# histogram2

Bivariate histogram plot

## Description

Bivariate histograms are a type of bar plot for numeric data that group the data into 2-D 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.

## Creation

### Description

example

histogram2(X,Y) creates a bivariate histogram plot of 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 3-D rectangular bars such that the height of each bar indicates the number of elements in the bin.

example

histogram2(X,Y,nbins) specifies the number of bins to use in each dimension of the histogram.

example

histogram2(X,Y,Xedges,Yedges) specifies the edges of the bins in each dimension using the vectors Xedges and Yedges.

histogram2('XBinEdges',Xedges,'YBinEdges',Yedges,'BinCounts',counts) manually specifies the bin counts. histogram2 plots the specified bin counts and does not do any data binning.

example

histogram2(___,Name,Value) specifies additional options with one or more Name,Value pair arguments using any of the previous syntaxes. For example, you can specify 'BinWidth' and a two-element 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(ax,___) plots into the axes specified by ax instead of into the current axes (gca). The option ax can precede any of the input argument combinations in the previous syntaxes.

example

h = histogram2(___) returns a Histogram2 object. Use this to inspect and adjust properties of the bivariate histogram. For a list of properties, see Histogram2 Properties.

### Input Arguments

expand all

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 2-D 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-by-2 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. Although NaN, Inf, and -Inf values are typically not plotted, they are still included in normalization calculations that include the total number of data elements, such as 'probability'.

Note

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

Number of bins in each dimension, specified as a positive scalar integer or two-element 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 x-dimension and nbins(2) specifies the number of bins in the y-dimension.

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.

Bin edges in x-dimension, specified as a vector. Xedges(1) is the first edge of the first bin in the x-dimension, 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 ith bin in the last row if Xedges(end-1)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

Bin edges in y-dimension, specified as a vector. Yedges(1) is the first edge of the first bin in the y-dimension, 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 ith bin in the last row if Xedges(end-1)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

Bin 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])

Axes object. If you do not specify an axes, then the histogram2 function uses the current axes (gca).

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: histogram2(X,Y,'BinWidth',[5 10])

The properties listed here are only a subset. For a complete list, see Histogram2 Properties.

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

ValueDescription
'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 jointly normally distributed. This rule is appropriate for most other distributions, as well. It uses a bin size of [3.5*std(X(:))*numel(X)^(-1/4), 3.5*std(Y(:))*numel(Y)^(-1/4)].

'fd'

The Freedman-Diaconis rule is less sensitive to outliers in the data, and might be more suitable for data with heavy-tailed distributions. It uses a bin size of [2*IQR(X(:))*numel(X)^(-1/4), 2*IQR(Y(:))*numel(Y)^(-1/4)], where IQR is the interquartile range.

'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 (210). If the data range for either dimension is greater than 1024, then the integer rule uses wider bins instead.

histogram2 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.

Note

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.

Width of bins in each dimension, specified as a two-element vector of positive integers, [xWidth yWidth].

If you specify BinWidth, then histogram2 can use a maximum of 1024 bins (210) 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.

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 3-D bars.

Example: histogram2(X,Y,'DisplayStyle','tile') plots the histogram as a rectangular array of tiles.

Transparency 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 semi-transparent bar edges.

Histogram edge color, specified as one of these values:

• 'none' — Edges are not drawn.

• 'auto' — Color of each edge is chosen automatically.

• RGB triplet, hexadecimal color code, or color name — Edges use the specified color.

RGB triplets and hexadecimal color codes are useful for specifying custom colors.

• An RGB triplet is a three-element 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].

• A hexadecimal color code is a character vector or a string scalar that starts with a hash symbol (#) followed by three or six hexadecimal digits, which can range from 0 to F. The values are not case sensitive. Thus, the color codes '#FF8800', '#ff8800', '#F80', and '#f80' are equivalent.

Alternatively, you can specify some common colors by name. This table lists the named color options, the equivalent RGB triplets, and hexadecimal color codes.

Color NameShort NameRGB TripletHexadecimal Color CodeAppearance
'red''r'[1 0 0]'#FF0000'

'green''g'[0 1 0]'#00FF00'

'blue''b'[0 0 1]'#0000FF'

'cyan' 'c'[0 1 1]'#00FFFF'

'magenta''m'[1 0 1]'#FF00FF'

'yellow''y'[1 1 0]'#FFFF00'

'black''k'[0 0 0]'#000000'

'white''w'[1 1 1]'#FFFFFF'

Here are the RGB triplets and hexadecimal color codes for the default colors MATLAB® uses in many types of plots.

[0 0.4470 0.7410]'#0072BD'

[0.8500 0.3250 0.0980]'#D95319'

[0.9290 0.6940 0.1250]'#EDB120'

[0.4940 0.1840 0.5560]'#7E2F8E'

[0.4660 0.6740 0.1880]'#77AC30'

[0.3010 0.7450 0.9330]'#4DBEEE'

[0.6350 0.0780 0.1840]'#A2142F'

Example: histogram2(X,Y,'EdgeColor','r') creates a 3-D histogram plot with red bar edges.

Transparency 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 semi-transparent bars.

Histogram 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, hexadecimal color code, or color name — Bars are filled with the specified color.

RGB triplets and hexadecimal color codes are useful for specifying custom colors.

• An RGB triplet is a three-element 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].

• A hexadecimal color code is a character vector or a string scalar that starts with a hash symbol (#) followed by three or six hexadecimal digits, which can range from 0 to F. The values are not case sensitive. Thus, the color codes '#FF8800', '#ff8800', '#F80', and '#f80' are equivalent.

Alternatively, you can specify some common colors by name. This table lists the named color options, the equivalent RGB triplets, and hexadecimal color codes.

Color NameShort NameRGB TripletHexadecimal Color CodeAppearance
'red''r'[1 0 0]'#FF0000'

'green''g'[0 1 0]'#00FF00'

'blue''b'[0 0 1]'#0000FF'

'cyan' 'c'[0 1 1]'#00FFFF'

'magenta''m'[1 0 1]'#FF00FF'

'yellow''y'[1 1 0]'#FFFF00'

'black''k'[0 0 0]'#000000'

'white''w'[1 1 1]'#FFFFFF'

Here are the RGB triplets and hexadecimal color codes for the default colors MATLAB uses in many types of plots.

[0 0.4470 0.7410]'#0072BD'

[0.8500 0.3250 0.0980]'#D95319'

[0.9290 0.6940 0.1250]'#EDB120'

[0.4940 0.1840 0.5560]'#7E2F8E'

[0.4660 0.6740 0.1880]'#77AC30'

[0.3010 0.7450 0.9330]'#4DBEEE'

[0.6350 0.0780 0.1840]'#A2142F'

If you specify DisplayStyle as 'stairs', then histogram2 does not use the FaceColor property.

Example: histogram2(X,Y,'FaceColor','g') creates a 3-D histogram plot with green bars.

Lighting effect on histogram bars, specified as one of the values in this table.

ValueDescription
'lit'

Histogram bars display a pseudo-lighting 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 DisplayStyle is 'bar3'.

'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.

FaceLighting can only be 'none' when DisplayStyle is 'tile'.

Example: histogram2(X,Y,'FaceLighting','none') turns off the lighting of the histogram bars.

Line style, specified as one of the options listed in this table.

Line StyleDescriptionResulting Line
'-'Solid line

'--'Dashed line

':'Dotted line

'-.'Dash-dotted line

'none'No lineNo line

Width 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

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.

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) and numel(y). The sum is less than numel(X) and numel(y) only when some of the input data is not included in the bins.

'countdensity'

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

• Count or frequency scaled by area of bin.

• The volume (height * area) of each bar is the number of observations in the bin. The sum of the bar volumes is less than or equal to numel(X) and numel(y).

'cumcount'

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

• Cumulative count. Each bin value is the cumulative number of observations in each bin and all previous bins in both the x and y dimensions.

• The height of the last bar is less than or equal to numel(X) and numel(Y).

'probability'

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

• Relative probability.

• The sum of the bar heights is less than or equal to 1.

'pdf'

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

• Probability density function estimate.

• The volume of each bar is the relative number of observations. The sum of the bar volumes is less than or equal to 1.

'cdf'

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

• Cumulative density function estimate.

• The height of each bar is equal to the cumulative relative number of observations in each bin and all previous bins in both the x and y dimensions. The height of the last bar is less than or equal to 1.

Example: histogram2(X,Y,'Normalization','pdf') plots an estimate of the probability density function for X and Y.

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.

Bin limits in x-dimension, specified as a two-element vector, [xbmin,xbmax]. The vector indicates the first and last bin edges in the x-dimension.

histogram2 only plots data that falls within the bin limits inclusively, Data(Data(:,1)>=xbmin & Data(:,1)<=xbmax).

Selection mode for bin limits in x-dimension, specified as 'auto' or 'manual'. The default value is 'auto', so that the bin limits automatically adjust to the data along the x-axis.

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.

Bin limits in y-dimension, specified as a two-element vector, [ybmin,ybmax]. The vector indicates the first and last bin edges in the y-dimension.

histogram2 only plots data that falls within the bin limits inclusively, Data(Data(:,2)>=ybmin & Data(:,2)<=ybmax).

Selection mode for bin limits in y-dimension, specified as 'auto' or 'manual'. The default value is 'auto', so that the bin limits automatically adjust to the data along the y-axis.

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.

### Output Arguments

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Bivariate histogram, returned as an object. For more information, see Histogram2 Properties.

## Properties

 Histogram2 Properties Histogram2 appearance and behavior

## Object Functions

 morebins Increase number of histogram bins fewerbins Decrease number of histogram bins

## Examples

collapse all

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 = 1×2

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 = 5×5

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 = 1×2

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 = 1×2

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 $|x|<2$.

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

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]

Show all properties

## Tips

• 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.