 Jeg hedder John Daneskov Samson fra Olbo University, og jeg vil gøre en præsentation her om, eller som en del af populistisk modning, der vil være mere også i morgen. Så der er tre mændige ting i denne præsentation, så det første er, at det er det mellem lukkende mål og nogle applicaterer af det. Det næste er nogle baser og togelser, siden baser og regressinanalyser. Og senere nogle ting om, hvordan man kan opdage en modul, en modul opdagerer. Det er nogle liter, som du mener her, som jeg vil komme ind i denne præsentation her. Så først, det er en mellem lukkende mål, at nogle af dem, som du altså ved, så det er en af de togelser, som er brugt til at opdage de distriktioner, i en given distriktion. Så det er en måde, at det er brugt. Der er andre tekniker, som er brugt til at opdage de distriktioner, som de borgmæssere, B-square-mæssere, baser og statistikker, og nogle af dem vil jeg også mænge lidt om, later i denne præsentation. Men først, de basic steps inusing de maximum likelihood method, det er simpelthen at opdage de likelihoodfunktioner, når du har nogle data, så de giver det bedste fit til de parameter, som du har, så du assume, at du har en given distributionfunktion, og du vil have en fit de parameters i den distributionfunktion. Og det er også noget, som mange af dig, probably already have learned, when you have a basic course in statistics. Så der vil vi have small attention of that in the next slides. So just to take an example, so if you look at the Bible distribution, which has this distribution function here, and we can obtain the density function then. So this is a pre-parameter Bible distribution with a tree parameter, so there is a scale parameter w, a scale parameter her, and a threshold gamma. And then we have this density function here, and then we simply write the likelihood function as a product here, over the number of data that we assume that we have. So here is assumed that it is end. This is not easy to read from the back. Or do you have the slides? Yeah. So we have this density function here, so we simply make a product where we take the value of the density function for each of the data values that we assume that we know, and we assume that they are independent. And what we do not know, that's the tree distribution parameters. So the likelihood function will then be a function of those. And then we can take the do-it for that one, and then we obtain the so-called dog likelihood function. So that becomes this a little bit along the expression here, which is nice because it's a summation. When we have taken the law written, then we get a summation that needs numerically much more stable than working with the product that we have here. So that's the main reason that is done. And then the whole idea is, when you have some data, that is to maximize this dog likelihood function with respect to the tree parameters that we want to obtain. So we simply want to maximize this one here, this dog likelihood function. And for that you can use a lot of optimizers available on your computer from MATLAB for example. There are also more dedicated programs that can do that, but in principle any nonlinear optimizer can do this. And then you obtain the tree parameters k, w, and tau. And then in principle you have done the fit via the maximum likelihood measures. But then we can also go one step further, and that is to obtain the uncertainty related to the parameters that we have estimated. And that's very much related to the statistical uncertainty. So if we have more than 25 to circuit data, then it can be shown that asymptotically the uncertainty related to the tree parameters here will be normal distributed with a mean value which corresponds to the solution that we have obtained here and with a covariance matrix that is obtained from this expression here, namely by obtaining the so-called Hessian matrix, containing the second or derivatives of the dog likelihood function. If that one is calculated in the point here where we have the maximum, then we obtain this h matrix here. And if you take the inverse of that and put a minus in front, then that becomes the covariance matrix of the tree parameters. So that means that in the diagonal here we have the standard activation squared, corresponding to ease of the tree parameters that we have estimated. And outside the diagonal we then have the correlation coefficients. So that means that we have a measure of how uncertain are the parameters that we have estimated by the maximum likelihood measure. And that's very useful when we want to take into account that we have this uncertainty related to the parameters. And I'll give you an example that shows that this can in fact have an influence on the estimates that we want to obtain. So if there are any questions that you should just ask. So this is clear, more or less. Good. Then we can use that on an example. And that's one of the examples that I think is used a lot for a little bit related to what Jochen said just before, related to fitting data, extreme data. So here's the example where it's related to wind data. And it's used here in connection with the so-called PGO threshold method, where we want to fit some data with a mindful distribution. It could also be another distribution, but this is just an example. And there are usually three steps that we want to follow if you use the PGO threshold method. Namely first we want to identify the data that we want to use for the analysis. And that we simply do by putting a threshold and then identify the venues here of the wind speed that is larger than this threshold. But in such a way that we make sure that the data that we obtain, that they are statistically independent. So if there is wind data, then it's typically done by saying that there should be at least one day between the data that we extract, such that each data represents the maximum of one storm. This could also be used for waves, wave heights. That's another set of data where this approach is used a lot. But when we then have extracted a number of data, and it's not so that we want to extract just one data per year, it could be more data per year, then we have end data. So these end largest wind speeds, and they have then been extracted from data series, which cover T-years. And then the next step is to fit a distribution function to these end data. And here we then fit a vital distribution, let's pre-permit a vital distribution. And then in principle we have a model for the extreme wind speed. And this one is in this example used to estimate wind speeds with a given return period. And the example here also shows how this can be used to include the uncertainty on the per meters, if we want the maximum light release method. So we do it with and without taking into account the statistical uncertainty. So the first step is, if you want to look at some data to identify these data that we want to extract, and then fit the distribution function via the maximum light release method. So this is in principle the same as we did before. We find the maximum of the clock light limit function. And then in this case, and that is something that is done quite often, when using the three parameter vital distribution, is that the threshold, the top value here is fixed. So it's not determined by the maximum light release. So in this case here we only obtain two per meters k and w. And that then means that they will have a larger uncertainty in principle than if we had all three parameters included. That also then means that if you have sufficient data, more than 25 data, then we can also obtain the statistical uncertainty, the covariance matrix, using this Hessian matrix, so that we obtain this matrix here with the standard deviations and coefficient of the two parameters in the vital distribution. And that then means that we assume that k and w, they have mean values equal to the parameter that they have estimated, some standard deviations and the correlation coefficient. And then the next step would be to fit or obtain the return pair of corresponding to, or the value of the wind speed corresponding to a given return pair, and that we can do first by establishing the distribution function for the annual maximum wind speed, and that we can do by taking the distribution function that we have fitted to the data, and then raise that to the power of k, where this is equal to n divided by the return pair here, so the number of data per year. Again, assuming that we have independence between the data that we have selected. So the distribution function for the maximum wind speed per year that follows this distribution here, where we have this number of data per year as the eight components here. And then if we do not take into account statistical uncertainty, then we can simply obtain the return pair value corresponding to a given return pair, by putting this distribution function we have here equal to 1 minus 1 divided by r. And that can also be simplified to one, to take the distribution function that we have as the basic one, and put it at 1 equal to 1 minus 1 divided by the number of data per year times the return pair. And then solve that equation for the value here, which is then the return pair for the wind speed. And this can then also be extended to take into account the parameter uncertainty, because the definition of the return pair is, or can also be written in this way here, maybe as the probability that we have a stochastic level modeling the maximum annual wind speed, and then of course the probability that the wind speed corresponding to a given return pair is larger than this value, that is the definition of the return pair value. But that can also be written in this way here, as the value that we want to obtain minus the inverse of the distribution function. And then we introduce a stochastic variable u1, which is a standard normal stochastic variable. And then we take the distribution function corresponding to that. So this is simply modeling the uncertainty related to the maximum annual wind speed. And then we have the two parameters k and w. And what we now do is that we assume that they are also uncertain, meaning that now we have here in this expression here three stochastic variables, namely u1, which is the one related to the wind speed, and then k and w related to the statistical uncertainty. And this can, we have a probability here that we want to define, and then we put that equal to this number here, which corresponds to the exceedance probability of the value that we want to obtain. Then in principle we can solve this expression here for the unknown value that we have here. And I think tomorrow you will learn how to do that, or at least in principle how to do that by, for eksempel, the reliability index technique, the first order reliability methods that can easily be used to solve this expression here, to obtain XR. Defining a limit state equation that corresponds to what we have here on the left hand side of the expression. But that is for tomorrow, and I think Jochen will say something about first order reliability methods. So you cannot do it, you cannot do it today, but you will get the tools to do it tomorrow. What that means now we have integrated also the statistical uncertainty. And there is a small numerical example to illustrate that. So here there are some wind speeds selected over a period of seven years. So that's some wind speeds that satisfy the requirement that they are from each storm during this period. And there are 38 data. They are fitted to this vibration distribution. And this is done by some software called STATRA. That some of you may have heard about before. But that is some software and I'm not sure that we have it available here. But it's not that difficult to apply. So you could take the data and put them into that program. Then there is a tool to obtain distribution parameters and fit them to data. And there you can choose to use the maximum likelihood method. And you can also choose to do that. So that you also obtain the uncertainty on the parameters by solving this optimization problem. So this is simply a print of what you get out of this program called STATRA. So we obtain the three parameters in the vibration distribution. And here I have fixed one of them to 23.9 meter per second. But the two others they are fitted. And then you also get the standard deviation of the parameters. So they are written here. And of course there is no uncertainty on the one that you have chosen yourself. And you also get the correlation coefficient matrix. So the correlation coefficient. So here we only have a correlation between the two parameters W1k. And this is 0.31 in this case. Of course this can be solved for the maximum likelihood. Because that's also on one of the next slides I see. So in fact, and it's also in the slides that you have got that this can also be solved by MATLAB. And as far as I remember it can also be done by accident. So it is not that difficult. So there is an example. You cannot read what it is here, but it is in the PowerPoints. So we can also solve this problem by MATLAB. Because there are also a number of optimizers in MATLAB that can be used to do this. So there is a description of here how to do it in MATLAB. And you obtain exactly the same result as the one shown here. And here is then also shown a plot of the fit of the data. So the distribution function that is filled. The vital distribution is the red line here. And the data shown by the blue dots that you can see here. So the fit is okay. This is a typical fit. It's maybe not that good all the ways. But again it's okay. And then based on that and using the expressions from before, it's all possible to calculate values of the width speed corresponding to different return pairs 2050, 100 year. And they are shown here. 31, 32 and 33 meter per second. If you do not take it into account statistically on certain parameters. But if you do that and you apply and that is what has been done here using the first order and I built a method. Then you obtain these values here for the width speed corresponding to these three return pairs. And we see here that there is an increase. Not that big, but there is some increase. And that is simply due to the uncertainty related to the parameters because we do not fit completely exact. And that is also a limited number of data. So this is a way to include the uncertainty on the parameters. And that is what should be done if you make an estimate of for eksempel 150 year width speed. You should include the uncertainty on the parameters that you have obtained from some data. Here we see that the number goes from 32 to 33.4. So that is a small increase. I understand that between the second numbers and uncertainty are taken into account. But I do not understand what is the operation to pass from one number to the other. So what the difference is between obtaining this number and this number here? I understand that in the number on the right and the statistical characteristics are taken into account. But I do not understand the operation to pass from one to the other. Okay, yeah, it was. Yeah. So let's go back to this slide here. So this was the first one. So if we do not take statistical uncertainty into account, so it's quite easy. Then it's just to find the inverse of the vital distribution to find the value. But then the next step, that is when we want to take the statistical uncertainty into account, then we know that the parameters in the distribution function, they are also uncertain. So in fact we have now three stochastic values, namely the wind speed itself and then the two parameters. And this additional uncertainty, of course, will have an effect that will have more uncertainty. And if you want to go to a certain level of exceedance, then that would have the effect that the wind speed would be larger. And the way it is done is to write simply this distribution function in the way that we normally do it, that this distribution function of the unknown value XR should be larger than or equal to 1 divided by lambda, divided by the return pair. And then what is done here is simply that that is solved with using the inverse distribution function. And then we simply get into the expression here the two parameters which are now also stochastic values. And the problem is then to solve this probability here because that's not one that you simply calculate because you now have three stochastic values. And that is what can be reformulated as a limited equation. That you, tomorrow, will see that you can use this first of all and I will be measured in a efficient way to obtain the often calculated. And then this is solved iteratively with respect to this example. Just to connect this to what I told you before, you remember John and you switch a little bit forward to the switchboard here. Yes. This is total probability theory. I introduced for this to use for harvest. Ja. And that's actually the integral here we want to solve. So this theta, these are the parameters from the Bible distribution. That's the theta, the vector. And then we want to integrate out and often that's very demanding to solve this integral. And therefore it can be reformulated as standard and you can use the reliability methods for at we look at more to solve this integral. And the next topic here which is related to patient statistics, patient regression analysis. Sorry, ja, I just thought of a question. Ja. We spoke this morning about using some logic as well as observations that might identify when a different distribution adequately fits a parameter. Ja. In terms of fitting a distribution to the answer to the distribution parameters of the initial distribution that's possibly a different distribution you're using now. Is that right? So this. So when we account the statistical uncertainty and we have this new distribution here, that's not necessarily any problem. Ja. This is not, this is the distribution function. So that's the one that's related to the data that we have extracted from the pre-go threshold technique. So we have a number of years, so very often we do not have a lot of years of data. So it could be, for example, if we are looking for wind turbines, then typically the number of years of data that's maybe five years or seven or something like that. And then you are asked to give an estimate of the 50-year value of the wind speed for the design of the wind turbine. And then if you just took the largest value each year, then you had seven data, and that would not. But actually there was a very interesting question because the maximum likelihood method is very nice because it treats uncertainties somewhat consistently, uncertainties that are related to lack of observations or finite data sets. But what we implicitly assume is that we know the probability distribution function to use. So when we use a probability distribution, that is totally off the phenomena, then we might get misleading results. Actually the scatter of the parameters we get out from the maximum likelihood is not reflecting the inability of our distribution function to reflect the phenomena. So we have to be careful very much. And also we have to be careful with a very low number of observations because you can imagine every distribution function has a degree of freedom, which is the number of parameters. And when we are near buying, for instance when we have a three-parameter distribution and only have five observations, then we might get a very good fit, but it's like fitting an elephant with a polyron. So we have to be careful about that. So the number of observations has to be always all of magnitude higher than the degrees of freedom on parameters of the distribution. But the implicit assumption is that we know the distribution function. We cannot... It's very hard, you know, like your ratio test and things like that, to test which distribution is better, but still we can go to very misleading results when the distribution function is very much off the phenomena. Something that happens typically when you look at data for waves is that if you look at the data and then plot it like what is done here, then it's quite often possible to see that the data they are coming in effect from two distributions. Maybe because there are some depth limitations on the maximum waves that can be observed. And then you will see that there is some kind of a kink in the data. And that's then also a sign that you have selected too many data. So you have to fit to the table. But of course going not too far out in the table with the data that you have. So therefore if it is possible to get 20-25 data, then that is absolutely too preferred. Ja. So Jochen, did you say anything about Bayesian statistics? I introduced the rule of Bayesian. I also introduced the formulation for continuous waves. Ja. So we have mentioned that prior distribution and posterior and predictive. I have in fact put one or I think two slides here at the end. Just to, because that is what is being used here. So maybe you have already said that but now I just... No, no. Okay. Because this is an assumption for this Bayesian regression analysis. That's the next topic. And in Bayesian statistics there are three main concepts or words that's important to know. And we often need to know how to handle this by Bayesian statistics, which is an extension you can say of Bayes rule. So this is also related to fitting data to a distribution function. So what can be assumed again here is that we have some observations. And they are also again assumed to be independent. And we have a density function that we want to fit. And that density function has some parameters. And they are here denoted by the vector Q. So that could be the three parameters we had for the Bible distribution before. And then the whole idea in Bayesian statistics is that we assume that we have a prior. We have some prior knowledge about these parameters, which we put into a density function. Det er noted as a prior density function for the parameters. And there is a simply one mark here to just to indicate that this is the prior for the parameters. Then we have the observations. We can have n observations. And then we obtain the posterior density function, the updated density function for the parameters Q. And that is the one that we have here. So there is a double mark here. And there is a vertical line here indicated that it is conditional on the data. So it is updated using the data. And that is done by this expression here, which is another way to write the base rule. And in that one we have the likelihood function, which is the same as we had before in the maximum likelihood principle, but now is used in the base approach here. So we have the likelihood function here. And you will see it on the next slide. Then we multiply it with the prior of the parameters. And the result we get here on the left hand side that is then an updated density function for the parameters given the parameters. So we simply take the order of the parameters here or the variables. We simply follow the base rule. And then to make sure that this is a real density function, then we have a normalization here, where we integrate all the values of the parameters Q. So this is the basic expression here to obtain the updated density function for the parameters in the density function. And maybe I should have them now. Heta is replaced by Q, otherwise it's the same, right? Ja, ja, so it's the same expression that we have here. And the likelihood function is also the same as we had before, namely the product using the density function. And we put in the data values that we know, but given the unknown or what we consider now as uncertain parameters Q. So it's the same as for the maximum likelihood principle. And then it's possible to calculate this one if we know the density function. The expression here looks complicated, but in many cases it's not that complicated. For example for a normal distribution function, where we know the standard deviation, then the expression here we assume this one is a normal distribution, then this one also becomes a normal distribution. And then it's also possible to go one step further and then obtain the so-called predictive density function. And that is the density function now for the stochastic variable that we are really interested in, maybe the x stochastic variable, but it's updated with respect to the data that we have used. So that's why it's written here like a conditional density function. And that is simply done by taking the density function we had from the beginning, the assumption or the one that we want to look at. And then we multiply with the posterior density function for the parameters and integrate all values of the parameters. And then of course Q disappears at a certain parameter and we only get this density function here, which is the predictive density function. So we have a prior that we, in a way, choose ourself based on experience or based on other data. And then we update that by data to obtain the posterior. And finally we can obtain the predictive density function, maybe this one here. And for reliability analysis, this one here would then typically be the one to use. And the other two, they are just a step in between. So this procedure here, which I've put here at the end, can now be used for Bayesian regression analysis. So it's more or less exactly the same procedure, but now used for a model, which we assume can be written in this way here, namely as a linear model, where we have some parameters that we know. So that's the X values here. And there are some parameters beta that we consider as unknown, so that's the regression parameters. And then we have added here an uncertainty term, which we assume is normal distributed with mean value equal to 0, and the standard deviation, which is written as 1 divided by the square root of h. So that means the parameters that are unknown here, that's the regression parameters beta, and it is the parameter h. And we assume that we have some observations, some data. And the data that we assume that we know, they are then denoted by Y with a hat on the top. So there are capital N of these data, and they are then connected to known values of the X parameters here. So that means that we have a whole matrix here of X values. So that is what we assume. And then one way to proceed is, again, also to write the likelihood function corresponding to the observed values that we have. So that means that now epsilon here is the stochastic variable, which is normally distributed. And that can then be applied to obtain the likelihood function, because we can write it as, again, a product over the end data sets that's available. And then we have the density function here, which is a density function related to epsilon. So in this case here, it will be a normal distribution, or density function that we have to apply here. And then we can simply apply the expression here, and that can be written as Y hat minus this summation here, corresponding to the regression model. And that means that we calculate this density function here for a given or conditional on a given value of the uncertain H parameter here. So that's the way to write the likelihood function in the case that we have such a regression model as this one here. And then we can, again, use the maximum likelihood method to obtain the best estimate of the parameters. And in connection with that, a number of parameters are calculated, and you will then see how they are applied in the next slides. So there is a matrix eta here, which is simply the matrix with the known x-values transport times itself. P is the rank of x, u is the number of observations we have minus p. We have here calculated using the best estimate of the parameters beta obtained by the maximum likelihood method, and that is the one that we have here in the case that we have the linear regression model. So all this is the result of using the maximum likelihood method. So now we then go one step further, and then we want to put this into a Bayesian description. So what we do now is that we consider the parameters beta and H stochastic variables in the same way that we had this vector Q before for the parameters in a density function. And we assume that we have some prior information about beta and H. And that could be in the form of having some data, a data set that we can use to obtain prior values. And the prior values are then given in the context of these parameters that we have here. So if we had some prior data, then we can calculate these parameters here, and then put a mark, we indicate that that's our prior. So that is what is done here. So we can see that the parameters from the slides before, but now we have a single mark on, because that's now considered as our prior if we have some data. If we do not have some data, but instead we have some idea about what is the mean value of beta, what is the mean value of H, and also some information about the covariance matrix of beta and the standard deviation of H. Then that can also be used to formulate the prior, and that is what is indicated by these equations here. So that's another way to obtain the prior parameters, but that is if you have such information available. So either we can have some data that we use to obtain the prior, or we can have some knowledge about basically the mean value and the standard deviation of the standard parameter and H. And then the idea is that we use that to follow the Bayesian procedure as before, and to do that we need not only to have these parameters as I have mentioned here, for these four parameters here, we also need a density function for the prior. And there is very convenient to use a so-called conjugate prior, and such a prior is one that if you use that for the prior, then the same density function will be the posterior. So these two will be the same. Then we say that they are conjugate. And it can be shown that in the case that's in the situation that's described here, that if we use a so-called normal gamma distribution, which is the one that is shown here, if that one is used as the prior, then it also becomes the posterior. So it is a combination of a normal distribution function and a gamma distribution function, or in fact an inverse gamma distribution function. This is not something that's easy to do by hand, but the computer can do this. And there are programs like this that is put into already programs, so you do not need to do that yourself. But if this is chosen as the prior with these parameters that we had before, and then we assume that we have some new data that we can use to update our prior, and now we know that this density function here also becomes the posterior, then the only thing we have to do is to obtain the updated values of the parameters, and they can be obtained by these four expressions that are shown here. Så eta, with a double mark, is then the parameter in the posterior. So that is simply obtained as the prior plus eta corresponding to the data. And then there are some little bit complicated equations here that I will not go into detail with, but you can see them and they can be calculated. If you have the data and you have the prior, then you obtain the posterior. And that one will then be this normal gamma distribution, which is a combination of a normal distribution function and a gamma 2 distribution. So then by that we have an updated distribution function for the parameters for beta and eta. But also here we can obtain a predictive distribution function, so that means an updated distribution function for the response, so if we go back then the response, that is what we have here on the left hand side, so we can also obtain an updated predictive density function for this parameter. And that one becomes a student t distribution as seen here. So this is the updated density function for the response, given the data, and that can be written by this long expression here, but that is a student t distribution with a parameter here, and at least a parameter that is t. So we follow more or less exactly the same principles as before, but the equations just become much more complicated. So that is what can be done by a subset regression analysis. And here it should be noted, as I write here, that if the standard deviation, so that means this h is assumed to be known, then much of this simplifies, and then it simplifies into normal distribution functions. So then things becomes much more simple, but it's not always that we know this h parameter. This is typically a difficult one to obtain. So in many cases it's considered that it's been uncertain, and we also want to update that one. But if it is known, then all these expressions become more simple. And in fact, those expressions, they are in the book by Michael Paver. So if you look at the pages related to this correlation model, then you will find those expressions given there. But what I have shown here is then the extension, we have that the h parameter is also considered as uncertain. Yes, question to this. There are some simple examples here to illustrate that it can be used in more practical applications. The first slide here is just an example without any data, but in fact this can be used if you want to fit some fatigue data. So if you have the s-n curve connecting the number of cycles to failure with a given stress plane s, and m is the number of cycles to failure, and we have the two parameters k and m, which are unknown if you want to do some statistics related to those, then that can be formulated also as a linear regression model and simply by taking ln to this expression here, then we have the relation here. And here we can consider the two unknown parameters ln k and m as the two beta 1 and beta 2 parameters in a linear regression model. And data, the x vector or that corresponding to that one would then be, so here we have x1 and x2 related to those, so that will be 1 and it will be ln s, so ln to the stress planes. And then what we will typically have available, that number of cycles to failure in, and it is the stress planes connected to that. So that means that we know the x values here and we know the result on the left hand side. And then we can simply do statistics as shown in the previous slides. So that's one way to apply it. And there is another example, not on this slide here, but on the next slides. This slide here, and the following one is just another example, which is much simpler, but just also to illustrate the steps into the Bayesian approach. But here applied to only one stochastic variable, namely the heat strings. And this is an example taken from the book here. So you can follow that. And what is shown here in the red box, that is simply the equations that just showed before. They are not on the slides. I will put them a little bit later, after I made the PDF version. But it is the same as you saw before. And you can also get a copy of these slides. So what is written here is that this is the prior, this is the posterior, and this is the credited. And that is of course what we can obtain here. So what it is assumed here is that we only consider the mean value to be unknown and assuming that the instance here can be modeled by a normal distribution function with a mean value and with a standard deviation. And the standard deviation we assume to be known equal to 17.5 megapascade. So only the mean value is unknown here. And that has a prior. And here with a mean value of 350 and the standard deviation that is 10. So the mean value here in the stochastic variable for the instance is then also in the self-concept table for the prior. And then we have some data. Here we have five data. And then we can simply follow a similar procedure and then obtain the posterior for the mean value. And that becomes this one here. The mean value goes a little bit up compared to the prior and the standard deviation goes down. And that is what is illustrated in the figure shown here because here we have the prior. So that is the full line here. And then what is also shown is the posterior. So that is the dashed line here. And you can see the mean value is a little bit larger and the standard deviation is smaller because it's more narrow. And what is also shown is the likelihood function. So that's the one that we ought to have here. But basically what we enter to show here is that we have a prior and a posterior for the mean value. And then we can then go one step further and obtain the predictive distribution function for the mean strength. And that is the one that is shown here together with the one that we started with the original one. But the updated one, the predictive one, is the dashed one here. A little bit to the right indicating that the mean value has increased a little bit by the data. And then the next example here is a little bit the same but now extended to a regression model. And here it's related to some data for Timber, also taken from the book of Michael Paper. So it's also written there. But it's applying the same principle as I showed before but assuming here that the mean value know that the h-permeter, the standard deviation is known. So in that way the expression becomes not so complicated. What we have here is that we have the tensile strength of the Timber denalded by X. And we have the modulus of elasticity which is denalded by Y. And then there is a linear regression model here saying that the modulus of elasticity is linear in the tensile strength but with two parameters beta 0 and beta 1. And we have some uncertainty which is assumed to be normal distributed with mean value 0 and unknown or standard deviation that we want to obtain. And what is assumed is that we have some data that they are shown here. And then we could have used the maximum likelihood method at obtain a first guess of beta and beta 1 and the standard deviation here. But that is not what is done in the book. There instead the least square method is used to obtain these parameters. So that's an alternative way to do it. In this case here the difference would be I think it would result in the same values. It's just another way to obtain the parameters. So here the parameters beta 0 and beta 1 they are obtained by least square technique to obtain these two numbers here and the standard deviation since this one here. And the corresponding covariance matrix that is the one shown also here. And then there is a patient updating assuming that there are some additional information available, namely two more data set shown here. And then assuming that the standard deviation is known we can consider what we had here as the prior for the two parameters beta 0 and beta 1. And then we can obtain a posterior using the data that's available and then we obtain the two values here for beta. So there is a double mark here whereas the single mark here indicated that that was the prior with the corresponding covariance matrix here. But here we then obtain the posterior and I have not shown the equation here but they are similar to those I showed before but a little bit more simple because we do not hear you that this standard deviation is unknown. But that doesn't mean that having these posterior parameters we can write an updated regression line simply by using these two parameters and that is then also what is shown here. So we have a prior based on four data points I think that's shown here and there is one more somewhere and they are then updated with the new results namely this one here and this one here and then we get the posterior which is this line here. So that was some aspect of linear regression analysis. So now we move to something different about completely different but something related to model uncertainties. So now I also think that Jürgen has said something about different types of uncertainty that we need to consider if we need to make a proper model to be used for decision analysis tomorrow and the day after tomorrow. So we have different types of uncertainty and I will just repeat it here. So we can have some physical uncertainty so that could be the yield strength or it can be the annual maximum width speed of some uncertainties that they will be there and they are not that easy to change. So they are denoted as physical uncertainties or aleatory uncertainties. But then we can have some other uncertainties that we also need to account for and that can be measurement uncertainty so if we have measurements and we have some uncertainty related to that we can have statistical uncertainty due to a limited number of observations or data but if we had a lot of data then that uncertainty here would disappear and the same with the measurement uncertainty if we had very precise instruments then that could also disappear so it's something we can control but we can also have model uncertainty because typically we would have a model for parameters that we want to obtain a mathematical expression for example if we want to model the load being capacity of a column with respect to stability then we put up a model that can say something about that but that model will have some uncertainty and that is model uncertainty and that is very important to model and of course the model uncertainty is connected exactly to that model that we consider. So this next slide here they are related how to obtain this model uncertainty and if we had a very perfect model then the model uncertainty would also disappear but then that could be very impractical to use very complicated to use so sometimes we choose a more simple model but then we will have some more uncertainty related to that so that's a balance and these uncertainties here they are also called systemic uncertainties because we in a way can control them ourselves if we do something better then we can reduce them and there are different ways to formulate and to make these models more and more complicated so what I will show here is in the relative simple end of how to obtain model uncertainty this can also be done much more complicated but I don't think there is a need to do that as a starting point so we will assume that we have a model and this is here denoted by f and it will in this first express in our model here be a function of some parameters x that we consider as a physical uncertainties so that would be the yield strength of steam for eksempel if it is the public capacity of steam column we are looking at then that would be one of the parameters that would go in so there would be some parameters here as physical uncertainties denoted here by x and then there can be some other parameters which are regression parameters someone that we want to fit in the mathematical model if here and then one way to model uncertainty is simply to multiply this expression here with a stochastic variable and other possibility could be to add a model uncertainty and then that would be even further possibilities but what is really often done at least when looking at what is done for structural engineering that is to multiply with a stochastic variable here and that model uncertainty is the one here in the middle den den den den den den den den den den den Щ delta which is the model uncertainty which is very often assumed to be pinlog normal med en nødvendig valg af 1 og den standardivation. Og så er der også en faktor b her, som er den bias af modelen. Så det er også en parameter, der er ondt og begyndt til modelens kærlighed. Så hvad er ondt, det er i princip, de parameters beta her. De kan så i nogle cases være nøde, men det er altså den bias b, og det er den standardivation her. I modelens kærlighed. Og så kan vi simpelthen bruge, igen, den maximale kærlighed, at få de parameters. Så det er en forskildelse. Så hvad vi betyder er, at vi har nogle data available. Så vi betyder, at vi kender de valgene af X. Så med fysikale kærlighed. Så vi betyder, at hvis vi tager en test, så measures vi de parameters. Og så betyder vi også, at vi measures de valgene her. Og vi betyder selvfølgelig også, at vi measures resultatet på left-hand side. De er nøde. Men det er meget vigtigt, at vi skal møde, at vi skal møde de valgene af fysikale kærlighed. Og det er vigtigt, fordi i enden, vi vil typisk bruge det som en del af valgene, eller som en del af en stor model, hvor vi vil account for de bedre kærlighed. Så hvad vi vil account for i dette yderligere af denne, så er det en kærlighed i X-factor her under fysikale kærlighed. Men vi vil også account for de bedre kærlighed. Så det er to bedre kærlighed. Og i en del af det, vil vi typisk også være interesseret i accounting for de statistiske kærlighed. Så det er to bedre kærlighed. Og hvis measurement kærlighed, så er det også de kærlighed. Men hvad det er i fokus her, det er faktisk at sætte op en model, hvor vi kan include fysikale kærlighed, og så estimere modelkærlighed her. Og det kan blive gjort via de maximale fysikale kærlighed, fordi de underfysikale kærlighed er klar, så der er en possibilitet, og der er en anden, så en anden possibilitet, at i hvert fald, hvis du vil bruge en unikode, er en approach, der er brugt meget. Namely en approach, der er beskyttet i unikode 0, så jeg har også included et par slags på denne, fordi det er brugt, jeg synes det er meget. I hvert fald, for nogle materialer, for at obtain de kærlighed her, og denne modelkærlighed her. I en systematisk måde, som kan blive brugt også for en lille analyser, og som en del af at sætte op de kærlighed. Men hvis vi, i den første step, bruger de maximale fysikale kærlighed, så er det præsideret exakt the same, som i de første applicater, som de maximale fysikale kærlighed Namely, at vi identifierer omkring kærlighed, så det kan være beta her, the bias B, og standardivationen af modelkærlighed. Og så formulerer vi de kærlighedfunktioner, og så er det bare en produkt herover, og så er det alle data sets, hvor vi bruger, at vi har betyder, at modelkærlighed er lovnormt distributet. Så det vil være denne densitetfunktion for det lovnormt distributet for modelkærlighed. Og så er det simpelthen et præsid for modelkærlighed, og så er det konditioneret under kærlighed. Og vi kan formule de lovkærlighedfunktioner, og vi kan maximale det, og i det måde, obtain de ondående kærlighed. Og i den måde, kan vi også include statistiske eller kærlighed kærlighed på de, hvis vi har en sufficient number of data om at obtain denne matrix, som jeg havde før, så de kan også være includet. Og så er vi præsident, fordi vi har det, vi vil have. Den anden approach, hvilke er, på en måde, det samme, men ikke exakt, er at bruge denne approach i Eurocode 0, i NXT, og jeg tror, at det er et godt procedure, som kan bare være brugt for mange forskellige sætter af data, og det er vigtigt, som det første, at obtain input til en kærlighed-analysisk. Så først er det af Figur her, illustrering også, hvad vi vil gøre, Namely, af Figur, der bruger valgen af modelen på den eneste, og eksperimentale, obtain data på den anden aksperimentale. Så vi ved, fra measurement, vi ved alle valgen af stokastisk verden her, exakt, fysisk, de er messet. Og så, hvad vi kan gøre, er, at vi kan simpelthen prøve alle data-sætter, som har været obtain via eksperimenter, og det er dotten, som jeg ser her, og hvis modelen er perfekt, så er alle dotten her på en slags line, med en slag. Så det, hvis data er perfekt, men det er ikke skat, så det vil være en bias, så det vil ikke være en her, og vi har nogle skadder, og skadder er messet via standardiseringen af modelen, eller appellentlig, kofisnerbaseringen af modelen. Så det er, hvad de næste, to slags det, er om, at det er et sted, for at være followt på denne NXD, for at obtain B her, og den kofisnerbaseringen af modelen. Det er måske ikke så vigtigt, at læse, men, at have copyviseringen, så vi kan se, hvordan det ser ud. Så vi simpelthen har de data-sætter, og vi har også de korrespondede valgene, obtainet fra modelen, og den første dag, vi stopper det parameter B, baser på linear-represseanalysisk. Og så er der nogle steder her, der kan være followt, at obtain den kofisnerbasering af modelen. Og der er nogle steder her, hvor vi tager LN, fordi modelen her, er, at du er lovnormat, distributet. Så det er en faktisk valg med standardiseringen af LN, til testdater, der er kagelig. Men endresultat er det, som vi ser her, i den første dag, at vi obtain den kofisnerbasering af modelen, og her er det bias. Og så er det også i nogle situationer, som er gammelt i LN, i LNXD, i Eurocodes, er det, der har med, at vi har den den minvarisering, og så kan du kalpe det altså kontribution til den tolste kurs i den kraftlige moden. Så først, du kan obtain den minvarisering, og du kan også obtain den kofisnerbasering, og det betyder, at vi kan også obtain den tolste kofisnerbasering af modelen, som vil have to kontributionser, en fra modelinserti og en fra modelinserti i X-spektoren i fysiske permeter. Men der er en forhold til modelen, som vi ser. Det er den, som er optaget her, baser på ligniseringen af modelen. Og vi kan også tilføje measurementinserti. Men det er en lille måde til at sætte osv. Og der er to eksempel her, som vi ser fra nogle data, som vi har tænkt på. Så der er en eksempel her, hvor de data er plottet. Så det er en eksempel, hvor der er en lille osv. Og altså en bias, der ikke er så stor, som 1.06. Og den kofist af asien er 12%. Men det er stille noget. Men det er meget mindre osv. Og det er optaget i den anden eksempel her, hvor der er en stor bias 2.5. Indikerer, at modelen er meget konservativ. Men på den anden side, modelen er meget osv. Fordi kofist af asien er 25%. Så de data, de har været optaget fra nogle konkrete valg, hvor kofistens kapacitet har været optaget og begyndt til modelen. Og de valg, de har, tydeligt, det er sjovt her. Og selvfølgelig er det vigtigt, for de, som har en stor bias, men på den anden side er det vigtigt at være skabt. Men kofist af asien her er meget vigtigt, for at få de to nommere her, hvad du skal bruge. Hvis du vil gå før, er det en relativt billig analys. Og tomorrow er jo så meget nummeret for modelen og servicier, der er så typisk lige ude for relativt billig analys, og for kofkalibrationer, og simpelthen, og problemer. Men det er vigtigt, at være vigtigt for modelen og servicier, fordi de er i mange case af det samme ordre, som de fiskale og servicier, som selvfølgelig er meget vigtigt. Og samtidig vil vi bare vise, at modelen og servicier har en kofist af asien af 10%. Men vi har en følelse, men vi ved ikke. Men hvis det er så muligt, så vil vi have nogle testdater for at bruge denne basis for at bruge denne moderne servicier. Denne redline har kommet fra det maximale brugte i destinationet. Ja. Denne redline har kommet fra dem, og så har du kommet fra modeller for det, og du har kommet fra det, som er baseret på din kofist. Ja, så testdaterne her er denne, som har kommet fra dataet. Ja, så det kunne være gjort med det maximale brugte, som har kommet fra Lisequare-technik. Okay. Det kan blive nærmest det samme. Hvad var din intervendation af denne? Denne? Ja. Her kan vi se, at modelen er meget konservativ, bedre til testdaterne. Så vi obtainter flere løbige kapaciteter, end hvad modelen predicter. Så det er derfor, at biasen er meget vigtig. Men på den anden side, den konservativ af modelen er også meget vigtig. Så, i en måde, du må betale for det, og det er virkelig gjort med de modeler, som de chooseer konservativ per meter. Så det var det, jeg havde. Og så er der nogle ting, som du kan lide. Det er også i en lille eksercise, eller eksempel, som du kan se, for at estimate de modeler. Så der er nogle data i en exel file, som jeg tror Sebastian kan distribuere. Så du kan prøve at applicere denne procedure her.