOfLogarithms(DoubleArrayList data,
                               int from,
                               int to)

      Returns the sum of logarithms of a data sequence, which is `Sum( Log(data[i])`.
      Parameters:
        `data` - the data sequence.
        `from` - the index of the first data element (inclusive).
        `to` - the index of the last data element (inclusive).

    - #### sumOfPowerDeviations

          public static double sumOfPowerDeviations(DoubleArrayList data,
                                    int k,
                                    double c)

      Returns `Sum( (data[i]-c)k )`; optimized for common parameters like `c == 0.0` and/or `k == -2 .. 4`.

    - #### sumOfPowerDeviations

          public static double sumOfPowerDeviations(DoubleArrayList data,
                                    int k,
                                    double c,
                                    int from,
                                    int to)

      Returns `Sum( (data[i]-c)k )` for all `i = from .. to`; optimized for common parameters like `c == 0.0` and/or `k == -2 .. 5`.

    - #### sumOfPowers

          public static double sumOfPowers(DoubleArrayList data,
                           int k)

      Returns the sum of powers of a data sequence, which is `Sum ( data[i]k )`.

    - #### sumOfSquaredDeviations

          public static double sumOfSquaredDeviations(int size,
                                      double variance)

      Returns the sum of squared mean deviation of of a data sequence. That is `variance * (size-1) == Sum( (data[i] - mean)^2 )`.
      Parameters:
        `size` - the number of elements of the data sequence.
        `variance` - the variance of the data sequence.

    - #### sumOfSquares

          public static double sumOfSquares(DoubleArrayList data)

      Returns the sum of squares of a data sequence. That is `Sum ( data[i]*data[i] )`.

    - #### trimmedMean

          public static double trimmedMean(DoubleArrayList sortedData,
                           double mean,
                           int left,
                           int right)

      Returns the trimmed mean of a sorted data sequence.
      Parameters:
        `sortedData` - the data sequence; must be sorted ascending.
        `mean` - the mean of the (full) sorted data sequence.

    - #### variance

          public static double variance(double standardDeviation)

      Returns the variance from a standard deviation.

    - #### variance

          public static double variance(int size,
                        double sum,
                        double sumOfSquares)

      Returns the variance of a data sequence. That is `(sumOfSquares - mean*sum) / size` with `mean = sum/size`.
      Parameters:
        `size` - the number of elements of the data sequence.
        `sum` - `== Sum( data[i] )`.
        `sumOfSquares` - `== Sum( data[i]*data[i] )`.

    - #### weightedMean

          public static double weightedMean(DoubleArrayList data,
                            DoubleArrayList weights)

      Returns the weighted mean of a data sequence. That is `Sum (data[i] * weights[i]) / Sum ( weights[i] )`.

    - #### weightedRMS

          public static double weightedRMS(double sumOfProducts,
                           double sumOfSquaredProducts)

      Returns the weighted RMS (Root-Mean-Square) of a data sequence. That is `Sum( data[i] * data[i] * weights[i]) / Sum( data[i] * weights[i] )`, or in other words `sumOfProducts / sumOfSquaredProducts`.
      Parameters:
        `sumOfProducts` - `== Sum( data[i] * weights[i] )`.
        `sumOfSquaredProducts` - `== Sum( data[i] * data[i] * weights[i] )`.

    - #### winsorizedMean

          public static double winsorizedMean(DoubleArrayList sortedData,
                              double mean,
                              int left,
                              int right)

      Returns the winsorized mean of a sorted data sequence.
      Parameters:
        `sortedData` - the data sequence; must be sorted ascending.
        `mean` - the mean of the (full) sorted data sequence.

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