## Inverse function as a method to generate rvs

In this post I attempt to explore some basic intuition about the inverse function as a method to generate random variables.

## PDF and CDF

Suppose there is a probability density function $f$. One can easily calculate the cumulative density function $F$ as:

$F(x) = \int_{-\infty}^x f(t) dt = g_x$

Where $g_x \in [0, 1]$.

### PDF – example

Let’s generate a probability density function $f$ based on a polynomial equation $k$.  Suppose we want this pdf to describe the probability of some variable $x \in r$, where $r$ is the region ${[-4, -1] \cup [+1, +4]}$. We define $k$ arbitrarily as:

$k = -(x^2 - 1)(x^2 - 16)$

It is necessary to adjust this polynomial function to a factor of $h$ defined by:

$h = \int_{-4}^{-1} k dx + \int_{1}^{4} k dx = \frac{1044}{5}$

Therefore, the probability density function $f$ is a piecewise function defined as:

$f = \begin{cases} 0 &\quad\text{if } x \in r^c \\ \frac{k}{h} &\quad\text{if } x \in r \ \end{cases}$

By doing so, we can be confident that:

$\int_{-\infty}^{\infty} f(t) dt = 1$

### CDF – example

Naturally, the cdf function is just an integral. But due to the fact that the pdf we are using is a pisewise function, we must divide the integrals across the region. So the cdf function $F$ would have the following form:

$F = \begin{cases} I_1 &\quad\text{if } x \leq -4 \\ I_2 &\quad\text{if } -4 < x \leq -1 \\ I_3 &\quad\text{if } -1 < x \leq 1 \\ I_4 &\quad\text{if } 1 < x \leq 4 \\ I_5 &\quad\text{if } 5 < x \ \end{cases}$

We can figure the value of $I_i$ intuitively.

$I_1 = 0$

$I_2 = \int_{-4}^x \frac{k}{h} dx = \frac{1}{3132} (1408 - 240 x +85 x^3 - 3 x^5)$

$I_3 = \int_{-4}^{-1} \frac{k}{h} dx = \frac{1}{2}$

$I_4 = \int_{-4}^{-1} \frac{k}{h} dx + \int_{1}^x \frac{k}{h} dx = \frac{1}{3132} (1724- 240 x +85 x^3 - 3 x^5)$

$I_5 = \int_{-4}^{-1} \frac{k}{h} dx + \int_{1}^4 \frac{k}{h} dx = 1$

As a result we have:

## Inverse function

The inverse function of $F$ is expressed as $F^{-1}$ such that:

$F^{-1} (g_x) = x$

In more general terms, if $y = f(x)$, the inverse function  $y^{-1} = f(y)$ must satisfy $y^{-1}(f(x))=x$.

### Example I

To begin with a simple example assume we have a pdf $f_2$ that describes the density function of an $x$ variable:

$f_2 = \begin{cases} \frac{6-3x}{6} &\quad\text{if } 0 < x \leq 2 \\ 0 &\quad\text{if } x \leq 0 \cup 2 < x \ \end{cases}$

The cdf is then:

$F_2 = \begin{cases} a &\quad\text{if } x \leq 0 \\ b &\quad\text{if } 0 < x \leq 2 \\ c &\quad\text{if } 2 < x \ \end{cases}$

where:

$a = 0$

$b = a +\frac{1}{6} \int_0^{x} (6-3x) dx = \frac{1}{6}(6x - \frac{3}{2} x^2)$

$c = a +\frac{1}{6} \int_0^{2} (6-3x) dx = 1$

The inverse function $f^{-1}$ can only be deduced on increasing $f$ functions. For this reason, only the $b$ component of the piecewise cdf is going to be calculated. We can easily find the solution with $x = \frac{-B \pm \sqrt{B^2 - 4 A C}}{2 A}$. The general approach is to define $y = f(x)$ and solve for $x$.

$b^{-1} = \begin{cases} 2(1-\sqrt{1-x}) \\ 2(1+\sqrt{1-x}) \ \end{cases}$

Since we expect that $b^{-1} (1) = 2$ and $b^{-1} (0) = 0$ we can rule out the second solution.

$F_2^{-1} = 2(1-\sqrt{1-x})$

### Example II

If we try to follow the same steps shown in Example I to calculate the inverse function of $f$, we would probably fail. High order polynomial function are difficult (if not impossible) to solve with analytical methods. Therefore, a numerical method is suggested for this problem.

At a first glance, switching from an analytical approach to a numeric method does not seem appealing. Nonetheless, in many cases we are better off working on a problem with a non-parametrical estimated density function (obtained from the empirical data) that with an analytical function determined by assumptions.

The first step is to have available the numeric data of the x-axis, pdf and cdf values.

The cdf function $F$ maps a variable in the x-axis to only one value in the y-axis. The inverse method $F^{-1}$ suggest the contrary; mapping the a variable form the y-axis into the x-axis.

So let’s suppose that we want to know the value of $F^{-1} (u)$. The first step is to find the position of the nearest value yet smaller than $u$ in the vector containing the data of the cdf. If the value is really close, the next step would be to use the position obtained to figure out what is the correspondent x-value. Pretty straight forward, right? There would not be problem with this if the “nearest value” is indeed significantly near to $u$. Due to the fact that this situation is not commonly true, an interpolation technique is recommended.

To maintain simplicity, linear interpolation is frequently used. Assume that the closest value to $u$ in the vector containing the data of the cdf is in the $i$-th position. Lets call the x-axis vector $x$ and the cdf vector $z$. Linear interpolation uses a simple linear model to estimate the corresponding x-value for $u$. In essence:

$\hat{x} = \frac{x[i+1]-x[i]}{z[i+1]- z[i]} (u - z[i]) + x[i]$

Since we know that $z[i] < u$ and $z[i]$ is the closest value smallest value to $u$ we also know that $u < z[i+1]$. By definition, cdf are increasing functions so $F^{-1}(z[i]) < F^{-1}(u) < F^{-1}(z[i+1])$ must also hold true. As a result $F^{-1}(u) \approx \hat{x}$.

## Inverse method

If there is a random variable $u$ drawn from a uniformly distribution $U[0,1]$, then $y$ is a random variable with $f$ distribution only if $y = F^{-1} (u)$.

Then by using random uniformly distributed variables we can simulate any distribution. The following code shows the implementation of the numeric method to calculate the inverse function.

As a result, the inverse method allowed us to generate 5000 random variables that apparently distribute according to $f$.

As it can be seen, the inverse method approach to generate random variables is a handy tool. The methodology allows us to apply this technique with analytical or numeric methods.

In further posts, we will explore how a density distribution can be estimated via non-parametric statistics. We will enhance monte-carlo simulations by doing so and hopefully build accurate predictive models. The inverse methodology is at the core of many high level simulations.

## Alcohol and radar plots in R with ggplot2

Radar plots may be an unusual way to represent data, but under the right circumstances they can provide meaningful visualizations. In this post I will present how to create and customize some basic parameters of radar plots in R programming language.

Make sure you have the following packages:

require(ggplot2)
require(tibble)

The easiest way to generate good-looking  yet simple data visualization is through the syntax of Grammar of Graphics, cleverly implemented by Hadley Wickham in his ggplot2 package. The tibble package, created by this same author, provides an alternative to the use of dataframes. In this post we’ll give it a try.

The database used in this example can be found at The World Health Organization Global Health Observatory data.

Radar plots are generally used to represent higher dimensional data in two dimensions. It does so by plotting each variable into a separate axis resembling polar coordinates. Each axis is arranged radially from the center at an equi-angular distance from each other. Then each observation is potted according to the value presented on each category, usually joined by a line forming a polygon.

Nevertheless, this way of representing data can be misleading. For instance, the use of many axis can interfere with the visualization. Furthermore, the use of different units among axis (and unit separations) is regarded as inappropriate. A general critique of radar plots can be found here.

Taking in account the possible downsides when using radial plots, we proceed to obtain and clean the data with the following code:

Cleaning is also important in the process of data exploration. In this case we get rid of some useless variables and add some rows concerning the average data.

Now we continue with the visualization. At first we plot all the observations we have. Then we move on to select specific cases to show how can radar plots can contribute to a better understanding of the data.

As a result, the following plots are generated:

In this plot we can see the 53 different countries in a radar form. The radar plot is not a good alternative to graph many observations. The shapes generated for each observation are indistinguishable  from each other. For this reason, it was decided to use the same color except in the average data. So at the end, this plot is merely illustrative.

The previous graph made obvious the need to reduce the number of observations. In this case 5 countries were randomly selected. Now we are able to see the observations and their consumption pattern.

To understand the nature of this radar plots, sometimes it is useful to use data you know. For this reason we selected some countries from which their alcoholic consumption preference are obvious. As stereotypes dictate, France shows a strong preference toward wine consumption, Germany toward beer and Russia toward spirits. We can clearly see and compare the consumption patterns for each country.

At a glance, the presentation of data in this format is really useful. The radar plot points toward the direction with more consumption per country. We can think of the shape of this plots as a “preference polygon” for each country. In this view the plots don’t overlap so we have a clean visualization. It provides an overall understanding of the consumption pattern of alcohol by country.

At the end, using radar plots is a tricky task. Not all data fits in the format, dimensions or number of observations that can make a radar plot interesting. As a rule of thumb, I would suggest only to use radar plots when you are confident that they can provide accurate meaning. Otherwise, the advise is to avoid them at all.

For further information:

This post was inspired by the publication From Parallel Plot to Radar Plot by Erwan Le Pennec.

Useful perspective offered by Graham Odds in A Critique of Radar Charts.