Histograms

# Title: Histograms

## 1 Class methods

`GSL::Histogram.new(n)`
`GSL::Histogram.new(n, [xmin, xmax])`
`GSL::Histogram.new(n, xmin, xmax)`
`GSL::Histogram.alloc(n)`
`GSL::Histogram.alloc(array)`
`GSL::Histogram.alloc(vector)`

  Constructor for a histogram object with n bins.

  Examples:

  1. With an integer:

         h = Histogram.alloc(4)  <--- Histogram of 4 bins.
                                      The range is not defined yet.

                   [ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )
                   |---------|---------|---------|---------|
                range[0]  range[1]  range[2]  range[3]  range[4]

  2. With an array or a vector:

         h = Histogram.new([1, 3, 7, 9, 20])  <--- Histogram of 4 bins.
                                                   The range is initialized as
                                                   range[0] = 1, range[1] = 3, ..., range[4] = 20.

  3. With size and the range [min, max]:

         irb(main):004:0> h = Histogram.alloc(5, [0, 5])
         irb(main):005:0> h.range
         => GSL::Histogram::Range:
         [ 0.000e+00 1.000e+00 2.000e+00 3.000e+00 4.000e+00 5.000e+00 ]
         irb(main):006:0> h.bin
         => GSL::Histogram::Bin:
         [ 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00 ]
         irb(main):007:0> h.increment(2.5)
         irb(main):008:0> h.bin
         => GSL::Histogram::Bin:
         [ 0.000e+00 0.000e+00 1.000e+00 0.000e+00 0.000e+00 ]

`GSL::Histogram.equal_bins_p(h1, h2)`
`GSL::Histogram.equal_bins(h1, h2)`
  Return 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.
`GSL::Histogram.equal_bins_p?(h1, h2)`
`GSL::Histogram.equal_bins?(h1, h2)`
  Return true if the all of the individual bin ranges of the two histograms are identical, and false otherwise.

## 2 Methods

`GSL::Histogram#set_ranges(v)`
  This sets the ranges of the existing histogram using a GSL::Vector object.
`GSL::Histogram#set_ranges_uniform(xmin, xmax)`

  This method sets the ranges of the existing histogram self to cover the range xmin to xmax uniformly. The values of the histogram bins are reset to zero. The bin ranges are shown as below,

      bin[0] corresponds to xmin <= x < xmin + d
      bin[1] corresponds to xmin + d <= x < xmin + 2 d
        ......
      bin[n-1] corresponds to xmin + (n-1)d <= x < xmax

  where d is the bin spacing, d = (xmax-xmin)/n.

`GSL::Histogram#increment(x, weight = 1)`
`GSL::Histogram#fill(x, weight = 1)`
`GSL::Histogram#accumulate(x, weight = 1)`
  These methods updates the histogram self by adding weight (default = 1) to the bin whose range contains the coordinate x. If x is less than (greater than) the lower limit (upper limit) of the histogram then none of bins are modified.
`GSL::Histogram#increment2(x, weight = 1)`
`GSL::Histogram#fill2(x, weight = 1)`
`GSL::Histogram#accumulate2(x, weight = 1)`
  These methods updates the histogram self by adding weight to the bin whose range contains the coordinate x. If x is less than the lower limit, the lowest bin is incremented. If x is greater than the upper limit, the highest bin is incremented.
`GSL::Histogram#get(i)`
`GSL::Histogram#[i]`
  These methods return the contents of the i-th bin of the histogram self.
`GSL::Hiatogram#get_range(i)`
  This method finds the upper and lower range limits of the i-th bin of the histogram self, and returns an array [lower, upper].
`GSL::Histogram#range`
  This returns a `Vector::View` object as a reference to the pointer `double *range` in the `gsl_histogram` struct.
`GSL::Histogram#bin`
  This returns a `Vector::View` object to access the pointer `double *bin` in the `gsl_histogram` struct.
`GSL::Histogram#max`
`GSL::Histogram#min`
`GSL::Histogram#bins`
  These methods return the maximum upper and minimum lower range limits and the number of bins of the histogram self.
`GSL::Histogram#reset`
  This method resets all the bins in the histogram self to zero.
`GSL::Histogram#find(x)`
  This method finds and sets the index i to the bin number which covers the coordinate x in the histogram self.
`GSL::Histogram#max_val`
  This returns the maximum value contained in the histogram bins.
`GSL::Histogram#max_bin`
  This returns the index of the bin containing the maximum value. In the case where several bins contain the same maximum value the smallest index is returned.
`GSL::Histogram#min_val`
  This returns the minimum value contained in the histogram bins.
`GSL::Histogram#min_bin`
  This returns the index of the bin containing the minimum value. In the case where several bins contain the same maximum value the smallest index is returned.
`GSL::Histogram#mean`
  This returns the mean of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.
`GSL::Histogram#sigma`
  This function returns the standard deviation of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.
`GSL::Histogram#sum(offset = 0, n = # of bins)`
  The sum of values of n bins from the offset offset. Negative bin values are included in the sum.
`GSL::Histogram#normalize`
  This methods scales the contents of the bins of histogram self by its maximum value.
`GSL::Histogram#rebin(m = 2)`

  This method creates a new histogram merging m bins in one in the histogram self. This method cannot be used for histograms of non-uniform bin size. If m is not an exact divider of the number of bins of self, the range of the rebinned histogram is extended not to lose the entries in the last m-1 (at most) bins.

  Example: a histogram h of size 5 with the range [0, 5), binned as

      GSL::Histogram::Range:
      [ 0.000e+00 1.000e+00 2.000e+00 3.000e+00 4.000e+00 5.000e+00 ]
      GSL::Histogram::Bin:
      [ 0.000e+00 3.000e+00 1.000e+00 1.000e+00 3.000e+00 ]

  When a new histogram is created merging two bins into one as `h2 = h.rebin`, then `h2` looks like

      GSL::Histogram::Range:
      [ 0.000e+00 2.000e+00 4.000e+00 6.000e+00 ]
      GSL::Histogram::Bin:
      [ 3.000e+00 2.000e+00 3.000e+00 ]

`GSL::Histogram#integrate`
  This method calculates cumulative counts of the histogram bins through the range, and returns a `GSL::Histogram::Integral` object. The i-th bin value of a `GSL::Histogram::Integral` object hi created from a `GSL::Histogram` h is given by `hi[i] = hi[i-1] + h[i]`.
`GSL::Histogram::Integral#differentiate`
`GSL::Histogram::Integral#diff`

### 2.1 Histogram Operations

`GSL::Histogram#add(h2)`
`GSL::Histogram#sub(h2)`
`GSL::Histogram#mul(h2)`
`GSL::Histogram#div(h2)`
`GSL::Histogram#scale(val)`
`GSL::Histogram#shift(val)`

### 2.2 Reading and writing histograms

`GSL::Histogram#fwrite(io)`
`GSL::Histogram#fwrite(filename)`
`GSL::Histogram#fread(io)`
`GSL::Histogram#fread(filename)`
`GSL::Histogram#fprintf(io, range_format = "%e", bin_format = "%e")`
`GSL::Histogram#fprintf(filename, range_format = "%e", bin_format = "%e")`
`GSL::Histogram#fscanf(io)`
`GSL::Histogram#fscanf(filename)`

### 2.3 Graphics

`GSL::Histogram#graph(options)`
  This method uses the GNU plotutils `graph` to draw the histogram self. The options as "-T X -C -l x" etc are given by a String.

### 2.4 Fitting

`GSL::Histogram#fit_exponential(binstart = 0, binend = n-1)`
  This method fits the histogram self to an exponential model `h[n] = a exp(b x[n])` using the bins of indices from binstart to binend. The result is returned as an Array of 6 elements, [a, b, erra, errb, sumsq, dof], where
  - a: scale factor
  - b: exponent
  - erra, errb: fitting errors
  - sumsq: fitting chi-squared (not reduced chi-squared)
  - dof: degree-of-freedom, the number of bins used minus the number of parameters (2)
`GSL::Histogram#fit_power(binstart = 0, binend = n-1)`
  This method fits the histogram self to a power-law model `h[n] = a x[n]^b` using the bins of indices from binstart to binend. The result is returned as an Array of 6 elements, [a, b, erra, errb, sumsq, dof].
`GSL::Histogram#fit_gaussian(binstart = 0, binend = n-1)`

  This method fits the histogram self to Gaussian distribution using the bins of indices from binstart to binend, and returns an Array of 8 elements, [sigma, mean, height, errsig, errmean, errhei, sumsq, dof].

  Example:

      #!/usr/bin/env ruby
      require("gsl")

      N = 10000
      MAX = 8
      rng1 = Rng.alloc
      hg = Histogram.alloc(100, [-MAX, MAX])

      for i in 0...N do
        r1 = rng1.gaussian(1.5) + 2
        hg.increment(r1)
      end

      result = hg.fit_gaussian
      sigma = result[0]
      mean = result[1]
      height = result[2]
      x = Vector.linspace(-MAX, MAX, 100)
      y = height*Ran::gaussian_pdf(x-mean, sigma)
      GSL::graph(hg, [x, y], "-T X -C -g 3")

## 3 The histogram probability distribution

The probability distribution function for a histogram consists of a set of bins which measure the probability of an event falling into a given range of a continuous variable x. A probability distribution function is defined by the following class, which actually stores the cumulative probability distribution function. This is the natural quantity for generating samples via the inverse transform method, because there is a one-to-one mapping between the cumulative probability distribution and the range [0,1]. It can be shown that by taking a uniform random number in this range and finding its corresponding coordinate in the cumulative probability distribution we obtain samples with the desired probability distribution.

### 3.1 GSL::Histogram::Pdf class

`GSL::Histogram::Pdf.new(n)`
`GSL::Histogram::Pdf.alloc(n)`
  Constructors
`GSL::Histogram::Pdf#init(h)`
  This initializes the probability distribution self with the contents of the histogram h.
`GSL::Histogram::Pdf#sample(r)`

  This method uses r, a uniform random number between zero and one, to compute a single random sample from the probability distribution self. The algorithm used to compute the sample s is given by the following formula,

      s = range[i] + delta * (range[i+1] - range[i])

  where i is the index which satisfies `sum[i] <= r < sum[i+1]` and `delta` is `(r - sum[i])/(sum[i+1] - sum[i])`.

`GSL::Histogram::Pdf#n`
  This returns the number of bins of the probability distribution function.
`GSL::Histogram:Pdf#range`
  This returns a `Vector::View` object as a reference to the pointer `double *range` in the `gsl_histogram_pdf` struct.
`GSL::Histogram:Pdf#sum`
  This returns a `Vector::View` object as a reference to the pointer `double *sum` in the `gsl_histogram_pdf` struct.

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