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Bivariate histogram plot
Bivariate histograms are a type of bar plot for numeric data that group the
data into 2D bins. After you create a Histogram2
object, you can
modify aspects of the histogram by changing its property values. This is particularly
useful for quickly modifying the properties of the bins or changing the
display.
histogram2(X,Y)
histogram2(X,Y,nbins)
histogram2(X,Y,Xedges,Yedges)
histogram2('XBinEdges',Xedges,'YBinEdges',Yedges,'BinCounts',counts)
histogram2(___,Name,Value)
histogram2(ax,___)
h = histogram2(___)
histogram2(
creates a
bivariate histogram plot of X,Y
)X
and Y
.
The histogram2
function uses an automatic binning
algorithm that returns bins with a uniform area, chosen to cover the range
of elements in X
and Y
and reveal the
underlying shape of the distribution. histogram2
displays the bins as 3D rectangular bars such that the height of each bar
indicates the number of elements in the bin.
histogram2(___,
specifies additional options with one or more Name,Value
)Name,Value
pair arguments using any of the previous syntaxes. For example, you can
specify 'BinWidth'
and a twoelement vector to adjust the
width of the bins in each dimension, or 'Normalization'
with a valid option ('count'
,
'probability'
, 'countdensity'
,
'pdf'
, 'cumcount'
, or
'cdf'
) to use a different type of normalization. For
a list of properties, see Histogram2 Properties.
histogram2(
plots into the axes specified by ax
,___)ax
instead of into the
current axes (gca
). The option ax
can
precede any of the input argument combinations in the previous
syntaxes.
returns a h
= histogram2(___)Histogram2
object. Use this to inspect and
adjust properties of the bivariate histogram. For a list of properties, see
Histogram2 Properties.
X,Y
— Data to distribute among bins (as separate arguments)Data to distribute among bins, specified as separate arguments of
vectors, matrices, or multidimensional arrays. X
and
Y
must be the same size. If X
and Y
are not vectors, then
histogram2
treats them as single column
vectors, X(:)
and Y(:)
, and plots
a single histogram.
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, but
histogram2
concatenates these inputs into a
single N
by2
matrix of the
dominant data type.
histogram2
ignores all NaN
values. Similarly, histogram2
ignores
Inf
and Inf
values, unless
the bin edges explicitly specify Inf
or
Inf
as a bin edge.
If X
or Y
contain integers
of type int64
or uint64
that
are larger than flintmax
, then it is recommended
that you explicitly specify the histogram bin
edges.histogram2
automatically bins the
input data using double precision, which lacks integer precision for
numbers greater than flintmax
.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
 logical
nbins
— Number of bins in each dimensionNumber of bins in each dimension, specified as a positive scalar
integer or twoelement vector of positive integers. If you do not
specify nbins
, then histogram2
automatically calculates how many bins to use based on the values in
X
and Y
.
If nbins
is a scalar, then
histogram2
uses that many bins in each
dimension.
If nbins
is a vector, then
nbins(1)
specifies the number of bins in
the xdimension and
nbins(2)
specifies the number of bins in
the ydimension.
Example: histogram2(X,Y,20)
uses 20 bins in each
dimension.
Example: histogram2(X,Y,[10 20])
uses 10 bins in the
x
dimension and 20 bins in the
y
dimension.
Xedges
— Bin edges in xdimensionBin 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.
The value [X(k),Y(k)]
is in the
(i,j)
th bin 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)
.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
 logical
Yedges
— Bin edges in ydimensionBin 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.
The value [X(k),Y(k)]
is in the
(i,j)
th bin 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)
.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
 logical
counts
— Bin countsBin counts, specified as a matrix. Use this input to pass bin counts
to histogram2
when the bin counts calculation is
performed separately and you do not want histogram2
to do any data binning.
counts
must be a matrix of size
[length(XBinEdges)1 length(YBinEdges)1]
so that
it specifies a bin count for each bin.
Example: histogram2('XBinEdges',1:1,'YBinEdges',2:2,'BinCounts',[1
2 3 4; 5 6 7 8])
ax
— Axes objectAxes object. If you do not specify an axes, then the
histogram2
function uses the current axes
(gca
).
Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside single quotes (' '
). You can
specify several name and value pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
histogram2(X,Y,'BinWidth',[5 10])
The properties listed here are only a subset. For a complete list, see Histogram2 Properties.
'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. 
If you set the NumBins
,
XBinEdges
, YBinEdges
,
BinWidth
, or BinLimits
property, then the BinMethod
property is set
to 'manual'
.
Example: histogram2(X,Y,'BinMethod','integers')
creates a bivariate histogram with the bins centered on pairs of
integers.
'BinWidth'
— Width of bins in each dimensionWidth of bins in each dimension, specified as a twoelement vector
of positive integers, [xWidth yWidth]
.
If you specify BinWidth
, then
histogram2
can use a maximum of 1024 bins (2^{10}) along each dimension. If instead the specified
bin width requires more bins, then histogram2
uses a larger bin width corresponding to the maximum number of
bins.
Example: histogram2(X,Y,'BinWidth',[5 10])
uses
bins with size 5
in the
x
dimension and size 10
in the
y
dimension.
'DisplayStyle'
— Histogram display style'bar3'
(default)  'tile'
Histogram display style, specified as either
'bar3'
or 'tile'
. Specify
'tile'
to display the histogram as a
rectangular array of tiles with colors indicating the bin
values.
The default value of 'bar3'
displays the
histogram using 3D bars.
Example: histogram2(X,Y,'DisplayStyle','tile')
plots the histogram as a rectangular array of tiles.
'EdgeAlpha'
— Transparency of histogram bar edges1
(default)  scalar value between 0
and
1
inclusiveTransparency of histogram bar edges, specified as a scalar value
between 0
and 1
inclusive. A
value of 1
means fully opaque and
0
means completely transparent
(invisible).
Example: histogram2(X,Y,'EdgeAlpha',0.5)
creates
a bivariate histogram plot with semitransparent bar
edges.
'EdgeColor'
— Histogram edge color[0.15 0.15 0.15]
(default)  'none'
 'auto'
 RGB triplet or color nameHistogram edge color, specified as one of these values:
'none'
— Edges are not
drawn.
'auto'
— Color of each edge is
chosen automatically.
RGB triplet or a color name — Edges use the specified color.
An RGB triplet is a threeelement row vector whose elements
specify the intensities of the red, green, and blue components of the color. The intensities
must be in the range [0,1]
; for example, [0.4 0.6 0.7]
.
Alternatively, you can specify some common colors by name. This table lists the long and short
color name options and the equivalent RGB triplet values.
Option  Description  Equivalent RGB Triplet 

'red' or 'r'  Red  [1 0 0] 
'green' or 'g'  Green  [0 1 0] 
'blue' or 'b'  Blue  [0 0 1] 
'yellow' or 'y'  Yellow  [1 1 0] 
'magenta' or 'm'  Magenta  [1 0 1] 
'cyan' or 'c'  Cyan  [0 1 1] 
'white' or 'w'  White  [1 1 1] 
'black' or 'k'  Black  [0 0 0] 
Example: histogram2(X,Y,'EdgeColor','r')
creates
a 3D histogram plot with red bar edges.
'FaceAlpha'
— Transparency of histogram bars1
(default)  scalar value between 0
and
1
inclusiveTransparency of histogram bars, specified as a scalar value
between 0
and 1
inclusive.
histogram2
uses the same transparency for all
the bars of the histogram. A value of 1
means
fully opaque and 0
means completely transparent
(invisible).
Example: histogram2(X,Y,'FaceAlpha',0.5)
creates
a bivariate histogram plot with semitransparent
bars.
'FaceColor'
— Histogram bar color'auto'
(default)  'flat'
 'none'
 RGB triplet or color nameHistogram bar color, specified as one of these values:
'none'
— Bars are not
filled.
'flat'
— Bar colors vary with
height. Bars with different height have different colors.
The colors are selected from the figure or axes
colormap.
'auto'
— Bar color is chosen
automatically (default).
RGB triplet or a color name — Bars are filled with the specified color.
An RGB triplet is a threeelement row vector whose elements
specify the intensities of the red, green, and blue components of the color. The intensities
must be in the range [0,1]
; for example, [0.4 0.6 0.7]
.
Alternatively, you can specify some common colors by name. This table lists the long and short
color name options and the equivalent RGB triplet values.
Option  Description  Equivalent RGB Triplet 

'red' or 'r'  Red  [1 0 0] 
'green' or 'g'  Green  [0 1 0] 
'blue' or 'b'  Blue  [0 0 1] 
'yellow' or 'y'  Yellow  [1 1 0] 
'magenta' or 'm'  Magenta  [1 0 1] 
'cyan' or 'c'  Cyan  [0 1 1] 
'white' or 'w'  White  [1 1 1] 
'black' or 'k'  Black  [0 0 0] 
If you specify DisplayStyle
as
'stairs'
, then
histogram2
does not use the
FaceColor
property.
Example: histogram2(X,Y,'FaceColor','g')
creates
a 3D histogram plot with green bars.
'FaceLighting'
— Lighting effect on histogram bars'lit'
(default)  'flat'
 'none'
Lighting effect on histogram bars, specified as one of the values in this table.
Value  Description 

'lit' 
Histogram bars display a pseudolighting effect, where the sides of the bars use darker colors relative to the tops. The bars are unaffected by other light sources in the axes. This is the default value when

'flat' 
Histogram bars are not lit automatically. In the presence of other light objects, the lighting effect is uniform across the bar faces. 
'none' 
Histogram bars are not lit automatically, and lights do not affect the histogram bars.

Example: histogram2(X,Y,'FaceLighting','none')
turns off the lighting of the histogram bars.
'LineStyle'
— Line style''
(default)  ''
 ':'
 '.'
 'none'
Line style, specified as one of the line styles listed in this table.
Line Style  Description  Resulting Line 

''  Solid line 

''  Dashed line 

':'  Dotted line 

'.'  Dashdotted line 

'none'  No line  No line 
'LineWidth'
— Width of bar outlines0.5
(default)  positive valueWidth of bar outlines, specified as a positive value in point units. One point equals 1/72 inch.
Example: 1.5
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
'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: histogram2(X,Y,'Normalization','pdf')
plots an estimate of the probability density function for
X
and Y
.
'ShowEmptyBins'
— Toggle display of empty bins'off'
(default)  'on'
Toggle display of empty bins, specified as either
'off'
or 'on'
. The default
value is 'off'
.
Example: histogram2(X,Y,'ShowEmptyBins','on')
turns on the display of empty bins.
'XBinLimits'
— Bin limits in xdimensionBin limits in xdimension, specified as a
twoelement vector, [xbmin,xbmax]
. The vector
indicates the first and last bin edges in the
xdimension.
histogram2
only plots data that falls within
the bin limits inclusively, Data(Data(:,1)>=xbmin &
Data(:,1)<=xbmax)
.
'XBinLimitsMode'
— Selection mode for bin limits in xdimension'auto'
(default)  'manual'
Selection mode for bin limits in xdimension,
specified as 'auto'
or
'manual'
. The default value is
'auto'
, so that the bin limits automatically
adjust to the data along the xaxis.
If you explicitly specify either XBinLimits
or
XBinEdges
, then
XBinLimitsMode
is set automatically to
'manual'
. In that case, specify
XBinLimitsMode
as 'auto'
to rescale the bin limits to the data.
'YBinLimits'
— Bin limits in ydimensionBin limits in ydimension, specified as a
twoelement vector, [ybmin,ybmax]
. The vector
indicates the first and last bin edges in the
ydimension.
histogram2
only plots data that falls within
the bin limits inclusively, Data(Data(:,2)>=ybmin &
Data(:,2)<=ybmax)
.
'YBinLimitsMode'
— Selection mode for bin limits in ydimension'auto'
(default)  'manual'
Selection mode for bin limits in ydimension,
specified as 'auto'
or
'manual'
. The default value is
'auto'
, so that the bin limits automatically
adjust to the data along the yaxis.
If you explicitly specify either YBinLimits
or
YBinEdges
, then
YBinLimitsMode
is set automatically to
'manual'
. In that case, specify
YBinLimitsMode
as 'auto'
to rescale the bin limits to the data.
h
— Bivariate histogramBivariate histogram, returned as an object. For more information, see
Histogram2
.
Histogram2 Properties  Control histogram2 appearance and behavior 
Generate 10,000 pairs of random numbers and create a bivariate histogram. The histogram2
function automatically chooses an appropriate number of bins to cover the range of values in x
and y
and show the shape of the underlying distribution.
x = randn(10000,1); y = randn(10000,1); h = histogram2(x,y)
h = Histogram2 with properties: Data: [10000x2 double] Values: [25x28 double] NumBins: [25 28] XBinEdges: [1x26 double] YBinEdges: [1x29 double] BinWidth: [0.3000 0.3000] Normalization: 'count' FaceColor: 'auto' EdgeColor: [0.1500 0.1500 0.1500] Show all properties
xlabel('x') ylabel('y')
When you specify an output argument to the histogram2
function, it returns a histogram2 object. You can use this object to inspect the properties of the histogram, such as the number of bins or the width of the bins.
Find the number of histogram bins in each dimension.
nXnY = h.NumBins
nXnY =
25 28
Plot a bivariate histogram of 1,000 pairs of random numbers sorted into 25 equally spaced bins, using 5 bins in each dimension.
x = randn(1000,1); y = randn(1000,1); nbins = 5; h = histogram2(x,y,nbins)
h = Histogram2 with properties: Data: [1000x2 double] Values: [5x5 double] NumBins: [5 5] XBinEdges: [4 2.4000 0.8000 0.8000 2.4000 4] YBinEdges: [4 2.4000 0.8000 0.8000 2.4000 4] BinWidth: [1.6000 1.6000] Normalization: 'count' FaceColor: 'auto' EdgeColor: [0.1500 0.1500 0.1500] Show all properties
Find the resulting bin counts.
counts = h.Values
counts =
0 2 3 1 0
2 40 124 47 4
1 119 341 109 10
1 32 117 33 1
0 4 8 1 0
Generate 1,000 pairs of random numbers and create a bivariate histogram.
x = randn(1000,1); y = randn(1000,1); h = histogram2(x,y)
h = Histogram2 with properties: Data: [1000x2 double] Values: [15x15 double] NumBins: [15 15] XBinEdges: [1x16 double] YBinEdges: [1x16 double] BinWidth: [0.5000 0.5000] Normalization: 'count' FaceColor: 'auto' EdgeColor: [0.1500 0.1500 0.1500] Show all properties
Use the morebins
function to coarsely adjust the number of bins in the x dimension.
nbins = morebins(h,'x'); nbins = morebins(h,'x')
nbins =
19 15
Use the fewerbins
function to adjust the number of bins in the y dimension.
nbins = fewerbins(h,'y'); nbins = fewerbins(h,'y')
nbins =
19 11
Adjust the number of bins at a fine grain level by explicitly setting the number of bins.
h.NumBins = [20 10];
Create a bivariate histogram using 1,000 normally distributed random numbers with 12 bins in each dimension. Specify FaceColor
as 'flat'
to color the histogram bars by height.
h = histogram2(randn(1000,1),randn(1000,1),[12 12],'FaceColor','flat'); colorbar
Generate random data and plot a bivariate tiled histogram. Display the empty bins by specifying ShowEmptyBins
as 'on'
.
x = 2*randn(1000,1)+2; y = 5*randn(1000,1)+3; h = histogram2(x,y,'DisplayStyle','tile','ShowEmptyBins','on');
Generate 1,000 pairs of random numbers and create a bivariate histogram. Specify the bin edges using two vectors, with infinitely wide bins on the boundary of the histogram to capture all outliers that do not satisfy .
x = randn(1000,1); y = randn(1000,1); Xedges = [Inf 2:0.4:2 Inf]; Yedges = [Inf 2:0.4:2 Inf]; h = histogram2(x,y,Xedges,Yedges)
h = Histogram2 with properties: Data: [1000x2 double] Values: [12x12 double] NumBins: [12 12] XBinEdges: [1x13 double] YBinEdges: [1x13 double] BinWidth: 'nonuniform' Normalization: 'count' FaceColor: 'auto' EdgeColor: [0.1500 0.1500 0.1500] Show all properties
When the bin edges are infinite, histogram2
displays each outlier bin (along the boundary of the histogram) as being double the width of the bin next to it.
Specify the Normalization
property as 'countdensity'
to remove the bins containing the outliers. Now the volume of each bin represents the frequency of observations in that interval.
h.Normalization = 'countdensity';
Generate 1,000 pairs of random numbers and create a bivariate histogram using the 'probability'
normalization.
x = randn(1000,1); y = randn(1000,1); h = histogram2(x,y,'Normalization','probability')
h = Histogram2 with properties: Data: [1000x2 double] Values: [15x15 double] NumBins: [15 15] XBinEdges: [1x16 double] YBinEdges: [1x16 double] BinWidth: [0.5000 0.5000] Normalization: 'probability' FaceColor: 'auto' EdgeColor: [0.1500 0.1500 0.1500] Show all properties
Compute the total sum of the bar heights. With this normalization, the height of each bar is equal to the probability of selecting an observation within that bin interval, and the heights of all of the bars sum to 1.
S = sum(h.Values(:))
S = 1.0000
Generate 1,000 pairs of random numbers and create a bivariate histogram. Return the histogram object to adjust the properties of the histogram without recreating the entire plot.
x = randn(1000,1); y = randn(1000,1); h = histogram2(x,y)
h = Histogram2 with properties: Data: [1000x2 double] Values: [15x15 double] NumBins: [15 15] XBinEdges: [1x16 double] YBinEdges: [1x16 double] BinWidth: [0.5000 0.5000] Normalization: 'count' FaceColor: 'auto' EdgeColor: [0.1500 0.1500 0.1500] Show all properties
Color the histogram bars by height.
h.FaceColor = 'flat';
Change the number of bins in each direction.
h.NumBins = [10 25];
Display the histogram as a tile plot.
h.DisplayStyle = 'tile';
view(2)
Use the savefig
function to save a histogram2 figure.
y = histogram2(randn(100,1),randn(100,1)); savefig('histogram2.fig'); clear all close all
Use openfig
to load the histogram figure back into MATLAB. openfig
also returns a handle to the figure, h
.
h = openfig('histogram2.fig');
Use the findobj
function to locate the correct object handle from the figure handle. This allows you to continue manipulating the original histogram object used to generate the figure.
y = findobj(h, 'type', 'histogram2')
y = Histogram2 with properties: Data: [100x2 double] Values: [7x6 double] NumBins: [7 6] XBinEdges: [3 2 1 0 1 2 3 4] YBinEdges: [3 2 1 0 1 2 3] BinWidth: [1 1] Normalization: 'count' FaceColor: 'auto' EdgeColor: [0.1500 0.1500 0.1500] Use GET to show all properties
Histogram plots created using histogram2
have a context
menu in plot edit mode that enables interactive manipulations in the figure
window. For example, you can use the context menu to interactively change the
number of bins, align multiple histograms, or change the display order.
This function supports tall arrays with the limitations:
Some input options are not supported. The allowed options are:
'BinWidth'
'XBinLimits'
'YBinLimits'
'Normalization'
'DisplayStyle'
'BinMethod'
— The 'auto'
and 'scott'
bin
methods are the same. The 'fd'
bin method is not
supported.
'EdgeAlpha'
'EdgeColor'
'FaceAlpha'
'FaceColor'
'LineStyle'
'LineWidth'
'Orientation'
Additionally, there is a cap on the maximum number of bars. The default maximum is 100.
The morebins
and fewerbins
methods
are not supported.
Editing properties of the histogram object that require recomputing the bins is not supported.
For more information, see Tall Arrays.
Histogram2 Properties  bar3
 discretize
 fewerbins
 histcounts
 histcounts2
 morebins
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