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# ecdfhist

Histogram based on empirical cumulative distribution function

## Description

example

[n,c] = ecdfhist(f,x) returns the heights, 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, hist produces bars with heights representing bin counts.

example

[n,c] = ecdfhist(f,x,m) returns the histogram bars using m bins.

example

n = ecdfhist(f,x,centers) returns the heights of the histogram bars with bin centers specified by centers.

example

ecdfhist(___) plots the histogram bars.

## Examples

expand all

### 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 =

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 =

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)
set(get(gca,'Children'),'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

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### f — Empirical cdf valuesvector

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 pointsvector

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 binsscalar

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 binsvector

Center points of bins, specified as a vector.

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

Data Types: single | double

## Output Arguments

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### n — Heights of histogram barsrow vector

Heights of histogram bars ecdfhist calculates based on the empirical cdf values, returned as a row vector.

### c — Position of bin centersrow vector

Position of bin centers, returned as a row vector.

## See Also

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