Science of Societal Safety pp 113-120 | Cite as

# Methods in Risk Analysis

## Abstract

To protect our lives and properties from disasters and accidents, we need to identify sources of risks, acknowledge weaknesses in our societies and ourselves, and be readily prepared in both hardware and software. We frequently express the odds of events taking place with probabilistic numbers; thus, we must have at least the minimum knowledge in mathematical statistics. This chapter first introduces quantitative evaluations of probabilities of events to cause damages to us and the magnitudes of the damages. We will then learn how to analyze and estimate risks using the evaluations. The chapter closes with decision-making methods in finding the best measures that minimize the risks.

## Keywords

Confidence interval Linear model Mathematical programming Random variables Relative risk## 10.1 Evaluation and Probabilities of Risks

*E*

_{1},

*E*

_{2}, the function

*μ*that satisfies the following conditions is called the fuzzy measure.

The first two conditions show that when we set the size of the set of all possible results to 1, fuzzy measure is a function that expresses the ratio of the results of interest occupy. The third condition is called monotonicity, and it shows that the evaluation goes up when more results are included.

*λ*is a real number larger than −1, a fuzzy measure that satisfies the next condition is called

*λ*-fuzzy measure.

*λ*-fuzzy measures with

*λ*= 0. Probabilities have complete additivity expressed with the following equation:

*λ*-fuzzy measure is a measure that generalizes complete additivity required to probability. In fact, when the parameter

*λ*takes a positive number, the measure shows the property called super-additivity with the expression:

*λ*takes a negative value expressed:

In classic probability theory, when a trial results in *N* equally probable cases, and the number of events of interest is *r*, the occurrence probability of the events is defined *r*/*N*. For example, rolling a cubic die and recording the result have six possible outcomes. With a proper die, all results have equal chances; thus, the possibility of getting 1 is 1/6. The occurrence probability of an event equals the relative frequency of the event occurrence with an infinite number of independent trials. Natural phenomena, however, like flood or earthquake often break out with some events in combination, and their occurrence usually depends on past records; thus, we need to proceed carefully in applying probabilistic models to their risks.

*p*, and the number of flights to

*n*, the probability of not being involved in an accident with a single flight is 1 −

*p*. The probability, thus, of not encountering an accident at all with

*n*flights is (1 −

*p*)

^{n}. Subtracting this probability from the entire probability of 1, we find the probability of being involved with at least one accident is 1 − (1 −

*p*)

^{n}. The frequency today of airplane accidents is about 0.3 times for each million flights; thus, if a person flies on an airplane everyday for 80 years, the probability we are looking for is:

A variable that changes its values probabilistically is called a random variable, and a random variable that changes its chances with time is called a stochastic process. When the set of values that a random variable can take, like in the case of amount of rainfall or earthquake magnitude, forms a section of continuous real values. In this case, the distribution function that defines the probability that the value is less than a threshold or its derivative, the probabilistic density function, gives an idea of what the probability distribution is like. The sum of many mutually independent random variables forms a normal distribution. Normal distributions are often used to model probabilistic random phenomena like measurement errors. The probabilistic density function for a normal distribution is bilaterally symmetric around the average value. A random variable with normal distribution is most likely to take a value near the average, and the probability drops as the value shifts away from the average.

On the other hand, for stochastic processes with increasing occurrence probability with time, like for the case of machine failure, the random variable that expresses the machine life or time to failure follows the Weibull distribution. If the distribution function or the probabilistic density function is known, we can find the confidence interval about characteristic values of the average or variance without having to observe the phenomena for an infinite number of times.

Odds is a measure that we often use in comparing the likelihood of events. We can calculate the odds by dividing the number of cases that the event of interest is taking place by the number of cases it is not. If the occurrence probability of event is known, the odds for the event is the occurrence probability divided by the probability the event does not occur. For example, assume there are ten test procedures for a certain illness and A was positive with five tests, whereas B had eight positive results. A’s odds is 5/(10–5) = 1 and B’s odds 8/(10–8) = 4. B’s odds is 4 times that of A and the possibility of B having the illness is 4 times that of A. In general, the odds of an event of interest divided by that of a reference event is called the odds ratio. As we will discuss in the next section, the odds ratio is often used in statistically estimating the magnitude of risk for an event.

Among objects lost with tsunami or fires, there are things like “photo album of memories” that are difficult to give monetary evaluation, i.e., things that cause big psychological pain when lost. In case it is difficult to directly calculate the absolute value of a property, comparing the relative value to evaluations of other properties can lead to absolute values. As we can easily verify, when there are *n* pieces of property, 1, …, *n*, with values *w*_{1},…,*w*_{n}, the *n* × *n* matrix with (*i*, *j*)-th entry *w*_{i}/*w*_{j} has the maximum eigenvalue *n*, and the corresponding eigenvector is (*w*_{1}, …, *w*_{n})^{T}. Then for each pair of (*i*, *j*) from the set *i*,* j* ∈ {1, …, *n*},*i* ≠ *j*, having the owner answer how many times property *i* is worth property *j* and set the answer to *a*_{ij}. The diagonal components of the matrix are 1, and the eigenvector corresponding to the maximum eigenvalue of this matrix is a multiple of vector (*w*_{1}, …, *w*_{n})^{T} of absolute values of properties 1, …, *n*.

## 10.2 Analysis and Forecast Models of Risks

*X*times while repeating the trial

*N*times, with a large enough

*N*and

*X*/

*N*at a reasonable value, the occurrence probability

*p*of the event of interest is within the following range with a 95% confidence level:

*s*is standard error expressed with the following equation:

In case an event of interest took place 38 times out of 380 independent trials, the 95% confidence interval for the occurrence probability is evaluated at 0.07–0.13. From the definition of standard error, quadrupling the number of trials *N* will halve the confidence interval for the occurrence probability *p*.

*n*independent trials of an event with occurrence probability

*p*, binominal distribution gives the probability of the event of interest to occur

*k*times:

For example, if we set the probability *p* of getting 1 with one throw of die at *p* = 1/6, we expect to get 1 once with six throws of the die. The probability of getting 1 exactly once, however, is only *B*_{6,1/6}(1) ≃ 0.402, twice *B*_{6,1/6}(2) ≃ 0.201, and never *B*_{6,1/6}(0) ≃ 0.335.

*n*is sufficiently large and

*p*is sufficiently small, the Poisson distribution approximates the probability of the event of interest occurring

*k*times at

*λ*is the expectation for the number of times the event of interest takes place and is calculated by

*λ*=

*np*. The constant

*e*is Napier’s constant, and it approximately equals 2.72. For example, with a machine that produces 1 defective product for every 500 pieces, the probability of finding at least 1 defective product in 1000 pieces produced by this machine is:

*a*/

*l*, whereas that without factor A is

*c*/(

*n*−

*l*); thus, the risk of facing result B when factor A is present is:

*q*, in general, is called relative risk or risk ratio.

Cross table of causal correlation

Result B | Total | |||
---|---|---|---|---|

Occurred | Did not occur | |||

Factor A | With | | | |

Without | | | | |

Total | | | |

*a*/

*c*, and that for the group without result B is

*b*/

*d*. Thus, the odds ratio of the former to the latter is:

When the probability of occurrence of result B is extremely low, i.e., when *a* ≪ *b*, *c* ≪ *d*, the odds ratio *r* is a good approximation of risk ratio *q*.

*n*samples, randomly picked out from the parent population, the odds ratio is within the range:

*s*is standard deviation defined with the following equation:

For example, if *a* = *b* = *c* = *d* = 460, *e*^{1.96s} ≃ 1.2 and accurate estimation of the odds ratio will require a fairly large sample size.

*p*of result B based on observations

*x*

_{1},…,

*x*

_{n}for factors A

_{1},…,A

_{n}, with the following equation:

The calculation of determining the coefficients *a*_{1},…,*a*_{n} and constant *h* from observations of factors A_{1},…,A_{n} with result B occurrence and those without result B occurrence is called logistic regression analysis. For generalized linear models, the odds *p*/(1 − *p*) of result B is *e*^{−z}; thus, when factor A_{i} increases by 1 unit, the risk of occurrence of result B is *e*^{−z} times in terms of odds ratio.

## 10.3 Decision-Making for Risk Minimization

Mathematical programming is one of the methodologies applied in rationally solving decision-making problems that we encounter in various fields of natural sciences and social sciences. Mathematical programming formulates the decision-making problem into a mathematical optimization problem to maximize or minimize the value of the objective function with variables subject to some constraints. Thus, by setting the policies for disaster management and accident prevention to decision variables, the physical and social conditions that govern the policies to constraints, and the sizes of possible risks under the policies to the objective function, we can solve the decision-making problem within the framework of mathematical programming.

For this set of formulae, * x* is the vector of decision variables,

*is the parameter matrix, and*

**A***and*

**b***are parameter vectors. Setting the sizes of risks with accidents or disasters is difficult, and parameters*

**c***,*

**A***, and*

**b***have uncertainties. Especially when parameters*

**c***and*

**A***have uncertainties, both sides of the equation*

**b**

**A***=*

**x***are uncertain; thus, it takes clarifying the constraints that require the two sides are equal for solving the linear programming problem. We then turn the problem into a “Chance Constraining Problem” that looks for the decision variables*

**b***that minimize the objective function*

**x**

**c**^{T}

*under the constraints that the probability of the equation*

**x**

**A***=*

**x***or its fuzzy measure is not less than a certain threshold or the “recourse problem” that adds the magnitude of the residual error*

**b**

**A***−*

**x***to the objective function instead of equation*

**b**

**A***=*

**x***.*

**b**If the range of parameters * A* and

*are known, the method to find decision variables*

**b***that minimize the objective function*

**x**

**c**^{T}

*while the equation is satisfied whatever values*

**x***and*

**A***take is another effective approach (“robust optimization”). In assessing the effectiveness of risk management, “worst case analysis” is also well practiced that identifies the case that maximizes the objective function*

**b**

**c**^{T}

*among optimized results of linear programming problems for all possible combinations of parameters*

**x***and*

**A***. Worst case analysis takes solving the two-level mathematical programming problem that have the original linear programming problem in its lower level.*

**b**## Copyright information

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