 Today's lecture is one of the most difficult lecture, let us please pay attention and wherever you want me to stop, just stop me, okay so we have been looking at this black box modelling and I introduced what is called as output error modelling, so I just showed you how to go about doing output error modelling for a second order system, we had four parameters to be estimated a1, a2, b1, b2 and whether we want to assume x1, x2 to be 0 or whether we want to estimate that was a choice, well the problem that was finally formulated was an optimization problem, so we said that estimate a1, a2, b1, b2 such that some norm of the residue, model residue, model residue is y measured minus y estimated or y predicted, some of the square of this model residue should be as small as possible, that was the idea, so choose these parameters such that this is one way of formulating the problem, one could say y square, I could minimize some of absolute values, no problem, you could minimize infinite norm which means you could minimize maximum error, no problem, it is a choice, we typically use two norm, because two norm comes with a lot of associated properties, may be if time permits I will just hint at those properties, we do not have time to get into everything, but analysis of the method is much easier using two norm, so we normally use two norm, okay we could choose, we could simplify by saying x1, x2 is 0 and we can identify the model parameters, I am showing you here this particular problem solved for the two tank system that we are considering, model parameters estimated are given here, numerator polynomial and denominator polynomial, I am looking at a CISO system, single input, single output okay, we will worry about multiple input, multiple output little later what to do, okay, I showed you this plot 2 at the end of my last lecture, so the blue line is the model prediction and I think the plus are the measurements, even though blue line is shown as a continuous actually it is not continuous, it samples, many times we just draw it continuous out of some kind of an habit, we should actually draw it points, the difference between these two is this vk signal vk, actually technically I should put v hat k not vk, estimate of vk, whatever is not explained by the known inputs, known inputs that are going outside my computer, okay, whatever is not explained by whatever is going outside my computer that is captured approximately in this vk, approximately because we have an estimate, we do not know the true value okay, now the problem is I want to bridge this gap, I want to model, I want to develop a model for vk, this is a very very very tough problem, I have a signal which is varying like a random function, it is drifting right, it is very difficult to come up with a structure for this signal, now moreover we just have measurement of this signal, okay let us assume that you know we have done this output error modeling and we have this signal with us now and now I want to develop a model for this signal, I want to uncover some kind of structure into this signal, I have a measurement of this signal I do not know what is the cause, the output error model we developed was a cause and effect model, u gives rise to change in x, x was the effect of u and then we found the model, here what is the cause, we do not know, okay, this is combined effect of everything that we do not know, it contains measurement noise, okay, it contains unknown disturbances, unknown fluctuations for example, I have a pump, pump may be receiving voltage supply which is fluctuating that results in flow fluctuations which results in fluctuations in my, see my input was the control wall position, the control wall position is known but the flow fluctuations caused by, even if my condition is steady, the flow fluctuations caused by fluctuations in the voltage, supply voltage is not measured, I am not going to measure that, I am not going to measure everything that happens, okay, so this effect is there in my, now as if this two were not enough, we also have errors coming because of approximation, we are approximating a non-linear dynamic system using a linear model, okay, so this vk has everything, okay and so it is driven by some kind of an unknown source, I want to cover, I want to attach a model, I want to come up with a model to discover this system, so measurements contain effects of, you know, unmeasured errors and unmeasured disturbances, measurement errors and unmeasured disturbances, in addition you have approximation errors that arise because of approximation that you come up with, okay, well there is one problem with output error modeling, why I want to move from output error modeling to some other modeling, noise modeling, because output error modeling can be used only when the system dynamics is open loop stable, okay, system dynamics is open loop stable, then the predictor is, if you start guessing an unstable pole, the predictor is unstable, the predictions can become unbounded and you cannot solve that problem numerically, it is not possible to solve, okay, so one requirement, one fundamental requirement is that for output error model development by this method which I described just now, you have to have the model says to be open loop stable, okay, which means pole should be inside unit circle, we have done this, what is open loop stable, okay, the discrete time system, pole should be inside the unit circle, okay, so now actually if you want to work with system that are unstable or if you have system which are subjected to significant amount of unmeasured disturbances which is more often the case in real systems, then you better model these unmeasured disturbances, so the tool that I am going to use is called as stochastic processes, okay, stochastic processes are sequence of random variables, okay, which are typically correlated in time, what happens now even though it appears random, has some relation to what has happened in the past, what is the relationship that is what you want to uncover, okay, well what I am going to do now this lecture is try to give a very very brief introduction to these stochastic processes, it is unrealistic to give introduction to a topic which has taken centuries to develop, it is very very complex business and try to put it in you know nutshell and put it in one or two hours or whatever three hours or four hours, it is very difficult, but you know even though it has taken centuries to develop, you know even language that we learn as a kid, language has taken centuries to develop, but start learning, right, we start syllables and words and sentences without understanding you know what it means, for example take word responsibility, you start using this word at age of 4 or 5 or 6, without understanding what is the meaning of responsibility, okay, you might know spelling in the beginning and as time evolves you know as you mature, you start getting the meaning, so I hope as you start working in this area more and more meaning will appear, so this brilliant theory was what we study today as formal theory of stochastic processes was developed by Russian mathematician Kolmogorov, around 1930s he formalized the structure of this stochastic process theory, the other mathematician who contributed to this I think he was Austrian, Wiener was these two people stand out as giants and so the formal theory that we have actually they sort of formalized this theory, the efforts were on since very long time, probability and statistics is an old topic and people have been looking at it for generations, but it took around 1930s, 40s where it took a shape that we study now, okay, now let us begin with what is a random variable, a random variable is a mapping which assigns outcome of a random experiment, okay, to what is called as a real number, okay, so there is some random experiment, for example the noise in the measurement, okay, I am going to assume is a random number, the noise that occurs now, the noise that will occur in the next instant or instant after that or that has happened something in the past, okay, may or may not have any relation, it is random, okay, if you open an instrument manual you will see you know something like what is the standard deviation of the measurements, it will tell you what is the spread, if you keep the measurement, measuring instrument into your, let us say I am measuring temperature, if I dip thermocouple into some constant temperature bath, you expect temperature to be constant, measurement to be constant, but you will never get constant measurement, I will show you actual data, you will get you know data which is fluctuated all over, okay, well maybe you can just have a look, okay, we will postpone looking at the data whenever the slide comes, this version of power point seems to have some trouble, okay, so there is a random experiment and we assume that outcome of that experiment belongs to what is called as sample space, sample space is set of all possible outcomes, okay, set of all possible outcomes and then we you know identify behavior of this variable through a probability density function, okay, you know about probability density functions, you have studied this at some point in your undergraduate or in your first year of your undergraduate studies or graduate studies and probability density function is a model, if you look at it very carefully, it is a model, okay, it is a model that explains how the randomness you know how the randomness is and this probability density function is a model for one random variable, right, one single random variable, okay, right now we are not talking of and then in books you will have these examples of coin tossing experiment and so on, right, coin tossing experiment, the sample space will be only two, was it what heads and tail and so on. Now this abstract concept called probability density function it actually is a some kind of you know a limiting notion which tries to mathematically conveniently express all possible outcomes of a random variable, okay. Now with this very quick review of random variables I am going to move what Kolmogorov did, stochastic processes, well this definition is hard, okay, so let me preempt you today you will understand this in this course but keep it in mind, you will understand it eventually. A discrete time stochastic process is a family of random variables, now I am going to look at a family of random variables not just one random variable, so vk where k here comes from an index set, in the context of digital control index set is time, okay, if you call current time as 0, minus 1 is 1 in the past, minus 2 is 2 in the past, plus 1 is 1 in future, okay, plus 2 is 2 in future, okay. So in context of control this is not the only way to define the stochastic process but in context of control, okay, k represents time typically we take some initial time 0 and we define a stochastic process going from 0 to infinity, okay, so that means it is a set of random numbers, okay, so at each time point, okay, there is one random number attached, okay, not a sequence, I take back, I need a collection, it is not a sequence, sequence is a specific sequence is called something else, I will come to sequence little later, no, no, no, it is not one random variable, one random variable when you take, it only has spatial dimension, that is a good question, see here I have two, this random, this stochastic process has two components, one is time, another is randomness, okay, so I am going to denote this stochastic process as k and zeta, zeta is the random variable, k is the time variable, okay, k goes from 0 to infinity, this set entire set is denoted by this random process vk zeta, okay, vk zeta is a set which is, okay, no, no, different values taken by this is called a realization, let us, no, no, no, no, do not confuse the two, okay, I will show you a picture, I will try to show a picture and then if I fix myself to zeta equal to zeta 0, then it is a ordinary time function, sequence of random variables, okay, which is called realization of a stochastic process, I will give examples, okay, then it will become clear and if I fix myself in time, it is a random variable, at any given time k, I have a random variable, okay, now let us look at pictures that will, you will understand what is trying to be communicated here, this is a random process, a random process may have or has multiple realizations in time, okay, what I have plotted here is multiple realizations in time, okay, see for example of a, this random process could be, you know, you are measuring heartbeat of a patient, okay, now of course in reality you do not measure it from 0 to infinity, but let us assume for a long time, you know, you are measuring heartbeat of a patient, okay, let us say you have a device which is placed in the shirt and then you know it can sense the heartbeat and sense you a signal, every one second you get or every five seconds or whatever interval, you get information about heartbeats, okay, what happens today, say morning 8 to night 8, okay, what happens tomorrow, okay, let us say I am plotting that, 12 hour behavior today to, today morning 8 to evening 8, so you know today I will get this plot, okay, tomorrow if I do the same thing, let us say one minute, at every one minute I am plotting heartbeat, information, okay, tomorrow between morning 8 to evening 8 I am not going to get the same random sequence, okay, I am not going to get the same random sequence, tomorrow I am going to get a different random sequence, okay, now if I fix myself to some time, let us say I fix myself to time 12 o'clock, okay and decide to watch, observe, collect information about the heartbeat of the patient at 12 o'clock in the afternoon, okay, you will see a random distribution, do you agree with me, you will see a random distribution, so if I fix myself to a time I have a random variable, but if I look at the sequence which is generated today between morning 8 to evening 8 I get a realization of a stochastic process, what is the stochastic process, heartbeat of the patient collected at every one minute interval, okay, another example which I many times have been giving in these lectures is let us say average temperature of the day at 12 o'clock in the afternoon, okay, I want to do this stochastic process, so collection of 365 average temperatures from January 1st to December 31st, I take a span of a year I look at average temperature of the day, okay, you go to Google weather and you say that Bombay temperature for last year, it will show you a plot, maybe it is taken at some particular point in the day say 4 o'clock or afternoon 12 or I do not know what reference point they use, but they will show you a plot, okay, the random sequence that you got this year is not the random sequence you got next year, I am trying to view this 365 variables, okay as a random process, they are connected temperature, they are physically connected collection of random variables, I am not saying temperature and pressure and you know any arbitrary combination, I am looking at temperature at 12 o'clock average temperature at 12 o'clock, so what is K here, day, today, tomorrow K plus 1 is tomorrow, K minus 1 is yesterday and so on, okay, so if I plot the data for year 1991, okay I will get one realization, if I plot data for 1992 I will get another realization, stochastic process is same, if I fix myself to you know 24th of January or 25th of January, it is a random variable, I start looking at data for average temperature on 25th of January for last 100 years, I will see a distribution, right, if I fix myself to 25th of January, so what is trying to communicate here is that if I fix myself to T1, okay, I will see a random distribution, each one of them can have a different distribution, okay, the distribution or the randomness associated with temperature, average temperature today may not be same as randomness associated with temperature tomorrow or March 15th or whatever, right, so it all depends upon, so this last sentence is important is if I fix myself to a time K equal to K naught, okay, then it is a random variable, okay, so when you define a probability density function for a process it has two things, at every K there is a separate probability density function, okay, at every time instant K is a separate probability density function, this is another picturization, so this is a stochastic process, okay, some stochastic process plotted here and you can view that at every time instant there is an associated probability density function, so this is my probability density function at this time, okay, because we are collecting, we are talking about a sequence, a collection of random variables, each one of them will have a different density function associated with it, okay, yeah, this one is a continuous stochastic process, actually the, so the number of random variables in a continuous stochastic process is infinity plus the, each random variable can take infinite values, so this plot is a little you know complex plot than what I am trying to explain, this is not a discrete process, what I have shown here as a picturization is not a discrete process, a discrete process will have only values at the finite time instance, right now I am talking about any one, one can talk about vector of, vector stochastic processes, let us not get into it right now, we will get into it at some other point, later point, okay, I am not talking about vector, okay, see for example you might want to say that why one temperature, you know morning 8 o'clock, afternoon 12, evening 4 and night 12, I want to you know look at this 4 average temperatures, okay, this vector, so it is a vector stochastic process, you can view that way, it is not possible, okay, or there are 4 patients attached to you know 4 ECG and you know you are getting 4 random sequences possible, okay, so this stochastic variables is used everywhere, okay, if you know about this you can enter into financial market and do you know stock market studies, exchange, you can model exchange rate fluctuations, conversion, currency conversions, you can model audio, videos, speech signals, you can, I mean just imagine when you are receiving data, okay, in your mobile let us say, it has signal, signal is the speech, okay, plus noise which starts picked up and you better model the noise if you want to clean it, right, you do not have to be electronic internet to understand, appreciate that the signal which you get here will have data, you listen to it you will get you know funny sounds which is noise and it is not able to reject the noise, so you get those funny sounds, okay, so medical data you know ECG and EEG, not EKG, ECG, blood pressure and temperature data or Brownian movement, okay, all these things can be modelled as stochastic process, I am just showing you here measurement of a temperature recorded in a experimental setup that I have, well there is no heating nothing, there is a sump, I am recording temperature of water in the sump, look at this data here, okay, it is going all over, the mean temperature you can talk about, can you say what is the temperature inside this or you talk about some kind of mean, okay, you talk about you know standard deviation around the mean, all those things you can talk about, okay, but now I collected this data for 400 samples, next 400 samples if you collect data will you get the same sequence, you will not, okay, now how do I model this, how do I model this, so the way I model it is by imagining that at every time point there is a random variable attached and I try to model each one of those random variables, how do I model random variables, probability density function, okay, so if there are 400 samples and if I can find out those 400 probability density functions I am done, well looks very simple while talking very difficult task, okay and coming with an idea that one could model such a random behavior with a sequence of or a stochastic process to generations, it is not at all easy, okay, so let us see whether we understand this, how do you characterize this, I am going to sort of jump and go in terms of concepts, what are the good books, well Astrom's book which I have listed in my last lecture notes, end of my lecture notes, Astrom and Wittenmark gives a good introduction, very brief introduction, the kind of introduction I am going to give because we do not have time to get into stochastic processes full blown, there is a book by Astrom on stochastic control systems which is also kind of a benchmark book, if you want to know more about stochastic processes much more in detail, in depth you should refer to Astrom's book on stochastic processes, stochastic controls, well we have to talk about mean variants, all the usual stuff and this operator E is going to haunt you for next few lectures, what is E operator, expectation, okay, expectation of a random variable, random variables, we can appreciate now for a random process there will not be one mean value, will there be one mean value because we are looking at you know collection of random variables, for each random variable there is a mean value, okay, so that is why this density function here if you look here it has two attributes, one is time attribute, other is this you know probability space attribute, let us call it zeta, so I am only integrating this over zeta, okay, so I get this mu k, I will get this mu k which is actually sequence of means, okay which is sequence of means, very very important, why do we plot mean temperature, why do we plot mean temperature, if I want to study you know what is the behavior of you know temperature at 12 o'clock in Bombay, what I will do is you know I will record data for last 100 years, okay, collect data for last 100 years on 25th of you know January at 12 o'clock what was the temperature, I will try to find a mean, today's mean will be different from tomorrow's mean, you can appreciate that, okay, I will try to find out a standard deviation, today's standard deviation is not same as tomorrow's standard deviation, okay, I might attach, I might say that well this behaves like a Gaussian distribution, this is a model which I am proposing, okay, it may explain my data, it may not explain my data but you know you can propose some kind of a model and find out its parameters, estimated parameters but the mean for each day is going to be different, okay, mean for each day is going to be different and the standard deviation is going to be different, if you take Gaussian, if you assume that each one of them is a Gaussian random variable, okay, simple model, Gaussian random variable requires only two things mean and standard deviation, okay, a simple way of modeling and then I will have a stochastic process which is Gaussian stochastic process, each one of them is a Gaussian variable and each one of them is characterized by its own mean and own standard deviation, okay, yeah, now this is mean of mean, I am talking about you take mean temperature, okay, mean temperature in the sense, what I mean here is, okay I should explain the difference, average temperature as in specially average temperature, average temperature in this room or outside, okay or in the tree or in the sun, which temperature, so I am talking about the average temperature not in time but physically is, you know what I will do if I want to find a mean temperature of Bombay, I will put temperature measurement at some 100 locations in Bombay and at 12 o'clock I will find an average, okay, so now I am talking of mean of mean that means I have every day, you know I am doing this business, okay, finding a mean temperature in this area, okay, because what is mean temperature in Bombay is also can be questioned further, is it in IIT or outside IIT, IIT temperatures are lower, right, so that is what I mean here by and then the next thing is of course to look at distribution of this mean temperature in time, okay, what are all possible outcomes, all possible outcomes are all possible values this mean can take, okay, I can collect a sample of mean temperature over last 100 years, okay and then yeah, so mean here is first mean which I meant was spatial mean, the second mean which I am talking about is that temporal mean, okay, now temporal mean will have a distribution, right, temporal mean will have a distribution, oh yeah, yeah let us not get into that right, let us not get into that, let us look at now a simple, well the example I get turned out to be more complex than what I thought, heartbeat example is better, you know, heartbeat example is not distributed, it is one person one heartbeat, right, yeah, yeah, yeah, at a particular time instant I am integrating over omega sample space, all possible values heartbeat can take, it cannot take negative values of course, that is why see typically when you go to got, you will put minus infinity to plus infinity, there is no minus infinity heartbeat, right, so it will be whatever 200, so the real sample space will be somewhere between, it is a discrete space of course in this case heartbeat, yeah, values which random variable can take, now comes the tricky thing, okay, I have a random variable, now I have a stochastic process not a random variable, I have collection of random variables, I want to find out where randomness today and randomness tomorrow are they related, okay, or randomness today and randomness yesterday if I am looking at this average temperature business or if I am looking at heartbeat, is the heartbeat now and heartbeat five samples before, is it related, valid question, if it is related how do I uncover this, how do I understand, you know, whether temperature today and temperature five days back, well you know from intuition that yeah, temperature today is related to what happened yesterday and what happened day for yesterday, you also know that well it is not quite related to what happened first of January, it is too much in past, but last three, four days there should be some connection, you do not know what it is, okay, but you want to find out, so the next question that I want to answer is, within a random process are there connections, are there correlations between the randomness at instant K and instant T, where K and T I am just abstracting what I said, okay, I said that temperature today and temperature five days back or is it related, abstract mathematical terms is VK and VT, two different time instance K and T, are they related, okay, that is what I want to do, well what we do here jumping again is that we use second moments to characterize or to answer this question, I am going to define something called as autocorrelation of a function, autocorrelation of a function, what is correlation, just leave aside, what is correlation between two, when you talk about correlation, there are two random variables, right and you can talk of expectation of jointly both of them occurring together, when there are two physical, like you talk of one coin and two coins, two coins being tossed simultaneously, okay and then you can talk about joint distribution and so on, now here I am trying to look at, you know correlation between instant K and instant T, okay, what I am going to do is I am going to define something called as autocorrelation, autocorrelation is expected value of VK VT, right now, so expected value means it would need joint distribution function of K and T to be integrated with, I am not writing all those definitions, okay because I am going to come with a simplified form little later, this is the most generic definition which I gave you just now, is expected value of VK VT, so I need joint distribution of VK VT, okay, integrated over omega 1, omega 2 or omega K, omega T, okay, then you will get this autocorrelation which relates between K and T, okay, so autocorrelation function actually quantifies dependence of dependence of random variables within the stochastic process, see I can ask this question, I have measurement noise, I just showed you data, temperature data in a sum, okay, I can ask this question whether the temperature measurement at 200 instant, does it have any relation with what happened three instances before, valid question, okay, so within a stochastic process, are there relationships that is answered through this autocorrelation function and then you can also ask the question whether two random processes are they correlated, okay, let me take humidity, average humidity, average spatial average humidity, let us say in IIT Bombay and average temperature in IIT Bombay, okay, if you start looking at this data, okay, as a sequence of random variables collected for 365 days, valid question to ask is the temperature stochastic process relate to the humidity stochastic process, there we know that from physics I know that humidity and temperature are correlated, right, so if that works in physics it should work in maths, the stochastic process which is coming out or defined for temperature should have some relationship with the humidity data, right, the two different stochastic processes are they correlated, I can ask this question, okay, yeah, no, no, actually I am going to further simplify, ideally yeah these are integrals, I am avoiding those definitions right now, okay, I am not going to evaluate integrals, I am going to come up with simplifications which only require some summations, okay, I am going to define an idealized stochastic process soon, okay, just wait for a while, okay, you can ask this question again, what do you say here is difficult, it is knowing this distribution function for each time instant and then it is not impossible but it is difficult, okay, in modeling for control we use some simplified versions, see for example I will give analogy, in real systems you have you know signals which are you know periodic in nature, right, oscillatory periodic signals but when you study control systems you study sign and cost, why you study sign and cost, those are some kind of idealized signals which help you to do understand mathematics or do your mathematics, you know in a very simplified way and then you develop you know understanding of the system, so likewise, okay, you know instead of modeling a signal which has multiple frequencies, this is like right now I am talking about a signal which is completely general, okay, each random variable has a different distribution, after some time I am going to come up with a simplification called as a stationary stochastic process, I am going to say a subclass, a simplified class of this is a stochastic process where random variables are equally distributed, all of them have same distribution, okay, once I have that then you know I will have some ways of handling data which is easier than this but if you write more general definitions yes what you say is true, you have to write double integral, okay, you have to write double integral and then find out, so cross correlation is relationship between random variables to different stochastic processes, an example I gave is humidity and temperature, okay and then I could talk about humidity 3 days back and temperature today, are they related, okay, so that is what I mean here by Wk, Vk and Wt, two different time instances, okay, is there a correlation, I can ask this question, if I suppose by some means I have modeled for these two stochastic processes I know density functions, I have some way calculated, okay, then actually I could compute these integrals and find out the value of cross correlation, okay, I could find out the value of cross correlation between humidity and temperature which are separated in time, right, I could do that, now it comes to the simplified version of this stochastic process which we use for controller in control systems, particularly linear control systems, okay, so analogy is same, instead of working with periodic signals with multi frequencies, you know we like to work with sign at one frequency, right, sinusoidal signal at one frequency or you know a real system probably there is never something like a step change, there might be you know sudden rise in some input but you never have an ideal step change, you cannot implement an ideal step change but we idealize signal, we define an idealized signal called step input, right, we define an idealized signal called step input, why we do that, maths becomes easy, we can understand the system behavior and then extend that idea as to you know understand the complex processes, so here too with the same analogy I am going to define an idealized stochastic process, okay, I am going to define an idealized stochastic process, well the first thing is that the mean of this idealized stochastic process is constant, mean does not change with time, all of them have same mean, okay, all of them have same mean value, can you say this about heartbeat of a patient, normal person, you take normal person, okay and I am monitoring his heartbeat after every 1 minute sampling, okay, for a normal person what do you expect, will the mean change, mean change and all will occur for a you know patient or somebody who has undergone a surgery and post surgery you might find some fluctuations and all that, that is different, take a normal person, okay, you have you are monitoring his heartbeat, mean now, mean after 1 minute, mean after 2 minutes, mean after 3 minutes, mean before 5 minutes from now is same, mean is not going to change, right, so a good model, a practical model for this scenario would be constant mean model, all of them collection of random variables, all of them have same mean, absolutely fine, very nice signal, okay, all of them have same mean, no, no, do not confuse with the reality and the model, this is the proposed model, no, no, no, 1 minute, you are confusing between the estimate of the mean and the mean, true mean, I am talking about a true mean of the distribution, estimate of the mean might change, if I take data for 1 hour, I am talking of a patient who is not doing too much of activities which are normal activities he is doing, not patient of a human being who is not jogging for an hour and after that sitting and writing, not like that, he has some normal activity and then you are monitoring his heartbeat, okay, so you are probably confusing between if I take data for 1 hour and if I take data for 5 hours the mean will be different, the mean is not different, estimate of mean might be different, I will come to the difference between the estimate and the mean, I am talking about a true mean, okay, why do you find in the medical book or in your class 10 book, heartbeat of human beings is 84, what does it mean, 84 per minute, is it like, 84 per minute or 72 per beats per minute, why do you get this number, you take any sample, will it ever be 72, no, you take all possible measurements at all possible times for human beings, 1 or 2 might come out to be exactly 72 but it will not be 72, right, okay, so this limiting value, this is the distribution mean which is 72, yeah, if you take all possible, not sufficiently high, if you take all possible values that can occur then you will get 72, that is what it means, okay, so here I am saying a stochastic process whose ensemble mean for all possible values is constant but constant means at every, take any time instant you have a stochastic process which has constant mean and which is same, which is not changing with time, okay, this is an idealized signal stochastic process, okay, in the same sense we use idealized deterministic signal like sin omega t cos omega t, okay, no, no, no, it is mean of that random variable, not over time because see what we are going to do, now it is a very good question, okay, because you know it is constant for all the variables, we are going to mix the two, we will take mean over time and say this is same as mean in the space, yeah, yeah, so because I make this assumption I will be able to use data in time to estimate mean in the space actually, I am saying that I am proposing a model that is a constant mean, okay, so if I take a sample, now if I want to estimate mean from the samples, if all of them have same distribution, if I take sample now or after next instant does not matter, I collect all of them and take a mean, okay, because all of them are coming from the same distribution, same mean, okay, so that actually helps me to simplify computations, so this is an idealized model, okay, there is one more thing required for this idealized process, okay, well you have a good question, so we have two distinctions here, okay, a strongly stochastic process and a weakly stochastic process, so a weakly stochastic process only first two movements are constant, okay, for a strongly stochastic process all movements are constant which means same distributions, so for practical purposes work with weakly stochastic processes, I am not getting into all these niceties in the first lecture, okay, so you have to jump and ask you know grade 10 questions when I am right now in you know grade 1, okay, second important thing for this weakly stochastic process, idealized process is that if I take autocorrelation between you know what happened now and what happened say 10 minutes before, okay, that should remain invariant, if I do it now in the afternoon, in the morning, in the night, if the patients you know if the model for the patients heartbeat is a stationary stochastic process model, okay, then what should matter is time gap between the two samples, okay, if I decide to say you know what is the relationship between the randomness at 12 o'clock and 11.50, okay, that should remain same as relationship between the randomness at 4 o'clock and 3.50, okay or 4.18 and 4.28 only what matters is that delta T, K minus T matters, okay, it should not matter where I am in time, okay, such a idealized process, such a idealized process, okay, is called as weakly stationary process, well I have already thanks to you I have already qualified you know strongly stochastic and strongly stationary and weakly stationary, weakly stationary is only two movements are satisfying these conditions, we cannot say anything about the higher movements, if first two movements satisfy this condition then it is called as a weakly stationary process, if you take all of them have Gaussian distribution with identical mean and variance and you know autocorrelation then it follows that you know identical distributions, what we work in practice you know I am starting with most general thing, what we work in practice is a Gaussian random variables which are identically distributed, okay, that stochastic process is very simple to handle and most useful in modeling, so what we use in practice is sequence of random variables which are identically distributed all of them have same mean, all of them have same distribution, whatever we many times assume that all of them are Gaussian, life is simple after making this, okay, probably when Kolmogorov or people who started looking at this, they started looking at these simple processes by the time it came to Kolmogorov he came up with a very general theory you know which is stochastic processes where randomness can change from, so if you start a book reading a book they will start from the most general thing and then it becomes formidable and you do not understand why you need such a you know complex maths, but anyway we will be using stationary processes mostly, okay, so that what you asked Smita asked is very good question because of this particular thing because all of them have same distribution, okay or all of them have same movement first two moments I can just pick samples in time and then estimate mean, estimate is that okay, everyone with me on this you have to stop me stop anywhere here, okay, now so what is the advantage I can estimate the statistical properties using samples collected in time, see what will happen can I find out average temperature of today by taking samples in time yesterday's temperature tomorrow's temperature I cannot, right, a stochastic process which is you know the first one which we are talking about you will have to collect average temperature on 25th of September 25th of January for last 100, 200, 300, 400 whatever those many years you cannot take time samples because this is not a stationary stochastic process, okay, but heartbeat model is a stationary stochastic process I can take heartbeat 10 minutes back now after 10 minutes after 15 minutes I can collect a sample of heart beats take a mean because mean is constant not changing with time, okay, so for that model I can use time data to here I have to do time data means I have to collect data you know in a different way, right you understand the difference, okay, well right now we have not talking about input output modeling we are just talking about random variable or a sequence of random variables all of which have same mean we have no clue there is no cause and effect modeling right now, okay, there is no attempt to assign any cause it is just see when I am saying if I develop a model for that temperature you know mean temperature as a function of time in a day using some data over 100, 200 years it is not what I am saying is it is a first of all it is a model that you have to understand it is a proposed model, okay and it is not you can you repeat your question I mean this constant means what it only says that expected value what is the best value that you can expect next time is constant, okay every time if you ask me to say what is the best guess you can give you will say 72 because that mean is not changing with time, okay that mean is not changing with time, so no but you have to understand that this is a proposed model, okay this is the model which fits see in chemical engineering we study CSTR right Sturt Tang reactor and PFR if you go to a real plant you will never find a CSTR I mean you will find something which is reality is between the CSTR and PFR there is something it might be close to this idealized thing called CSTR it might be close to idealized thing like PFR but they may not be you know a perfect okay, so perfect mixing based reactor is an idealization, okay so constant mean over time, okay set of random variables whose all of which have same mean is a idealization, okay this idealization does it answer your question we can see as we go further you know you can probably ask me again this question yeah, yeah yeah all of them have same mean, any every minute see I have one random variable at each minute I am proposing a model I am proposing a model understand this that all of them have same mean okay it does not change with time it does not change with time, so because of that you know what happens next instant is drawn from a realization of a random variable which has same mean okay let us assume that it has same distribution if it has same distribution then I can use a sample at this time sample at next time and then okay this is not possible for the stochastic process where temperature is taken over a year I cannot use data of yesterday because yesterday's mean is not constant is not same as today's mean is not same as tomorrow's mean okay it is different okay, so one can of course propose a constant mean model for the temperature variation but it will be a bad model it will not fit the reality okay, now some interpretation of this will start using this on the real data when you will understand this now I have a problem of chicken and egg I have to first define the language and then start using it okay, so I am defining all these terms which I am going to use subsequently autocorrelation correlation function and you know so what is expected value at when k and t both are equal to 0 what will you get see what is expected value of we have this random variable vk okay for the time being okay let us consider expected value of vk will be mean right okay what is expected value of vk square is variance right is variance this is nothing but expected value of vk, vk is or vk into vk sorry not vk, vk into vk, so which is same as see how did we define correlation function okay we defined the correlation function here which is difference between k minus t okay just a minute let us look at here yeah, so see what is what is expected value of vk with vk I can ask this see we are talking about vk with vk minus 1 okay vk with vk minus 2 what about autocorrelation of vk with vk will give you what will it give you variance is single variable it will give you variance okay we are talking about one we are not talking about vector of random variables we are talking about one random variable yeah, so it will be v square you know just go back here no correlation variance will be v minus mu yeah v minus mu correlation yeah correlation yeah it will be well I should not say covariance correlation function this is here well I do not want to modify here this machine is not so correlation function at tau equal to 0 will be same as expected value of vk square and then it tells you how about the variability if the variable is 0 mean it will be variance right the variable is 0 it will be variance it will be same as variance and the standard deviation will be square root of that will give you standard deviation what we normally do is that we to study you know correlation in time we normalize this we normalize rv tau by r0 okay and then plot okay I will show you this plots and normally what we know is that r0 is the maximum value there can be no value greater than r0 okay so if you within a process stochastic process you can show this I am not going into the derivation you can refer to books on so I will just show you this is okay let us go back and come here okay this is the signal you remember this signal that we had okay I am now going to apply these ideas to this signal I have this signal here okay I am looking at this as a stochastic process okay there are 250 samples I think there are 250 samples yeah okay so this is a stochastic process I want to ask question whether vk and vk-5 vk and vk-6 vk and vk-1 are they correlated okay I am going to estimate this using this sample data okay I have not yet come to the point how to estimate these numbers from data samples but let us say there is a I am going to talk about it soon but before that I want to show you how this correlations look like okay so a model see I am trying to construct a model now by saying is there relationship between randomness at time k and k-1 k and k-2 k and k-3 does not matter if I go at 150 you know 150 and 150 at 145 okay or you know 100 and 100 and 100 and 100 and 95 or 95 and 90 are they related okay I can estimate this using this autocorrelation function okay autocorrelation function can be estimated directly from this data okay now I am going to go you know funny way I am going to show you the plot of autocorrelation first and then I will come to how to compute it okay so let us get first physical insight into what is autocorrelation so what I want to uncover here is that is there relationship between you know you take any point here let us say this point and 5 samples before or 10 samples before or 20 samples before are they related okay what I am going to do is to look at autocorrelation within this stochastic process okay that will tell me how does the autocorrelation plot look like you see this graph here I have plotted 2 things here one is autocorrelation that is vk vk-1 so this is what is this first point vk vk lag 0 okay what is the second point this is r0 the first point here the first point here is r0 what is this r1 what is r1 correlation between vk-vk-1 this has been normalized using the variance at 0 okay so that is why maximum value it takes is at instant 1 instant 0 lag 0 is at lag 0 and it is 1 okay but you can see that there is a relationship between randomness at k and k-1 k and k-2 k and k-3 okay so this is a so this simple calculation of autocorrelation tells me well I am assuming that this is a stationary process okay this is a stationary process I am modulated as a stationary process and then I am trying to find out correlation within this randomness okay so that gives me this particular plot so it tells me that this is not completely random what is happening at k has relation to what is happening at k-1 lag 2 k-2 k-3 in fact you can see even if you go 25 instance in the past there is still relationship and very strong relationship okay what about relationship between you and what are the two stochastic processes I am looking at input which is given to the excitation as a signal and trying to find correlation between the vk and uk that is the second plot we will not interpret it right now just look at this if this correlation coefficient okay if it is close to 1 that means there is a strong correlation okay if this correlation coefficient is close to 0 which means there is no correlation okay if this correlation coefficient is positive then there is a positive correlation the correlation coefficient is negative is a negative correlation okay and autocorrelation can be of different shapes I will show you another okay this is example I am going to talk about this white noise after some time so this white noise is something in which there is no time correlation what happens now has no relation to what happens after some time this is data collected from a real temperature bath constant temperature bath you can see samples are all over right they are not constant and if I find autocorrelation function it is very close to 0 and it is for any other lag it is close to 1 for lag 0 yeah for lag 0 it is 1 and for all other lags it is close to 0 okay so this simple calculation of autocorrelation tells me whether randomness now and randomness sometime in the past are they related this is the model that we are proposing okay.