Fuzzy Probability Theory---(1)Fuzzy Sets

Introduction

Example: $X=\{x_1,x_2,x_3\}$ with $P(x_i)=a_i,~\sum^{3}_{i=1}a_i=1$. In practice all the $a_i$ values must be known exactly through estimatimation or provided by experts.

Now assume that some of these $a_i$ values are uncertain and we will model this uncertainty using intervals. Suppose we estimate $a_1\in[0.1,0.3],~a_2\in[0.3,0.7],~a_3\in[0.2,0.4]$.

Then $P(\{x_1,x_2\})=\{a_1+a_2|a_1\in[0.1,0.3],~a_2\in[0.3,0.7],~a_3\in[0.2,0.4],~\sum^{3}_{i=1}a_i=1\}$=[0.6,0.8] which is not the sum of the two intervals [0.1,0.3]+[0.3,0.7] = [0.4,1.0]. Because the constraint that the probabilities must add to one. 

Interval probabilities can be easily extended to fuzzy probabilities because a fuzzy probability is a fuzzy number composed of a nested collection of intervals constructed by taking $\alpha$-cuts. The interval $P(\{x_1,x_2\})$=[0.6,0.8] will just only be one of the alpha-cuts of the fuzzy probability of event $\{x_1,x_2\}$  which will be a fuzzy number.

When all the probabilities are fuzzy we will still insist that the sum of all the individual probabilities is one. This will produce what we call restricted fuzzy arithmetic  $\sum^{n}_{i=1}a_i=1$.

A fuzzy random variable is just a random variable with a fuzzy probability mass function (discrete case such as $b(n,\bar{p})$ where $\bar{p}$ is a fuzzy number), or a fuzzy probability density function ( the continuous case such as $N(\bar{u},\bar{\sigma}^2)$ where $\bar{u},~\bar{\sigma}$ are fuzzy numbers).

Fuzzy Sets

If $\Omega$ is some set, then a fuzzy subset $\bar{A}$ of $\Omega$ is defined by its membership function $\bar{A}(x) \in [0,1]$ for all $x$ in $\Omega$. When $\bar{A}(x)$ is always equal to one or zero we obtain a crisp(non-fuzzy) subset of $\Omega$.

Fuzzy Numbers

Fuzzy number is a special type of fuzzy sets. Here we only involve triangular (shaped), or trapezoidal(shaped) fuzzy numbers. We will be using fuzzy numbers in this book to describe uncertainty.

$\alpha$-cuts

Alpha-cuts are slices through a fuzzy set producing regular (non-fuzzy) sets $$\bar{A}[\alpha]=\{x\in\Omega|\bar{A}(x)\geq \alpha\}$$. For any fuzzy number $\bar{A}$ we know that $\bar{A}[\alpha]$ is a closed, bounded, interval for $0\leq \alpha \leq 1$. We will write this as $$\bar{A}[\alpha]=[q_1(\alpha),q_2(\alpha)]$$. More properties about $q_i(\alpha)$, please see the image below.

For example, if $L(x)=0.5+\frac{1}{1+e^{1-x}},~x\in[-\infty,1]$, we get $x=1-In(3-2y)+In(2y-1),~y\in[0.5,1]$ then $q_1(\alpha)=1-In(3-2\alpha)+In(2\alpha-1)$.

Fuzzy Arithmetic

There are 2 ways to compute fuzzy numbers: (1)Extension principle (2)$alpha$-cuts and interval arithmetic.

(1)Extension principle. Denote $\bar{*}$ as a 2-Dim fuzzy arithmetic operator between $\bar{A},~\bar{B}$, then the result of $\bar{*}$ $\bar{C}$ is

$$\bar{C}(z)=sup\{min(\bar{A}(x),\bar{B}(y))~|~x~\bar{*}~y=z\}$$.

Obviously, given $\bar{A}$ and $\bar{B}$, the above equation is quite complicated. So, we now present an equivalent procedure based on a-cuts and interval arithmetic. 

(2-1)Interval Arithmetic. Let $[a_1,b_1]$ and $[a_2,b_2]$ be two closed, bounded, and intervals of real numbers. If $\bar{*}$ denotes addition, subtraction, multiplication, or division, then $[a_1,b_1]~\bar{*}~[a_2,b_2]=[\alpha,\beta]$ where $$[\alpha,\beta]=\{a~\bar{*}~b~|~a\in[a_1,b_1],~b\in[a_2,b_2]\}$$

(2-2)Fuzzy Arithmetic. 

 Again we have two fuzzy numbers $\bar{A},~\bar{B}$. We know $\alpha$-cuts are closed, bounded, intervals so let $\bar{A}[\alpha]=[a_1(\alpha),a_2(\alpha)]$ and $\bar{B}[\alpha]=[b_1(\alpha),b_2(\alpha)]$, then $\bar{C}[\alpha]=\bar{A}~\bar{*}~\bar{B}$. Therefore fuzzy arithmetic becomes interval arithemetic. This method is equivalent to the extension principle method of fuzzy arithmetic. Obviously, this procedure, of $\alpha$-cuts plus interval arithmetic, is more user(and computer) friendly.

Fuzzy Functions

A fuzzy function is a mapping from fuzzy numbers into fuzzy numbers such as $H(\bar{X},\bar{Y})=\bar{Z}$. Where do these fuzzy functions come from? They are usually extensions of real-valued functions. For example, $H(\bar{X},\bar{Y})=\bar{Z}$ comes from $h:[a,b]\times[c,d]\to R$. There are 2 ways of extending $h$ to $H$:(1)Extension principle (2)using $\alpha$-cuts and interval arithmetic.

Extension Principle

Let $z=h(x_1,x_2,...,x_n)$ for $x_i$ in $[a_i,b_i]$. We extend $h$ to $H(\bar{X_1},\bar{X_2},...,\bar{X_n})=\bar{Z}$ as  $$\bar{Z}(z)=sup\{min(\bar{X_1}(x_1),\bar{X_2}(x_2),...,\bar{X_n}(x_n))~|~h(x_1,x_2,...,x_n)=z\}$$.

z = peaks(25);
figure
mesh(z);
surfl(z);
shading interp;
hold on;

x=0:.1:30;
[X,Y] = meshgrid(x);
a=0; b=0; c=1; d=1.5;
Z=(d-a * X - b * Y)/c;
surf(X,Y,Z);
shading flat;
xlabel('x'); ylabel('y'); zlabel('z=h(x,y)');

Alpha-Cuts and Interval Arithmetic

For $\alpha$-cuts of $\bar{Z}=H(\bar{X},\bar{Y})$, assuming $h$ is continuous, we have $$\bar{Z}[\alpha]=[z_1(\alpha),z_2(\alpha)]=[min\{P(\alpha)\},max\{P(\alpha)\}]$$ where $P(\alpha)=\{h(x,y)~|~x\in\bar{X}[\alpha],~y\in\bar{Y}[\alpha]\}$. Let's look at the $z=h(x,y)$

 

 

N = 5.0;
x=linspace(-N, N);
y=x;
[X,Y]=meshgrid(x,y);
z=(1000/sqrt(2*pi).*exp(-(X.^2/2)-(Y.^2/2)));
surf(X,Y,z);
%shading interp
%axis tight

%figure(1);
% Use hold on and hold off to plot multiple cubes
hold on;
% Call the function to plot a cube with dimension of X, Y, Z, at point [x,y,z].
cube_plot([-0.5,-0.5,0],1.4,1,370,'r');
% Figure configurations
% Define the range of x-axis, y-axis, and z-axis in form of
% [xmin,xmax,ymin,ymax,zmin,zmax].
%axis([1,2,0,1,0,1]);
% Set the axis with equal unit.
%axis equal;
% Show grids on the plot
%grid on;
% Set the lable and the font size
%xlabel('X','FontSize',18);
%ylabel('Y','FontSize',18);
%zlabel('Z','FontSize',18)
% Control the ticks on the axises
%h = gca; % Get the handle of the figure
% h.XTick = 0:0.5:1;
% h.YTick = 0:0.5:1;
% h.ZTick = 0:0.5:1;
% Set the color as transparient
%material metal
%alpha('color');
%alphamap('rampup');
% Set the view point
%view(30,30);

Extension Principle VS. ($\alpha$-cuts, Interval Arithmetic)

 Actually they are different! Let us write $\bar{Z}^{*}=H(\bar{X})$ for the extension principle method of extending $h$ to $H$, while $\bar{Z}=H(\bar{X})$ denote for the $\alpha$-cut and interval arithmetic extension of $h$. It has been known that $\bar{Z}$ may be something larger than $\bar{Z}^{*}$. More specifically, if each fuzzy number appears only once in the fuzzy expression, the two methods produce the same results. However, if a fuzzy number is used more than once, the 2 procedures can give different results.