4. UNCERTAINTY MODELLING 76 - DNVresearch.dnv.com/skj/OffGuide/CH4.pdf · 4.1 Steps in Uncertainty Modelling In general, ... uncertainty can be determined by employing simulation

Guideline for Offshore Structural Reliability Analysis - General Page No._____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Report No. 95-2018 Chapter 4

Skjong,R, E.B.Gregersen, E.Cramer, A.Croker, Ø.Hagen, G.Korneliussen, S.Lacasse, I.Lotsberg, F.Nadim,K.O.Ronold (1995)“Guideline for Offshore Structural Reliability Analysis-General”, DNV:95-2018

JANUARY 3, 1995

4. UNCERTAINTY MODELLING 76

4.1 Steps in Uncertainty Modelling 76

4.2 Types of Uncertainties 764.2.1 Aleatory Uncertainty 764.2.2 Epistemic Uncertainty 76

4.3 Types of Distributions 784.3.1 Univariate Discrete Distributions 784.3.2 Univariate Continuous Distributions 79

4.3.2.1 Non-Parametric Models 794.3.2.2 Parametric Models 814.3.2.3 Extreme by Power N 84

4.3.3 Joint Description of Variables 844.3.3.1 Multidimensional Models 844.3.3.2 Sequence of Conditional Distributions 854.3.3.3 Nataf Correlation Model 85

4.4 Choice of Distribution Model 864.4.1 General 864.4.2 Well Known Stochastic Experiments 904.4.3 Probability Paper 904.4.4 Skewness and Kurtosis 904.4.5 TTT plot 91

4.5 Methods for Estimation of Distribution Parameters 924.5.1 Plot of Data on Probability Paper (Graphic Procedure) 924.5.2 Least-Squares Fit Methods 924.5.3 Maximum Likelihood Method 944.5.4 Method of Moments 954.5.5 Bayes-estimation 954.5.6 The Bootstrap Estimate of Standard Error 96

4.6 Verification of Fitted Distributions 984.6.1 Subjective Judgement 984.6.2 χ2 test 984.6.3 Kolmogorov-Smirnov Test 99

REFERENCES 100




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4. Uncertainty Modelling

4.1 Steps in Uncertainty Modelling

In general, the steps to be taken in order to model the uncertainty of the stochastic variablesentering the probabilistic model(s) are:

• collect data

• evaluate the dataset(s)- types of uncertainties- independent or dependent stochastic variables?- independent observations?- combination of uncertainties

• choose probability distribution(s) to represent the data- evaluate the underlying generating mechanisms- parametric or nonparametric model?- if parametric, which family of distributions?

• estimate the distribution parameters (only parametric models)- what accuracy is needed in which parts of the distribution?

• verify the selected distribution models

4.2 Types of Uncertainties

Reliability analysis requires that all relevant uncertainties in the analysis procedure be taken intoaccount. Uncertainties associated with an engineering problem, due to the different sources, canbe divided into two groups: aleatory (natural) uncertainty and epistemic (knowledge) uncertainty.This grouping of uncertainty sources is usually adequate. However, one shall be aware that othertypes of uncertainties may be present, such as human gross errors which are not covered here.

The information about uncertainties should be introduced in reliability analyses in the form ofrandom (stochastic) variables.

4.2.1 Aleatory Uncertainty

Aleatory uncertainty is a natural randomness of a quantity, also known as intrinsic or inherentuncertainty, e.g., soil natural variability from point to point, or the variability in wave and windloading over time. Aleatory uncertainty cannot be reduced or eliminated.

4.2.2 Epistemic Uncertainty




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Epistemic (knowledge) uncertainty represents errors which can be reduced by collecting moreinformation about a considered quantity and improving the methods of measuring it. Thisuncertainty may be classified into:

Measurement Uncertainty Uncertainty due to imperfection of an instrument used to registera quantity.

Measurement uncertainty may be described in terms of accuracy, see Bitner-Gregersen andHagen (1990). To characterise the accuracy of a measurement process it is necessary to indicateboth its systematic error (bias) and its precision (random error). Measurement uncertainty isusually given by a manufacturer of an instrument. It can also be evaluated by a laboratory test orfull scale test.If a considered quantity is not obtained directly from the measurements, but some estimationprocess is interposed, e.g., the significant wave height, then the measurement uncertainty must becombined with the estimation or model uncertainty, by appropriate means.

Statistical Uncertainty Uncertainty due to limited information such as a limited numberof observations of a quantity.

Statistical uncertainty may be referred to as estimation uncertainty. It is present not only becausethe distribution parameters are estimated from a limited set of data, but it is also affected by thetype of estimation technique applied for evaluation of the distribution parameters. Statisticaluncertainty can be determined by employing simulation techniques or by use of the maximumlikelihood method as asymptotic results are available with this method.

Model Uncertainty Uncertainty due to imperfections and idealisations made inphysical model formulations for load and resistance as well as inchoices of probability distribution types for representation ofuncertainties.

Model uncertainties in a physical model for representation of load and/or resistance quantitiescan be described by random factors, each defined as the ratio between the true quantity and thequantity as predicted by the model. A mean value not equal to 1.0 expresses a bias in the model.The standard deviation expresses the variability of the predictions by the model. An adequateassessment of a model uncertainty factor may be available from sets of laboratory or fieldmeasurements and predictions. Subjective choices of the distribution of a model uncertaintyfactor will, however, often be necessary. The importance of a model uncertainty may vary fromcase to case and can be studied by interpretation of parametric sensitivities.

Combination of uncertaintiesSeveral sources of uncertainty may exist for a stochastic variable. If the various sources areindependent, the standard deviation of the total uncertainty may be calculated as

σ σtotal ii

=

21 2/

(4. 1)

where σi is the standard deviation of uncertainty source i.




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4.3 Types of Distributions

4.3.1 Univariate Discrete Distributions

If a random variable can only have an enumerable number of values it is called discrete and itsdistribution is called discrete distribution. Some of the most important theoretical models are:

• Binomial• Poisson• Negative Binomial• Hypergeometric

A number of other distributions exist, among these are combinations or truncated versions of thedistributions listed above and discrete versions of continuous distributions. Refer to Johnson andKotz (1969) for more details.

If the underlying generating mechanism of the random variable is not known, or satisfactoryestimates for the point probabilities of the possible realisations exists, the data (estimates of thepoint probabilities) may be used directly.

Binomial DistributionIf N independent trials are made, and in each there is a probability p that the outcome E willoccur, then the number of trials in which E occurs may be represented as a Binomial distributionwith parameters N, p.

In a situation where the assumptions of independence and constant probability is not strictlycorrect the model may still give a sufficient accurate representation.

Poisson DistributionIn addition to having N independent trials and constancy of the probability p from trial to trial, asfor the binomial distribution, it is assumed that N is large and p is small. The number of trials inwhich the event of interest occurs may then be represented as a Poisson distribution withparameter θ = Np . The distribution may be suitable when modelling the number of occurrencesof an event within a time interval or in a limited part of space.

Negative Binomial DistributionThe Negative Binomial distribution is frequently used as a substitute for the Poisson distributionwhen it is doubtful whether the strict requirements, particularly independence, will be satisfied.Among specific fields where negative binomial distributions have provided usefulrepresentations are accident statistics, birth-and-death processes and psychological data.

One simple model leading to the Negative Binomial distribution is that representing the numberof independent trials necessary to obtain m occurrences of an event which has constantprobability p of occurring in each trial.




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Hypergeometric DistributionThere are two natural ways in which a Hypergeometric distribution arises naturally:1. An urn contains N balls, X are white and (N-X) are black. n balls are drawn from the urn

without replacing any of the balls. The number of white balls among the n balls will thenhave a Hypergeometric distribution with parameters n,X,N.

2. Consider two independent random samples of sizes n1 and n2 drawn from a population inwhich a measured character has a continuous distribution. The number of observed values inthe second sample exceeding at least (n1-m+1) of the values in the first sample will have aHypergeometric distribution.

4.3.2 Univariate Continuous Distributions

A random variable X is continuous and has a continuous distribution if:

(i) F(x) is continuous

(ii)ddx

F x f x( ) ( )= exists for all x except for a finite number of values

(iii) f(x) is piecewise continuous

Under these assumptions the domain of X may consist of one or more finite or infinite intervals.

4.3.2.1 Non-Parametric Models

Assume n observations being realisations of n stochastic variables X1,...,Xn with somesimultaneous distribution function. If the functional form of this simultaneous distribution is notknown, the model is called non-parametric.

Empirical DistributionThe empirical cumulative distribution is defined as:

F x observations xnX ( ) #= ≤

(4. 2)

where n is the total number of observations.

Splined DistributionA splined distribution interpolates or approximates data on some cumulative distributionfunction. Arge and Dælen (1989) describes a spline model using linear combinations of B-splinesand constrained least squares approximation. B-splines are piecewise polynomials with localsupport. For details see Arge and Dælen (1989), de Boor (1978) and Shumacher (1982). Variousconstraints are forced on the spline function to ensure the properties of a distribution function. Ifthe density is forced to be unimodal, i.e., to have only one maximum, the spline function must bethree times continuously differentiable implying polynomials of order k = 5. If there is nodemand of unimodality order k = 4 is sufficient.

The distribution function F xX ( ) is represented by a linear combination of B-splines:




80

F x c BX ii

n

i k( ) ,==

1

(4. 3)

The density spline function is found from:

f x c B k c ct tX i

i

n

i k ii i

i k i

( ) , ( )',

'= = − −−=

−−

+ −

21

1

1

1 c (4. 4)

t = (t1,t2,...,tn+k) is the knotvector which generates the B-splines.

Some general constraints are forced on the spline function to ensure the properties of adistribution. Lower and upper bounds, a and b, must be specified:

F a F bX X( ) , ( )= =0 1 (4. 5)

f x a x bX ( ) ,> < <0 (4. 6)

Additional constraints may be included to ensure:• unimodal density• at most one inflection point on each side of the top point• vanishing or equal density at the endpoints

Figure 4. 1 shows an empirical distribution (histogram scaled to have area equal to 1) and twosplined distributions fitted to the empirical. The solid line represents a spline fit forced to beunimodal and have at most 1 inflection points on each side of the top point. The dashed linerepresents a spline fit which is only forced to have vanishing end points.

Figure 4. 1 Nonparametric Distributions




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4.3.2.2 Parametric Models

Assume n observations being realisations of n stochastic variables X1,...,Xn with somesimultaneous distribution function. Based on prior knowledge one may assume some knownfunctional form of the distribution:

F x xX X n rn1 1 1,..., ( ,..., ; ,..., )θ θ (4. 7)

where θ θ1,... r are unknown parameters.

Some of the most commonly used parametric models are briefly described below. Truncatedversions exist for all non-bounded distributions. They should be used when the stochasticvariable for some reason is known to have lower and/or upper limits.

Beta DistributionThe beta distribution is a flexible tool for modelling a distribution of a bounded variable. It isused, e.g., in economical applications of the reliability methods, for modelling of random phaseand other angular variables over the range (0,2π), and for representation of long-term distributionof spectral width over the range (0,1).

Birnbaum-Saunders DistributionThis distribution has been used as a lifetime distribution in fatigue. It is based on a stochasticinterpretation of the Miner-Palmgrens method.

Chi-square (χ2 ) DistributionOne important application of the distribution is its use in hypothesis testing. The χ2 goodness-of-fit test examines the goodness of fit of a set of data to a specific probability distribution.

Exponential DistributionThe exponential distribution is an important distribution with widespread use in statisticalprocedures. Currently among the most prominent applications are those in the field of life-testing. The lifetime (or life characteristic) can often be usefully represented by an exponentialrandom variable, with (usually) a relatively simple associated theory. The time intervals betweenevents in a Poisson process are exponentially distributed.

F-DistributionFisher's F-distribution is a special form of Pearsons Type VI distribution. The most commonapplication of the F-distribution is in standard tests associated with the analysis of variance, e.g.,testing equality of variances of two normal populations.

Gamma Distribution




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The gamma distribution appears naturally in the theory associated with normally distributedrandom variables, as the distribution of the sum of squares of independent unit normal variables(a χ2 distribution).

The use of the gamma distribution to approximate the distribution of quadratic forms(particularly positive definite forms) in multinormally distributed variables is well-establishedand widespread.

In applied work, gamma distributions give useful representations in theory of random countersand other topics associated with random processes in time, in particular in meteorology.

Generalised Gamma DistributionThe generalised-gamma distribution is based on the ordinary gamma distribution. Thegeneralisation is obtained by a power transformation, see Appendix A. The convenience of thegeneralisation lies mainly in the fact that cumulative and exceedance probabilities may becalculated by the same routines as the ordinary gamma distribution, and that the moments may beexpressed as analytical functions. The generalised gamma distribution may also be extended tomore variables.

The generalised gamma distribution has some convenient properties with important applicationsin long term extreme value and fatigue failure prediction.

Gumbel DistributionThe Gumbel distribution is also called the Type I asymptotic extreme value distribution of largestvalues. It is used to model the extreme values of variables which have initial distributionsexponential or of the exponential type. It is applied to representation of extreme environmentalconditions and extreme environmental loads, and is also used in response analysis.

Hermite Transformation ModelThe Hermite Transformation Model is applied to data which show weakly non-normal behaviour.The model is based on the first four statistical moments. It is more flexible and has the ability toreflect wider ranges of non-linearities than the commonly used Gram-Charlier and Edgeworthseries.

Inverse Gaussian DistributionAlso known as Wald's distribution. It has its origin in studies of Brownian motion and randomwalk. Time to travel a fixed distance is Inverse Gaussian, distance travelled in a fixed time isGaussian.

Currently this distribution is being used as the distribution of the lifetime for a component, whosefailure rate (event rate, hazard rate) will increase until maximum is reached and then decreaseasymptotically towards a fixed non-zero value as the lifetime approaches infinity. (This is incontradiction to lognormally distributed lifetimes whose failure rates decrease towards zero forlarge lifetimes, and to exponentially distributed lifetimes whose failure rates are constant.)

Lognormal Distribution




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The lognormal distribution is a widely applied distribution in practical statistical work.Application of the distribution is almost always based on empirical observations. In some rarecases it can be supported by theoretical argument - the lognormal distribution is appropriate if thevariable is the product of a large number of independent quantities.

Normal DistributionIf some variable is the sum of a large number of independent variables, a normal distribution isappropriate, according to the central limit theorem. The normal distribution is applied to describelinear physical phenomena (e.g., linear waves, linear response) as well as additive independenterrors.

An important property of the normal distribution and its related distributions (χ2 , t −,lognormal, etc.) is their mathematical convenience.

Oval DistributionThe oval distribution was introduced by Det Norske Veritas Sesam (1994) in order to modelmispositioning of a foundation template.

Rayleigh DistributionThe square root of the sum of squares of two independent standard normal variables is a Rayleighvariable (i.e., the square root of a χ2 -variable with two degrees of freedom). The Rayleighdistribution thus describes the amplitudes of a linear Gaussian process, since each spectralcomponent consists of a sine and a cosine term. The Rayleigh distribution is derived based on anassumption that the considered process is narrow banded. This assumption leads to conservativeresults when applying the Rayleigh distribution to broad banded processes.

Student's-t DistributionThe Student's-t distribution is applied to construction of tests and confidence intervals relating toestimation of the expected values of normal distributions.

Uniform DistributionThe uniform distribution appears in models of several physical phenomena, e.g., approximationof tidal water variations, directional distribution of swell, and random phase in a Gaussian signal.If information about some quantity with evident, physical bounds does not exist, the uniformdistribution is often used as a prior distribution of that quantity. (Otherwise the normaldistribution is often used.)

Weibull DistributionThe Weibull distribution is used to fit empirical data, especially long term values. It is applied indifferent fields, e.g. oceanography, hydrodynamics, fatigue. The shift parameter γ1, which givesa three-parameter Weibull distribution as described in Appendix A, gives better possibilities to fitempirical data than the two-parameter Weibull distribution (withγ1 0= ) and the lognormaldistribution.




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4.3.2.3 Extreme by Power N

This distribution is used to define extreme distributions based on information from an underlyingparent distribution.

X1,...,Xn are n independent identical distributed stochastic variables with distribution functionFX(x) and density fX(x). Define V and U as the maximum and minimum values of the Xj's:

V X j nj j= =max , ,...,1 (4. 8)

U X j nj j= =min , ,...,1 (4. 9)

The distribution of the maximum value may then be obtained from:

F v F vV Xn( ) [ ( )]= (4. 10)

f v n F v f vV Xn

X( ) [ ( )] ( )= −1 (4. 11)

and the distribution of the minimum value from:

F u F uU Xn( ) [ ( )]= − −1 1 (4. 12)

f u n F u f uU Xn

X( ) [ ( )] ( )= − −1 1 (4. 13)

When n → ∞ limiting distributions may be obtained. These distributions are frequently referredto as three families: Fisher-Tippett Type 1, 2, and 3. For normally distributed variables, themaximum and minimum values are both Type 1 (Gumbel) distributed. For lognormallydistributed variables, the maximum value is Type 3 (Weibull) distributed. The Type 3 (Weibull)distribution is otherwise used as a distribution of smallest values, e.g., phenomena produced by aweakest-link mechanism. The Type 2 (Frechet) distribution is its own maximum distribution, andso is the Type 1 (Gumbel) distribution.

4.3.3 Joint Description of Variables

A simultaneous description of two or more random variables will often be needed in a reliabilitycalculation. If the variables are independent, the joint distribution is obtained as the product ofthe marginal distributions. In general the involved variables will be mutually dependent.Subsequently, three approaches modelling the joint distribution are mentioned. The choiceamong these will depend on the nature of the correlation and of the available backgroundinformation.

4.3.3.1 Multidimensional Models




85

Multidimensional versions exist for most of the Univariate distributions listed in Sections 4.3.1and 4.3.2. The marginal distributions of individual variables correspond mostly to such knowndistributions or combinations of them. Some of the most important models are:

• Multinomial distribution• Multinormal distribution• Multivariate Beta, Gamma, Extreme Value and Exponential distributions

4.3.3.2 Sequence of Conditional Distributions

The explicit analytic form of the joint distribution may be very complicated. If a sufficientamount of simultaneous data are available, the joint distribution of two random variables mayconveniently be modelled as

f x y f x f y xXY X x x Y X y y( , ) ( ; , ) ( | ; , )|= α β α β (4. 14)

where the mutual dependency is accounted for by modelling the parameters of the conditionaldistribution of Y given X as function of x, i.e.: α αy y x= ( ) and β βy y x= ( ).

Example 4.1: Model a two dimensional stochastic variable ( , )X Y with uniform distribution on a circular diskwith radius R by combining an Oval distributed variable and an Uniform distributed variable:

X Oval Mean Scale R~ ( , )= =0Y Uniform Lower R x Upper R x~ ( , )= − − = −2 2 2 2

Example 4.2: Model a two dimensional stochastic variable wind (W), characterised by its speed (S) anddirection(D), given windrose data:

D Splined~ ( )marginal distributionS Weibull f d g d~ ( ( ), ( ), )δ β γ= = =0

For each wind rose direction estimate the Weibull parameters δ and β, for instance by using themaximum likelihood method described in Section 5.5.4, and generate interpolating orapproximating functions f and g.

Example 4.3: The stochastic variable X have a normal distribution with uncertain mean value Μ (with normaldistribution) and known standard deviation σ :

X N~ ( , )Μ σ2

Μ ~ ( , )N δ ξ2

where δ and ξ are known.

4.3.3.3 Nataf Correlation Model

In some cases one will only know the marginal distribution of variables and the correlationcoefficient between them. In such cases the Nataf distribution model is a convenient approach.




86

In general, there is an infinite number of joint distributions that correspond to the samecorrelation coefficient between the variables. Therefore, it is not possible to point out onedistribution and state that this is the correct one. The probabilistic model information is generallyincomplete.

To be able to treat models with incomplete information, one has to choose the type of jointdistribution. Der Kiureghian and Liu (1986) suggest the Nataf model, Nataf (1962), and show fora variety of cases that the model yields good results. The basic idea of the Nataf correlationmodel is that if each random variable is mapped onto a standard normal random variable, thejoint distribution is a multivariate normal distribution. Thus, if the distributions of two randomvariables are F x iX ii

( ), ,..,=1 2 and

V F x ii X ii= =−Φ 1 1 2( ( )), ,.., (4. 15)

the joint distribution expressed in terms of v-space variables is

F v v v v v v dv dvV Vv v

1 2 1 2 2 12

22 1 2

2 1 21

2 112

21

( , ) exp ( )=−

− + −−π ρ ρ

(4. 16)

A general procedure for computing ρv from the correlation ρx of X1 and X2 and vice versa, hasbeen developed, Winterstein et al. (1989), using an expansion in Hermite polynomials.

If the x-space variables are Gaussian, the Nataf model corresponds to the multinormaldistribution. If the x-space variables are random lognormal, the Nataf model corresponds to themultilognormal distribution.

4.4 Choice of Distribution Model

4.4.1 General

In order to describe statistical nature of loads, material properties and geometrical parameters,distribution functions need to be assigned to these quantities. It is often no theoretical preferencewhen it comes to deciding on probabilistic models for external action parameters, materialparameters, and geometrical parameters. The actual choices have therefore to be made on anempirical basis and engineering judgement. There should, if possible, be a logical basis for thechoice of the probabilistic model, and the model should be flexible and have a sufficient numberof adjustable parameters to fit the empirical data.

Future codes of practice may standardise the distribution types to be used in authorised reliabilityanalyses (refer to Chapters 5, 6, and 7). Section 4.3 lists some distribution functions commonlyused in reliability analysis. A more detailed description of the distributions is given in AppendixA.

Figure 4. 2 shows three different distributions (Exponential, Lognormal and Weibull) fitted toPOD (Probability Of Detection) data for crack detection, the empirical data are marked with (+).




87

The distributions were fitted using the least square method described in Section 4.5.2. The choiceof distribution to be used may have large impact on the analysis results.

Figure 4. 2 Three curves fitted to POD data

As far as possible a standard recommended distribution should be used. Such standardrecommendations should be based on some general consensus concerning their adequacy.

If no recommendations are available, the approaches described in the following subsections maybe useful in order to select a proper model. Given a set of possible distributions, one mayestimate the distribution parameters, using the methods described in Section 4.5, and selectmodel using the verification methods listed in Section 4.6.

Stratified dataFigure 4. 3 shows an example of a stratified dataset (plot of cumulative data). The situation mayoccur if the observations covers e.g. more than one physical phenomenon. In this case thereseems to be one statistical behaviour for values less than 10 and another for values greater than20. Here there were also no observations between 10 and 20. The data may be represented in twoways:

1. Estimate one distribution for the whole sample (Figure 4. 4). This may lead to poor estimates,due to difficulties in obtaining parametric models fitting the data. Further, the distributionmay not be well suited for doing reliability calculations with methods other than simulationmethods (e.g. due to multimodality of the density function).

2. Split the sample into two groups and estimate one distribution for each (Figure 4. 5 andFigure 4. 6). Then use a series system representation to account for the stochastic variablehaving outcomes from either group.




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Figure 4. 3 Stratified data

Figure 4. 4 Stratified data - one distribution




89

Figure 4. 5 Stratified data - lower part

Figure 4. 6 Stratified data - upper part




90

4.4.2 Well Known Stochastic Experiments

At first one should always consider the underlying generating mechanisms in order to possiblyidentify a known stochastic experiment or a function of a known stochastic experiment. If this isthe case, model uncertainties due to imperfect choice of probability distribution to represent thedata may be reduced to a minimum. Well known stochastic experiments are typically:

• the additive mechanism (Gaussian distribution)• the multiplicative mechanism (lognormal distribution)• Poisson process (Poisson distribution, exponential/gamma distribution)• asymptotic extreme values (extreme value distributions)

4.4.3 Probability Paper

A possible background for an empirical selection of a distribution is to plot the empiricaldistribution on a probability paper, e.g. normal or lognormal paper, Weibull paper or Gumbelpaper. Some commonly adopted probability papers are included in Appendix C.

By considering the behaviour of the empirical distribution in an actual probability paper, onemay often recognise a certain similarity to well known probabilistic models.

4.4.4 Skewness and Kurtosis

If a rather large sample is available, reasonable estimates for the coefficient of skewness γ s , andcoefficient of kurtosis γ k are easily estimated (Appendix A). By considering γ s

2 versus γk forthe various probabilistic models, one may establish a graph as shown in Figure 4. 7. It is seen thatthe normal and the exponential distributions just correspond to a point each in this coordinatesystem, the lognormal distribution corresponds to a single curve, while the beta distributioncorresponds to an area in this diagram. By considering the empirical coefficients in view of sucha figure, one may often select a class of reasonable models or exclude non-reasonable models.




91

Figure 4. 7 γ s2 versus γk for various probabilistic models, Hahn & Shapiro (1988).

4.4.5 TTT plot

TTT (Total Time on Test) plots relate to the modelling of lifetime distributions, i.e., thedistribution of the time until some defined event occurs. For these distributions it is essential thatthe event intensity z t( ) (also denoted failure rate or hazard rate) is adequately modelled.

z t f tF t

( ) ( )( )

=−1

(4. 17)

TTT plots assist in deciding whether the lifetime distribution is IFR (Increasing Failure Rate),DFR (Decreasing Failure Rate), or a combination of the two. Assume that n sample time pointsare arranged in increasing order. The Total Time on Test at time x is defined as:

T x X n i xjj

i

( ) ( )( )= + −=

1

(4. 18)

where i satisfies:




92

X x Xi i( ) ( )≤ ≤ +1 (4. 19)

The TTT plot is now obtained by plotting in

T XT X

i

n

,( )( )

( )

( )

. The diagonal from ( , )0 0 to ( , )1 1

represents a constant failure rate, corresponding to an exponentially distributed lifetime. Theempirical lifetime distribution is IFR when the plotted points fall above the diagonal and DFRwhen the points are located below the diagonal.

4.5 Methods for Estimation of Distribution Parameters

Assume that a specific family of distribution functions f xX( ; )θ are considered to be appropriatefor a statistical representation of the data set at hand. The estimation of the parameters θ from aset of independent, identically distributed observed values Xi, i=1,..,n, may be carried out in anumber of ways depending on the size of the data set, the accuracy needed and the toolsavailable. The most commonly used methods for estimation are:

• graphic procedure (plot of data on a probability paper)• least-squares fit methods• maximum likelihood method• moment method• Bayes-estimation

The effects of selecting different methods for estimation of the distribution parameters may besignificant.

4.5.1 Plot of Data on Probability Paper (Graphic Procedure)

In some cases it may be convenient, as a first approximation, to estimate distribution parametersby a graphic procedure. The idea of this method is to graph functions in transformed scales suchthat the form of the function and the parameter values may immediately be read from the graph.The most commonly used probability papers are: Weibull, Gumbel, Frechet, normal, lognormal.These are shown in Appendix C.

This method can only be used if a plot of the cumulative probability distribution of some randomvariable fluctuates closely about a straight line on a probability paper.

4.5.2 Least-Squares Fit Methods

Assume that the observations are organised as a set of quantiles and cumulative probabilities(xi,yi), i=1,..,n, possibly with standard deviations σi assigned to each set. In general, the Least-Squares fit method for estimating the parameters θ involves optimising the quadratic objectivefunction:




93

y F xi X i

ii

n −

=

( ; )θσ1

2

(4. 20)

where FX is the cumulative distribution function. One may, alternatively, also minimise thesquared error in x instead of y , F y xX i

− −1 2( ; )θ , where FX

−1 denotes the inverse cumulativedistribution function. However, this alternative approach may not necessarily lead to exactly thesame estimate for θ as the minimization of the squared error of the cumulative distributionfunction.

The terms 1 2σ i in Eq. (4. 20) may be interpreted as weights assigned to each data set. Thismakes it possible to put greater weight on large than on small empirical values in order toimprove the extrapolation to long-term extremes. Emphasis on the distribution tail is mostimportant for the reliability analysis (e.g., lower tail for resistances and upper tail for loads). Alimiting case of this weighting procedure occurs when one single point in the empiricaldistribution is considered to be certain. It is then required that the curve of the analyticaldistribution contains this point.

If a plot of the data on probability paper (cfr. the previous section) forms an approximatelystraight line, the distribution parameters may be found by simple linear regression. That means,one shall find a line

y x= +θ θ0 1 (4. 21)

which minimises the squared error

[ ]Q y xi ii

n

( , )θ θ θ θ0 1 0 12

1

= − −= (4. 22)

Solution of the normal equations lead to the following estimators for θ0 and θ1:

( )

( )θ1

1

2

1

=−

−

=

=

x x y

x x

i ji

n

ii

n (4. 23)

θ θ0 1= −y x (4. 24)

where xx

n

ii

n

= =

1 and yy

n

ii

n

= =

1 .

The interpretation of the variables x and y, and the parameters θ0 and θ1 will be different for thedifferent probability distributions.




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4.5.3 Maximum Likelihood Method

The maximum likelihood method is the best-known method for estimation of distributionparameters. For some distributions, the maximum likelihood equations leads back to the momentequations. In other cases the maximum likelihood equations must be solved by some numericaliteration method. Thus, a disadvantage of maximum likelihood estimation is that some morecomputational effort may be required to obtain the parameters than when using the method ofmoments for some distributions.

The principle applied is to maximise the likelihood function:

l f xX ii

n

==∏ ( ; )θ

1

(4. 25)

where fX is the density function for the chosen distribution type and θ its parameters. Someintuitive feeling for the likelihood function may, perhaps, be achieved by realising that thisproduct of probability densities will tend to be large when both data sample and distributionfunction correspond to each other.

In practice, it is often more convenient to work with the logarithm of the likelihood function. Toobtain parameter estimates, the supremum of the likelihood function is sought, by taking thederivatives of the likelihood function, and finding their zeroes:

∂∂θln , ,..,l j r

j

= =0 1 (4. 26)

Maximum likelihood estimators are given for a number of distributions in Appendix A. Refer toKendall and Stuart (1977), Cox and Hinkley (1974) or Johnson and Kotz (1969, 1970a, 1970b,1972) for more details on the maximum likelihood estimators.

Results are available with the maximum likelihood method, which state that:

(i) The maximum likelihood estimators are asymptotically unbiased

(ii) Asymptotically the covariance matrix of the estimated parameters may be obtained fromthe information matrix defined below.

(iii) Asymptotically the estimated parameters are jointly normal.

(iv) The method is asymptotically efficient; i.e., other estimation methods are not moreefficient, in the sense of obtaining a lesser variance for the parameter estimates (undercertain regularity conditions).

Returning to item (ii) above, an element of the information matrix A is given by




95

a E f xij

X

i j

=

∂ θ∂θ ∂θ

2 ln ( ; )(4. 27)

Asymptotically the covariance matrix for the estimated parameters is obtained from the inverseof the information matrix

C Cov A= =→∞

−[ ]θn n

1 1 (4. 28)

4.5.4 Method of Moments

If the use of the maximum likelihood method becomes troublesome, the method of moments canbe used to evaluate distribution parameters by equating analytical moments (see Appendix A) tosample moments. We need to consider as many orders of moments as there are unknownparameters in the distribution formula. The first four moment estimators are (µ:Mean, σ:Standarddeviation, δ:Skewness, κ:Kurtosis):

µ ==

11n

xii

n

(4. 29)

( )/

σ µ= −

=

1 2

1

1 2

nxi

i

n

(4. 30)

( )

δ

µ

σ=

−

=

1 3

13

nxi

i

n

(4. 31)

( )

κ

µ

σ=

−

=

1 4

14

nxi

i

n

(4. 32)

The method of moments should be used with care when the sample size n is small.

4.5.5 Bayes-estimation

In the previous sections the parameters to be estimated has, priorly, been assumed to belong to asubset Ω of the r-dimensional Euclidian space. However, within Ω no vectors have beenconsidered to be more likely than others.




96

Bayes-estimation assumes that a parameter θ is a stochastic variable Θ with some knowndistribution function fΘ ( )θ denoted the prior distribution. This distribution reflects knowledgeabout the parameters of the distribution of the considered random variable (or vector) X beforesome new independent data are available (usually in the form of an outcome of the vector (X1 ,..,Xn) with all X i mutually independent and distributed like X).

The simultaneous density for X and Θ are:

f x f x fX X, |( , ) ( | ) ( )Θ Θ Θθ θ θ= (4. 33)

The posterior distribution is a conditional distribution of the parameters given the priorinformation and the sample data:

f xf x

f xXX

XΘ

Θ|

,( | )( , )( )

θθ

= (4. 34)

where f xX ( ) is the marginal density for X:

f x f x d f x f dX X X( ) ( , ) ( | ) ( ), |= = Θ Θ ΘΩΩ

θ θ θ θ θ (4. 35)

The Bayes estimator ( )θ X is defined to be the one that minimise [ ]E X ( )θ − Θ2. This implies

choosing the mean value of the posterior distribution as the estimator, see Høyland (1986).

[ ]( ) |θ X E X= Θ (4. 36)

Example 4.4: Assume X Xn1,.., are independent and identically distributed N( , )θ σ2 where σ2 is known.

The prior distribution for Θ is N( , )µ τ 2 where µ and τ 2 are both known. The Bayes estimatorfor Θ will then be:

( ,.., )θ σσ τ

τσ τ

µX X nn

Xnn1

2

2 2

2

2 2=+

⋅ ++

⋅−

− −

−

− − (4. 37)

Example 4.4 illustrates the general property of a Bayes estimator that it can be expressed as aweighted average of the estimator one would use if no prior information was available ( X ) andthe estimator one would use if we only had prior information (µ ).

4.5.6 The Bootstrap Estimate of Standard Error




97

Having chosen an estimator for an unknown parameter θ , the next question to answer is howaccurate the estimate is. The bootstrap is a general methodology for answering this question.

Assume that the observed data y = ( , ,..., )x x xn1 2 consists of independent and identicallydistributed observations X X Xn1 2, ,..., with unknown distribution F. We have the statistic of

interest, (θ y) , to which we wish to assign an estimated standard error. If, for instance, the

statistic of interest is the mean value, then ( ) ( , ,..., )θ θy x x xn

xn ii

n

= ==1 2

1

1

Let σ( )F indicate the standard error of θ , as a function of the unknown sampling distribution F

[ ]σ θ( ) (/

F VarF= y)1 2

(4. 38)

The bootstrap estimate of the standard error is

( )σ σ= F (4. 39)

where F is the empirical distribution putting probability 1 n on each observed datapoint xi . AMonte Carlo simulation algorithm may be used to estimate σ :

(i) Use a random generator to independently draw a large number B of samples (calledbootstrap samples) y y y* * *( ), ( ),..., ( )1 2 B . Each bootstrap sample y*( )b is generated bydrawing n times with replacement from the originally observed data set y .

(ii) For each bootstrap sample y* ( )b evaluate the statistic of interest

θ θ* *( ) ( ( )), , ,...,b b b B= =y 1 2 (4. 40)

(iii) Calculate the standard sample deviation of the ( )θ b values:

( ) ( )* */

σθ θ

Bb

B

b

B=

− ⋅

−

=

2

1

1 2

1(4. 41)

where the mean value of the evaluation of the statistic of interest is

( )

( )*

*

θθ

⋅ = = b

Bb

B

1 (4. 42)

For most situations B in the range of 50 to 200 is adequate.




98

The bootstrap is one of several resampling methods for estimation of standard error. For moredetails on the bootstrap, the jackknife and other resampling methods, reference is made to Efronand Tibshirani (1993).

4.6 Verification of Fitted Distributions

The adequacy of a fitted model can be indicated by objective methods or by subjectivejudgement. The most commonly adopted objective methods are:

• χ2 test• Kolmogorov-Smirnov test

The actual data often consist of correlated observations. In such cases, the statistics of the testvariable is not known, and objective tests may then only be applicable for relative comparisonsbetween alternative choices of distributions.

4.6.1 Subjective Judgement

A subjective judgement by visual inspection of a probability plot is often the most convenientverification approach. Such a verification is carried out by plotting both the empirical and thefitted distribution function preferably in a probability paper which are constructed so that thefitted model appear as a straight line.

4.6.2 χ2 test

The χ2 test may be a useful method, provided a data sample of independent observations areavailable and one is primarily interested in the fit in the central range of the distribution. Mostoften, however, one is interested in adequacy of the tail behaviour of the model and this excludesthe chi-square test. (Note that one should be aware of the problem of overfitting in the tail wherestatistical uncertainty is particularly large.)

In Bendat and Piersol (1971) the test variable is defined by:

ZF x F x

F xi X i

X ii

k

=−

=

( ) ( )( )

2

1

(4. 43)

where xi is a quantile, F xi( ) the corresponding empirical cumulative probability, F xX ( ) thefitted cumulative distribution and k the number of comparison points. If F x F xi X i( ) ( )− isassumed to be normal distributed, Z will be χ2 distributed with k −3 degrees of freedom.




99

4.6.3 Kolmogorov-Smirnov Test

This method tests if at least one value on the empirical distribution function is not generated bythe fitted distribution. The test variable D is defined as the largest vertical difference between thefitted cumulative distribution and the empirical cumulative distribution:

D F x F xx X= −sup ( ) ( ) (4. 44)

where F xX ( ) is the fitted distribution and F x( ) is the empirical distribution. The fitteddistribution is not accepted if D is larger than some defined value.

The distribution function for D as the number of observations n → ∞ may be obtained from thefollowing relations:

lim ( )n

P Dn

R→∞

<

=λ λ (4. 45)

R ej j

j

( ) ( )λ λ= − −

=−∞

∞

1 2 2 2

(4. 46)

See Johnson and Kotz (1970b) and Sobczak (1970) for more details on the distribution of D anda more general description of R( )λ .




100

References

Arge, E., and M. Dæhlen (1989), "Estimation of Cumulative Distribution Functions using B-splines", Research Report No. 129, Dept. of Informatics, University of Oslo, Oslo, Norway.

Bendat, J.S., and A.G. Piersol (1971), Random Data: Analysis and Measurement Procedures,John Wiley and Sons, New York, N.Y.

Bitner-Gregersen, E., and Ø. Hagen (1990), "Uncertainties in Data for the OffshoreEnvironment", Structural Safety, Vol. 7.

Cox, D.R., and D.V. Hinkley (1974), Theoretical Statistics, Chapman & Hall, London, England.

de Boor (1978), A Practical Guide to Splines, Springer-Verlag, New York.

Der Kiureghian, A. and P.-L. Liu (1986), Structural Reliability under Incomplete ProbabilityInformation, Journal of Engineering Mechanics, ASCE, 112(1),85-104

Efron, B., and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, New York,N.Y., 1993.

Hahn and Shapiro (1988), Statistical Models in Engineering, John Wiley and Sons, New York.

Holen A.T., A. Høyland, and M. Rausand (1988), Pålitelighetsanalyse , 2. utgave, TAPIR forlag,Trondheim, Norway.

Høyland, A. (1986), Sannsynlighetsregning og Statistisk Metodelære 1 og 2, 4. utgave, TAPIRforlag, Trondheim, Norway.

Johnson N.L. and S. Kotz (1969), Discrete Distributions, Wiley Series in Probability andMathematical Statistics, John Wiley and Sons, New York.

Johnson N.L and S. Kotz (1970a), Continuous Univariate Distributions 1, Wiley Series inProbability and Mathematical Statistics, John Wiley and Sons, New York.

Johnson N.L and S. Kotz (1970b), Continuous Univariate Distributions 2, Wiley Series inProbability and Mathematical Statistics, John Wiley and Sons, New York.

Johnson N.L. and S. Kotz (1972), Continuous Multivariate Distributions, Wiley Series inProbability and Mathematical Statistics, John Wiley and Sons, New York.

Kendall, M.G., and A. Stuart (1977), The Advanced Theory of Statistics, Vol. 2, Inference andRelationship, 4th ed., Griffin, London, England.

Det Norske Veritas Sesam (1994), "SESAM User's Manual. PROBAN Distribution Manual."

Nataf, A. (1962), Determination des Distribution dont les Marges sont Donnees, ComptesRendus l'Academie des Sciences, Vol. 225, Paris, France, pp. 42-43.

Shumacher (1982), Spline Functions: Basic Theory, John Wiley and Sons, New York.




101

Sobczak, W. (1970), Metody statystycne w elektronice, Wydawnictwa Naukowo-Techniczne,Warsaw, Poland.

Winterstein, S.R, R.S. De, and P. Bjerager (1989), "Correlated Non-Gaussian Models in OffshoreStructural Reliability", Proceedings, 5th International Conference on Structural Safety andReliability, San Francisco, California.

—A—Accident statistics, 78Aleatory uncertainty, 76

—B—Bias, 77Bootstrap, 97, 98

—C—Central limit theorem, 83Conditional distributions, 85, 96

sequence of, 85Confidence

interval, 83Cumulative distribution function, 79, 93

—D—Distributions

continuous, 78, 79discrete, 78, 100nonparametric, 80

—E—Empirical distribution, 80, 90, 93, 97, 99Error

gross, 76systematic, 77

Estimationmaximum likelihood, 94method of moments, 92, 94, 95probability paper, 90, 92, 93, 98

Estimator, 96, 97Exponential distribution, 81, 90

—F—Failure rate, 82, 91, 92Fatigue, 81, 83

—G—Gaussian process, 83Gross error, 76

—H—Hermite polynomial, 86Human error

gross, 76

—I—Inherent uncertainty, 76Inspection, 98

—J—Jackknife, 98

—K—Kurtosis, 90, 95

—L—Least-squares estimation, 92Linear regression, 93Load, 77Lognormal distribution, 83, 90

—M—Maximum likelihood estimation, 94Method of moments, 92, 94, 95Model, 76, 77, 78, 79, 82, 85, 86, 87, 90, 98

uncertainty, 77Monte Carlo simulation, 97

—N—Nataf distribution model, 85, 86

—O—Objective function, 92Observations, 76, 77, 79, 81, 83, 87, 92, 97, 98, 99

—P—Poisson distribution, 78, 90Poisson process, 90Probability paper, 90, 92, 93, 98




76

—R—Rayleigh distribution, 83Resampling, 98

bootstrap, 97, 98jackknife, 98

Response, 82, 83

—S—Significant wave height, 77Simulation, 77, 87, 97

Monte Carlo, 97Simulation methods, 87Skewness, 90Splined distribution, 79, 80State, 86Statistical uncertainty, 98Stochastic process

normal, 83Poisson, 90

Stratified data, 87Swell, 83System

series, 87

—T—TTT plot, 91, 92

—W—Wave height, 77

significant, 77Weibull distribution, 83, 84Wind, 85Wind rose, 85

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