 And I have to say that my my my internet, it always happens for some reason this last to this is unstable. Okay, the most important is what what can I what can I do. So okay, so let's start with the with the first presentation. This would be project 10 with the title very fine interaction among different types of discrete events by using the multivariate Hawkes process. This was done by a thank you Carlos Carlos Velázquez and the supervisor were a young ganzuan and a master owner. So Carlos please go ahead. Yes, thank you thank you for the presentation. So my project is the number 10 as was said very fine interaction among different types of discrete events by using multivariate Hawkes process. And in particularly, we will focus on epidemic type after shock sequence model that is kind of Hawkes process. So this is the agenda. Let me start a Hawkes process help us to understand temporal point process in particularly. It can be so characterized by its conditional intensity that is the expected number of events that happen in a certain time interval given a history of this point process in particularly Hawkes process so the point process as a cell exciting and in this sense can be the process can be thinking. Yes, don't worry. So the conditional intensity can be thought as a bad one rate and a triggering rate. So this bad one rate is that the the normal behavior of the point process and this point process could produce some some offsprings and this behavior can be seen in this diagram. So we observe a temporal point process and the point process have a background behavior and the these background events can produce some offsprings and such offspring can produce and others of offsprings in the final stage. The Hawkes process we will have a level background process and triggering events. So we also can extend the temporal process to include the spatial characteristics. So in this sense we will have the expectation number of events that happens in a temporal time interval and in a spatial interval. So all of this is given a certain history time. So in the case of cell exciting spatial temporal the triggered function will depend on the spatial coordinates and temporal coordinates. And for simplicity is recommended to to split such function as a separable function in space and temporal characteristics. So the time the spatial temporal form let us introduce a market characteristics. So we will have that for each event we'll have a spatial coordinate temporal coordinate and also a market coordinate and example of this would be the earthquakes that they have a epicenter time of occurrence and the magnitude. So this is a kind of market temporal spatial temporal Hawkes process and in particularly we call multivariate point process for when the space is a finite set of market possibilities of each event. Examples of multivariate point process are for example crime reports. So in policy stations can classify different crime reports and they will have different coordinates and also in time. So we can produce a model with this kind of Hawkes processes. Also for social media when we post something it will have a location and a temporal location and maybe different person could retweet or share in this in the sense that Hawkes process could simulate the behavior in social media. Also for pandemic situation like this one that is present can be modeled with Hawkes process and as I said seismicity activity could be modeled with Hawkes process. So the main idea is based on the waiting time of these processes. So the cumulative probability distribution of waiting time that the next event happened from certain time t can be described by the question 4 and so its condition and its probability density function will be the function 5. Also the survival function is introduced that is the equation 6 and is the probability that an event from t on from the waiting time greater than u and also we can introduce the hazard function that is the probability that the next event occurs between time u and u plus differential u make sure respect to a probability waiting time that is greater than time waiting time u. So it can be seen as equation 7 and from equation 7 we can solve in order to have equation 8. So the survival function is related to the hazard function by this last equation. So then given process n in a temporal interval s t we can introduce the likelihood function l as the joint probability density of waiting time to each of these events. So this likelihood function can be seen as the product of the probability for each event in an order in temporal order. So we have the multiplication of these the products of probabilities and can be written as a density probability function of each one of them and finally we could obtain the equation 9 because by definition the hazard function is related to the probability density function that I told you in the last slide. So the hazard function is basically the definition of the conditional intensity rate. So we can find equation 9 in terms of the conditional intensity rate and so the likelihood function we can take the the logarithm that is a very usual way to to use this function and it will help us to find a non-regular parameters that will be introduced or used for the model of our Hawke's process and we will use the the very use maximum likelihood estimate in order to find some parameters of the model. So in the case of the spatial time it has model that is epidemic time after shock sequence suggests that the intentional conditional rate could be so as the mu that is the background rate and also we will have the contribution of the trigger n function. So in this case the spatial background rate is thinking as a rate of time homogeneous Poisson process and the trigger n function is the contribution of the seismic hazard due to trigger n effects of the of each one of earthquakes. So the explicit functional form of this conditional density is presented to this slide. So in this model we will interested in find the parameters new a c alpha p d q and gamma that are positive parameters and u is a non-special function that we will find by this model. So we can find the probability that the events were generated by the background process and is this will be the relation to us and the probability of if the event is triggered from previous event that will be described by the equation trees 13. So the Hawke's process suggests an interactive process until we will find a certain convergence in order to find the values of the parameters that we want to find. So in this sense we decided to apply this model to Mexico seismicity. So we use the Mexico catalog in order to apply this model. The new Mexico catalog can be described by the the Gutama Richter law that and we describes the the frequency cumulative of the earthquakes but the according to the magnitude. So as we can see we have we find what from which from what magnitude the catalog consider complete and from this plot we can see that the maximum frequency is close to 3.8. So we consider just magnitudes greater than this magnitude that will be considered as the completeness magnitude for the catalog. So when we apply the Hawke's the etas model we find different step of iterations and we can see that the convergence of these parameters is observed quickly. So the etas model is a good proposal in order to find the parameters of the model used in order to find the triggered function and the background rate. With this we can find the probability that the events are of background. So in this sense we can separate the events of the background and also what events were produced by other and the classical case is the aftershock sequences of main earthquake. So also we can find the probability that an event j is triggered by even i and so we will find this matrix of the probability transitions. So we can represent this behavior with complex networks because a complex network could be identified by a adjacency matrix. In this sense we can set a different threshold in order that the earthquakes will be connected by the probability that they were triggered by another. So we study how is this probability function and of course we need to set a probability threshold in order to to consider what event we will be connected. So in this sense for different values of this raw threshold we find that the earthquake, the number of earthquakes will decrease because you will separate the background from the triggered events. The complex network, this is an example of the complex network for these parameters of thresholds and also we can find different characteristics in complex networks. The most common characteristic is the degree distribution that indicates how are the the the nodes of the complex network with different number of links between them. Also we can find the hierarchy structure for for some earthquakes and the offspring of these that will be the aftershocks and also these aftershocks will produce another aftershocks. So in this sense we we have the the main the main objective of the TAS model on Hocus processes. So it depends of the level of the magnitude threshold of the sorry the probability threshold in order to find different complex networks. So from the complex network we can find different measures and these measures could describe the topology that will characterize different events. So I show you a little example of the complex network that is cluster events and we we can find for each node or of each earthquake will have different characteristics. For example the degree is the number of links to each one node. So as we can see the green dot is the main shock and this main shock will produce aftershocks. Also we can find the clustering coefficient that describe triangles produces in the complex network. Also the diameter and so we can apply different measures in order to to characterize these cluster events. So as a conclusion, detailed models let us identify background and triggered events of point process. Stochastic declustering allows identify the aftershock sequence and it's very important but because when we analyze earthquake activity we need to separate what what are the background events at what what are the aftershocks and that aftershock will describe the characteristics of the tectonic plates and their root tour in in in the in the behavior of the location of the main shock. A detailed model requires a long catalogs in order to have a better results so we can use three graph representation that will help us to identify offspring generations and finally complex network could characterize seismic regions by means of their complexity measures. So we will continue with this study in order to to find different radioanalysations from Mexico because Mexico is a very great country so we need to apply in a local places these these analyses and for your rotation thank you. Thank you Carlos for this very interesting award congratulations for that and also congratulations to Max and Jan. So there are time for questions because you you finished too soon so we have time for any questions so if anybody has any questions please directly either ah now I can see race hands in the icon so come on please go ahead. Hi Carlos really really nice work loved it and I was just wondering I mean have you tried to combine the network representation with some sort of spatial graph because I guess somehow that's that that that's really the kind of interesting question as to how these how local or long range these sort of triggered events are is that something you've kind of looked at or would be able to do in this model? Yes of course as I told you we can from the complete network yes for the complete network we can take the biggest component and the biggest component we are thinking that this biggest component will be a cluster part of the network and this cluster should be a main shock so we will have different main components of the complex network so in order to to to choose the isolated after chucks so in this sense we can do and analyze the topological structure so this is a very interesting idea and it's a new new way to to to study these there are few works on them but it's it's very good apply this to Mexico because in Mexico there is few works about that. Carlos if I can add on Colin I think that the I think Colin was asking or maybe I have a different question or the same question as Colin was asking whether you can wrap this complex network in the in the geographical yeah map of of Mexico yes yes if you can from from your answer it was not clear like for instance if you plot this or maybe put another graph yes please sorry yes this this distributed along the Mexico so along the Mexico what uh map yeah but I think it would be very cool to have the map of Mexico and then this network on the geographical map of Mexico that would be a maybe made for analyzing it and analyzing it maybe doesn't give you an information but from a visual point of view I think it would be much better right yes of course I think that was Colin's question yeah I think so because I guess I'm just I'm wondering how long range the triggering mechanism is and it probably varies I guess there's probably a lot of additional information in this network and you know little bit so people draw these sort of networks for climate problems as well to try to find teleconnections you know but basically by trying to represent kind of long-range interactions it's can be a useful way to kind of combine the network and the spatial yes I agree very good uh more questions yeah I have a I have a question um go go ahead Carlos it how how do you decide whether a shock that happens or actually any sort of event but let's let's stick to the earthquake example how do you decide whether a shock that happens um soon after another shock is a is an offspring shock or whether it's just a new parent shock yes my means of the it is model the it is model calculates the probability of some events are are are from background so this is an internal the procedure of the it is model when we calculate a the just this one the probability that an event is generated by the background and what events are are are offsprings and I'm from what from yes from what event comes this offspring so that is in an internal iteration steps of the it as model okay can you can you then does that allow you to presumably that only allows your probabilistic identification of real data as either parent or offspring so you're starting probabilities okay understood very good uh more more questions if not I don't see the hands being raised either the human hands or the or the icon hands so if no more questions shall we thank Carlos for this fantastic talk thank you very much Carlos thank you so now let me stop the record