# ecdfhist

Histogram based on empirical cumulative distribution function

## Description

`[`

returns
the heights, `n`

,`c`

] =
ecdfhist(`f`

,`x`

)`n`

, of histogram bars for 10 equally
spaced bins and the position of the bin centers, `c`

.

`ecdfhist`

computes the bar heights from the
increases in the empirical cumulative distribution function, `f`

,
at evaluation points, `x`

. It normalizes the bar
heights so that the area of the histogram is equal to 1. In contrast, `histogram`

produces bars with heights
representing bin counts.

`ecdfhist(___)`

plots the histogram
bars.

## Examples

### Return Histogram Bar Heights and Bin Centers

Compute the histogram bar heights based on the empirical cumulative distribution function.

Generate failure times from a Birnbaum-Saunders distribution.

rng('default') % for reproducibility failuretime = random('birnbaumsaunders',0.3,1,100,1);

Assuming that the end of the study is at time 0.9, mark the generated failure times that are larger than 0.9 as censored data and store that information in a vector.

T = 0.9; cens = (failuretime>T);

Compute the empirical cumulative distribution function for the data.

`[f,x] = ecdf(failuretime,'censoring',cens);`

Now, find the bar heights of the histogram using the cumulative distribution function estimate.

[n,c] = ecdfhist(f,x); [n' c']

`ans = `*10×2*
2.3529 0.0715
1.7647 0.1565
1.4117 0.2415
1.5294 0.3265
1.0588 0.4115
0.4706 0.4965
0.4706 0.5815
0.9412 0.6665
0.2353 0.7515
0.2353 0.8365

### Return Bar Heights and Bin Centers for a Given Number of Bins

Compute the bar heights for six bins using the empirical cumulative distribution function and also return the bin centers.

Generate failure times from a Birnbaum-Saunders distribution.

rng('default') % for reproducibility failuretime = random('birnbaumsaunders',0.3,1,100,1);

Assuming that the end of the study is at time 0.9, mark the generated failure times that are larger than 0.9 as censored data and store that information in a vector.

T = 0.9; cens = (failuretime>T);

First, compute the empirical cumulative distribution function for the data.

`[f,x] = ecdf(failuretime,'censoring',cens);`

Now, estimate the histogram with six bins using the cumulative distribution function estimate.

[n,c] = ecdfhist(f,x,6); [n' c']

`ans = `*6×2*
1.9764 0.0998
1.7647 0.2415
1.1294 0.3831
0.4235 0.5248
0.7764 0.6665
0.2118 0.8081

### Draw Histogram for Given Bin Centers

Draw the histogram of the empirical cumulative distribution histogram for specified bin centers.

Generate failure times from a Birnbaum-Saunders distribution.

rng default; % For reproducibility failuretime = random('birnbaumsaunders',0.3,1,100,1);

Assuming that the end of the study is at time 0.9, mark the generated failure times that are larger than 0.9 as censored data and store that information in a vector.

T = 0.9; cens = (failuretime>T);

Define bin centers.

centers = 0.1:0.1:1;

Compute the empirical cumulative distribution function for the data and draw the histogram for specified bin centers.

```
[f,x] = ecdf(failuretime,'censoring',cens);
ecdfhist(f,x,centers)
axis([0 1 0 2.5])
```

### Compare Histogram with Known Probability Distribution Function

Generate right-censored survival data and compare the histogram from cumulative distribution function with the known probability distribution function.

Generate failure times from an exponential distribution with mean failure time of 15.

rng default; % For reproducibility y = exprnd(15,75,1);

Generate drop-out times from an exponential distribution with mean failure time of 30.

d = exprnd(30,75,1);

Record the minimum of these times as the observed failure times.

t = min(y,d);

Generate censoring by finding the generated failure times that are greater than the drop-out times.

censored = (y>d);

Calculate the empirical cdf and plot a histogram using the empirical cumulative distribution function.

[f,x] = ecdf(t,'censoring',censored); ecdfhist(f,x) h = findobj(gca,'Type','patch'); h.FaceColor = [.8 .8 1]; hold on

Superimpose a plot of the known population pdf.

xx = 0:.1:max(t); yy = exp(-xx/15)/15; plot(xx,yy,'r-','LineWidth',2) hold off

## Input Arguments

`f`

— Empirical cdf values

vector

Empirical cdf values at given evaluation points, `x`

,
specified as a vector.

For instance, you can use `ecdf`

to obtain
the empirical cdf values and enter them in `ecdfhist`

as
follows.

**Example: **`[f,x] = ecdf(failure); ecdfhist(f,x);`

**Data Types: **`single`

| `double`

`x`

— Evaluation points

vector

Evaluation points at which empirical cdf values, `f`

,
are calculated, specified as a vector.

For instance, you can use `ecdf`

to obtain
the empirical cdf values and enter them in `ecdfhist`

as
follows.

**Example: **`[f,x] = ecdf(failure); ecdfhist(f,x);`

**Data Types: **`single`

| `double`

`m`

— Number of bins

scalar

Number of bins, specified as a scalar.

For instance, you can draw a histogram with 8 bins as follows.

**Example: **`ecdfhist(f,x,8)`

**Data Types: **`single`

| `double`

`centers`

— Center points of bins

vector

Center points of bins, specified as a vector.

**Example: **`centers = 2:2:10; ecdfhist(f,x,centers);`

**Data Types: **`single`

| `double`

## Output Arguments

`n`

— Heights of histogram bars

row vector

Heights of histogram bars `ecdfhist`

calculates
based on the empirical cdf values, returned as a row vector.

`c`

— Position of bin centers

row vector

Position of bin centers, returned as a row vector.

## Version History

**Introduced before R2006a**

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