0.9 Lab 7a - discrete-time random processes (part 1)  (Page 5/5)

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Notice that we cannot form a density estimate by simply differentiating the empirical CDF, since this function contains discontinuities at thesample locations ${X}_{i}$ . Rather, we need to estimate the probability that a random variable willfall within a particular interval of the real axis. In this section, we will describe a common method known as the histogram .

The histogram

Our goal is to estimate an arbitrary probability density function, ${f}_{X}\left(x\right)$ , within a finite region of the $x$ -axis. We will do this by partitioning the region into $L$ equally spaced subintervals, or “bins”,and forming an approximation for ${f}_{X}\left(x\right)$ within each bin. Let our region of support start at the value ${x}_{0}$ , and end at ${x}_{L}$ . Our $L$ subintervals of this region will be $\left[{x}_{0},{x}_{1}\right]$ , $\left({x}_{1},{x}_{2}\right]$ , ..., $\left({x}_{L-1},{x}_{L}\right]$ . To simplify our notation we will define $bin\left(k\right)$ to represent the interval $\left({x}_{k-1},{x}_{k}\right]$ , $k=1,2,\cdots ,L$ , and define the quantity $\Delta$ to be the length of each subinterval.

$\begin{array}{ccc}\hfill bin\left(k\right)& =& \left({x}_{k-1},{x}_{k}\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k=1,2,\cdots ,L\hfill \\ \hfill \Delta & =& \frac{{x}_{L}-{x}_{0}}{L}\hfill \end{array}$

We will also define $\stackrel{˜}{f}\left(k\right)$ to be the probability that $X$ falls into $bin\left(k\right)$ .

$\begin{array}{ccc}\hfill \stackrel{˜}{f}\left(k\right)& =& P\left(X\in bin\left(k\right)\right)\hfill \\ & =& {\int }_{{x}_{k-1}}^{{x}_{k}}{f}_{X}\left(x\right)dx\hfill \end{array}$
$\begin{array}{ccc}& \approx & {f}_{X}\left(x\right)\Delta \phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}x\in bin\left(k\right)\hfill \end{array}$

The approximation in [link] only holds for an appropriately small bin width $\Delta$ .

Next we introduce the concept of a histogram of a collection of i.i.d. random variables $\left\{{X}_{1},{X}_{2},\cdots ,{X}_{N}\right\}$ . Let us start by defining a function that will indicate whether ornot the random variable ${X}_{n}$ falls within $bin\left(k\right)$ .

${I}_{n}\left(k\right)=\left\{\begin{array}{cc}1,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}{X}_{n}\in bin\left(k\right)\hfill \\ 0,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}{X}_{n}\notin bin\left(k\right)\hfill \end{array}\right)$

The histogram of ${X}_{n}$ at $bin\left(k\right)$ , denoted as $H\left(k\right)$ , is simply the number of random variables that fall within $bin\left(k\right)$ . This can be written as

$H\left(k\right)=\sum _{n=1}^{N}{I}_{n}\left(k\right)\phantom{\rule{4pt}{0ex}}.$

We can show that the normalized histogram, $H\left(k\right)/N$ , is an unbiased estimate of the probability of $X$ falling in $bin\left(k\right)$ . Let us compute the expected value of the normalized histogram.

$\begin{array}{ccc}\hfill E\left[\frac{H\left(k\right)}{N}\right]& =& \frac{1}{N}\sum _{n=1}^{N}E\left[{I}_{n}\left(k\right)\right]\hfill \\ & =& \frac{1}{N}\sum _{n=1}^{N}\left\{1·P\left({X}_{n}\in bin\left(k\right)\right)+0·P\left({X}_{n}\notin bin\left(k\right)\right)\right\}\hfill \\ & =& \stackrel{˜}{f}\left(k\right)\hfill \end{array}$

The last equality results from the definition of $\stackrel{˜}{f}\left(k\right)$ , and from the assumption that the ${X}_{n}$ 's have the same distribution. A similar argument may be used to show that the variance of $H\left(k\right)$ is given by

$\begin{array}{c}\hfill Var\left[\frac{H\left(k\right)}{N}\right]=\frac{1}{N}\stackrel{˜}{f}\left(k\right)\left(1-\stackrel{˜}{f}\left(k\right)\right)\phantom{\rule{4pt}{0ex}}.\end{array}$

Therefore, as $N$ grows large, the bin probabilities $\stackrel{˜}{f}\left(k\right)$ can be approximated by the normalized histogram $H\left(k\right)/N$ .

$\stackrel{˜}{f}\left(k\right)\approx \frac{H\left(k\right)}{N}$

Using [link] , we may then approximate the density function ${f}_{X}\left(x\right)$ within $bin\left(k\right)$ by

${f}_{X}\left(x\right)\approx \frac{H\left(k\right)}{N\Delta }\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}x\in bin\left(k\right)\phantom{\rule{4pt}{0ex}}.$

Notice this estimate is a staircase function of $x$ which is constant over each interval $bin\left(k\right)$ . It can also easily be verified that this density estimate integrates to 1.

Exercise

Let $U$ be a uniformly distributed random variable on the interval [0,1]with the following cumulative probability distribution, ${F}_{U}\left(u\right)$ :

${F}_{U}\left(u\right)=\left\{\begin{array}{cc}0,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}u<0\hfill \\ u,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}0\le u\le 1\hfill \\ 1,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}u>1\hfill \end{array}\right)$

We can calculate the cumulative probability distribution for the new random variable $X={U}^{\frac{1}{3}}$ .

$\begin{array}{ccc}\hfill {F}_{X}\left(x\right)& =& P\left(X\le x\right)\hfill \\ & =& P\left({U}^{\frac{1}{3}}\le x\right)\hfill \\ & =& P\left(U\le {x}^{3}\right)\hfill \\ & =& {\left({F}_{U},\left(u\right)|}_{u={x}^{3}}\hfill \\ & =& \left\{\begin{array}{cc}0,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}x<0\hfill \\ {x}^{3},\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}0\le x\le 1\hfill \\ 1,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}x>1\hfill \end{array}\right)\hfill \end{array}$

Plot ${F}_{X}\left(x\right)$ for $x\in \left[0,1\right]$ . Also, analytically calculate the probability density ${f}_{X}\left(x\right)$ , and plot it for $x\in \left[0,1\right]$ .

Using $L=20$ , ${x}_{0}=0$ and ${x}_{L}=1$ , use Matlab to compute $\stackrel{˜}{f}\left(k\right)$ , the probability of $X$ falling into $bin\left(k\right)$ .

Use the fact that $\stackrel{˜}{f}\left(k\right)={F}_{X}\left({x}_{k}\right)-{F}_{X}\left({x}_{k-1}\right)$ .
Plot $\stackrel{˜}{f}\left(k\right)$ for $k=1,\cdots ,L$ using the stem function.

Inlab report

1. Submit your plots of ${F}_{X}\left(x\right)$ , ${f}_{X}\left(x\right)$ and $\stackrel{˜}{f}\left(k\right)$ . Use stem to plot $\stackrel{˜}{f}\left(k\right)$ , and put all three plots on a single figure using subplot .
2. Show (mathematically) how ${f}_{X}\left(x\right)$ and $\stackrel{˜}{f}\left(k\right)$ are related.

Generate 1000 samples of a random variable $U$ that is uniformly distributed between 0 and 1 (using the rand command). Then form the random vector $X$ by computing $X={U}^{\frac{1}{3}}$ .

Use the Matlab function hist to plot a normalized histogram for your samples of $X$ , using 20 bins uniformly spaced on the interval $\left[0,1\right]$ .

Use the Matlab command H=hist(X,(0.5:19.5)/20) to obtain the histogram, and then normalize H .
Use the stem command to plot the normalized histogram $H\left(k\right)/N$ and $\stackrel{˜}{f}\left(k\right)$ together on the same figure using subplot .

Inlab report

1. Submit your two stem plots of $H\left(k\right)/N$ and $\stackrel{˜}{f}\left(k\right)$ . How do these plots compare?
2. Discuss the tradeoffs (advantages and the disadvantages) between selecting a very large or very small bin-width.

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