Statistics and Probability Theory in Civil Surveying and

Statistics and Probability. Theory in Civil, Surveying and Environmental Engineering Prof. Dr. Michael Havbro Faber Swiss Federal Institute of Technology ETH Zurich, Switzerland Swiss Federal Institute of Technology 1 / 40

Contentsof Todays Lecture • Presentation on the result of the classroom assessment • What is a random variable? • The decision context! • What are we doing today? • Details will follow Swiss Federal Institute of Technology 2 / 40

Lecture Swiss Federal Institute of Technology 3 / 40

Subjects Swiss Federal Institute of Technology 4 / 40

What is a random variable? • Let us considera very simple structuralengineeringproblem! • We want to designa steel beam – and assume– based on experiencethat the design controllingload effect is the midspan bendingmoment. M - the designvariablebeing the momentof resistance. W of the cross section - the load p and the yield stress sy of the beam are associatedwith uncertainty Mid span cross-section Swiss Federal Institute of Technology 5 / 40

What is a random variable? • The momentcapacityof the cross-section. RM and the mid span moment. M are calculatedas: Mid span cross-section Swiss Federal Institute of Technology 6 / 40

What is a random variable? • We can now establisha design equationas: The engineer must now select W, or rather b and h such that the design equation is fulfilled But as p and sy are associated with uncertainty – she/he must take this uncertainty into account ! Mid span cross-section Swiss Federal Institute of Technology 7 / 40

What is a random variable? • The uncertaintyis accountedfor by representingp and sy in the design equationas two randomvariables. The randomvariable. P representsthe randomvariabilityof the load p duringa periodof one year The randomvariable. Sy representsthe randomvariabilityof the steel yield stress sy - producedby an unknownsteel producer. Swiss Federal Institute of Technology 8 / 40

What is a random variable? • As the load and yield stress are uncertain the design equation cannot be fulfilled with certainty – independent on the choice of b and h. • However, it can be fulfilledwith probability! • The beam can be designedsuch that the probabilityof failureis less or equal to a given number– the requirementto safety. Swiss Federal Institute of Technology 9 / 40

What is a random variable? • Let us assume that the load and yield stress are given as: we can now write the event of failure as: This is called a safety margin! let us further assume that l=5000 mm andb=50 mm • Let us now determine h such that the annual probability of -3 failure is equal to 10 Swiss Federal Institute of Technology 10 / 40

What is a random variable? • We have already learned that a linear combination of Normal distributed random variables is also Normal distributed The expected valueof S is equal to: The varianceof S is equal to: The probability of failure is now easily determined from the standard Normal cumulative distribution function Swiss Federal Institute of Technology 11 / 40

What is a random variable? • Calculating the probability of failure as a functionh of we get: The height of the beam must thus be equal to 258 mm! Swiss Federal Institute of Technology 12 / 40

The decisioncontext! • Why uncertaintymodeling? Uncertain phenomenon Randomvariables Randomprocesses Swiss Federal Institute of Technology 13 / 40

What are we doing today? • We have already introduced random variables as a means of representing uncertainties which we may quantify based on observations – related to time frames from which we have experience and observations! • In many real problems of decision making we need to take into account what might happen in the far future – exceeding the time frames for which we have experience! - 475 year design earthquake! - 100 year storm/flood - 100 year maximum truck load - etc. . Thus we need to developmodelswhich can supportus in the modelingof extremes of uncertain/random phenomena! Swiss Federal Institute of Technology 14 / 40

What are we doing today? • We have already introduced random variables as a means of representing uncertainties which we may quantify based on observations. • Often we use random variables to represent uncertainties which do not vary in time: - Model uncertainties (lack of knowledge) - Statistical uncertainties (lack of data). • Or we use such random variables to represent the random variations which can be observed within a given reference period. Swiss Federal Institute of Technology 15 / 40

What are we doing today? Discreteevent of flood • In many engineering problems we need to be able to describe the random variations in time more specifically: The occurrences of events at random discrete points in time (rock-fall, earthquakes, accidents, queues, failures, etc. ) - Poisson process, exponential and Gamma distribution Continuous stress The random values of events occurring variations due to waves continuously in time (wind pressures, wave loads, temperatures, etc. ) - Continuous random processes (Normal process) Swiss Federal Institute of Technology 16 / 40

What are we doing today? Extremewaterlevel • However, we are also interested in modeling extreme events such as: the maximum value of an uncertain quantity within a given reference period - extreme value distributions the expected value of the time till the occurrence of an event exceeding a certain severity Maximum wave load - return period Swiss Federal Institute of Technology 17 / 40

What are we doing today? • In summary we will look at: - Random sequences. Poisson ( process ) - Waiting time between events. Exponential ( and Gamma distributions ) - Continuous random processes (the Normal process) - Criteria for extrapolation of extremes stationarity ( and ergodicity) - The maximum value within a reference period extreme ( value distributions ) - Expected value of the time till the occurrence of an event exceeding a certain severityreturn ( period ) Swiss Federal Institute of Technology 18 / 40

Random. Sequences • The Poisson counting process was originally invented by Poisson “life is only good for two things: to do mathematics and to teach it” (Boyer 1968, p. 569) Poisson, Siméon-Denis (1781 -1840) Student of Laplace Former law clerk Poissonwas originally interested in applying probability theory for the improvement of procedures of law Swiss Federal Institute of Technology 19 / 40

Random. Sequences • The Poisson counting process is one of the most commonly applied families of probability distributions applied in reliability theory The Poisson process provides a model for representing rare events – counting the number of events over time Swiss Federal Institute of Technology 20 / 40

Random. Sequences • The Poisson counting process is one of the most commonly applied families of probability distributions applied in reliability theory The process. N(t) denoting the number of events in a (time) interval (t, t+Dt[ is called a. Poisson processif the following conditions are fulfilled: 1) the probability of one event in the interval t, t+Dt[ ( is asymptotically proportional to Dt. 2) the probability of more than one event in the interval (t, t+Dt[ is a function of higher order of Dt for Dt→ 0. 3) events in disjoint intervals are mutually independent. Swiss Federal Institute of Technology 21 / 40

Random. Sequences • The Poisson process can be described completely by its intensity n(t) if n(t) = constant, the Poisson process is said to be homogeneous , otherwise it isinhomogeneous. The probability ofn events in the time interval (0, t[ is: Homogeneous case ! Swiss Federal Institute of Technology 22 / 40

Random. Sequences • Early applicationsincludethe studiesby: Ladislaus. Bortkiewicz(1868 -1931) - horse kick death in the Prussian cavalry - child suicide William Sealy. Gosset (“Student”) (1876 -1937) - small sample testing of beer productions (Guinness) RD Clarke - study of distribution of V 1/V 2 hits under the London Raid Swiss Federal Institute of Technology 23 / 40

Random. Sequences • The mean value and variance of the random variable describing the number of events N in a given time interval (0, t[ are given as: Inhomogeneous case ! Homogeneous case ! Swiss Federal Institute of Technology 24 / 40

Random. Sequences • The Exponential distribution The probability ofno events(N=0) in a given time interval (0, t[ is often of special interest in engineering problems - no severe storms in 10 years - no failure of a structure in 100 years - no earthquake next year - ……. This probability is directly achieved as: Homogeneous case ! Swiss Federal Institute of Technology 25 / 40

Random. Sequences • The probability distribution function of the (waiting) timetill the first event. T 1 is now easily derived recognizing that the probability of. T 1 >t is equal to. P 0(t) we get: Homogeneous case ! Exponential cumulative distribution Exponential probability density Swiss Federal Institute of Technology 26 / 40

Random. Sequences The Exponentialprobability density and cumulative distribution functions Swiss Federal Institute of Technology 27 / 40

Random. Sequences • The exponential distribution is frequently applied in the modeling of waiting times - time till failure time till next earthquake time till traffic accident …. The expected value and variance of an exponentially distributed random variable T 1 are: Swiss Federal Institute of Technology 28 / 40

Random. Sequences • Sometimes also the time T till the n’th event is of interest in engineering modeling: - repair events - flood events - arrival of cars at a roadway crossing If Ti, i=1, 2, . . n are independent exponentially distributed waiting times, then the sum. T i. e. : follows a. Gamma distribution : This followsfrom repeateduse of the result of the distribution of the sum of two randomvariables Swiss Federal Institute of Technology 29 / 40

Random. Sequences The Gamma probability density function Exponential Gamma Swiss Federal Institute of Technology 30 / 40

Random. Processes • Continuous random processes A continuous random process is a random process which has realizations continuously over time and for which the realizations belong to a continuous sample space. Variations of; water levels wind speed rain fall. . . Realization of continuous scalar valued random process Swiss Federal Institute of Technology 31 / 40

Random. Processes • Continuous random processes The mean valueof the possible realizations of a random process is given as: Function of time ! The correlationbetween realizations at any two points in time is given as: Auto-correlation function – refers to a scalar valued random process Swiss Federal Institute of Technology 32 / 40

Random. Processes • Continuous random processes The auto-covariance function is defined as: for t 1=t 2=t the auto-covariance function becomes the covariance function: Standard deviation function Swiss Federal Institute of Technology 33 / 40

Random. Processes • Continuous random processes A vector valued random process is a random process with two or more components: with covariance functions : auto-covariance functions cross-covariance functions The correlation coefficient function is defined as: Swiss Federal Institute of Technology 34 / 40

Random. Processes • Normal or Gauss process A random process. X(t) is said to be. Normalif: for any set; X(t 1), X(t 2), …, X(tj) the joint probability distribution X(t of 1), X(t 2), …, X(tj) is the Normal distribution. Swiss Federal Institute of Technology 35 / 40

Random. Processes • Stationarityand ergodicity A random process is said to be strictly stationaryif all its moments are invariant to a shift in time. A random process is said to be weakly stationary if the first two moments i. e. the mean value function and the autocorrelation function are invariant to a shift in time Weakly stationary Swiss Federal Institute of Technology 36 / 40

Random. Processes • Stationarityand ergodicity - A random process is said to be strictlyergodicif it is strictly stationary and in addition all its moments may be determined on the basis of one realization of the process. - A random process is said to be weaklyergodicif it is weakly stationaryand in additionits first two moments may be determined on the basis of one realization of the process. • The assumptions in regard to stationarityand ergodicityare often very useful in engineering applications. - If a random process is ergodicwe can extrapolate probabilistic models of extreme events within short reference periods to any longer reference period. Swiss Federal Institute of Technology 37 / 40

Extreme. Value Distributions In engineeringwe are often interestedin extremevalues i. e. the smallestor the largestvalue of a certainquantitywithina certaintime intervale. g. : The largestearthquakein 1 year The highestwave in a winterseason The largestrainfallin 100 years Swiss Federal Institute of Technology 38 / 40

Extreme. Value Distributions We could also be interestedin the smallestor the largestvalue of a certainquantitywithina certainvolume or area unit e. g. : The largestconcentrationof pesticidesin a volumeof soil The weakestlink in achain The smallestthicknessof concretecover Swiss Federal Institute of Technology 39 / 40

Extreme. Value Distributions Observedmonthly extremes Observedannual extremes Observed 5 -year extremes Swiss Federal Institute of Technology 40 / 40

Extreme. Value Distributions If the extremeswithinthe period T of an ergodicrandom process. X(t) are independent andfollowthe distribution : Then the extremes ofthe same processwithinthe period: will followthe distribution : Swiss Federal Institute of Technology 41 / 40

Extreme. Value Distributions Extreme Type I –Gumbel Max When the upper tail of the probabilitydensityfunctionfalls off exponentially(exponential, Normal and Gamma distribution ) then the maximumin the time interval. T is said to be Type I extremedistributed For increasing time intervals the variance is constant but the mean value increases as: Swiss Federal Institute of Technology 42 / 40

Extreme. Value Distributions Extreme Type II –Frechet. Max When a probability density function is downwards limited at zero and upwards falls off in the form then the maximum in the time interval T is said to be Type II extreme distributed Mean value and standard deviation Swiss Federal Institute of Technology 43 / 40

Extreme. Value Distributions Extreme Type III – Weibull. Min When a probabilitydensityfunctionis downwardslimitedat e and the lowertail falls offtowardse in the form then the minimumin the time interval. T is said to be Type III extremedistributed Mean value and standard deviation Swiss Federal Institute of Technology 44 / 40

Return. Period The return period for extremeevents TR may be definedas: Example: Let us assumethat - accordingto the cumulativeprobability distributionof the annual maximumtrafficload - the annual probabilitythat a truck load larger than 100 ton is equal to 0. 02 – then the return periodof such heavy truck events is: Swiss Federal Institute of Technology 45 / 40