**4. Method of moments for uncertainty propagation in FTA and ETA**

The method of moments uses first and second moments of the input parameters (mean and variance) to estimate the mean and variance of the output function using propagation of variance or coefficient of variation. As a measure of uncertainty, the coefficient of variation is defined as a ratio of the standard deviation to the mean, which indicates the relative dispersion of uncertain data around the mean. The uncertainty measure is a readily interpretable and dimensionless measure of error, differently for standard deviation, which is not dimensionless [27].

In PRA, the method of moments can be used to propagate the uncertainties of the inputs (i.e., event probabilities) and propagate the uncertainty for the outputs. The probability density functions (pdfs) for the inputs can be estimated from reliability data of gathered components or from historical records of undesired events. Hypothesizing that the events (or basic events) are independent, probabilistic approaches for propagating uncertainties in FTs and ETs are given as follows in Sections 4.1 and 4.2, respectively [28].

#### **4.1 Method of moments applied to FTA**

The uncertainty propagation in a fault tree begins with the propagation of uncertainties of basic events through "OR" and "AND" gates, until it reaches the top event. The fault tree should be represented by MCSs in order to avoid direct dependence between intermediate events, facilitating probabilistic calculations.

For an "OR" gate of a fault tree, the probability of the output event, *Por*, is given by Eq. (4):

$$P\_{or} = \mathbf{1} - \prod\_{i=1}^{n} (\mathbf{1} - P\_i),\tag{4}$$

where *Pi* denotes the probability of ith (i = 1, 2, 3, …*, n*) independent events (or basic events) and *n* is the number of input events.

The uncertainty propagation through the "OR" gate is given by Eq. (5) that calculates the coefficient of variation of output, *C*<sup>0</sup> *or*, as function of the coefficients of variation of inputs, *C*<sup>0</sup> *i* , according to Eqs. (6) and (7) [29]:

$$\mathbf{1} + \mathbf{C}'^2\_{or} = \prod\_{i=1}^{n} \left(\mathbf{1} + \mathbf{C}'^2\_i\right),\tag{5}$$

$$C\_{or}' = \frac{s\_{or}}{1 - P\_{or}},\tag{6}$$

*Treatment of Uncertainties in Probabilistic Risk Assessment DOI: http://dx.doi.org/10.5772/intechopen.83541*

estimation. It is applicable only when the uncertainties in the basic events of the model are log-normally distributed. FW estimates are most accurate in the central range, and the tails of the distributions are poorly represented. The Wilks method requires relatively few samples and is computationally inexpensive. It is useful for providing an upper bound (conservative) for the percentiles of the uncertainty distribution. However, its calculated values are less accurate than the FW

estimates over practically the entire range of the distribution. For both Wilks and FW methods, the greatest errors are found in the low tails of the distributions, but in almost all reliability applications the high tails are of more interest than the

**4. Method of moments for uncertainty propagation in FTA and ETA**

(mean and variance) to estimate the mean and variance of the output function using propagation of variance or coefficient of variation. As a measure of uncertainty, the coefficient of variation is defined as a ratio of the standard deviation to the mean, which indicates the relative dispersion of uncertain data around the mean. The uncertainty measure is a readily interpretable and dimensionless measure of error, differently for standard deviation, which is not dimensionless [27]. In PRA, the method of moments can be used to propagate the uncertainties of the inputs (i.e., event probabilities) and propagate the uncertainty for the outputs. The probability density functions (pdfs) for the inputs can be estimated from reliability data of gathered components or from historical records of undesired events. Hypothesizing that the events (or basic events) are independent, probabilistic approaches for propagating uncertainties in FTs and ETs are given as follows

The uncertainty propagation in a fault tree begins with the propagation of uncertainties of basic events through "OR" and "AND" gates, until it reaches the top event. The fault tree should be represented by MCSs in order to avoid direct dependence between intermediate events, facilitating probabilistic calculations. For an "OR" gate of a fault tree, the probability of the output event, *Por*, is given

*Por* <sup>¼</sup> <sup>1</sup> � <sup>Y</sup>*<sup>n</sup>*

(or basic events) and *n* is the number of input events.

1 þ *C*´ 2 *or* <sup>¼</sup> <sup>Y</sup>*<sup>n</sup> i*¼1

*C*0

calculates the coefficient of variation of output, *C*<sup>0</sup>

*i*

*i*¼1

where *Pi* denotes the probability of ith (i = 1, 2, 3, …*, n*) independent events

The uncertainty propagation through the "OR" gate is given by Eq. (5) that

*or* <sup>¼</sup> *sor* 1 � *Por*

, according to Eqs. (6) and (7) [29]:

1 þ *C*´ 2 *i*

ð Þ 1 � *Pi ,* (4)

� �*,* (5)

*,* (6)

*or*, as function of the coefficients

The method of moments uses first and second moments of the input parameters

low tails [26].

by Eq. (4):

**132**

of variation of inputs, *C*<sup>0</sup>

in Sections 4.1 and 4.2, respectively [28].

*Reliability and Maintenance - An Overview of Cases*

**4.1 Method of moments applied to FTA**

$$C'\_i = \frac{s\_i}{1 - P\_i},\tag{7}$$

where *si* denotes the standard deviations of ith (i = 1, 2, 3, …*, n*) input, *n* is the number of input events, and *sor* is the standard deviation of the output of "OR" gate.

For an "AND" gate of a fault tree, the probability of output event, *Pand*, is given by Eq. (8):

$$P\_{and} = \prod\_{i=1}^{n} P\_{i\bullet} \tag{8}$$

where *Pi* denotes the probability of ith (i = 1, 2, 3, …, *n*) independent events (or basic events) and *n* is the number of input events.

The uncertainty propagation through the "AND" gate is given by Eq. (9). It calculates the coefficient of variation of output, *Cand*, as function of the coefficients of variation of inputs, *Ci*, according to Eqs. (10) and (11) [29]:

$$\mathbf{1} + \mathbf{C}\_{and}^2 = \prod\_{i=1}^n \left(\mathbf{1} + \mathbf{C}\_i^2\right),\tag{9}$$

$$C\_{and} = \frac{s\_{and}}{P\_{and}},\tag{10}$$

$$\mathbf{C}\_{i} = \frac{s\_{i}}{P\_{i}},\tag{11}$$

where *si* denotes the standard deviations of ith (i = 1, 2, 3, …, *n*) input, *n* is the number of input events, and *sand* is the standard deviation of output of the "AND" gate.

#### **4.2 Method of moments applied to ETA**

Uncertainty propagation in an event tree is similar (or analogous) to uncertainty propagation of an "AND" gate of a fault tree. The frequency of occurrence of each accident scenario, *Fseq*, is given by Eq. (12),

$$F\_{\varkappa q} = \lambda \times \prod\_{i=1}^{n} P\_{i\nu} \tag{12}$$

where λ is the frequency of occurrence of the initiating event and *Pi* denotes the probabilities of ith (i = 1, 2, 3, …, *n*) subsequent independent events leading to the accident scenario and *n* is the number of input events. These values can be obtained from fault trees constructed for each ith event or system failure of the event tree.

The uncertainty propagation through the accident sequence is given by Eq. (13) that provides the coefficient of variation of accident sequence, *Cseq*, as function of the coefficients of variation of subsequent events, *Ci*, according to Eqs. (14) and (15), respectively:

$$\mathbf{1} + \mathbf{C}\_{seq}^2 = \prod\_{i=1}^n \left(\mathbf{1} + \mathbf{C}\_i^2\right),\tag{13}$$

$$\mathbf{C}\_{seq} = \frac{\mathbf{s}\_{seq}}{F\_{seq}},\tag{14}$$

$$\mathbf{C}\_{i} = \frac{s\_{i}}{P\_{i}},\tag{15}$$

X(7)), characterized by the log-normal distributions. This simple example was chosen in order to compare the results of the method of moments with the uncer-

The log-normal distributions assigned to the basic events (represented by median and mean values of probabilities, error factors, and standard deviations) are shown in **Table 2**. An analysis of the fault tree shows that its minimal cut sets (MCSs) are X(1), X(6), X(7), X(2)X(4), X(2)X(5), X(3)X(4), and X(3)X(5), which are used to estimate the top event probability and propagate the uncertainties. The application of the method of moments is carried out in a bottom-up approach. Starting from basic events of the fault tree, the coefficients of variation of the intermediate events are estimated using Eqs. (4)–(7) for "OR" gates and

Eqs. (8)–(11) for "AND" gates. This procedure is repeated interactively until the top event is reached, and its standard deviation is obtained. Considering that, in the same way as the basic events, the top event has also a log-normal distributions, Eqs. (16)–(20) are used to estimate the 5th percentile, median, and 95th percentile for the top event, as shown in **Table 3**. These estimates are slightly lower than the values obtained by Chang et al. [8] with the Monte Carlo simulation (percent

*Fault tree analysis for a generic top event "system failure" (adapted from Chang et al. [8]).*

*EF* **of lognormal pdf**

X(1) 1.00 <sup>10</sup><sup>3</sup> <sup>3</sup> 1.25 <sup>10</sup><sup>3</sup> 9.37 <sup>10</sup><sup>4</sup> X(2) 3.00 <sup>10</sup><sup>2</sup> <sup>3</sup> 3.75 <sup>10</sup><sup>2</sup> 2.81 <sup>10</sup><sup>2</sup> X(3) 1.00 <sup>10</sup><sup>2</sup> <sup>3</sup> 1.25 <sup>10</sup><sup>2</sup> 9.37 <sup>10</sup><sup>3</sup> X(4) 3.00 <sup>10</sup><sup>3</sup> <sup>3</sup> 3.75 <sup>10</sup><sup>3</sup> 2.81 <sup>10</sup><sup>3</sup> X(5) 1.00 <sup>10</sup><sup>2</sup> <sup>3</sup> 1.25 <sup>10</sup><sup>2</sup> 9.37 <sup>10</sup><sup>3</sup> X(6) 3.00 <sup>10</sup><sup>3</sup> <sup>3</sup> 3.75 <sup>10</sup><sup>3</sup> 2.81 <sup>10</sup><sup>3</sup> X(7) 1.00 <sup>10</sup><sup>3</sup> <sup>3</sup> 1.25 <sup>10</sup><sup>3</sup> 9.37 <sup>10</sup><sup>4</sup>

*Basic event distribution for a generic top event "system failure" (χ<sup>50</sup> and* EF *values were taken from Ref. [8]).*

**Mean of lognormal pdf** *(P)* **Standard deviation of lognormal pdf (***s***)**

tainty propagation analyses using Monte Carlo simulation.

*Treatment of Uncertainties in Probabilistic Risk Assessment*

*DOI: http://dx.doi.org/10.5772/intechopen.83541*

**Figure 4.**

**Basic events**

**Table 2.**

**135**

**Median of lognormal pdf** *(***χ50***)*

where *si* denotes the standard deviations of ith (i = 1, 2, 3, …*, n*) subsequent event of the sequence, *n* is the number of input events, and *sseq* is the standard deviation of the accident sequence.

## **4.3 Propagation of log-normal distributions**

Many uncertainty distributions associated with the basic events of fault trees (reliability or failure probability data) often can be approximated in reliability and safety studies by log-normal functions. If a random variable ln(*x*) has a normal distribution, the variable *x* has then a log-normal distribution. The log-normal probability density function (pdf), *f(x)* is then given by Eq. (16) [30]:

$$f(\mathbf{x}) = \frac{1}{\mathbf{x}\sigma\sqrt{2\pi}} \exp\left(\frac{-\left(\ln\left(\mathbf{x}\right) - \mu\right)^2}{2\sigma^2}\right),\tag{16}$$

where μ and σ are the mean and the standard deviation of ln(*x*), respectively (i.e., these are the parameters of the "underlying" normal distribution).

The error factor, *EF*, of a log-normal pdf is defined as Eq. (17):

$$EF = \frac{\chi\_{\mathfrak{H}}}{\chi\_{\mathfrak{H}}} = \frac{\chi\_{\mathfrak{H}}}{\chi\_{\mathfrak{H}}},\tag{17}$$

where χ95, χ50, and χ<sup>5</sup> are the 95th, 50th (median), and 5th percentiles, respectively.

*EF* is often used as an alternative to the standard deviation of "underlying" normal distribution, σ, for characterizing the spread of a log-normal distribution, and these two quantities are related by Eq. (18):

$$EF = \exp\left(\mathbf{1}.645\,\sigma\right). \tag{18}$$

The mean, *P*, and standard deviation, *s*, of the log-normal variable, *x*, can be given by the following Eqs. (19) and (20), respectively:

$$P = \exp\left(\mu + \frac{\sigma^2}{2}\right),\tag{19}$$

$$s = \sqrt{\exp\left(2\mu + \sigma^2\right) \left[\exp\left(\sigma^2\right) - 1\right]}\tag{20}$$

Eqs. (4)–(20) are used for uncertainty propagation of log-normal pdf in fault and event trees, as illustrated in the following examples.
