 At the beginning, when I presented the first paper with network, one way to translate the fact that the network can change over time, it was to have a short film with different network represented, and then showing that really what you observe in terms of network, the visual representation of the network with the changing color, the changing ages, and so on, was something really able to represent what was happening that time in the financial markets. Some colleagues told me that it was also the case that at the beginning there was the images, then there was music, and someone told me, you can also add dance. I was thinking to this now, where I was waiting, but not able to do that. It was just to. Since really at the beginning was first the images, then some music, since my co-author Andrew Law at that time decided that it was the case to put out some music. Here we are, thank you. And then someone suggested also some dance. OK. OK, here we are. We can start again. Thank you. And then as I said, thank you for confirming also that now you can see. As I said, I mean, then introducing new time variant dynamic network, then in terms of method and models, then the way in which it is possible to work with array data, and application, then clearly application on network data. Then the two papers I said are these two papers, where the first that I will present with a little bit more detail is a multi-layer networks with dynamic, but real valued edges. And then here we consider a smooth dynamics, and then also some inputs response function will be done. And the other paper, companion paper, where again it is a multi-layer network, but in this case with binary edges. And there the attention is more on discrete switching dynamics, then on the file that there can be breaks and switch. Then how it is possible to exploit that information from the structure of the data? I mean, in the literature, a part of you certainly know that when also you can work with a matrix or more complicated data, one way is to go through vectorize everything, and then to translate something that is a little bit more complicated in a more or less standard way. But doing this, what we lose is the information on the structure of the data. And then this is, I mean, when I show you before the networks in terms of connectivity, but when we analyze the impact of the topology, what we are saying is that the structural matter is not simply the fact to put together the edges in such a way to see what happened, or the noses to see what happened. Then the important thing is to maintain and to use the information in the structure of the data. Then this means to work with tensor data without, I mean, transforming the tensor data in other things. And clearly, there is a lot of data to treat. A part of them only is relevant. Then it is important to be able to account for sparsity, since this is the only way in which you can proceed. Then what we propose is dynamic models for tensor data. We account for a different type of data dynamics, as I said before, and in particular, I mean, there is also this part on the ability to explore dynamics using shock propagation, then the impulse response function. And what we practically propose, I mean, on one side is to use tensor, then working with tensor, and then I am obliged to introduce some operation representation of tensor in such a way to be able to see what is possible to do. And in terms of estimation procedure, after we use a Bayesian approach, and we use a global local hierarchical prior distribution. And this is the fact that, in some way, we need to use a similar structure, the share of information among the different priors, and at the same time, we need to take care of sparsity, since, as we said before, otherwise it is impossible to work. As I said, why not vectorize? Since the estimation becomes quickly feasible, since the dimension becomes twig to treat. But we are losing the information, as I said. And more than that, I mean, since we need to impose some and to work with some sparsity restriction, it is difficult to understand the way in which this is possible to do when you have a vectorized form. While working directly on the matrix, on the tensor form, it is easier. Then the estimation becomes feasible. This is something that you have to see. It is possible then to maintain the data structure and then to use to exploit this data structure. And it is possible to use tensor also in terms of the type of the composition that you can consider in the operators that you can work. Practically, the general model that we propose is that it appears, it is a generalization of the linear regression models to the tensor framework. And we, as I said, we need to introduce parsimony in terms of model specification, but at the same time, we need to also learn sparsity patterns from the data in such a way to put together the parsimony on one side, but also the sparsity. And clearly working with sufficient reflection prior definition and using efficient posterior computation, we can obtain the result. Here are a few and a quick view on the two paper. This is what we will see later on here. As I said before, we work with real-valued networks, graphs. And we also want to analyze the shock propagation in time and in space. Then this is the model. As I said, from really a general view and also the way in which it is written there, you see it is a linear regression. But in these cases, for tensor time series data, essentially we generalize a multivariate linear regression. And we consider tensor-valued impulse response analysis. And what is really important, and this is the common aspect also with the other paper, is to use the para-fact tensor, the composition. Clearly, we will see the detail later. And in such a way to have a parsimony, as I said, and to use a Yashica global local shrinkage prior in order to have a sparse coefficients. And the application that I would see is a two-layer network on international trade and capital flow. And as I said, some analysis of the hedge-shock propagation. The other paper, companion paper, means part from the same observation, let me say. But in this case, it is binary network, then instead of real-valued network. And there, the idea is how is possible to analyze and to formalize the structural breaks in the natural structure. Again, we need to deal with sparsity. And then here, the model is something that appear a little bit more complicated in the writing. But in practice, it is a zero-inflated logic for each entry. And for the parameters, we consider a Markov switching dynamics. And again, the estimation is through Bayesian inference. Here, we need to use a particular type of data augmentation, the polyagama. But after, I mean, the two main points are, again, the para-fact tensor, the composition, in order to have the parsimony. And the Yashica global local shrinkage prior, in order to be able to induce and to obtain sparsity for the coefficients. There, the application is for a financial network, in this case, for the European institution. And in some way, it is an exercise related to the beginning of the history when we work with network for the financial system. And the impact of risk factors and network topology, in this case, on edge probability and on the presence of an edge or not. Now, let's start more seriously than on the main presentation today, than this paper. But I thought it was important to give, I mean, an idea of the old history also on the reason for which we started with, on one side, financial and economic questions and the way in which it was possible to represent through network, and then how it was the case to treat the data. Data that nowadays are increasing in size, then the high dimensionality is a topic. There is a situation in which it would be the case. It was also in the past, but nowadays it appears more important to treat contemporaries in multiple data sources than to have different layers and to treat them together. Example of tensor value data are already there in the literature. As I said, the second point is the paper that I cited at the beginning. There, it was a temporal network that the relation between a manga and the subjects were observed t times. Then it is, if you see in such a way, it is a three-order tensor, but also in other contexts, not only in economics and finance, but even more on medical data. I mean, it is quite easy to have this type of data. Then here it is the data, which we will work later, and then is a liar for trade. And then it is the com-trade of the import-export among 10 countries, in this case, the way in which the edge is traced means the direction. And then it goes from left to right. And the color it is, if it is import or export. And this is the way in which you represent. Also, the thickness of the edges give you the relevance and importance of the export and imports. The second liar is the financial. In this case, not exchanged since our financial linkages in terms of the Bank of International Assetments data. And then what is the idea is to represent a layer for the real part and the trade, and a second layer for the financial part. It is just to have an idea of the way in which it appear most similar. I mean, this goes from 2004 to 2016. I mean, it is not a long period, but this can also make clear also the way in which it is possible to work, where clearly the dimension is much higher than the sample dimension in this case. And I mean, looking more carefully to the different pictures, it is easy. It is possible to see that there is change in patterns. Looking in this way, it appear more similar. But after, we will see also the result. As I said, I mean, the question is, how to model a time series? And this is the main point of TensorFlow value data. And then, how to be able to account for the fact that first, many variables, few of them are relevant than to introduce sparsity and maintaining, using, exploiting the information that comes from the data. What we propose is a dynamic model. And we also explore dynamics in terms of shock propagation. What we use is tensor algebra. The use of, as I said, the global local hierarchical prior in order to be able to work with the Bayesian method after, and the impulse response analysis. Then here, the tensor. Looking such a way is not so complicated, since it is a three. It is a tensor, what is represented of order three. Clearly, the order can be even higher. In this case, it is difficult to give an idea of what we are discussing. But clearly, the way which you can also describe here, the tensor is thinking in terms of slice. Then horizontal slice, lateral slice, frontal slice. It depends on the way we should look at the tensor. And also, I mean, what we have the habit to say, but it becomes more generally if we talk about fibers, then it is column and rows. But then they can be tube and fibers in general, just to describe the pieces of the tensor. What the tensor algebra do is to generalize the matrix algebra to multiple dimensions. And once you are inside the tensor analysis, you see quite easily the extension. At the beginning, it is a little bit more complicated. Let's start from the matrixization. The words say what we are thinking is how you translate, you transform a tensor in a matrix. And clearly, the way it is formalized, the way in which you have to do this means to cut the tensor into slices. You have to give the dimension of these slices. And then to put them horizontally in such a way to transform the tensor, you first cut in slices. And then you put the slices together. At the end, you have a matrix instead of a tensor. Clearly, there is only one way the dimension change. Since you have the matrixization, you say the mode K matrixization, OK, then the dimension change. But the way we show to proceed must be the same every time. After there is some tensor operation, then the mode N product. The mode N product, then we have always these calligraphic kicks. That is the tensor of order N, just to say that we go further three. And then you can have a matrix or a vector. The mode N product is defined in such a way, but it is practically impossible to understand what is written there. Matteo, we know. But the idea is, I mean, you have to, you compute the inner product of each mode N fiber with the matrix of the vector. In practice, what you obtain, what you obtain is that you change the nth dimension since it is a mode N product, then the N, this means, OK. Then you change the nth dimension of the tensor or reduce it over one of one, OK, when you multiply mode N product with a vector, OK. Other operations are, from this point of view, easier since they are performed in the usual way. But, I mean, it is just to move over the matrix. And essentially, clearly, you have more possibility since the way in which you can reduce, you can transform the tensor. I mean, you have more possibility than the matrix. But essentially, I mean, you are there. I mean, you are not really doing different things. When you see the contracted product, it appear more difficult, but probably easier to read since what you say is that you have the calligraphic X tensor. Sorry, this is the result. Then you have two, sorry, X. It is the calligraphic. X is the tensor of dimension of order K plus N. And the other is the tensor of order N plus M. When you have the contracted product among them, the result is a tensor of dimension K plus M. Then the N inside disappear. And this is what you probably understand a little bit easier. I mean, from a technical point of view, it is more complicated. But it is the way where you take the product of two matrices that the inner dimension disappear. This is what happened. Clearly, for tensor, it is a little bit more complicated. But essentially, it is the same. Now, this is the first part. The first part to say that we are working with tensor. And of order 3 you see, since it is a cube of bigger order, it becomes a little bit more difficult. What we have learned is that there is possibility to work with the tensor using similar way with the matrix algebra. As I said, the fact that you have order bigger than 3 give you more possibility. And this is what you manage with the different operators. But now we move to the fact that the tensor admit also several interesting representation, the composition. Since it remain a quite complicated tool. And in particular, what we consider as the interesting part is this parafax decomposition. Parafax of rank capital R here. And this means that you can decompose the tensor, in this case, of order n. In once you define the rank, it is the outer product of, and then the sum, that you first, for each component, you define the, you see in the representation below. I mean, you first, in some way, see for the cube, you produce first smaller, sorry, simplest cube. And then you sum of them. The way which you represent overall is then using the rank. And then for each dimension, you have the component that is clearly as the same order of the tensor. And at the end, you obtain. Then once we have this element, I need to go much quicker now, you can have a tensor representation. Then for each entry of the response tensor, then you can represent through a vector, in this case, beta i. And then the vector representing another tensor. Clearly, if you put in a compact way, what you see to appear is the tensor on the left as the response tensor. The input tensor becomes a vector, since it is vectorized. But what is in the middle, then the coefficient, it is a tensor of a dimension of one dimension more than the response tensor. Then in this case, it is a tensor of dimension of order n plus 1. OK, and then you add the error term. The error term, the calligraphic epsilon noise, that is a tensor that we introduce as a tensor normal distribution that exists. And then here, it is possible to manage why a calligraphic kicks of tensor with different order. But it's also possible to include other regressors in the analysis. Here, then, we can go to, I mean, it is possible to see the vectorized form, as we said. But it is, I mean, you can manage. But after from a practical point of view, this is less of interest. Also, if for analyzing the properties of the resulting model, it is useful also to be able to work with this vectorized form. Then, special cases clearly are the univariate regression, but nothing new and the multivariate regression. This means that it's possible to include the example as the var, the vacuum, the panel var, and the SURE model. Where it is interesting for us is to move to the new special cases. And then this is the tensor autoregressive. And the tensor autoregressive is to not simply have the tensor regression, but to have the dynamics there. And then to have the calligraphic yt that can explain by the past calligraphic yt. And then to be able to have dynamics on this. Some examples of matrix variate models are already in the literature, treated in a different way. Here they are put together in, as we say, this tensor autoregression. And here it is the general representation of order one. But clearly it can be moved to the order p. And as I said before, it is possible to add other variable, other regressors. And then it's such a way that it is not only an autoregressive, but an autoregressive with exogenous variable. In the paper, you find the properties of the autoregressive tensor that it can be analyzed in terms of the contracted product. And since it has this possible representation and where you remain simply using the tensor instead of working with the vectorized, at least for a part, for the input tensor, working under mild condition, it is possible to see under which condition the process is weakly stationary and thus then an infinite moving average representation. A sufficient condition can be tested on the associated var model. OK. And then here, I mean, it is the properties that the days are on the paper, then for the stationarity. Here it is the equivalent var representation, as I said. OK. The proposed parametrization, the proposed parametrization, if we consider the restricted var then to translate the vectorized form, this means that the number of parameters that appear is really important, especially in terms of covariances. OK. Then using the tensor, the normal tensor, allow us to reduce first the dimension of the covariances. And the second, using the para-factor, the composition that I have been introduced before, means that we can really move from an infeasible number of parameters to consider to something that is feasible. Difficult to see, but probably in this graph you can understand better. The restricted var is what is exploding in terms of number of parameters. If you work with the para-factor representation and then also the tensor representation of the error term, clearly it depends on the rank that you are considering. But the number of parameters are highly reduced, but also increased in a quite linear way. Then after what we have done, that we can work with tensor, in particular the parameters that have to be estimated is a tensor. And then the way in which we reduce this dimension is using the para-factor, the composition. Then there can be some issues in the way in which the different marginals of the para-factor and the composition are used and considered. Then we need to impose some scaling variance, permutation and variance. I mean, you have to work in such a way that the product that we consider, I mean, give us something that is meaningful. And what is relevant is that we have no specific interest in the marginals. But what we are interested in is the coefficient tensor. And these are always identified. Then the problem, the identification problem for the marginals, they're not translating identification problem for the coefficients. And then it is possible to work in a reduced way. OK. Then here it is the example of the vectorized form of the matrix autoregressive of order 1. But since I have two minutes or zero minutes, 1. OK. I go directly to the very quickly to the prior specification, then also for the prior specification. Here it is what is used and it is the para-factor. OK. The para-factor, it becomes difficult with the para-factor to use other type of priors since, I mean, working the vectorized it is impossible. As we said, OK, the para-factor is helpful since you're reducing. But, I mean, among the possibility to consider in terms of prior specification, the Iyashica global local shrinkage prior appears to the way which we can proceed. It is the way which we can proceed since we have different level of commonality of information. If you want, this is the global part. Then there is the component part that is related to the component of the para-factor composition and the local part that is specific for each entry. They are combined. And here it is, you are also on the paper, the way which they are combined. And they give the prior structure of the model. And then it is possible to work. And the posterior is through a Gibbs sample. There is some aspects that need to be taken care during the Gibbs sample. But essentially, you work in such a way. And then the nice things, I'm going to go just to the end. It is the application. Here it is the matrix. This is simple, since it was the first example, you can have the same for the two layers. And then this is the matrix size, all three matrix size tensor. Then it is only the single-layer network. Then it is a 10 by 10, since you have the double dimension, you go to 100 by 100, since it is 10 by 10 and 10 by 10. And what you can see is, in terms of color, each entry of this matrix size tensor give the impact of edge j in t minus 1 to the edge i at time t. Then the regularity that you see that can appear comes from the fact that the transaction in t minus 1 can have a similar impact on all the transaction at time t. This is the way in which you can read. OK, there is the impulse response function. But at the moment, I used my time till here. Thank you, thank you. Thank you, Monica. So questions from the floor or from the Webex? So, hello, Monica, you made a relatively tight connection between the Tenzo models and the networks they imply. So I was wondering whether you've also considered applications to financial data, where these tensors also come in naturally, like yield curves or option prices or anything where there are multiple countries and data over time, but also then a maturity dimension so that there's always sort of three dimensional matrices. But one wouldn't naturally think of these as networks or implied networks, but still there could be interesting applications. She'll collect first two more questions. I think Luca had one. So Monica, sorry for making your questions to you. I hope you will not kill me. Now the fact is that when you're doing the paraphark, you use R and it's something that is fixed by you. Is it possible to estimate it also, that kind of R or not? Or is too much complicated? Let's say also since the tensor structure is really heavy and uncomplicated. OK, so thanks both of you for the questions. So in reverse order, yes, R is a kind of tuning parameter. You can consider that as modeling how flexible you are because the higher it is, the more flexible you are, the more parameters you have. And so this kind of trade-off that you can have. What we did is to use information criteria to decide the value of R. It's very simple. So around several kind of estimation just fix the best one. You can estimate the answer is yes, it's going to be complicated. So you can do it parametrically or non-parametrically. Parametrically you go through reversible jump. Typically, non-parametrically can be much, much more computational cost. But in principle, it is possible. And for the data, fully agree. So of course this presentation was in the motivation for the paper starts from networks. But the methodology is it just requires you to have a multidimensional real value array, whatever is the meaning of that. So you can typically use that for the kind of data from finance that you mentioned. So option prices or anything else is perfectly fine. We didn't do there because we were concerned with the dynamic networks. But you can definitely work with them without problems, both in this paper and the other one. If it is a binary, you can just use it. So network is the application, but it is not an ingredient which is key for the methodology to work. Thank you. Marta. So the first one was Pernschwapp. Next one is Marta Bambura, both from the beginning. Ah, so Bernthi. Okay. Now I wanted to ask, because it's not so easy to understand this notation, but I understand that this part of fact somehow introduces some homogeneity or some restrictions, how sort of like things propagate through the system. And I was wondering whether for a smaller system, you did some testing, like whether these restrictions are sort of like too much for the data. So like if you reduce the system, I don't know, to one or two or three variables, and you try your part of fact versus like a vectorized unrestricted system, like whether you have a sense, how much of a restriction you actually put through this. Okay. Then thank you. I mean, the question is also related on the way in which the rank is chosen for the part of fact. Okay. What clearly, this cannot be too small. It cannot be one, two, three. I mean, it must be, we usually work at the way, the five, five, six, we also try till 10. Okay. But on the specific case to compare the fact to remain vectorized instead to go through the part of fact. The part of fact, I mean, give you a way to work with a tensor. Okay. It is not really, since all the parameters that appear in the tensor are estimated, there is no restriction on, it is the way in which they are represented. But I mean, it remained the full tensor, remain there. Okay. It is not reduced. It is reduced, it is a sort of rank reduction if you want. This way you can read the part of fact, the composition. But clearly, if you have a small one, then also the, I mean, the interesting in having the tensor and then the part of fact, the composition disappear. Since clearly this is useful when you have important dimension. Okay. Here, the example is a time by time that we have worked in the simulation, it was till 50 by 50 and was still feasible. But we also tried, Matteo better, tried also to go from 100 and 100. But you see that in that case you need to have a way to reduce the impact of the dimensionality. In terms of application, I mean, clearly, we really started as a, I make the history just to tell that this was the idea where we started. After it has been different research lines, I mean, also looking at the more methodological aspects. But really at the beginning it was how we can work with network and then their representation and then to translate these in models and then a better understanding of what is happening. Okay. And also in term structure, we are working on this. There is some of the application is also on this. Here, clearly the multi-lier dimension is important. But also the linkages among the different layers is important. And then, I mean, it is the dimension but you have more constrain in some way that can go through the different layers. But yeah, it is one of the application that we are already seeing. Thank you. Thank you. I also have a question. And my question is to link this stream of research to the topic of the conference which is forecasting at risk. And I see that you talk about pants or normal distribution. So the first question is how easy is to introduce skew distributions? And the second question is how can it help us to estimate tails of the distribution of this complex object you're after? Let me take the question from another point of view. I mean, here the application has been seen for network obtain from a trade or the relationship among the different performances if you want in all the return on the market. But you can also see the network as they have been structured from the covariances. And then directly on the way in which you represent risk and then you can work directly on a different way to see also the skewness in terms of the distribution of the risk. OK. And then I mean, this is a way in which you can treat then instead of working on the first moment, let me say, work on the second moment to the extraction and the analysis are part of the work. The work with Daniele has been, for example, for the network on the variances and not on the first moment, just to understand. OK, then it is a way to represent directly the risk. OK, clearly, if you maintain the times for regression, then you are looking at the first moment. I mean, to introduce a more complicated form of the error term, OK, this need to have also the times for representation. And this is a little bit more complicated. Then to deal with the forecast of risk, I see better the way to move directly on the network from the covariances than on the second order moment. In that case, you can manage different things. OK, and also to understand the way in which you have spill over, you have a part of the application is also in this direction. I hope to have more or less answered it. Yeah, thank you. Actually, it comes to mind a section from the Konometica paper by Debal and Cotters, which was called Var for Var, which then we have utilized for a paper of mine. And that is exactly what you were saying. They were just estimating this vector of regression on the valued risk, on the risk estimates directly. Thank you.