# Class Statistics

Object
Statistics

All Implemented Interfaces:
  `Serializable, Cloneable, DoubleConsumer, LongConsumer`

public class Statistics extends Object implements DoubleConsumer, LongConsumer, Cloneable, Serializable
Holds some statistics derived from a series of sample values. Given a series of y₀, y₁, y₂, y₃, etc… samples, this class computes the minimum, maximum, mean, root mean square and standard deviation of the given samples.

In addition to the statistics on the sample values, this class can optionally compute statistics on the differences between consecutive sample values, i.e. the statistics on y₁-y₀, y₂-y₁, y₃-y₂, etc…, Those statistics can be fetched by a call to `differences()`. They are useful for verifying if the interval between sample values is approximately constant.

If the samples are (at least conceptually) the result of some y=f(x) function for x values increasing or decreasing at a constant interval Δx, then one can get the statistics on the discrete derivatives by a call to `differences().scale(1/Δx)`.

Statistics are computed on the fly using the Kahan summation algorithm for reducing the numerical errors; the sample values are never stored in memory.

An instance of `Statistics` is initially empty: the count of values is set to zero, and all above-cited statistical values are set to `NaN`. The statistics are updated every time an `accept(double)` method is invoked with a non-NaN value.

## Examples

The following examples assume that a y=f(x) function is defined. A simple usage is:

    Statistics stats = new Statistics("y");
    for (int i=0; i<numberOfValues; i++) {
        stats.accept(f(i));
    }
    System.out.println(stats);

Following example computes the statistics on the first and second derivatives in addition to the statistics on the sample values:

    final double x₀ = ...; // Put here the x value at i=0
    final double Δx = ...; // Put here the interval between x values
    Statistics stats = Statistics.forSeries("y", "∂y/∂x", "∂²y/∂x²");
    for (int i=0; i<numberOfValues; i++) {
        stats.accept(f(x₀ + i*Δx));
    }
    stats.differences().scale(1/Δx);
    System.out.println(stats);

Since:
  0.3
See Also:
  - Serialized Form

- ## Constructor Summary

  Constructors
  Constructor
  Description
  `Statistics(CharSequence name)`
  Constructs an initially empty set of statistics.
  `Statistics(CharSequence name, int countNaN, int count, double minimum, double maximum, double mean, double standardDeviation, boolean allPopulation)`
  Constructs a set of statistics initialized to the given values.

- ## Method Summary

  Modifier and Type
  Method
  Description
  `void`
  `accept(double sample)`
  Updates statistics for the specified floating-point sample value.
  `void`
  `accept(long sample)`
  Updates statistics for the specified integer sample value.
  `Statistics`
  `clone()`
  Returns a clone of this statistics.
  `void`
  `combine(Statistics stats)`
  Updates statistics with all samples from the specified `stats`.
  `int`
  `count()`
  Returns the number of samples, excluding `NaN` values.
  `int`
  `countNaN()`
  Returns the number of `NaN` samples.
  `Statistics`
  `differences()`
  Returns the statistics on the differences between sample values, or `null` if none.
  `boolean`
  `equals(Object object)`
  Compares this statistics with the specified object for equality.
  `static Statistics`
  `forSeries(CharSequence name, CharSequence... differenceNames)`
  Constructs a new `Statistics` object which will also compute finite differences up to the given order.
  `int`
  `hashCode()`
  Returns a hash code value for this statistics.
  `double`
  `maximum()`
  Returns the maximum sample value, or `NaN` if none.
  `double`
  `mean()`
  Returns the mean value, or `NaN` if none.
  `double`
  `minimum()`
  Returns the minimum sample value, or `NaN` if none.
  `InternationalString`
  `name()`
  Returns the name of the phenomenon for which this object is collecting statistics.
  `void`
  `reset()`
  Resets this object state as if it was just created.
  `double`
  `rms()`
  Returns the root mean square, or `NaN` if none.
  `void`
  `scale(double factor)`
  Multiplies the statistics by the given factor.
  `double`
  `span()`
  Equivalents to maximum - minimum.
  `double`
  `standardDeviation(boolean allPopulation)`
  Returns the standard deviation.
  `double`
  `sum()`
  Returns the sum, or 0 if none.
  `String`
  `toString()`
  Returns a string representation of this statistics.

  ### Methods inherited from class Object

  `finalize, getClass, notify, notifyAll, wait, wait, wait`

  ### Methods inherited from interface DoubleConsumer

  `andThen`

  ### Methods inherited from interface LongConsumer

  `andThen`

- ## Constructor Details

  - ### Statistics

    public Statistics(CharSequence name)
    Constructs an initially empty set of statistics. The count() and the `sum()` are initialized to zero and all other statistical values are initialized to `Double​.NaN`.

    Instances created by this constructor do not compute differences between sample values. If differences or discrete derivatives are wanted, use the `forSeries(…)` method instead.

    Parameters:
      `name` - the phenomenon for which this object is collecting statistics, or `null` if none. If non-null, it will be shown as column header in the table formatted by `StatisticsFormat`.

  - ### Statistics

    public Statistics(CharSequence name, int countNaN, int count, double minimum, double maximum, double mean, double standardDeviation, boolean allPopulation)
    Constructs a set of statistics initialized to the given values. The `countNaN` and `count` arguments must be positive. If `count` is 0, all following `double` arguments are ignored. Otherwise the following restrictions apply:
    - `minimum` and `maximum` arguments are mandatory and cannot be `NaN`.
    - `mean` argument is mandatory (cannot be NaN) if `standardDeviation` is not NaN.
    - `mean` and `standardDeviation` arguments can be both `NaN` if unknown, but statistics initialized that way will always return NaN from `sum()`, `mean()`, `rms()` and `standardDeviation(boolean)` methods.
    Parameters:
      `name` - the phenomenon for which this object is collecting statistics, or `null` if none.
      `countNaN` - the number of `NaN` samples.
      `count` - the number of samples, excluding `NaN` values.
      `minimum` - the minimum sample value. Ignored if `count` is zero.
      `maximum` - the maximum sample value. Ignored if `count` is zero.
      `mean` - the mean value. Ignored if `count` is zero.
      `standardDeviation` - the standard deviation. Ignored if `count` is zero.
      `allPopulation` - `true` if sample values were the totality of the population under study, or `false` if they were only a sampling.
    Since:
      1.2

- ## Method Details

  - ### forSeries

    public static Statistics forSeries(CharSequence name, CharSequence... differenceNames)
    Constructs a new `Statistics` object which will also compute finite differences up to the given order. If the values to be given to the `accept(…)` methods are the y values of some y=f(x) function for x values increasing or decreasing at a constant interval Δx, then the finite differences are proportional to discrete derivatives.

    The `Statistics` object created by this method know nothing about the Δx interval. In order to get the discrete derivatives, the following method needs to be invoked after all sample values have been added:

        statistics.differences().scale(1/Δx);

    The maximal "derivative" order is determined by the length of the `differenceNames` array:
    - 0 if no differences are needed (equivalent to direct instantiation of a new `Statistics` object).
    - 1 for computing the statistics on the differences between consecutive samples (proportional to the statistics on the first discrete derivatives) in addition to the sample statistics.
    - 2 for computing also the statistics on the differences between consecutive differences (proportional to the statistics on the second discrete derivatives) in addition to the above.
    - etc.
    Parameters:
      `name` - the phenomenon for which this object is collecting statistics, or `null` if none. If non-null, then this name will be shown as column header in the table formatted by `StatisticsFormat`.
      `differenceNames` - the names of the statistics on differences. The given array cannot be null, but can contain null elements.
    Returns:
      the newly constructed, initially empty, set of statistics.
    See Also:
      - `differences()`

  - ### name

    public InternationalString name()
    Returns the name of the phenomenon for which this object is collecting statistics. If non-null, then this name will be shown as column header in the table formatted by `StatisticsFormat`.
    Returns:
      the phenomenon for which this object is collecting statistics, or `null` if none.

  - ### reset

    public void reset()
    Resets this object state as if it was just created. The count() and the `sum()` are set to zero and all other statistical values are set to `Double​.NaN`.

  - ### accept

    public void accept(double sample)
    Updates statistics for the specified floating-point sample value. `NaN` values increment the NaN count, but are otherwise ignored.
    Specified by:
      `accept` in interface `DoubleConsumer`
    Parameters:
      `sample` - the sample value (may be NaN).
    See Also:
      - `accept(long)`
      - `combine(Statistics)`

  - ### accept

    public void accept(long sample)
    Updates statistics for the specified integer sample value. For very large integer values (greater than 252 in magnitude), this method may be more accurate than the `accept(double)` version.
    Specified by:
      `accept` in interface `LongConsumer`
    Parameters:
      `sample` - the sample value.
    See Also:
      - `accept(double)`
      - `combine(Statistics)`

  - ### combine

    public void combine(Statistics stats)
    Updates statistics with all samples from the specified `stats`. Invoking this method is equivalent (except for rounding errors) to invoking `accept(…)` for all samples that were added to `stats`.
    Parameters:
      `stats` - the statistics to be added to `this`.

  - ### scale

    public void scale(double factor)
    Multiplies the statistics by the given factor. The given scale factor is also applied recursively on the differences statistics, if any. Invoking this method transforms the statistics as if every values given to the `accept(…)` had been first multiplied by the given factor.

    This method is useful for computing discrete derivatives from the differences between sample values. See `differences()` or `forSeries(…)` for more information.

    Parameters:
      `factor` - the factor by which to multiply the statistics.

  - ### countNaN

    public int countNaN()
    Returns the number of `NaN` samples. `NaN` samples are ignored in all other statistical computation. This method count them for information purpose only.
    Returns:
      the number of NaN values.

  - ### count

    public int count()
    Returns the number of samples, excluding `NaN` values.
    Returns:
      the number of sample values, excluding NaN.

  - ### minimum

    public double minimum()
    Returns the minimum sample value, or `NaN` if none.
    Returns:
      the minimum sample value, or NaN if none.

  - ### maximum

    public double maximum()
    Returns the maximum sample value, or `NaN` if none.
    Returns:
      the maximum sample value, or NaN if none.

  - ### span

    public double span()
    Equivalents to maximum - minimum. If no samples were added, then returns `NaN`.
    Returns:
      the span of sample values, or NaN if none.

  - ### sum

    public double sum()
    Returns the sum, or 0 if none. May also be NaN if that value was explicitly specified to the constructor.
    Returns:
      the sum, or 0 if none.

  - ### mean

    public double mean()
    Returns the mean value, or `NaN` if none.
    Returns:
      the mean value, or NaN if none.

  - ### rms

    public double rms()
    Returns the root mean square, or `NaN` if none.
    Returns:
      the root mean square, or NaN if none.

  - ### standardDeviation

    public double standardDeviation(boolean allPopulation)
    Returns the standard deviation. If the sample values given to the `accept(…)` methods have a uniform distribution, then the returned value should be close to `sqrt(span² / 12)`. If they have a Gaussian distribution (which is the most common case), then the returned value is related to the error function.

    As a reminder, the table below gives the probability for a sample value to be inside the mean ± n × deviation range, assuming that the distribution is Gaussian (first column) or assuming that the distribution is uniform (second column).

    Probability values for some standard deviations
    n | Gaussian | uniform
    --- | --- | ---
    0.5 | 69.1% | 28.9%
    1.0 | 84.2% | 57.7%
    1.5 | 93.3% | 86.6%
    2.0 | 97.7% | 100%
    3.0 | 99.9% | 100%

    Parameters:
      `allPopulation` - `true` if sample values given to `accept(…)` methods were the totality of the population under study, or `false` if they were only a sampling.
    Returns:
      the standard deviation.

  - ### differences

    public Statistics differences()
    Returns the statistics on the differences between sample values, or `null` if none. For example if the sample values given to the `accept(…)` methods were y₀, y₁, y₂ and y₃, then this method returns statistics on y₁-y₀, y₂-y₁ and y₃-y₂.

    The differences between sample values are related to the discrete derivatives as below, where Δx is the constant interval between the x values of the y=f(x) function:

        Statistics derivative = statistics.differences();
        derivative.scale(1/Δx); // Shall be invoked only once.
        Statistics secondDerivative = derivative.differences();
        // Do not invoke scale(1/Δx) again.

    This method returns a non-null value only if this `Statistics` instance has been created by a call to the `forSeries(…)` method with a non-empty `differenceNames` array. More generally, calls to this method can be chained up to `differenceNames​.length` times for fetching second or higher order derivatives, as in the above example.
    Returns:
      the statistics on the differences between consecutive sample values, or `null` if not calculated by this object.
    See Also:
      - `forSeries(CharSequence, CharSequence[])`
      - `scale(double)`

  - ### toString

    public String toString()
    Returns a string representation of this statistics. This string will span multiple lines, one for each statistical value. For example:

            Number of values:     8726
            Minimum value:       6.853
            Maximum value:       8.259
            Mean value:          7.421
            Root Mean Square:    7.846
            Standard deviation:  6.489

    Overrides:
      `toString` in class `Object`
    Returns:
      a string representation of this statistics object.
    See Also:
      - `StatisticsFormat`

  - ### clone

    public Statistics clone()
    Returns a clone of this statistics.
    Overrides:
      `clone` in class `Object`
    Returns:
      a clone of this statistics.

  - ### hashCode

    public int hashCode()
    Returns a hash code value for this statistics.
    Overrides:
      `hashCode` in class `Object`

  - ### equals

    public boolean equals(Object object)
    Compares this statistics with the specified object for equality.
    Overrides:
      `equals` in class `Object`
    Parameters:
      `object` - the object to compare with.
    Returns:
      `true` if both objects are equal.