 So for this project, project seven, the title, as you can see, there is learning random dynamic systems from data. So it's going to be two parts, first from Toruwani and then from Johnson and the supervisors where you know, and I think also as well, okay. So Toruwani, thank you very much. Please go ahead. Good day, everyone. I'm Toruwani and I'm working on project seven, learning random dynamic systems from data. So for advice by Professor Lam, I studied a class of maps from your man build maps and described by the relation given above here. And this is a sample plot of the map for when alpha is equal to two. This class of maps were first studied in 1980 in the paper by Formu and Manubil cited here. The main motivation for studying these maps are number one, the similarity between the dynamics of these maps and turbulent flow in fluid dynamic. And this is because this map show intermittent behavior and intermittent behavior in dynamical system is when I mean a dynamical system we have some points where the orbit seems to be well behaved. There seems to be no chaos and the orbit seems to move slowly but then at some other points in the system, the orbit of some other points in the system are chaotic and the system suddenly becomes chaotic and begins to exhibit sensitive dependence on initial conditions and other properties of chaotic systems. And this is similar to what we see in turbulent fluid flow where at some point the fluid and the flow is smooth and there seems to be nothing going on but at the point the flow becomes very turbulent and so understanding this kind of maps could give some insight into what are the techniques of turbulent fluid flow and the second motivation is that these maps are almost uniformly expanding. In fact they are uniformly expanding at every point because at almost every point except at zero. The derivative of these maps is greater than one at every other point except at zero where we have derivative one and although this seems like a little variation it brings about a large difference between how this class of maps are studied and how uniformly expanding maps are studied. To study this kind of maps usually the induced map is introduced and induced map is a map that is obtained from this map which has better understood behavior and from the properties of the induced map we can deduce the properties of these homeoman field maps. And now I move to discuss a bit about numerical approximation errors that I encountered when studying this kind of maps. Because of sensitive dependence on initial conditions and computer random error that is not possible to get exact point of the orbit for this map because by sensitive dependence on initial conditions I mean two points very close together end up having completely different orbits. So a small difference in computation or a small computational error or a small approximation error could lead to completely different results for the orbit. For example if we have a computer in a very good scenario that has an accuracy of up to 100 decimal places. So let's say we have a point x and we have and the computer gives an approximation of x plus delta where delta is 10 raised to power minus 100 that is a seemingly very small value. After n iterations we have that by the mean value theorem we have that the difference between these points become this value. If we assume that f prime of ci is 1.1 just a value a bit greater than 1 for each ci then we see that after n iterations we have this difference but after 2,416 iterations 1.1 raise to power 2,416 is greater than 10 raise to power 100 already. So for n greater than 2,416 we have a difference greater than 1 and this is a very large error already. So this makes it difficult to compute the orbit or to compute the trajectory of the system by using numerical simulation and this is another example. This is the first return map for the homeoman build map for alpha equals g. And by construction this should be a full branch map but because of computational error we don't have a full branch map. And by a full branch map I mean each branch should map to the entire interval 01. But to see that after some time we begin to get some errors and we don't have a full branch map. So these are some of the limitations of trying to compute the trajectory of this system using numerical methods. Then we ask how do we study this map then if we cannot compute the orbit. But the interesting thing is that despite sensitive dependence on initial conditions the statistics of this class of maps are very well behaved and we can study this using physical measures. And we measure mu is a physical measure if for almost every x not the proportion of points of the orbit of x not the line a converges to the measure of a. So the physical measure tells us the probability that the orbit falls in different subset of the ambient space so it gives us an understanding of the statistics of the system asymptotically after a long period of time. So if a physical measure exists then we have information about the orbit of almost every point of the system. And so we have the diploma man build map a physical measure yes it does. And a well known theorem that the formula man build map has a physical measure for how far between zero and one. So quickly go through the proof the proof is majorly done by constructing the first return map, which is defined here the first return matches full map is full branch and has bounded distortion and a full branch map with bounded distortion has a physical measure so the first return map has a physical measure. And if we have, if you are to the measure on the first return map, we can easily extend the measure to the original map the original formula man build map by this relation. And if you are to the physical measure mu is also a physical measure. So we have a physical measure on the original system also. Now this is some numerical experiment. Here I plotted an histogram by divining the interval 01 into 100 days and computing the orbit of two differently, two different randomly chosen. I actually computed your bit for more than two points but because of time I will show just to this is for 0.1 and this is for the point 0.4. And we see that they have, they seem to have similar distribution after this is about 200,000 iterations, and we see that they seem to have a similar distribution and we ask, and why do we get similar plots for the difference of it and also, are these plot plots accurate because like we said earlier this kind of mouse are difficult to study because are difficult to compute because of sensitive dependence on initial condition but we see that we have something similar so we ask the question can we rely on this result despite the numerical approximation errors that we are likely to encounter. Now I moved on briefly to talk about to discuss the results obtained by Stefan Ogallatolo and Isayani Soli in their paper. They actually use some rigorous computation and functional analytic methods to compute the density of the invariance measure and formula and billmarks up to a given error. And they found that yes it is possible to compute this density up to a given error and they use Eulam's method which involves discretizing the transfer operator and then computing the fixed points which will be the invariant density of the system and once we have the invariant density we have the measure because by Radon-Nickering's theorem, since new is an absolutely continuous measure, we can easily obtain the measure of any sets using this relation and we see that we have and this is the plot they obtained in your paper after the using their method, the rigorous computation of the density, this is the plot of the density. However, we see that this is similar to what was obtained by just relying on the computational result and plot in the histogram and we ask why is this reliable? Why is this correct? Because we know that there is a lot of computational errors and this brings us to this conclusion that PATOP's maps with perturbations of given small magnitude have an invariant density that is close to the invariant density for the own PATOP is wasn't found in the system and this is a result obtained by M. Tanzi et al. cited here and this is something that I hope to look forward to more to see exactly why this is so. Thank you. Thank you very much. So the next one would be a Johnson. Are you ready? It's for the same project, but they do two different things. Can you share? Yes, I'm doing it. Good. Can you see my screen? Yes, that's perfect. So please go ahead. Good afternoon, everybody. I looked at the same project as Tolu, learning random dynamic assistance from data, but I looked at this from a different perspective. We like to answer the question that given a partial observation of the system, can we actually determine the underlying dynamic assistance? Like you can see the time series on the left hand side. It's a partial observation of the given system on the right, but assuming that we don't know that the dynamical system on the right is that of the partial observation on the left, is there a way that we can just reconstruct this dynamical system or have some knowledge that this is the actual dynamical system does from partial observations. And then the thing about the dynamical properties of these trajectories, we'll be able to, for example, do a model in low dimensional space with noise, instead of doing modeling of the underlying dynamical system in dimensional space. And thereby we are saving cost of computation. So next I talk about the modeling of random dynamical systems. There are different ways by which random dynamical systems can be defined, either by using stochastic differential equations, or by adding the additive noise to some discrete time mapping. Like you can see in the first set of equation above. This is the Lorentz system. It's a discrete mapping, but by introducing some noise and some function that belongs to some probability distribution space, we can construct a random dynamical system. Now suppose that we have partial observation of a random dynamical system. And then we have a model that we can learn about the dynamics. Yes, we are sure by the taking embedding. It gives us the condition on which this reconstruction or reproduction of the underlying dynamical system can be constructed. This is the, we have access to some data or time series, why won't to why, up to why N. The taking embedding. If we choose some delay coordinates, that is twice as much, at least twice as much as the coordinate of the underlying dynamical system, and we choose our embedding coordinate and delay correctly. We will be able to do this and be able to find, for example, the function that connects, why N to why N minus tau, and so on and so forth. So, this by this way, we will be able to learn the dynamic system and be able to reproduce it. Dynamic taking some bedding from a geometric perspective. I have some time series in a some time series below, but it actually belongs to an n dimensional manifold, but I'm not aware of these that it belongs to an n dimensional So taking embedding tells me that I can construct a function this file that adds on some historic on some embedding of the time series and then reconstruct this dynamic system in an n dimensional manifold space, which is topological equivalent to the original system, which I don't know. So for experiment purpose, we have used neural network alongside taking some bedding. So after we have used taking some bedding to construct dynamic system, we want to use neural network to learn what the taking some bedding does, and then use the results of its learning to make predictions and the neural network actually learns the transfer function and to determine to determine the solution of the nonlinear system. We describe some nonlinear transfer function between nodes, and then we are taking as input, the partial observation at SK and output at partial observation at SK plus one and in between there are some hidden layers, specifically for the purpose of this work we have used three hidden layers with 10 nodes in each layer. So these are the results from this definition of a random Lorentz system. So I took a partial observation of the random Lorentz system. So what does partial observation means the random Lorentz system as x component, y component and z component. Because I have access to say the x component alone, and I want to be able to reproduce the entire system. Then I applied the taking embedding to the partial observation and figure B is the embedding and starting from 100 different initial conditions. We also did something similar and got the embedded trajectories, but for this to actually work to get the right embedding, we need to get the right embedding dimension and delay so this can be used using ideas of mutual information for nearest neighbors to determine what is the correct dimension and the correct delay, and then this will work. So after the taking embedding has learned or has reconstructed the underlying dynamic system, we now take the neural network, train the neural networks on the taking embedding reconstruction. So these are the results of the simulation of the predictions done. On the left, on the right down corner, you can see, just a moment. On the right down corner, you can see the prediction by the neural network in dashed lines and the prediction by the takings embedding in, in thick lines. So the results seems to converge. And so this tells us that we can actually learn the behavior of random dynamic assistance from data. And this add the evaluation results of the learning the loss function is of the other 10 raised to power of minus two, which is significant enough. And this is the time series comparison simulation one and simulation to add the component twice time series. Relation between the neural network and the random dynamic assistant. When we started this project, our goal was to know whether taking some bedding can actually learn random dynamic assistant and have an idea of the noise. And this we have implemented here at some of their references that I consulted while doing this work. Thank you very much. Thank you. Thank you. Thank you. For the presentation. I think we have. We have time to actually the minutes or so for questions so colleagues and students of course go ahead and you have any questions. I think I saw some, did I see some hands raised from maybe they were just a pool of pools. Anything but somebody shared please stop sharing the presentation. I have a question but I'm happy to wait for getting student questions first. I don't think there are any hands raised Mark so go ahead. Okay, so I think these problem of learning a random dynamic system is one of the most important problems because these random dynamic systems are used in so many disciplines to model real world data generating processes for example also in economics financial time series and stuff like that. So I wonder whether the results that you found are specific. For example that learning for more money will type maps is is is possible with fewer observations, or whether there are some, some random dynamic systems which are for example harder to learn. I imagine that it's, it's not a very general statement that one will find there so for example is logistic map type of maps are these harder to learn than other maps or something like this. For example, if you have a map which stays for a long time in a in a in a constrained part of the face space and then jumps to somewhere as I guess it's pretty hard to learn the correct dynamics here. So I hope to look into that, but I focus on for me or mine. Maybe I can say something if if I'm allowed if no one's the student. I think Mark you you're you're you're pretty much right and I'm very much onto this topic for that reason that I think is very important I think actually the students have tried to do something. It's not, it's not easy. So what people do learn if you have data people learn deterministic rules where so people try to learn deterministic dynamics based on observations and you have to hope that that's possible so that's the usual targets and betting. So people try to learn efforts properties of of stochastic processes that's also people do so they try to kind of somehow estimate average properties, but what's very little done or not done as far as I know is that actually anyone tries to find out what's going. And random dynamics so, and it's my perception that a lot of processes are very well modeled by random dynamical systems and not just on this kind of average procedure but actually basically on the trajectory base. And I don't know of anyone that actually really looks into this and I think going for the average is of course nice. But it's you lose a lot of information and I think someone meant much of the information is very important to understand actually what's really going on. And it's, it's a long term objective the project is basically maybe a little bit hard and the objective is a little bit impossible to try to explore what one can, what one could do in this direction somehow. But I think it's, it's really important because I think random dynamical systems are very important people don't study them very much in some sense, because people don't try to learn from data whether random dynamical systems are good models for such data and I think that as soon as we start doing that we will find out that a lot of processes that are of interest. I believe will be very well modeled by random dynamical systems, not just on the, you know, on a kind of basis of averages and statistics, but basically also on the base of trajectories, but people, but this is quite essential because there's a there's an account of a gap in in what people do. And I think that's why the study of random dynamical systems also somewhat under under under studied because people say well okay how you know then people start looking for the code of theory and average properties and stuff like that and people say well we don't need all this detail, because it's not in the model usually know. And with these average properties you mean something like mean square deviation of these properties a lot deviation theory vehicle for footballs other stuff this is of course very well known that you can kind of extract this, but these are only dimensions of dynamical properties this is not. There's lots of dynamical data that you cannot reconstruct from from academic theory in some sense. So I think people are a little bit, I would say lazy but they just got so used to the fact that you just look at these averages, and average properties that define detail which gets washed out by averaging. And it gets lost in the process somehow and I think that actually if you look at these details in on this detail level you will find lots of very interesting things. Now sometimes we don't have access to these details so it's not relevant, but say in saying life sciences or medical data you see people have very detailed times here so why would you average out all the fluctuations and to just get some kind of like average properties you just lose information. And it's not on a Gothic. But there's lots of issues there somehow yeah, whatever you call it in some sense but I think the dynamics the time series themselves are very important. There was actually, there was an interesting talk at the 21 conference where somebody from the intensive care unit actually showed these biomedical markers, and that they have non-negotiate properties so it's actually a big problem if you create for example some scoring rules for triaging based on these average properties which then are never the case in any single individual. Now it is of course due to the success of statistical mechanics in some sense is that people are so used in statistics is it people so used to average everything. There's a lot of things to cover but I think one thing is important that we can actually identify from observation certain processes that we can actually find out for instance low dimensional models with noise are very good models for to represent such data on a pragmatic point of view and then this is not existent at the moment but I hope that it will come in a few years. It's an interesting thing probably also for scientific. Mark, this is interesting. Oh, sorry. Sorry. Liner wants to have a personal so Liner go ahead. Yeah, just very briefly, because you're touching on a very important point here. First of all, whether you were able to reconstruct or not. Has mentioned it may depend on the properties of the underlying system, right, whether you have strong chaos or weak chaos, meaning a positively up on an exponent or not. And then it noise and if I may make a brief advertisement of a talk tomorrow actually that is related to this tomorrow. I will talk about another random dynamic system, which in fact exhibits a transition under the strengths of the random perturbation from nice simple random dynamic system to another type of dynamics where you have actually a good the city breaking. This is also what you're on said that you can really generate totally new type of random dynamics here so they are very very subtle issues here. So that's also more tomorrow if you're interested but there's another talk which comes into play here. Thank you. Thank you, Liner. We have a couple of more minutes sorry to interrupt but I wanted to go ahead. Yeah, that's that's fine. I'm just wondering whether or at least becomes clear to me now that probably in terms of scientific hygiene, if I see the next time that somebody models some empirically stylized facts with a random system or with some dynamic a system that I would also like to see the story the other way around so would you be able to actually detect this random dynamical system among a set of possible models, given the observation so I just found this a very interesting and nice topic as a student project and in a universe of many, many other learning learning project. Thank you, Mark. Again, I apologize for interrupting from my part and I wanted to say from my from from my side that I think it depends a bit on the on the research area where you can see where this thing has been studied or not. Okay, I can point out but maybe we can discuss this later in certain areas where so you try to do an inference based on the trajectory for dynamical stochastic system. Okay, and these are some cases for instance in the calibration of optical tweezers or even a model model selection okay for a stochastic dynamics of running and type particles. Anyway, so let me stop.