histcounts2
Bivariate histogram bin counts
Syntax
Description
[
partitions the
values in N
,Xedges
,Yedges
]
= histcounts2(X,Y
)X
and Y
into 2D bins,
and returns the bin counts, as well as the bin edges in each dimension.
The histcounts2
function uses an automatic binning
algorithm that returns uniform bins chosen to cover the range of values
in X
and Y
and reveal the underlying
shape of the distribution.
[
partitions N
,Xedges
,Yedges
]
= histcounts2(X,Y
,Xedges
,Yedges
)X
and Y
into
bins with the bin edges specified by Xedges
and Yedges
.
N(i,j)
counts the value [X(k),Y(k)]
if Xedges(i)
≤ X(k)
< Xedges(i+1)
and Yedges(j)
≤ Y(k)
< Yedges(j+1)
.
The last bins in each dimension also include the last (outer) edge.
For example, [X(k),Y(k)]
falls into the i
th
bin in the last row if Xedges(end1)
≤ X(k)
≤ Xedges(end)
and Yedges(i)
≤ Y(k)
< Yedges(i+1)
.
[
uses
additional options specified by one or more N
,Xedges
,Yedges
]
= histcounts2(___,Name,Value
)Name,Value
pair
arguments using any of the input arguments in previous syntaxes. For
example, you can specify 'BinWidth'
and a twoelement
vector to adjust the width of the bins in each dimension.
[
also returns index arrays N
,Xedges
,Yedges
,binX
,binY
]
= histcounts2(___)binX
and binY
,
using any of the previous syntaxes. binX
and binY
are
arrays of the same size as X
and Y
whose
elements are the bin indices for the corresponding elements in X
and Y
.
The number of elements in the (i,j)
th bin is equal
to nnz(binX==i & binY==j)
, which is the same
as N(i,j)
if Normalization
is 'count'
.
Examples
Bin Counts and Bin Edges
Distribute 100 pairs of random numbers into bins. histcounts2
automatically chooses an appropriate bin width to reveal the underlying distribution of the data.
x = randn(100,1); y = randn(100,1); [N,Xedges,Yedges] = histcounts2(x,y)
N = 7×6
0 0 0 2 0 0
1 2 10 4 0 0
1 4 9 9 5 0
1 4 10 11 5 1
1 4 6 3 1 1
0 0 1 2 0 0
0 0 1 0 1 0
Xedges = 1×8
3 2 1 0 1 2 3 4
Yedges = 1×7
3 2 1 0 1 2 3
Specify Number of Bins in Each Dimension
Distribute 10 pairs of numbers into 12 bins. Specify 3 bins in the xdimension, and 4 bins in the ydimension.
x = [1 1 2 3 2 2 1 1 2 3]; y = [5 6 3 8 9 1 2 7 5 1]; nbins = [3 4]; [N,Xedges,Yedges] = histcounts2(x,y,nbins)
N = 3×4
1 0 2 1
1 1 1 1
1 0 0 1
Xedges = 1×4
0.6000 1.4000 2.2000 3.0000
Yedges = 1×5
0 2.3000 4.6000 6.9000 9.2000
Specify Bin Edges
Distribute 1,000 pairs of random numbers into bins. Define the bin edges with two vectors: one each for the x and y dimensions. The first element in each vector specifies the first edge of the first bin, and the last element is the last edge of the last bin.
x = randn(1000,1); y = randn(1000,1); Xedges = 5:5; Yedges = [5 4 2 1 0.5 0 0.5 1 2 4 5]; N = histcounts2(x,y,Xedges,Yedges)
N = 10×10
0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 0 0 0
0 0 5 5 3 5 1 2 0 0
0 2 19 23 29 25 26 20 5 0
0 10 36 51 59 71 54 46 10 0
0 7 43 46 79 64 60 46 9 0
0 3 12 18 21 23 19 9 6 0
0 0 5 3 2 8 2 2 0 0
0 0 0 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0
Normalized Bin Counts
Distribute 1,000 pairs of random numbers into bins. Specify Normalization
as 'probability'
to normalize the bin counts such that sum(N(:))
is 1
. That is, each bin count represents the probability that an observation falls within that bin.
x = randn(1000,1); y = randn(1000,1); [N,Xedges,Yedges] = histcounts2(x,y,6,'Normalization','probability')
N = 6×6
0 0 0.0020 0.0020 0 0
0 0.0110 0.0320 0.0260 0.0070 0.0010
0.0010 0.0260 0.1410 0.1750 0.0430 0.0060
0 0.0360 0.1620 0.1940 0.0370 0.0040
0 0.0040 0.0300 0.0370 0.0100 0.0010
0 0.0030 0.0040 0.0040 0.0010 0
Xedges = 1×7
4.0000 2.7000 1.4000 0.1000 1.2000 2.5000 3.8000
Yedges = 1×7
4.0000 2.7000 1.4000 0.1000 1.2000 2.5000 3.8000
Determine Bin Placement
Distribute 1,000 random integer pairs between 10 and 10 into bins, and specify BinMethod
as 'integers'
to use unitwidth bins centered on integers. Specify five outputs for histcounts2
to return vectors representing the bin placement of the data.
x = randi([10,10],1000,1); y = randi([10,10],1000,1); [N,Xedges,Yedges,binX,binY] = histcounts2(x,y,'BinMethod','integers');
Determine which bin the value (x(3),y(3))
falls into.
[x(3),y(3)]
ans = 1×2
8 10
bin = [binX(3) binY(3)]
bin = 1×2
3 21
Input Arguments
X,Y
— Data to distribute among bins (as separate arguments)
vectors  matrices  multidimensional arrays
Data to distribute among bins, specified as separate arguments
of vectors, matrices, or multidimensional arrays. X
and Y
must
have the same size.
Corresponding elements in X
and Y
specify
the x and y coordinates of 2D
data points, [X(k),Y(k)]
. The data types of X
and Y
can
be different.
histcounts2
ignores all NaN
values.
Similarly, histcounts2
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
nbins
— Number of bins in each dimension
scalar  vector
Number of bins in each dimension, specified as a positive scalar
integer or twoelement vector of positive integers. If you do not
specify nbins
, then histcounts2
automatically
calculates how many bins to use based on the values in X
and Y
:
If
nbins
is a scalar, thenhistcounts2
uses that many bins in each dimension.If
nbins
is a vector, thennbins(1)
specifies the number of bins in the xdimension andnbins(2)
specifies the number of bins in the ydimension.
Example: [N,Xedges,Yedges] = histcounts2(X,Y,[15 20])
uses
15 bins in the x
dimension and 20 bins in the y
dimension.
Xedges
— Bin edges in xdimension
vector
Bin edges in xdimension, specified as a
vector. Xedges(1)
is the first edge of the first
bin in the xdimension, and Xedges(end)
is
the outer edge of the last bin.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
 logical
Yedges
— Bin edges in ydimension
vector
Bin edges in ydimension, specified as a
vector. Yedges(1)
is the first edge of the first
bin in the ydimension, and Yedges(end)
is
the outer edge of the last bin.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
 logical
NameValue Arguments
Specify optional
commaseparated 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
.
[N,Xedges,Yedges] = histcounts2(X,Y,'Normalization','probability')
normalizes
the bin counts in N
, such that sum(N)
is
1.BinMethod
— Binning algorithm
'auto'
(default)  'scott'
 'fd'
 'integers'
Binning algorithm, specified as one of the values in this table.
Value  Description 

'auto'  The default 
'scott'  Scott’s rule is optimal if the data is close to
being jointly normally distributed. This rule is appropriate for most
other distributions, as well. It uses a bin size of 
'fd'  The FreedmanDiaconis rule is less sensitive to outliers
in the data, and might be more suitable for data with heavytailed
distributions. It uses a bin size of 
'integers'  The integer rule is useful with integer data, as it creates bins centered on pairs of integers. It uses a bin width of 1 for each dimension and places bin edges halfway between integers. To avoid accidentally creating too many bins, you can use this rule to create a limit of 1024 bins (2^{10}). If the data range for either dimension is greater than 1024, then the integer rule uses wider bins instead. 
histcounts2
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.
Example: [N,Xedges,Yedges] = histcounts2(X,Y,'BinMethod','integers')
uses
2D bins centered on each pair of integers.
BinWidth
— Width of bins in each dimension
vector
Width of bins in each dimension, specified as a twoelement
vector of positive integers, [xWidth yWidth]
.
If you specify BinWidth
, then histcounts2
can
use a maximum of 1024 bins (2^{10})
along each dimension. If the specified bin width requires more bins,
then histcounts2
uses a larger bin width corresponding
to the maximum number of bins.
Example: [N,Xedges,Yedges] = histcounts2(X,Y,'BinWidth',[5
10])
uses bins with size 5
in the x
dimension
and size 10
in the y
dimension.
XBinLimits
— Bin limits in xdimension
twoelement vector
Bin limits in xdimension, specified as a
twoelement vector, [xbmin,xbmax]
. The vector indicates
the first and last bin edges in the xdimension.
This option only bins data that falls within the bin limits
inclusively, X>=xbmin & X<=xbmax
.
YBinLimits
— Bin limits in ydimension
twoelement vector
Bin limits in ydimension, specified as a
twoelement vector, [ybmin,ybmax]
. The vector indicates
the first and last bin edges in the ydimension.
This option only bins data that falls within the bin limits
inclusively, Y>=ybmin & Y<=ybmax
.
Normalization
— Type of normalization
'count'
(default)  'probability'
 'countdensity'
 'pdf'
 'cumcount'
 'cdf'
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.
$${A}_{i}={w}_{xi}\cdot {w}_{yi}$$ is the area of each bin, computed using the x and y bin widths.
$$N$$ is the number of elements in the input data. This value can be greater than the binned data if the data contains
NaN
values, or if some of the data lies outside the bin limits.
Value  Bin Values  Notes 

'count' (default) 
$${v}_{i}={c}_{i}$$ 

'countdensity' 
$${v}_{i}=\frac{{c}_{i}}{{A}_{i}}$$ 

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

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

'pdf' 
$${v}_{i}=\frac{{c}_{i}}{N\cdot {A}_{i}}$$ 

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

Example: [N,Xedges,Yedges] = histcounts2(X,Y,'Normalization','pdf')
bins
the data using the probability density function estimate for X
and Y
.
Output Arguments
N
— Bin counts
array
Bin counts, returned as a numeric array.
The bin inclusion scheme for the different numbered bins in N
,
as well as their relative orientation to the xaxis
and yaxis, is
For example, the (1,1)
bin includes values
that fall on the first edge in each dimension, and the last bin in
the bottom right includes values that fall on any of its edges.
Xedges
— Bin edges in xdimension
vector
Bin edges in xdimension, returned as a vector. Xedges(1)
is
the first bin edge in the xdimension and Xedges(end)
is
the last bin edge.
Yedges
— Bin edges in ydimension
vector
Bin edges in ydimension, returned as a vector. Yedges(1)
is
the first bin edge in the ydimension and Yedges(end)
is
the last bin edge.
binX
— Bin index in xdimension
array
Bin index in xdimension, returned as a numeric
array of the same size as X
. Corresponding elements
in binX
and binY
describe which
numbered bin contains the corresponding values in X
and Y
.
A value of 0
in binX
or binY
indicates
an element that does not belong to any of the bins (such as a NaN
value).
For example, binX(1)
and binY(1)
describe
the bin placement for the value [X(1),Y(1)]
.
binY
— Bin index in ydimension
array
Bin index in ydimension, returned as a numeric
array of the same size as Y
. Corresponding elements
in binX
and binY
describe which
numbered bin contains the corresponding values in X
and Y
.
A value of 0
in binX
or binY
indicates
an element that does not belong to any of the bins (such as a NaN
value).
For example, binX(1)
and binY(1)
describe
the bin placement for the value [X(1),Y(1)]
.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
Code generation does not support sparse matrix inputs for this function.
If you do not supply bin edges, then code generation might require variablesize arrays and dynamic memory allocation.
ThreadBased Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports threadbased environments. For more information, see Run MATLAB Functions in ThreadBased Environment.
See Also
histogram
 histcounts
 discretize
 histogram2
 morebins
 fewerbins
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