 Welcome to this course on data simulation. We are going to provide a very broad background about the theoretical aspects of data simulation. Our first lecture will be on providing an introduction and an overview. I would like to start with the question what is a simulation? The concept of a simulation in general conjures up various meanings in different disciplines. In biology for example, the human animal plant body assimilates or absorbs food. So, here the word assimilation refers to the process of absorption of food by living organism be human animal or plant. So, here assimilation refers to a biological process. Assimilation also happens within the context of sociology. Immigrants refugees from one country they try to rehabilitate in another country by assimilating into the prevailing culture. In this context assimilation refers to the sociological process. As an example, United States is considered to be an example of social experiment. It is a melting part of sorts where there is continuous simulation of all cultures from all around the world. However, within the context of physical science as an engineering, assimilation refers to the process of fitting models to data. So, in this process the key players are model, data and the process of fitting data to the model. So, this involves 3 different disciplines. One is science of model building. Second one is ability to observe nature and the observations give rise to data. Data is available, model is available, but we need to bring them together. This bringing together that is the assimilation process. That process is also called fitting model to data. Looking from this perspective data assimilation is a very broad and a vast discipline. It is very broad, it is very deep. We would like to develop a very holistic view by looking at various practices that go with different names, but the underlying process is always related to data assimilation. Recall the old saying the whole is greater than the some other parts. So, the process of fitting models to data, the process of various types of assimilation, these are all various parts that come to be called the science of data assimilation. The whole is much bigger than many of its parts. First I would like to talk about data assimilation as a curve fitting or a regression process. The notion of fitting in data assimilation is the same as that is used in deterministic curve fitting or interpolation that is often used in numerical analysis. It is also intimately related to statistical regression analysis. To further understand the relation between data assimilation and other practices, we need to be able to introduce different classifications of models. Please recall models data and the process of fitting are the three components. Now we are going to look into properties of various kinds of model that occur in almost all of sciences and engineering. Models in general can be classified into two types. Type one is called process specific model, type two is called data specific models. Let us first concentrate on type one models. Here we understand the underlying scientific process that is at play. These are developed based on causality. If there is a force there is a reaction to that. There is always a cause and effect relation. In sciences many of these causal relations are nicely captured in the form of conservation laws. Here are some examples of models of type one. The harmonic oscillator that describes the motion of a pendulum in a friction free environment. The shallow water model that can be used to describe the motion of waves in a ocean. The primitive equation model that is the basis for many types of weather forecasting. Geo-strophic vorticity equation in particular. One of the equations which is very generic and has applications. In particular it is the basis for hurricane prediction. So you can see models come from various shapes and forms. Each one of them are developed based on fundamental scientific principles. Causality based, process based. These are essentially statements of conservation laws. As opposed to these the second type of model they are called data specific models. Here there is no underlying causation. There is no cause or effect that is well established. All that we have is the availability of a bunch of data. These data are observations of mother nature. Given a bunch of observations of mother nature's behavior at various times it is up to us to be able to mine the underlying information and that mining is done with respect to finding the correlation between the data. This is also called correlation based or similarity based. These models are essentially empirical as opposed to based on physical principles. These models can be either an explicit model or an implicit model. For example I have a time series. The time series could be value of IBM stock over each day. The maximum temperature in Bangalore each day. The unemployment rate in the state of Karnataka every month. This is an example of a time series. We would like to be able to build models for using these time series data to be able to predict how the unemployment will be in a year from now. What will be the maximum temperature in Bangalore in middle of January. These are called explicit models. These are empirical models. These are essentially based on correlation. Machine learning provides again lots of opportunities for model building based on data. Neural network is an example of a specific kind of machine learning process where we try to build a specific neural network to be able to undertake a specific task of being able to classify. Neural network models are essentially implicit models. You give an output. You try to tune the model until you get the required output. You probably may not be able to explain why and how except that if you did the right thing you can make it work. Again these are data specific. For example the time series model that is used for IBM stock price may not be the same type that can be used to predict tomorrow's maximum temperature in Bangalore or unemployment percentage so on and so forth. So each one of these data set has to be looked at separately. For each one of these data set we have to uncover the underlying serial correlation. Here I would like to make a statement about the definite need for correlation. If a collection of data are not correlated means one does not influence the other. If one does not influence the other we cannot we may not be able to predict anything at all. So ultimate aim in model building is to be able to predict. Once you have model I can pull the model solution forward to be able to make predictions. So correlation is very fundamental attribute of data specific models and it is using this correlation the model is able to extrapolate make predictions. Pattern recognition is another example of explicit models that are again data specific. Now another dimension to data simulation I would like to bring to fourth. Data the development of type 1 models and data simulation models for these models are separate processes. For example a physicist may try to develop a primitive equation model. An observation specialist may try to get data from satellites or radar. A data simulation person then comes into play to be able to understand the properties the model understand the data set and to be able to bring them together. So model development and data simulation process are two separate processes. It is the data simulation person is the one who sits in the middle who tries to talk to both the model and the data. The structure of the model in terms of spatio temporal evolution relation among the state variables are specified by the details of the model. Yet many of these models may have several unknowns. For example in a primitive equation model we may have an initial condition we may have a boundary condition. Depending on the type of processes involved different kinds of parameters may also become part of this equation. I know certain class of primitive equation model helps you to predict whether in a particular situation. But to be able to initialize the model to run the model forward I would need initial conditions I would need boundary conditions I would need parameters. So models are specified model low initial conditions boundary conditions and parameters. To be able to run the model forward we need the values of these and how do we get to know the values of these parameters that is where we use the data. So in here the goal of data simulation is to be able to estimate the unknown from the knowledge of the available data that is supposed to contain information about the unknowns. I would like to explain this a little bit further. For example a quasi-geostrophic water city equation is often used to be able to predict the movement of a hurricane. But that is a very generic kind of equation. If I want to be able to use that model to be able to predict the movement of a hurricane I need to know where it is today and what are the coordinates of that what are the pressure differences what are the various other attributes. So somebody has to fly planes into these hurricanes they collect data they bring the data to the office. So that tells you observations about the phenomenon I understand the science the physics behind the hurricane we would like to be able to bring these two together in order to be able to estimate the unknowns initial conditions boundary conditions and parameters. Therefore you can readily see data simulation is closely related to the classical estimation problem. This is fitting the observations of hurricane to the hurricane forecast models that was developed earlier this aspect is related to estimation theory. There is another point of view one can take data simulation has also lot in common with systems identification as used in systems engineering. In systems engineering estimation of parameters of a system to maximize the performance is known as system identification or adaptive identification. So let us look at the picture here this one is an engineering system that is a physical plant the physical plant may be a chemical plant it could be an aircraft it could be a ship or it could be any kind of engineering device it has input it has parameters which are knobs that is meant to control meant to change the behavior of the system. For example if this were to be chemical plant the input could be raw materials the parameters could be the presence of catalyst it could be temperature it could be pressure it could be concentration. So these are all various parameters that one can control. So the output of the plant depends on the physical properties of the engineering system along with parameters and the input I would like to be able to understand the dependence of output of the input. So we try to express that relation in the form of a cost function we would like to be able to maximize or minimize if it is cost is always minimization if it is profit it is generally written as a maximization it is an optimization problem. So we would like to be able to change the input of the parameters in a feedback loop such that I try to maximize the functioning of this plant this often occurs in many of the engineering system you can really see in here data simulation is done online your physical plant is operating it provides an output that is a function of the input I do not know what is the maximum output possible I am going to learn the maximum output by sequentially changing the input and the parameters in the loop based on a pre-specified criterion called the cost function this often happens in all branches of engineering. Now let us come to physically occurring systems such as meteorology such as geophysical sciences there are observations of geophysical sciences we often do the observations geophysical sciences come from satellites radars balloons ships collect information airplane collect information and these days we have ground base stations all around over the ocean we have lots of buoys. So observation comes in various shapes and forms the observation contain secrets about the functioning of the nature the modelers believe that mother nature behaves in a particular way they encapsulate the behavior of the model based on their understanding in the form of the process based models that we talked about the process based models in general can be a dynamic model if it is a dynamic model it can be based on ordinary differential equations or partial differential equations these models have initial conditions boundary conditions and parameters we only know the model describes the overall observations in some specific ways but we would like to be able to fit the model to the data so that a specific model can be used to accommodate the specific set of observation to be able to generate predictions about a particular hurricane a particular tornado a particular natural occurring event so this is how we use the data which are contained in the observation and the model fitting model to data. So this is an alternate view this is the view that often occurs within the geophysical domain oceanography hydrology atmospheric sciences and so on now I would like to talk about how data simulation takes place in empirical models in empirical model there is nothing we do not have physics all that we have is a bunch of data so I have to use this data in two ways so we have a twin task one is to be able to build the model and then once we have a general understanding of the class of models that we can use then we have to do data simulation on these models using the data so let us look at the data mining part the data mining part is the process by which I would like to be able to understand the structure that underlie the creation of the data. So what is that we do we compute the correlations we compute the similarity measures in a given data set based on the correlation we can narrow down a potential class of models to a small subset as an example in the case of time series we may say we are looking at the correlation structure if the correlation persists for a long time that means it has a long memory if the correlation comes down to low values very fast if it decreases for example a correlation could be a correlation structure could be like this here the correlation does not decrease too fast so this these kinds of process supposed to have long memory another could be the correlation comes down to 0 very quickly. So the class of models for this this is one kind of model this is a second kind of model there are guidelines to choose certain kinds of models for correlations of type 1 for correlations of type 2 even here one has to contain with the orders of the model but just by looking at it we cannot say this model work better that model work better so we have to be we have to be flexible try to be able to have at least couple of different ways of looking at it. So once we have narrowed down the size of models it is like having a barotropic vorticity equation it is having a primitive equation so we have to come to the same level as in the process based models by looking at the data to do data mining process we will talk a lot about data mining little later in this lecture series in this introductory chapter now once the model is available again in the case of time series models the models are dynamic given any dynamic model we will have initial conditions we have parameters to be able to pull the model forward we need to be able to know the values of these parameters again so once the set of models are selected we can use the same data set to estimate the unknown parameters by fitting each of the models to the data. So now you can readily see independent of the type of models the process of data assimilation requires the presence of a model the presence of an associated set of data and then the process of fitting that story must come pretty clear by now so here is an added difficulty in the case of empirical thing in the case of process based models for hurricane we know what class of models to use for certain time of atmospheric motions we know what certain type of models to use for oceanography they have a pretty good understanding what type of models to use so a given process has associated with the already understood class of models in the case of empirical model building models of types too there is no such clear cut data association there is no clear cut way to be able to decide this model will work better than that so we have to always content with a class of models so within this context the notion of model selection becomes a very important and a fundamental task there are several methods one can look into to be able to develop and compare the quality of each of these models to a particular given situation and we are not going to indulge into the model selection process those of us who are working in the empirical model building has to be cognizant of the requirement of being able to use very good criteria to be able to select good models for a particular given kind of data set so now you have seen two aspects data mining and data simulation I am going to use time series analysis again to be able to reinforce some of the basic thought process we have already discussed this is an example of an explicit model here data mining step what is the data mining step here given a time series first compute and plot the correlogram and what is the correlogram correlogram is the one that we already saw I am going to give another example of a correlogram this is the time index k this is rho of k so what does k refers to k refers to the separation between two data set time t and time t plus k how are the data are time t and time t plus k are associated with each other for example if I have a maximum temperature tomorrow today how is today's maximum temperature related to yesterday's maximum temperature day before yesterday's 10 days ago you can readily see today's maximum temperature may be related to yesterday's but to a lesser degree day before yesterday to a lesser degree to a week almost no link to a month from today so this correlation of maximum temperature one can think of it like this if you think of again unemployment percentage unemployment percentage do not vary day by day they will repeat of month so today's unemployment is very much related to yesterday's unemployment so what is the correlation between unemployment in time separated by one day one month six months one year where the basic units of time could be one day so k refers to the number of days that separates two time epochs between which I am interested in correlation so this is what is called a correlogram plot of rho of k versus k rho of k is a correlation between data that are case steps apart then what is that we do so this is data dependent so this created from a particular data so this is some sort of a summary of what this data tells us then how do you utilize this mathematicians have helped to create several different types of models for time series analysis these models were called a REMA AR for autoregressive I for integrated MA for moving so it is a three different families of models is called AR model integrated model moving average model ARREMA models these ARREMA models are a very broad class of potential model that one can build mathematicians have helped us to be able to build all these models ahead of time and they have analyzed the underlying correlation properties of each of these models and have cataloged in fact they are they have developed all of them and all of them of properties of correlations of various types of models for example an AR model could be an first order model second order model an MA model could be a first order model second order model and ARMA first order AR second order MA so in general one can have ARMA PQ so P refers to the AR type Q refers to the MA type by changing P is an integer Q is an integer by changing P and Q I can get a whole family of models if each of these models are created I can mathematically compute what their correlation should be and I can plot these correlations and create an album this album is the fundamental basis for almost all of time series analysis how do I use this album you have already cranked out the correlation for a specific data set then you visually compare the given picture the photo with the album that you already have you narrow down which of the pictures in the album close symbol the one that you have that depicts the particular data set you have and that helps you to narrow down maybe it looks like AR 2 and 3 maybe it is AR MA 11 you do not simply say this is it you look at the things that are closed you narrowed on the model to 234 each order will have different kinds of parameterization so use the same data to be able to estimate the various parameters of each of the models on once you fit different models to the same data then you can compute what is called the error in the forecast the error in the model use that error to further select much more finally the appropriate model that could be used in this particular case so the data assimilation step now so that is the so the first one is analysis of data second one is comparison of the given data with the album once a particular data has been narrowed down then you come back to the whole step called the data assimilation step estimate the parameters of the different models using the same data set so here I would like to emphasize the difference in the process based models models are created from by the scientists observations are created by measurement scientist they do not talk to each other but our job is to bring them together so the model building and data assimilation product separate but the case of empirical models while you are given is only the data based on the data you have to develop the model once you develop the model use the same data to be able to assimilate the data into the model and again model selection becomes an important situation in these cases here comparing the album I would like to bring in analogy here all of us happen to go to the doctor once a while in order to understand what happens with your health doctors order different kinds of observations on your blood test just x-ray maybe an MRI or maybe other chemical analysis of your blood once a chemical analysis the blood x-ray MRI all the things are made available a doctor hangs them all together and he looks at various he or she looks at various signals that is observations providers then they go to the anatomy book the anatomy book describes what an ideal body should be they compare the present situation with an ideal situation oh this looks right this looks right oh this looks not that good they try to narrow down a particular possibility and then do for the steps do for the steps do for the steps once they understand then they go for the treatment so to to label you have this disease to label you have that disease you to label you have this problem that is a prediction problem to be able to predict I need to analyze the model and how do they use the model they use them the models or to them the the anatomy textbook so there is a data there is an album they compare the data with the album to be able to generate to be able to make a forecast once the forecast is done with respect to medical diagnosis forecast then leads to a treatment and then recovery in the case of a hurricane forecast we make a forecast then we tell the public that is going to be a 20 drain tomorrow move people up up to the to the to the higher grounds and so on and so forth so the aim of data simulation in general is to be able to fit models to data so that we can generate useful forecast and the forecast can be used in many different ways for public consumption so that is the overall underlying philosophy that will be pursuing in this course so data mining and data simulation dm for data mining da for data simulation they are also used in machine learning in this case the models are implicit we are not going to do much of machine learning in this class we are not going to be doing much of times it in this class that is why I would like to spend a little bit longer time to start with to be able to see the other relations to data simulation so goal is to train a machine say an artificial neural network to classify samples into two classes given a set of for example medical diagnosis for example blood analysis a blood chemist gives the blood chemistry the whole idea is that can you feed this blood chemistry information into a neural network you train the neural network so well to be able to predict yes you have hepatitis a no you have hepatitis b no you have hepatitis c 50 years ago we did not know that there was a disease called hepatitis then they know that there is a disease called hepatitis then they found out hepatitis is not one kind it is a multiple kind a and b then they found out it is not a and b that is also a c have we discovered all aspects of hepatitis the answer is no we never know that could be c d e f g h so once you know that these are the characteristics of the blood that corresponds to a b c we would like to be able to automate the decision process so neural network is essentially a classificatory mechanism into which you feed this chemical information it prints out the particular kind of disease that corresponds to that that is a classification for a typical classification problem another classification problem is in a post office I would like to be able to develop a machine that can read the addresses to be able to sort them human can sort but I would like to automate the sorting process by machines so that is a pattern recognition process but I write in a very different from yours but the machine cannot handle all the ways of handwriting so we are going to say hey if you want to be able to use the machine to classify you have to type in a particular form once you have typed in a particular format a machine can be taught to be able to classify the addresses in different groups that is artificial network making this network do these processes is called learning phase so given a set of samples and their associated classes so what does that mean these corresponds to type a hepatitis these correspond to type b hepatitis these correspond to these type 3 c hepatitis so there must be an expert who already knows a particular analysis also the classification he has labeled them so once I have a sample with their associated classes then what do I do you divide the samples into two groups one large subset called the training data another small subset called the testing data so we use the larger training data to be able to train the machine that is called learning with the teacher we always learn with the teacher there are two times of learning learning with the teacher learning without a teacher in the in case of learning with the teacher you learn by recognizing that you made a mistake in the case of learning with the teacher I already know for this input this must be the output I know for specific so you can train using this data where the known classification an artificial neural network to classify so this is essentially a fitting process in the learning this is called the learning with the teacher during this phase I make sure on this data set this machine behaves in the best possible way then I would like to understand whether the machine will be able to do things that it has not already seen so if I use the same data that I use to teach and then certify that cheating so I would like to be able to test it with the set of data that was not used in the training phase and that is why we have talked about two subsets a smaller subset now you feed the data set that the machine has not seen earlier if on this data set the machine does very well then you have succeeded in making the machine learn though this is from a higher angle you can look at it as a data simulation process we are fitting the behavior of the network to suit the classifications of a given data set so you can see machine learning in data simulation have a very strong interconnection data mining and data simulation in artificial neural network is an implicit model again this is a quick summary of what I already described the data mining phase here what is the data is the choice of the structure of the neural network I simply talked about neural network how do I define the network it has number of stages number of neurons in a given stage the total number of inputs the total number of outputs these essentially describe the topology of the underlying artificial neural network this essentially comes from experience this is the data mining phase the mining relates to a lot of experience that one has should I have two layers should I have one layer should I have five layers should I have one output should I have two output so these are all the decisions that engineer has to make in the design of the networks once the network is designed the structure of the network is designed we are the same level as in the Armagh case I have picked the model we are the same level as I had picked a hurricane I have picked the hurricane model so the model is already fixed in here the data assimilation phase is to fit the chosen structure to the training data by minimizing the error in classification the testing phase is the prediction phase to see whether the artificial neural network handles the new situations very well so you can see the data assimilation is intimately associated with with learning my own background my PhD work was in essentially in machine learning that's why I could jump into data assimilation part rather easily data assimilation is the basis for any and any kind of learning mechanisms or learning devices so different facts about da all in one place now da can be looked at curve fitting da can be looked at estimation theory da can be looked at statistical regression da can also be looked at system identification adaptive optimization so looking at da is like 5 blind man looking at an elephant a person in curve fitting looks at curve fitting only a person in size statistical regression record statistical regression only our job is to be able to bring the big elephant this is what I was trying to tell in the very first slide the sum is greater than the parts the sum the data assimilation to further understand the richness of data assimilation we need to introduce further classifications of both models and data and that's what we are now going to look at until now we classified models only at a broad angle namely process based models or empirical models or data specific models that's the classification at the highest top most level now I need to bring down finer nuances into the classification so models in general can be classified along different dimensions a model can be static or dynamic a model can be deterministic or stochastic a model can be linear or it can be nonlinear a model can be perfect or it can be imperfect almost all the models are imperfect I don't think there is any model that is perfect except few I'm not saying there is no nonexistence for example the model that governs the motion of the planet earth around the sun it's very nearly perfect why how do we know that we are able to predict the loner solar eclipses for the next 100 years and sure enough when we say there will be a lunar eclipse you know like what happens but there are very few systems where we can make such accurate predictions so perfect models are few imperfect models are large a model can operate in discrete time or in continuous time a model can work in discrete space or in continuous space a model can be finite dimensional or model can be infinite dimensional for example a person working in static regression he may be interested only in static deterministic linear nonlinear case a person in system identification may be interested in a dynamic deterministic models linear nonlinear a person working in time series analysis would be interested at dynamics stochastic linear nonlinear imperfect models that operate in discrete time most of the models in geophysical sciences they work in continuous time and continuous space if a model is given by partial differential equation that is infinite dimensional in nature if models are given by ODE they are finite dimensional so you can see data simulation looks at the entire domain of modeling with all its nuances with all its abilities to classify models along these many different dimensions and in our course we are going to take such a global view so what does it mean with this course you can do data simulation in several different walks you can handle any kind of different models so we are looking for the ultimate generality that underlie the notion of what data simulation what models and what is the holistic view of data simulation that is our purpose examples deterministic dynamic continuous time infinite dimensional models ODE XT so I am now going to introduce some notations X is the state of the system T is the continuous time Rn is the set of all vectors of size in real vectors of size n when I say Xt belongs to Rn Xt consists of n components each one of them are functions of time Xn is considered the state of the system for example in meteorological setup what is the current temperature pressure in downtown Bangalore say so time midnight midday 8 o'clock so what is the state how do I describe the state of an atmospheric system temperature pressure humidity wind speed all these constitute different components of the vector and and that's called the state of the system the model also can have a set of parameters alpha that could be p number of them we will use n to be the size of the vector which represents the state p to be the size of the vector that represents the parameter we will use alpha the Greek letter for the parameter the English of X as as as as our our state so the what is the linear model X dot is equal to ax so this is a differential equation X dot dx by dt the rate of change of the system a is a matrix X is a vector X dot is a vector matrix is real matrix of size n by n it needs an initial condition X not so that is the ODE now PDU T is the time variable X is a space variable in the case of ODE there is no space only time P is a function of X and Y that is essentially space no time so P is a function of space but no time two dimensional U is a two dimensional but one is time another space a linear partial differential equation could be a Poisson's equation double derivative of P with respect to X the second derivative of P with respect to Y is equal to minus X X X comma Y it can be it has to be solved under certain kind of boundary conditions a standard nonlinear model is U t of X the time derivative X plus U times the space derivative X must be 0 that is called the burgers equation with a suitable initial or boundary condition oh gosh I think there is a spelling error is called burgers B U R G E R then there is an integral equation f of X 0 to t k X of t M of t dt here f and f and k are given our job is to find yeah so given k k is called the kernel f is called the forcing my job is to be able to find yeah so you can see models of deterministic dynamic continuous time infinite dimensional can occur either as an ODE or a PDE or an integral equation examples of deterministic dynamic discrete time infinite dimensional models replacing the time derivative with suitable discrete time approximation I can express differential equation into a difference equation by replacing an integral by a sum in the standard way that we do in numerical integration we can get a variety of discrete time models the integral equation gives raise to a linear equation A X is equal to B A is known B is known my job is to find X the that is a static model the dynamic model is X k plus 1 is equal to M times X k there is a linear discrete time model X not as initial condition a non-linear model X k plus 1 is equal to M of X k alpha as a parameter in this case I have initial condition I have parameters so I am now giving life to a representation for different kinds of model that we talked about in general now I would like to bring the distinction between model state versus observables observables are related to quantities that I can observe directly for example vorticity but I cannot measure vorticity directly has to be measured indirectly volatility the price changes that is called volatility volatility is a quantity that affects us but I can't measure it directly so there are observables but there are states not every state as observable for example if the observable is pressure I can directly measure pressure if the observable I am sorry if the state is vorticity I cannot measure vorticity directly I have to measure indirectly so I would like to be able to relate the states of the model and what is being observed and relate them so I would like to be able to bring the distinction between the two modeling in spite of all the effort that are gone on for centuries is still an art as well as a science a modeler has full freedom limited only by his her imagination to concoct newer relation between variables of various types so the novelty in modeling relates to the ability limited only by the imagination to be able to describe various relation that are alive between quantities that describes the system so here we can discuss the variables into two types direct variables the right variables direct variables are directly observed pressure temperature the right variables the model needs a direct variable but is not directly observable examples height position velocity temperature energy stock prices interest rate unemployment these are all direct variables which are directly observable vorticity entropy tape volatility inflation some of you may come from meteorological geophysical sciences but my interest comes from applied mathematics to me analysis of time series models in the financial setting is no different from analysis of time series models to analyze climate data underlying mathematics are same that's why I'm trying to give you a broader explanation of variables so vorticity entropy cake this is the convective available potential energy which is very important in severe storms volatility is very fundamental to many things we do in life for example the current deluge in in in Madras in Chennai is because of a low pressure storm they that came and sat in on the top of Madras didn't move at all and dumped it's a very rare event so if you look at the frequency of these events if you look at the intensity of these events this event were a high volatility it's the the the the the the effect was way too much same thing inflation inflation was very high when I when I went to United States in the late 70s we had 14 or 15% interest rate today is close to 0 so inflation in those days was very high inflation there's virtually none inflation affects everybody but inflation something we cannot be measured it has been for so inflation is a derived variable volatility is a derived variable entropy is a derived variable but my model may need these derived variables to be able to make analysis so here I'm going to talk about the relation between state variables and observables dynamics of sea surface temperature in equatorial Pacific that's a state variable this rate variable is a fundamental interest in predicting a linear as we all know we are in a very severe grip we are under the very severe grip with the linear very strong they say the high-temperature they the average temperature is more than 3 to 4 very close to 4 degrees and it is affecting whether in different parts of the world varticity dynamics is another state variable for varticity equation the total water content in a cloud system that's very much needed in in cloud physics speed of a car in a cruise control today we are talking about driverless car google is developing driverless car so I need to be able to measure the speed of a car very accurately to be able to feed back to the control elements all avionics your flight starts from Bangalore airport at 2 a.m. goes to London in 8 hours you cannot see anything everything is automated so the autopilot has to be has to be able to adjust the speed based on various measurements so your plane has tons and tons of observables relating to the situation where it flies the models of the dynamics is already programmed to the autopilot so the autopilot model varies the various control devices based on the observations is that a head when is that a tail when and so on and so forth so to be able to control I need to understand the state variable it could be cease first temperature it could be water dynamics the total water content it could be speed of a car in a cruise control observables how we measure the equatorial Pacific temperature thousand miles west of Hawaii nobody can go it is very difficult to develop a network of buoy systems so we have to measure the temperature essentially from satellites satellite measure only the thermal energy that was radiated in the infrared domain so we have to estimate the temperature based on the thermal energy received by the satellite and invert using the very well-known radiation physics laws Stefan's law Max Planck's law and so on what is the dynamics would use the observables are the prevailing when the UVW component from which you have to essentially compute the water city the total water content in a cloud there is no way to directly measure it is measured through the reflectivity of the radar the raindrops are hanging in the cloud once you send the beam the the radar beam it gets reflected by the the the water droplets so the intensity of the returned reflected beam is displayed in various colors and we can estimate the amount of water that inherent in the cloud by looking at the reflected energy so that is the reflectivity is observable but it is related to the state variable called total water content voltage generated which is proportional to the speed and that is what cruise control uses you sit the speed for 70 if this because the road condition if the speed comes to 68 the accelerator knows that there is a difference it pushes the car forward if the road is smooth if you have set 70 and the current setting it may go 72 it shuts off a little bit to be able to control so I need to be able to measure the speed to be able to to be able to do a good cruise control with the interest in driverless cars this notion of models observation data simulation becomes very fundamental to replace a driver by a computer because we need to know the dynamics of the model we need to know different road conditions so a car is fitted with lots of observing devices measuring device a non-boat computer has to process all the devices almost instantaneously to be able to steer the car in the particular lane and they have to a limited they have obtained a limited success in doing this it's a long way to go but it is the intellectually stimulating project where modeling observations bringing the model almost online to be able to use and to be able to steer the decision there is to be able to steer the decision meteorologist to be able to predict and so you can see the underlying aspects of data simulation again coming along from different directions now I like to we talked a lot about a model I have not talked much about the observations so I am now going to talk a little bit more about observation because it is a one of the three click players in the system observations of physical variables just temperature pressure wind are subject to measurement errors these measurement errors are noise are called noise for example what I may read as 2.5 you may say it is 2.3 one might say is 2.6 so observational errors when humans measure certain things even instruments if you look at any instrument they will say the following so if you buy an ultimate or they will say well it can read from 0 to 100 the accuracy is 10% what does it mean if it says 60 it is plus or minus 10% of 60 and where that error comes in the engineering aspect of the of the design of the instrument itself so that is one way the error can come in the other way the error can come in is how do I calibrate the the the meter shows 60 is it really 60 we have to calibrate a meter against standards so it could be design errors within our design differ there is no design error there is a design differences there are manufacturing differences there is calibration differences then the human being being able to read these all these together lumped into one phase called measurement noise or the measurement error following Gauss one of the very well known mathematicians of all time who lived between 1777 and 1855 he was the first one to be able to analyze the properties of observation noise in fact in his time the only kinds of data that was available was measurements about very many celestial objects made by humans using very simple telescopes that is all that they had they had nothing else so he was to be able to make sense out of these observations and these observations are not consistent these observations contain a lot of errors so by analyzing a given set of observation he modeled the statistical properties of the observation errors and developed the so-called bell curve so what did he show he showed the following if this is 0 the errors from a bell shaped curve on an average the error is 0 but on a given circumstance the probability of an error could be either positive or negative it took a bell shaped curve this is what is called the Gaussian curve and we all know this curve is given by ffx is proportional to e to the power of minus x square by 2 that is a constant of proportionality that comes into play so this curve is a bell shaped curve what one of the most fundamental contributions of Gauss is to be able to establish that observation noise essentially followed this bell shaped curve and we use this to to this date and that is a very enduring aspect of Gauss's discovery so this kind of noise it is also called the white noise what does it mean I measure the temperature today I measure the temperature tomorrow I measure the temperature day after tomorrow by the same instrument the error today is not correlated with the error tomorrow the error is not correlated with the with the error day after tomorrow so that is that there is no correlation when sequence are not correlated there is a particular way to characterize this that is called white Gaussian noise Gaussian refers to the bell shaped curve noise is the error white means they are uncorrelated white noise means errors are uncorrelated these noise are also have a known correlation structure for example if you buy a voltmeter they say a 10% accuracy 15% accuracy 20% accuracy you can measure you calibrate and you can compute the error so observation noise has to be associated with the error properties the error properties have to be related to the known covariance structures of the errors in economics stock prices the interest rate foreign exchange rate they are all intrinsically random pressure temperature there is a variability from night and day that comes because of the phases of the day and night the phases of the moon summer winter autumn spring so these are all the variations induced by season variations induced by the day and night December 31st the maximum temperature in downtown Bangalore this year next year over the past 100 years if you if you plot them there is a natural variability so things could have natural variability things could have underlying stochastic properties so when we talk about random the randomness can come from either from the noise or from intrinsically random variations we will stop here at this moment we will continue these topics in our next lecture thank you.