**4.4 Illustrative example 2 - Simulation result**

Can the results from example 1 be proved by a simulation, if the number of measured values is n=30?

For both cases a) 0.3, 0.5 and b) 0.05, 0.99 the values of *k* were found in Fig. 1 and Fig. 2 respectively:


In MATLAB, a set of 1000 samples of normal distribution with zero mean and standard deviations equal to one was generated. From the first 30 samples the mean value and standard deviation were estimated (16). Then the interval a aa a ks, ks was selected in accordance to (26) and the probability of falling into interval was computed from the whole set of 1000 samples - which is a procedure of how to compute measuring system's reliability.

The above mentioned procedure was repeated 5 000 times, and the probability of exceeding the predefined measuring system's reliability limit was computed - thus the test of measuring system's dependability was performed.

The obtained results for measuring system's dependability through simulation will be summarized as follows:

a. 0.553

b. 0.9864

The test was repeated many times and the predefined parameters for measuring system's reliability and dependability were achieved in all experiments.

Fig. 1. Dependence of parameter k on number of measurements 0.3. 0.5

Fig. 2. Dependence of parameter k on number of measurements 0.05. 0.99

k-prameter

0 5 10 15 20 25 30 35 40 45 50

Number of measurements

k-prameter

0 5 10 15 20 25 30 35 40 45 50

Number of measurements

Fig. 2. Dependence of parameter k on number of measurements 0.05. 0.99

Fig. 1. Dependence of parameter k on number of measurements 0.3. 0.5

1

2

4

6

8

10

Value of k

12

14

16

1.2

1.4

1.6

1.8

Value of k

2

2.2

2.4

2.6

2.8

### **5. Assessment of safety performance parameters**

We can suppose N sensors data available where the probability of right error detection is marked as PRD and the probability of non-correct error detection as PFD . Because of the enormous safety and economical impact in case of non-correct error alert, the method of filtering *"M from N"* will be presented.

Let us have N sensors and for simplicity let us suppose the same probabilities of correct PRD and non-correct error detection PFD on each sensor. If this assumption is not fulfilled the method can be easily extended to a more general case.

As mentioned above, the hypothesis H0 represents perfect system behavior (non system error, no sensor error) and hypothesis H1 as a state with detected error (error of system, or error of sensors).

In the next equation, the probability of error detection on *k* sensors of *N* sensors (*N-k* sensors do not detect errors) is given in case the system does not display any error (conditioned by hypothesis H0 ):

$$\mathbf{P}\begin{bmatrix}\mathbf{k} \end{bmatrix} \cdot \mathbf{H}\_0\mathbf{j} = \begin{pmatrix} \mathbf{N} \\ \mathbf{k} \end{pmatrix} \cdot \mathbf{P}\_{\rm FD}^{\rm k} \cdot \left(\mathbf{1} - \mathbf{P}\_{\rm FD}\right)^{\rm N-k} \tag{33}$$

In the same way, the probability of error detection by *k* of *N* sensors is given in case the system is in an error state (conditioned by hypothesis H1 ):

$$\mathbf{P}\begin{bmatrix}\mathbf{k} \mid & \mathbf{H}\_1 \end{bmatrix} = \begin{pmatrix} \mathbf{N} \\ \mathbf{k} \end{pmatrix} \cdot \mathbf{P}\_{\rm RD}^{\rm k} \cdot \left(1 - \mathbf{P}\_{\rm RD}\right)^{\rm N-k} \tag{34}$$

The main idea of "M from N" filtering is in selection of value M (threshold) defining the minimum number of sensors that detected error. If M sensors detect error then this error is taken as the real system error and the system starts sending error alert signals. The threshold M should be selected with respect to the following probabilities:

$$\begin{aligned} \mathbf{P\_F} &= \sum\_{\mathbf{k=M}}^{\mathbf{N}} \binom{\mathbf{N}}{\mathbf{k}} \cdot \mathbf{P\_{RD}^{\mathbf{k}}} \cdot \left(1 - \mathbf{P\_{RD}}\right)^{\mathbf{N}-\mathbf{k}}\\ \mathbf{P\_D} &= \sum\_{\mathbf{k=M}}^{\mathbf{N}} \binom{\mathbf{N}}{\mathbf{k}} \cdot \mathbf{P\_{FD}^{\mathbf{k}}} \cdot \left(1 - \mathbf{P\_{FD}}\right)^{\mathbf{N}-\mathbf{k}} \end{aligned} \tag{35}$$

where P ,P F D means probability of a *false alert* (an error is detected but the system works without any errors) and the probability of the *right detection* (the system error is correctly detected).

The number of detectors N and the threshold M can be chosen based on sensors parameters PRD , PFD and required probabilities P ,P F D .

Methods of data fusion and comparison are the main tools for estimation of system performance parameters (accuracy, reliability, integrity, continuity, etc.) and can be used for a derivation of an exact definition of false alert and right detection probabilities.

#### **5.1 Illustrative example - Geo-object detection**

In this example the measurement data comparison will be used as a tool for better geo-object detection in, e.g., electronic tolling application.

We can suppose N available position measurements of a geo-object where the probability of the right geo-object detection is marked as PRD and the probability of non-correct (false) geo-object detection as PFD . Let the hypothesis H0 represent the assumption of a perfect geo-object detection (no detection error reported). The hypothesis H1 represents a non-correct geo-object detection (error caused, for example, by wrong position accuracy, etc.).

The probability of *k* non-correct geo-object detections of *N* measurements for the final assumption that the geo-object is perfectly detected (conditioned on the hypothesis H0 ) can be given:

$$\mathbf{P}\begin{bmatrix}\mathbf{k} \mid \ \mathbf{H}\_0\end{bmatrix} = \begin{pmatrix} \mathbf{N} \\ \mathbf{k} \end{pmatrix} \cdot \mathbf{P}\_{\rm FD}^{\rm k} \cdot \left(1 - \mathbf{P}\_{\rm FD}\right)^{\rm N-k} \tag{36}$$

The probability of *k* correct geo-object detections of N measurements for the final assumption of non-correct geo-object detection (conditioned on hypothesis H1 ) is given:

$$\mathbf{P}\begin{bmatrix}\mathbf{k} \end{bmatrix} \cdot \mathbf{H}\_1\mathbf{l} = \begin{pmatrix} \mathbf{N} \\ \mathbf{k} \end{pmatrix} \cdot \mathbf{P}\_{\rm RD}^{\rm k} \cdot \left(\mathbf{1} - \mathbf{P}\_{\rm RD}\right)^{\rm N-k} \tag{37}$$

The main idea of "M matches from N measurements" principle is in the selection of the threshold M with respect to the following probabilities:

$$\begin{aligned} \mathbf{P\_F} &= \sum\_{\mathbf{k=M}}^{\mathbf{N}} \binom{\mathbf{N}}{\mathbf{k}} \cdot \mathbf{P\_{FD}^{\mathbf{k}}} \cdot \left(\mathbf{1} - \mathbf{P\_{FD}}\right)^{\mathbf{N}-\mathbf{k}}\\ \mathbf{P\_D} &= \sum\_{\mathbf{k=M}}^{\mathbf{N}} \binom{\mathbf{N}}{\mathbf{k}} \cdot \mathbf{P\_{RD}^{\mathbf{k}}} \cdot \left(\mathbf{1} - \mathbf{P\_{RD}}\right)^{\mathbf{N}-\mathbf{k}} \end{aligned} \tag{38}$$

where P ,P F D means the probability of a *false alert of geo-object detection* (the geo-object is detected even though the vehicle did not go through it) and the probability of a *right geoobject detection* (the right geo-object is detected based on the measured data, and the vehicle went through it).

The number of measurements N and the threshold M can be chosen based on the position probabilities PFD , PRD and the required probabilities P ,P F D . Further discussion will be presented within an illustrative example below.

There are two parallel roads (one under tolling, the other one free of charge) and the distance D between them of 20 meters, as it is shown in Fig 3. The length L is supposed to be 1 kilometer.

In this example, we will try to tune the parameter *M* to increase the probability of the correct toll road detection in order to reach the expected value of more than 99%.

In this example the measurement data comparison will be used as a tool for better geo-object

We can suppose N available position measurements of a geo-object where the probability of the right geo-object detection is marked as PRD and the probability of non-correct (false) geo-object detection as PFD . Let the hypothesis H0 represent the assumption of a perfect geo-object detection (no detection error reported). The hypothesis H1 represents a non-correct geo-object detection (error caused, for example, by wrong position accuracy,

The probability of *k* non-correct geo-object detections of *N* measurements for the final assumption that the geo-object is perfectly detected (conditioned on the hypothesis H0 ) can

> N k <sup>k</sup> 0 FD FD

The probability of *k* correct geo-object detections of N measurements for the final assumption of non-correct geo-object detection (conditioned on hypothesis H1 ) is given:

> N k <sup>k</sup> 1 RD RD

The main idea of "M matches from N measurements" principle is in the selection of the

F FD FD

 

N P P 1P k N P P 1P k

D RD RD

where P ,P F D means the probability of a *false alert of geo-object detection* (the geo-object is detected even though the vehicle did not go through it) and the probability of a *right geoobject detection* (the right geo-object is detected based on the measured data, and the vehicle

The number of measurements N and the threshold M can be chosen based on the position probabilities PFD , PRD and the required probabilities P ,P F D . Further discussion will be

There are two parallel roads (one under tolling, the other one free of charge) and the distance D between them of 20 meters, as it is shown in Fig 3. The length L is supposed to be

In this example, we will try to tune the parameter *M* to increase the probability of the correct

 

k M

k M

toll road detection in order to reach the expected value of more than 99%.

<sup>N</sup> N k <sup>k</sup>

<sup>N</sup> N k <sup>k</sup>

 

 

(36)

(37)

(38)

N P k| H P 1 P <sup>k</sup>

N P k| H P 1 P <sup>k</sup>

**5.1 Illustrative example - Geo-object detection** 

detection in, e.g., electronic tolling application.

threshold M with respect to the following probabilities:

presented within an illustrative example below.

etc.).

be given:

went through it).

1 kilometer.

We expect a maximum vehicle speed of 200 km/h or 55 m/s. If the length is 1000 m and the GPS receiver monitors the position every second, we can obtain as many as 18 position measurements per one road. The road can be distinguished by GPS received with probability app. 70% (we can assign the measurement to the right road, if the error is lower than D/2 which is in our case 10 meter - this accuracy is typically achieved by a GPS receiver at a probability level of 70%).

Fig. 3. Two parallel roads and Toll detection being on one of the roads

Based on the above mentioned assumptions, we can summarize the following parameters:

$$P\_{\rm RD} = 0.7, \ P\_{\rm FD} = 0.3, \ N = 18$$

Using the equations (38), the probabilities P ,P F D for different parameters M will be as given in Tab.1.


Table 1. Probabilities P ,P F D and their dependence on parameter M

If the parameter M is 6 or 8, we can achieve the requested probability of the geo-object detection higher than 99%. On the other hand, for M=6 the probability of lost vehicles is higher (the vehicles used the toll road, but the system did not detect them). For M=8 we can achieve a better balance between both probabilities P ,P F D . If the user needs to minimize the loss of a vehicle and to keep the acceptable detection probability, the variant M=10 could be a good compromise.
