 I'm going to university. My presentation today is about evaluating the value of SHM with longitudinal performance indicators and hazard functions using Bayesian dynamic predictions. There are mainly four parts of this presentation. The first is about the objective of evaluating the value of SHM. And the second is about the joint model of longitudinal data and hazard function. And the noise, yeah, it's working now. So the second is about joint model for the longitudinal data and the hazard function. Based on the hazard function, we can determine the maintenance plan and the value of SHM. And the final is the conclusion. The first objective is to determine whether monitoring or not, because civil engineering structures are subjected to time-dependent degrading processes which require considerations of a wide range of uncertainties. Then it is required to make decisions and these uncertainties, acquiring more information prior to making the decision is really crucial. And the SHM can provide information to reduce the uncertainty, so it is widely used. But it comes at a cost that is not always justified by its benefits. So we need to evaluate the value of SHM before its implementation. And the second objective is to determine the difference of implementing an SHM strategy. Because provided more information by SHM, this decentralized maker will have a totally different inspection planning for this structure. This will lead to a change of this expected total life cycle cost. And the difference can be calculated as the value of SHM. The calculation of the expected total life cycle cost is based on the inspection or repair plan. And here we use the hazard function or a failure rate as a basis to determine this inspection plan. So following, I will introduce the joint model of longitudinal data and hazard function. Actually, this joint modeling is an active topic in the medical research field. There are two types of data recorded for a group of patients subjected to the same disease. The first is the longitudinal data, or in other words, the time theory response measurement for each patient in this group. And the second is their time to test data. And based on the joint model fitted to this recorded data, the objective is to predict the death of a patient with the same disease. It is quite clear that similarities could be a draw from the patients and the civil engineering. So it might be interesting to apply this procedure in the structural engineering. So here we also define two sub-models. The first is the structural performance time series. Where this yt is observation outcome consists of the underlying structural states with the random effect b characterized by the variance parameter d. And this epsilon t is the observation error, which is time independent and is normally distributed. And for the second sub-model, it is the survival process defined the hazard function. This hazard function is defined as the limits of the probability of failure during the time interval s conditioned by the structure still surviving at time t and averaged over the same time interval. And actually in the civil engineering field, this functional form of the hazard function is normally unavailable. So here we make some assumptions of this functional form. We assume that it consists of two parts. The first is the general baseline hazard, which is H0t here. And the second term is associated with structural states characterized by the random effects. To put it more detail, for the baseline hazard, we assume a variable baseline, which in the red box of this equation, and it is an accelerated failure time model. And for the association structure, we assume that the failure rate of structure is related to the current value of its structural state and its changing rate. Normally, it is the case for degrading structures. Because for example, in this figure, if the structure has a larger structural states, mt, and a lower decreasing rate, it will lead to a lower failure probability. So with this functional form available, we can do some parameter estimation. But here I will not go to details about this. Here, MCMC measures can be used to do the parameter estimation. And there are also our packages available, named GMBase, to do the parameter estimation. So to sum up this part for the joint modeling, the objective is to derive the hazard function, which is dependent on the random effect. The different probabilistic model of the random effect will lead to a different hazard function. And as soon as this hazard function is available, we can do the hazard-based maintenance planning and also calculate the value of SHM. So for calculating the value of SHM, there are five notations. There are Z and X. There are inspection and monitoring outcomes variables. And Z is the structural state with prior information distribution of that. And A is maintenance action. And E is the inspection decision. And here it is the decision tree for planning these inspections and repairs. And M0 is not implementing SHM, and M1 is implementing a certain SHM strategy. So for M0, the expected total life cycle cost can be calculated based on the prior information of the structural state, which means the inspection and repair plan is based on the hazard function derived from the prior random effect. While if there is SHM available, we can update this random effect, such as this right curve in this feature. And we can see the uncertainty is less. So it will lead to a drop of this hazard function in return. The inspection and repair plan are changed. And these two updates of this expected total life cycle cost. So this difference actually can be calculated as the value of SHM. It is the kind of expected value of sample information. And this distribution of estimated SHM cost affects it is used to update the structural state. So for calculating the expected total life cycle cost, this part I mainly refer to the PhD thesis of professor's job and the thorns. And the total cost consists of the cost of failure, inspection, repair, and monitoring, if there are any. And for example, this is a functional form for the expected cost of the failure. And we can see that there are several probabilities required to calculate this. And these probabilities are related to the decision tree. Namely, there are four. The first is the hazard function, which we can derive from the drum model. And the second is a failure probability during time t, conditioned that there is no repair before t. And this is calculated based on the hazard function. And there are two methods to do that. One is by this. It is the integration. It's an integration. And the other is by variable distribution, because the hazard function has a form of variable distribution. And for the probability of damage detection and the probability of repair, I refer to the journal paper here. And for detail, we can refer to that paper. And for the risk acceptance, criteria, and decision rules, the maximum acceptable hazard is corresponding to the maximum allowable yearly failure rate, according to the GCSS model. And for the inspection planning, we use the stress-host approach. That is, the inspection is planned in the year before the stress-host value is crossed. So for different stress-host value here, we will have a different inspection plan. To sum up in this flow chart, with the John modeling, the hazard function can be derived. But with the SHM outcome distribution modeled, the structural states can be updated. So this will lead to a calculation of this value of SHM in the function of this hazard function, like this. So the last is the conclusions. First, a joint model of the time-dependent structural performance and hazard function is first introduced. And then based on the derived hazard function, it's used as a tool for determining the inspection and repair plan. And then the entity related to the SHM outcomes are considered and incorporated in the joint model, leading to updates of the inspection and repair planning and expected total life cycle cost. Finally, the difference between the prior and posterior expected total life cycle cost is defined as the value of SHM here. So thank you for your attention.