histc

Input arguments

Data

vector, matrix or hypermatrix of encoded integers, decimal numbers, complex numbers, polynomials, or texts. Sparse-encoded matrices are accepted.

• Data must have at least 2 components. histc([],..) returns [] for every output argument.
• Numerical Data may include Infinite or NaN values. However, NaN values are never binned in the histogram; Infinite values can be binned only in categorial histograms.
• Textual Data may include empty texts "" or extended-ascii or UTF-8 characters.

Binning:

histc allows to define the set of histogram bins in several ways depending on the Data type and on the need. Two major binning types / histogram modes can be used:

• continuous contiguous ranging bins :this is meaningful whether Data values are sortable. This is the case for encoded integers, decimal numbers, and texts.

  histc() continuously bins complex numbers considering only their real parts. Any number with either a real or imaginary part set to %nan, -%inf, or to +%inf is excluded from bins and from the histogram. For sparse-encoded Data, the zero value is not taken into account to define the whole binning range.

  In this case, bins are defined by their edges. For a given bin, any data value being between the bin's edges belongs to it.

• discrete / categorial binning mode :this can be used for any Data type. It is the only binning mode available for polynomial data.

  A categorial bin -- aka category -- is defined by its value: any data belongs to the bin if its value is equal to the bin's value.

  Any Data or bin's value being NaN is canceled before computing the categorial histogram.

(default)

When no binning specification is provided,

• For integers, decimal, or complex numbers, the "sqrt" binning algorithm is used See here-below for more informations.
• For texts and polynomials: the histogram is computed in "discrete" mode, with as many bins as there are distinct data entries.

nbins

single positive integer: required number of contiguous bins of equal widths covering the whole range of non-infinite Data values.

This binning specification can't be used for texts Data

binsWidth

Single decimal number > 0 specifying the bins width for all bins. Its opposite -binsWidth < 0 must be provided in input (to not get confused with nbins that is already a single positive number).

binsAlgo

Single text word among the ones described here-below. These automatic binning modes can be used for encoded integers, decimal, or complex numbers. None of them can be used for texts or polynomial data.

For these 3 modes, the whole range of data values is shared into nB bins of equal widths. nB is set according to the chosen algorithm as follows.

"sqrt":	nB is set to the square-root of the number Nvalid of valid data in Data, in such a way that there are as many bins as the average number of counts in bins. The vertical average relative resolution 1 count / nB counts = 1/nB of the histogram is then similar to the horizontal one binWidth/range = (range/nB)/range = 1/nB However, for encoded integers data, if the data range dR=max(Data)-min(Data)+1 is narrower than nB, nB is then set to dR, so setting the bins width to 1. Bins are then automatically centered on integer values in the range.
"freediac":	Freedmann - Diaconis binning criterion: nB = round(strange(Data)/binWidth) with binsWidth = 2*iqr(Data)* Nvalid^(-1/3).
"sturges":	Sturges binning criterion: nB = ceil(1 + log2(Nvalid))

binsEdges

Vector of values sorted in strict increasing order (without duplicates). N bins edges define N-1 bins. For encoded integers Data, binsEdges can be decimal numbers. For complex numbers Data, decimal numbers are expected in binsEdges: only the distribution of real parts is considered.

• First bin: Any non-infinite Data component belonging to the closed interval [binsEdges(1), binsEdges(2)] belongs to the first bin and is accounted in the Heights(1) count.
• Next bins # i>1: Any non-infinite Data component belonging to the semi-open interval ]binsEdges(i), binsEdges(i+1)] belongs to the bin #i and is accounted in the Heights(i) count.

Marginal bins:

For numerical and text Data, the first or/and the last binsEdges components may be set to collect and count in marginal bins all non-infinite Data components remaining in the left and right wings of the complete histogram:

• Left wing: set
  • binsEdges(1) = -%inf , or
  • binsEdges(1) = ""
  Then,
  • Data entries such that Data < binsEdges(2) are counted in Heights(1).
  • The actual bins(1) edge is set to min(Data).

• Right wing: set
  • binsEdges($) = %inf , or
  • binsEdges($) = "~~" (for texts in standard ascii, ascii(126)=="~" is the last printable character)
  Then,
  • Data entries such that Data > binsEdges($-1) are counted in Heights($).
  • The actual bins($) edge is set to max(Data).

binsValues

For polynomial Data or when the "discrete" option is used, binsValues provides values whose occurrences in Data must be counted.

• Duplicates and %nan values are priorly removed from binsValues.
• binsValues may include some %inf values. However, for encoded integers Data, any %inf value is removed before processing.
• Components of binsValues may be unsorted: the order of binsValues components is kept as is in the Heights output vector.

Options

Options is either a vector of textual flags, or equivalently a single word of comma-separated concatenated flags, or both. All flags are case-insensitive and can be specified in any order.

Examples: The following options specifications are equivalent: ["discrete" "countsNorm" "normWith: Out Inf"], or ["countsNORM" "NORMwith: inf out" "Discrete" ], or ["normWith: INF OUT", "discrete, countsNorm" ], or simply "discrete,countsNorm,normWith: inf out".

"discrete"

This flag must be used when a discrete / categorial histogram is required. Then, the vector provided in argument #2 with at least 2 components sets bins values instead of bins edges (by default).

Presently, polynomial Data are always processed in a categorial way. The "discrete" flag looks then useless. However, in a future release, polynomials could become sortable. Using the "discrete" flag does not hurt and would avoid future back-compatibility issues.

Histogram scale:

"counts"	This mode is the default one: Whatever is each bin's width, the height of the bin is equal to the number of Data components falling in it.
"countsNorm"	Whatever is each bin's width and position, the height of the bin is equal to the relative number of Data components falling in it, over all counted components. Then, unless the "normWith:.." option is used, the cumulated bins heights is equal to 1: sum(Heights)==1.
"density"	The area of each bin is equal to the number of Data components falling in it. This scaling mode is meaningless and ignored in case of categorial histogram.
"densityNorm"	The area of each bin is equal to the relative number of Data components falling in it. Then, unless the "normWith:.." option is used, the whole area of the histogram is equal to 1: This scaling mode is meaningless and ignored in case of categorial histogram.

"normWith:.."

When the "countsNorm" or "densityNorm" option is used, it is possible to provide additional informations about which components of Data out of bins should be considered for the total number N of counts over which the normalization is computed.

After the "normWith:" option's header, a space-separated list of case-insensitive flags can be provided in any order. If several concurrent flags are provided, only the last specified one is taken into account. Unrelevant flags for the given Data type are ignored. Available flags and their relative priorities are described here-below. Examples: "normWith: all", "normWith: out inf", "normWith: Nan inf", "normWith: rightout inf", etc.

"all"	All components of Data are considered: N = size(Data,"*"). If "all" is used, all other "normWith:.." options are ignored.
"out"	All Data out of bins that are not Nan or Inf or "" are accounted. If Data is sparse-encoded, zeros remain excluded unless the option "normWith: zeros" is used. If "out" is used, "leftout" and "rightout" options are ignored.
"leftout"	As with "out", but only for Data < binsEdges(1). This flag is ignored in discrete/categorial mode.
"rightout"	As with "out", but only for Data > binsEdges($). This flag is ignored in discrete/categorial mode.
"NaN"	NaN data are accounted, in addition to other ones.
"Inf"	Inf data are accounted, in addition to other ones. In discrete/categorial mode, Inf values are not specific and are processed as other ones. This flag is then ignored.
"zeros"	If Data is sparse-encoded, by default only non-zero elements are considered (otherwise, zeros are not specific and are processed as other values). Nevertheless, it's possible to take them into account in the normalization by using this "normWith: zeros" flag. Using this flag does not credit the Heights of the bin covering the zero value (if any).
"empty"	"" empty texts in Data are accounted, in addition to other ones.

Results

Heights

vector of decimal numbers whose values depend on the histogram scaling mode set with each dedicated option. See the description of the Histogram scales options here-above. In brief:

• "counts" mode: Heights(i) is the number of Data components equal to the bins(i) value (categorial), or belonging to the ]bins(i), bins(i+1)] interval (continuous histogram).
• "countsNorm" mode: Heights(i) is as for "counts", divided by the total number N of considered Data components. N is the sum of counts in all bins, plus possibly the number of counts of some special jokers values (%inf, %nan, 0, ""), according to the normWith: option used.

In continuous mode, statistical densities may be returned in the vector Heights instead of integer numbers of counts: Let's call counts(i) the number of counts in the bin #i defined by its edges. Then

• In "density" mode: Heights(i) is set such that the area of the bin is equal to its population: Heights(i) * (binsEdges(i+1) - binsEdges(i)) == counts(i).
• In "densityNorm" mode: the "density" results are divided by the total number N of considered counts (see "countsNorm").

jokers

Row vector of 1 to 5 decimal numbers indicating the frequency of special values in Data. Let's define the following numbers:

• Nnan: number of NaN objects in Data.
• Ninf: number of Inf objects in Data.
• Nzeros: number of null values in Data.
• Nempty: number of empty texts "" in Data.
• Nleftout: number of Data components not equal to -%inf nor to "", such that Data < binsEdges(1).
• Nrightout: number of Data components not equal to %inf such that Data > binsEdges($).
• Nout: number of Data components out of bins, non-infinite, not being Nan, not being empty text "", and for sparse Data: not equal to zero.

In unnormalized "counts" and "density" histogram scales, jokers returns the integer counts numbers of special values.

In normalized "countsNorm" and "densityNorm" histogram scales, jokers returns countsNorm frequencies of special values.

Then, according to the Data type and the continuous or categorial histogram mode, jokers is made of the following:

1. Encoded integers:
   • continuous: [Nleftout, Nrightout]
   • categorial: [Nout]
2. Decimal or complex numbers, full or sparse:
   • continuous: [Nleftout, Nrightout, Nzeros, Nnan, Ninf]
   • categorial: [Nout, 0, Nzeros, Nnan, Ninf]
3. Polynomials: [Nout, 0, 0, Nnan, Ninf]
4. Texts:
   • continuous: [Nleftout, Nrightout, Nempty]
   • categorial: [Nout, 0, Nempty]

bins

Row vector of bins edges or of bins values actually used to build the histogram. histc() allows using many semi-automatic or automatic binning modes for which no explicit or incomplete binsEdges or binsValues vector is provided as input.

• Continuous binning mode:
  • The actual binsEdges is returned in bins. It has the Heights number of components, + 1 (position of the closing edge).

  • For encoded integers, decimal numbers, and complex numbers Data, bins is of decimal type. For text Data, bins is of type text as well.

  • When marginal bins are required (see the binsEdges description) bins(1) and bins($) return the actual boundaries of the whole binning range used.

• Discrete categorial mode:

  For polynomial Data, or for other Data types used with the "discrete" option: if no explicit binsValues vector is provided, histc() sets it to unique(Data)(:)' and returns it as bins.

inBin

Array of decimal integers having the sizes of Data. If Data is sparse-encoded, inBin is so as well.

inBin(i,j) returns the index of the bins which Data(i,j) belongs to. If the value of Data(i,j) is out of bins, inBin(i,j)=0. Otherwise, Data(i,j) increments the Heights(inBin(i,j)) counts by one unit.

with decimal numbers:

data = [1 1 1 2 2 3 4 4 5 5 5 6 6 7 8 8 9 9 9];
N = size(data,"*")   // ==19

// Default binning; "sqrt": sqrt(19) => 4. .. => 4 bins
[h, j, b, i] = histc(data)
// expected: h = [6 5 3 5] = href
// expected: b = [1 3 5 7 9] bins edges
// expected: i = [1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 4 4 4]  d memberships to bins
histc(data, , "countsNorm")   // Expected: href/N
histc(data, , "density")      // Expected: href/2, 2 being the bins width
histc(data, , "densityNorm")  // Expected: href/N/2

// Automatic Sturges binning
[h, j, b, i] = histc(data,"sturges")    // h = [5 1 5 2 1 5]
                                        // b = [3 7 11 15 19 23 27] / 3
                                        // i = [1 1 1 1 1 2 3 3 3 3 3 4 4 5 6 6 6 6 6]

// Explicit bins edges, with marginal bins
// ---------------------------------------
data = [1 1 1 2 2 3 4 4 5 5 5 6 6 7 8 8 9 9 9];
be = [-%inf 3 5 7 %inf];
[href, j, b, i] = histc(data, be)   // href = [6 5 3 5] => sum N = 19
                                    // b = [1 3 5 7 9]  // bins completed with actual data bounds
                                    // i = [1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 4 4 4]
histc(data, be, "countsNorm")       // href/N
histc(data, be, "density")          // href/2   bins width = 2: see b
histc(data, be, "densityNorm")      // href/N/2

// Explicit bins edges, with outsiders
// -----------------------------------
data = [1 1 1 2 2 3 4 4 5 5 5 6 6 7 8 8 9 9 9]; // still the same
be = [2, 5.5, 7];   // Bins edges (2 bins)
[href, jref, b, i] = histc(d, be)   // href = [8 3]   jref = [3 5 0 0 0] = [leftout, rightout, ..]
                                    // i = [0 0 0 1 1 1 1 1 1 1 1 2 2 2 0 0 0 0 0]
histc(data, be, "countsNorm")                       // href / 11
histc(data, be, "countsNorm, normWith: leftout")    // href / 14
histc(data, be, "countsNorm, normWith: rightout")   // href / 16
histc(data, be, "countsNorm, normWith: out")        // href / 19
histc(data, be, "density")                          // href ./ diff(be)
histc(data, be, "densityNorm")                      // href ./ diff(be) / 11
histc(data, be, "densityNorm, normWith: leftout")   // href ./ diff(be) / 14
histc(data, be, "densityNorm, normWith: rightout")  // href ./ diff(be) / 16
histc(data, be, "densityNorm, normWith: all");      // href ./ diff(be) / 19

// With Nan and Inf values
// -----------------------
data = [1 1 1 2 2 3 4 4 5 5 5 6 6 7 8 8 9 9 9];
data = [%nan %inf, data, %nan %nan -%inf];
N = size(data,"*");         // 24
be = [2, 4.5, 7];           // Set bins edges (2 bins)
[href, jref, b, iref] = histc(data, be) //  href = [5 6]  jref = [3 5 0 3 2];
                            // continuous mode: jokers = [leftout, rightout, zeros, nan, inf]
                            // iref = [0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0]  memberships
[h, j] = histc(data, be, "countsNorm")                          // Expected: href/11, jref/11
[h, j] = histc(data, be, "countsNorm, normWith: nan")           // Expected: href/14, jref/14
[h, j] = histc(data, be, "countsNorm, normWith: inf")           // Expected: href/13, jref/13
[h, j] = histc(data, be, "countsNorm, normWith: inf nan")       // Expected: href/16, jref/16
[h, j] = histc(data, be, "countsNorm, normWith: leftout nan")   // Expected: href/17, jref/17
[h, j] = histc(data, be, "countsNorm, normWith: rightout inf")  // Expected: href/18, jref/18
[h, j] = histc(data, be, "countsNorm, normWith: out inf")       // Expected: href/21, jref/21
[h, j] = histc(data, be, "countsNorm, normWith: all")           // Expected: href/24, jref/24

// Normalized densities over a Bins width = 2.5 (see be)
[h, j] = histc(data, be, "densityNorm")                         // Expected: href/11/2.5, jref/11
[h, j] = histc(data, be, "densityNorm, normWith: nan")          // Expected: href/14/2.5, jref/14
[h, j] = histc(data, be, "densityNorm, normWith: inf")          // Expected: href/13/2.5, jref/13
[h, j] = histc(data, be, "densityNorm, normWith: inf nan")      // Expected: href/16/2.5, jref/16
[h, j] = histc(data, be, "densityNorm, normWith: leftout nan")  // Expected: href/17/2.5, jref/17
[h, j] = histc(data, be, "densityNorm, normWith: rightout inf") // Expected: href/18/2.5, jref/18
[h, j] = histc(data, be, "densityNorm, normWith: all")          // Expected: href/24/2.5, jref/24

with texts:

histc(["a" "c" "a" "a" "b" "c"])    //  [3 1 2]

t = [
"c" "n" "h" "i" "b" "i" "f" "i" "p" "l" "p" "d" "f" "i" "l"
"b" "m" "e" "o" "o" "f" "p" "o" "h" "f" "h" "h" "c" "k" "o"
"p" "f" "k" "a" "j" "o" "j" "d" "h" "h" "n" "m" "o" "l" "n"
"h" "b" "o" "l" "j" "n" "o" "i" "g" "i" "a" "a" "j" "d" "p"
 ];
// With default discrete bins
// --------------------------
[h,j,b,i] = histc(t)  // h = [3  3 2 3 1 5 1 7 6 4 2 4 2 4 8 5]
                      // b = "a" b c d e f g h i j k l m n o p
iref = [
3   14  8   9   2   9   6   9   16  12  16  4   6   9   12
2   13  5   15  15  6   16  15  8   6   8   8   3   11  15
16  6   11  1   10  15  10  4   8   8   14  13  15  12  14
8   2   15  12  10  14  15  9   7   9   1   1   10  4   16
];

// With given discrete bins WITHOUT "" bins
// ----------------------------------------
t2 = t;
t2([7 13 19 26 32 39 43]) = "";
// --> t2  =
// c  n  h     b  i  f  i  p  l  p  d  f  i  l
// b  m  e  o  o  f     o  h  f  h  h  c  k  o
// p     k  a     o  j  d  h        m  o  l  n
// h  b  o  l  j  n  o     g  i  a  a  j  d  p
//
// b =  '' a  b  c  d  e  f  g  h  i  j  k  l  m  n  o  p
// h =  7  3  3  2  3  1  4  1  6  4  3  2  4  2  3  8  4

[h, j, b, i] = histc(t2, ["a" "e" "i" "o"], "discrete")
                                            // h = [3 1 4 8];  N = 16
                                            // j = [37 0 7] = [out, 0, #""]
// i = [    // memberships
// 0  0  0  0  0  3  0  3  0  0  0  0  0  3  0
// 0  0  2  4  4  0  0  4  0  0  0  0  0  0  4
// 0  0  0  1  0  4  0  0  0  0  0  0  4  0  0
// 0  0  4  0  0  0  4  0  0  3  1  1  0  0  0
// ];

// With continuous and marginal bins: "" <=> -inf , "~~" <=> Inf (regular ascii)
// -----------------------------------------------------------------------------
[h,j,b,i] = histc(t, ["" "c" "e" "g" "i" "k" "m" "~~"])
                    // h = [8 4 6 13 6 6 17]    j = [0 0 0]
// i = [    // memberships
// 1  7  4  4  1  4  3  4  7  6  7  2  3  4  6
// 1  6  2  7  7  3  7  7  4  3  4  4  1  5  7
// 7  3  5  1  5  7  5  2  4  4  7  6  7  6  7
// 4  1  7  6  5  7  7  4  3  4  1  1  5  2  7
// ];

// Continuous bins. Data WITH ""
// -----------------------------
// t2  =
// c  n  h     b  i  f  i  p  l  p  d  f  i  l
// b  m  e  o  o  f     o  h  f  h  h  c  k  o
// p     k  a     o  j  d  h        m  o  l  n
// h  b  o  l  j  n  o     g  i  a  a  j  d  p
//
// b =  '' a  b  c  d  e  f  g  h  i  j  k  l  m  n  o  p
// h =  7  3  3  2  3  1  4  1  6  4  3  2  4  2  3  8  4
binsEdges = ["e" "f" "g" "h" "i" "j"];
[href, jref, b, i] = histc(t2, binsEdges)   // href=[5 1 6 4 3]; N = sum(href) = 19
                                            // jref=[11  23  7]; [leftout rightout ""]

[h,j,b,i] = histc(t2, binsEdges, "countsNorm,normWith: leftout")
                                            // h = href / (N+jref(1)),   j = jref / (N+jref(1))
[h,j,b,i] = histc(t2, binsEdges, "countsNorm,normWith: rightout")
                                            // h = href / (N+jref(2)),  j = jref / (N+jref(2))
[h,j,b,i] = histc(t2, binsEdges, "countsNorm,normWith: out");
                                            // h = href / sum([N jref(1:2)]), j = jref / sum([N jref(1:2)])
[h,j,b,i] = histc(t2, binsEdges, "countsNorm,normWith: empty")
                                            // h = href / (N+jref(3)), j = jref/(N+jref(3))
[h,j,b,i] = histc(t2, binsEdges,"countsNorm,normWith: out empty")
                                            // h = href / sum([N jref]), j = jref / sum([N jref])
[h,j,b,i] = histc(t2, binsEdges, "countsNorm,normWith: all")
                                            // h = href / sum([N jref]), j = jref/sum([N jref])

with polynomials:

histc([%z 2+%z %z])                                     // [2 1]
histc([%z 2+%z %z],, "countsnorm")                      // [2 1] / 3
histc([%z 2+%z %z %nan],, "countsnorm")                 // [2 1] / 3
histc([%z 2+%z %z %nan],, "countsnorm, normWith: Nan")  // [2 1] / 4
// Data order is kept:
histc([2+%z %z %z ]) == [1 2]