histcounts

Histogram bin counts

Description

example

[N,edges] = histcounts(X) partitions the X values into bins, and returns the count in each bin, as well as the bin edges. The histcounts function uses an automatic binning algorithm that returns bins with a uniform width, chosen to cover the range of elements in X and reveal the underlying shape of the distribution.

example

[N,edges] = histcounts(X,nbins) uses a number of bins specified by the scalar, nbins.

example

[N,edges] = histcounts(X,edges) sorts X into bins with the bin edges specified by the vector, edges. The value X(i) is in the kth bin if edges(k)X(i) < edges(k+1). The last bin also includes the right bin edge, so that it contains X(i) if edges(end-1)X(i)edges(end).

example

[N,edges,bin] = histcounts(___) also returns an index array, bin, using any of the previous syntaxes. bin is an array of the same size as X whose elements are the bin indices for the corresponding elements in X. The number of elements in the kth bin is nnz(bin==k), which is the same as N(k).

example

N = histcounts(C), where C is a categorical array, returns a vector, N, that indicates the number of elements in C whose value is equal to each of C’s categories. N has one element for each category in C.

N = histcounts(C,Categories) counts only the elements in C whose value is equal to the subset of categories specified by Categories.

example

[N,Categories] = histcounts(___) also returns the categories that correspond to each count in N using either of the previous syntaxes for categorical arrays.

example

[___] = histcounts(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments using any of the input or output argument combinations in previous syntaxes. For example, you can specify 'BinWidth' and a scalar to adjust the width of the bins for numeric data. For categorical data, you can specify 'Normalization' and either 'count', 'countdensity', 'probability', 'pdf', 'cumcount', or 'cdf'.

Examples

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Distribute 100 random values into bins. histcounts automatically chooses an appropriate bin width to reveal the underlying distribution of the data.

X = randn(100,1);
[N,edges] = histcounts(X)
N = 1×7

2    17    28    32    16     3     2

edges = 1×8

-3    -2    -1     0     1     2     3     4

Distribute 10 numbers into 6 equally spaced bins.

X = [2 3 5 7 11 13 17 19 23 29];
[N,edges] = histcounts(X,6)
N = 1×6

2     2     2     2     1     1

edges = 1×7

0    4.9000    9.8000   14.7000   19.6000   24.5000   29.4000

Distribute 1,000 random numbers into bins. Define the bin edges with a vector, where the first element is the left edge of the first bin, and the last element is the right edge of the last bin.

X = randn(1000,1);
edges = [-5 -4 -2 -1 -0.5 0 0.5 1 2 4 5];
N = histcounts(X,edges)
N = 1×10

0    24   149   142   195   200   154   111    25     0

Distribute all of the prime numbers less than 100 into bins. Specify 'Normalization' as 'probability' to normalize the bin counts so that sum(N) is 1. That is, each bin count represents the probability that an observation falls within that bin.

X = primes(100);
[N,edges] = histcounts(X, 'Normalization', 'probability')
N = 1×4

0.4000    0.2800    0.2800    0.0400

edges = 1×5

0    30    60    90   120

Distribute 100 random integers between -5 and 5 into bins, and specify 'BinMethod' as 'integers' to use unit-width bins centered on integers. Specify a third output for histcounts to return a vector representing the bin indices of the data.

X = randi([-5,5],100,1);
[N,edges,bin] = histcounts(X,'BinMethod','integers');

Find the bin count for the third bin by counting the occurrences of the number 3 in the bin index vector, bin. The result is the same as N(3).

count = nnz(bin==3)
count = 8

Create a categorical vector that represents votes. The categories in the vector are 'yes', 'no', or 'undecided'.

A = [0 0 1 1 1 0 0 0 0 NaN NaN 1 0 0 0 1 0 1 0 1 0 0 0 1 1 1 1];
C = categorical(A,[1 0 NaN],{'yes','no','undecided'})
C = 1x27 categorical
Columns 1 through 9

no      no      yes      yes      yes      no      no      no      no

Columns 10 through 16

undecided      undecided      yes      no      no      no      yes

Columns 17 through 25

no      yes      no      yes      no      no      no      yes      yes

Columns 26 through 27

yes      yes

Determine the number of elements that fall into each category.

[N,Categories] = histcounts(C)
N = 1×3

11    14     2

Categories = 1x3 cell
{'yes'}    {'no'}    {'undecided'}

Input Arguments

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Data to distribute among bins, specified as a vector, matrix, or multidimensional array. If X is not a vector, then histcounts treats it as a single column vector, X(:).

histcounts ignores all NaN values. Similarly, histcounts ignores Inf and -Inf values unless the bin edges explicitly specify Inf or -Inf as a bin edge.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | datetime | duration

Categorical data, specified as a categorical array. histcounts ignores undefined categorical values.

Data Types: categorical

Number of bins, specified as a positive integer. If you do not specify nbins, then histcounts automatically calculates how many bins to use based on the values in X.

Example: [N,edges] = histcounts(X,15) uses 15 bins.

Bin edges, specified as a vector. edges(1) is the left edge of the first bin, and edges(end) is the right edge of the last bin.

For datetime and duration data, edges must be a datetime or duration vector in monotonically increasing order.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | datetime | duration

Categories included in count, specified as a cell vector of character vectors or a categorical vector. By default, histcounts uses a bin for each category in categorical array C. Use Categories to specify a unique subset of the categories instead.

Example: h = histcounts(C,{'Large','Small'}) counts only the categorical data in the categories 'Large' and 'Small'.

Data Types: cell | categorical

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [N,edges] = histcounts(X,'Normalization','probability') normalizes the bin counts in N, such that sum(N) is 1.

Bin limits, specified as a two-element vector, [bmin,bmax]. This option bins only the values in X that fall between bmin and bmax inclusive; that is, X(X>=bmin & X<=bmax).

This option does not apply to categorical data.

Example: [N,edges] = histcounts(X,'BinLimits',[1,10]) bins only the values in X that are between 1 and 10 inclusive.

Binning algorithm, specified as one of the values in this table.

Value

Description

'auto'

The default 'auto' algorithm chooses a bin width to cover the data range and reveal the shape of the underlying distribution.

'scott'

Scott’s rule is optimal if the data is close to being normally distributed, but is also appropriate for most other distributions. It uses a bin width of 3.5*std(X(:))*numel(X)^(-1/3).

'fd'

The Freedman-Diaconis rule is less sensitive to outliers in the data, and may be more suitable for data with heavy-tailed distributions. It uses a bin width of 2*IQR(X(:))*numel(X)^(-1/3), where IQR is the interquartile range of X.

'integers'

The integer rule is useful with integer data, as it creates a bin for each integer. It uses a bin width of 1 and places bin edges halfway between integers. To prevent from accidentally creating too many bins, a limit of 65536 bins (216) can be created with this rule. If the data range is greater than 65536, then wider bins are used instead.

Note

'integers' does not support datetime or duration data.

'sturges'

Sturges’ rule is a simple rule that is popular due to its simplicity. It chooses the number of bins to be ceil(1 + log2(numel(X))).

'sqrt'

The Square Root rule is another simple rule widely used in other software packages. It chooses the number of bins to be ceil(sqrt(numel(X))).

histcounts does not always choose the number of bins using these exact formulas. Sometimes the number of bins is adjusted slightly so that the bin edges fall on "nice" numbers.

For datetime data, the bin method can be one of these units of time:

 'second' 'month' 'minute' 'quarter' 'hour' 'year' 'day' 'decade' 'week' 'century'

For duration data, the bin method can be one of these units of time:

 'second' 'day' 'minute' 'year' 'hour'

If you specify BinMethod with datetime or duration data, then histcounts can use a maximum of 65,536 bins (or 216). If the specified bin duration requires more bins, then histcounts uses a larger bin width corresponding to the maximum number of bins.

This option does not apply to categorical data.

Example: [N,edges] = histcounts(X,'BinMethod','integers') uses bins centered on integers.

Width of bins, specified as a scalar. If you specify BinWidth, then histcounts can use a maximum of 65,536 bins (or 216). If the specified bin width requires more bins, then histcounts uses a larger bin width corresponding to the maximum number of bins.

For datetime and duration data, the value of 'BinWidth' can be a scalar duration or calendar duration.

This option does not apply to categorical data.

Example: [N,edges] = histcounts(X,'BinWidth',5) uses bins with a width of 5.

Type of normalization, specified as one of the values in this table. For each bin i:

• ${v}_{i}$ is the bin value.

• ${c}_{i}$ is the number of elements in the bin.

• ${w}_{i}$ is the width of the bin.

• $N$ is the number of elements in the input data. This value can be greater than the binned data if the data contains NaN, NaT, or <undefined> values, or if some of the data lies outside the bin limits.

ValueBin ValuesNotes
'count' (default)

${v}_{i}={c}_{i}$

• Count or frequency of observations.

• Sum of bin values is less than or equal to numel(X). The sum is less than numel(X) only when some of the input data is not included in the bins.

• For categorical data, sum of bin values is either numel(X) or sum(ismember(X(:),Categories)).

'countdensity'

${v}_{i}=\frac{{c}_{i}}{{w}_{i}}$

• Count or frequency scaled by width of bin.

• For categorical data, this the same as 'count'.

Note

'countdensity' does not support datetime or duration data.

'cumcount'

${v}_{i}=\sum _{j=1}^{i}{c}_{j}$

• Cumulative count. Each bin value is the cumulative number of observations in that bin and all previous bins.

• The value of the last bin is less than or equal to numel(X).

• For categorical data, the value of the last bin is less than or equal to numel(X) or sum(ismember(X(:),Categories)).

'probability'

${v}_{i}=\frac{{c}_{i}}{N}$

• Relative probability.

• The sum of the bin values is less than or equal to 1.

'pdf'

${v}_{i}=\frac{{c}_{i}}{N\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{w}_{i}}$

• Probability density function estimate.

• For categorical data, this is the same as 'probability'.

Note

'pdf' does not support datetime or duration data.

'cdf'

${v}_{i}=\sum _{j=1}^{i}\text{\hspace{0.17em}}\frac{{c}_{j}}{N}$

• Cumulative density function estimate.

• N(end) is less than or equal to 1.

Example: [N,edges] = histcounts(X,'Normalization','pdf') bins the data using the probability density function estimate.

Output Arguments

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Bin counts, returned as a row vector.

Bin edges, returned as a vector. edges(1) is the left edge of the first bin, and edges(end) is the right edge of the last bin.

Bin indices, returned as an array of the same size as X. Each element in bin describes which numbered bin contains the corresponding element in X.

A value of 0 in bin indicates an element which does not belong to any of the bins (for example, a NaN value).

Categories included in count, returned as a cell vector of character vectors. Categories contains the categories in C that correspond to each count in N.

Tips

• The behavior of histcounts is similar to that of the discretize function. Use histcounts to find the number of elements in each bin. On the other hand, use discretize to find which bin each element belongs to (without counting).