HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-991102-A

Connections between the distance- and probability models

To avoid the dangerous situations we have to know in every condition, how close the system will be to the danger after the next step. In real life usually we know only the probability of a transition between conditions, the distance is unknown. Thus, it is useful to find a connection between the two models, and for us now it is more important the transition, which makes distance from probability. So we are looking for a function $\mu: (0,1] \rightarrow \mathbf{R}_0^+$, which has the following properties:

(i)

continuous,

(ii)

strictly monotonously decreasing,

$$p_1 < p_2 \Rightarrow \mu(p_1) > \mu(p_2),$$

(iii)

$$\mu(1)=0 \textrm{ and } \lim_{p \rightarrow 0} \mu(p)=\infty,$$

(iv)

$$\mu(p_1 \cdot p_2) = \mu(p_1) + \mu(p_2),$$

(v)

for paralell ways we have

$$\mu(p_1 + p_2) = \mu(p_1) \amalg \mu(p_2),$$

where $\amalg$ is a paralell composition operator.

Considering properties (i)-(iii) we have more different function-candidates, e.g.

$$d_{i \rightarrow j} \sim \frac{1}{p_{i \rightarrow j}}-1 \t... ...log\frac{1}{p_{i \rightarrow j}}= - \log p_{i \rightarrow j}.$$

However, from property (iv), which can be rewritten in the form

$$d_{ik} = d_{ij} + d_{jk}$$

follows, that the solution can only be some kind of logarithmic function ([3]).

EXAMPLE Let us investigate the problem with the candidate $d_{i \rightarrow j} \sim \frac{1}{p_{i \rightarrow j}}-1$. In this case we search the solution in the form $d_{i \rightarrow j} = c \cdot \left(\frac{1}{p_{i \rightarrow j}}-1\right)+ d$. From property (iv) the following equality holds:

$$c \cdot \left(\frac{1}{p_{i \rightarrow j}}-1\right)+ d + c ... ...{1}{p_{i \rightarrow j} \cdot p_{j \rightarrow k}}-1\right)+ d,$$

from which with simple transformations

$$\frac{c}{p_{i \rightarrow j}}+ \frac{c}{p_{j \rightarrow k}}... ... \frac{c}{p_{i \rightarrow j} \cdot p_{j \rightarrow k}} -c +d,$$

$$\frac{c p_{i \rightarrow j} + c p_{j \rightarrow k}} {p_{i \... ...c +d= \frac{c}{p_{i \rightarrow j} \cdot p_{j \rightarrow k}}.$$

Thus, we are not able to choose $c$ and $d$ independently from $p_{i\rightarrow j}$ and $p_{j \rightarrow k}$, so this function is not appropriate.

So for our function $d_{i \rightarrow j} \sim \log \frac{1}{p_{i \rightarrow j}}= - \log p_{i \rightarrow j}$. Knowing that $0 \le p_{i \rightarrow j} \le 1$, we have $\infty \ge - \log p_{i \rightarrow j} \ge 0$. Obviously

$$- \log p_{i \rightarrow j} - \log p_{j \rightarrow k} = - \log (p_{i \rightarrow j} \cdot p_{j \rightarrow k}),$$

and assuming the form $c ( - \log p_{i \rightarrow j}) + d$ we can choose $d=0$. The base of the logarithm can be an arbitrary number $a$, with $a>1$ from property (ii). Thus, we have the desired connection between the two models. We can specify the distance from the danger (starting from the probability model) in the following manner:

a)

Starting from a given condition we specify the probability of reaching the danger(ous conditions).

b)

Using the logarithmics proportionality we change to the distance model, getting so the distance from the danger (finally, we apply a constant multiplier if needed).

EXAMPLE Let us assume, that in a whole-type probability model from a given condition $c_1$ we can go directly into three conditions, from which two are dangerous ($a_1$ and $a_2$). How far are we from the danger?

System:

$$\begin{eqnarray*} c_2 \longleftarrow & c_1 & \longrightarrow a_1\\ & \downarrow & \\ & c_2 & \end{eqnarray*}$$

where $p_{c_1 \rightarrow c_2}=0.85$, $p_{c_1 \rightarrow a_1}=0.1$, $p_{c_1 \rightarrow a_2}=0.05$. Then $p_{a_1 \rightarrow v}=0.1 + 0.05 = 0.15$, $d_{a_1 v}= - \log 0.15$. Applying a probability estimate

$$0.85^4 \approx 0.5220 > 0.5 > 0.85^5 \approx 0.4437,$$

i.e. the system runs into danger in $4-5$ steps, prospectively. Since $- \textrm{ln} 0.15 \approx 1.8971$, using the function ln it is suitable to apply a constant multiplier $c=2$ to get the correct distance.

Let us denote the inverse of function $\mu$ with $\pi:\mathbf{R}_0^+ \rightarrow [0,1]$. Then $\pi$ assures way through from distance model to probability model. Its required properties can be written similarly, as those of function $\mu$:

(i)

continuous,

(ii)

strictly monotonously decreasing,

$$d_1 < d_2 \Rightarrow \pi(d_1) > \pi(d_2),$$

(iii)

$$\pi(0)=1 \textrm{ and } \lim_{d \rightarrow \infty} \pi(d)=0,$$

(iv)

$$\pi(d_1 + d_2) = \pi(d_1) \cdot \pi(d_2),$$

(v)

for paralell ways we have

$$\pi(d_1 \amalg d_2) = \pi(d_1) + \pi(d_2).$$

Similarly as above it can be proved, that $\pi$ is some kind of exponential function. Usually it can be written in the form $a^{-d}$, where $a>1$ from property (ii).

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-991102-A