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Fax varied conditional

this session analogy statistics 240 looking at stochastic simulation in particularly focuses on young Gaussian type of simulation models so I refer here to the NPS book there's a good summary in Chapter there on a garcia simulation or assimilation using random functions and then of course also the covariance book if you'd like to read some more detail about the theory okay so why do we need simulation so no previous few sessions we looked at Korean as a method for spatial interpolation so here such an example we have some data typically we would calculate their model their ground and they would use that Vera Graham in greeting to create a map of interpolated values and in addition we can get the creating variance so one thing we notice immediately is that this map is rather smooth certainly compared to say for example if I would reveal to you the actual reference data from which this is 29 are extracted you notice this very smooth map so why is that while creaking is an regression model so in regression models we tried to minimize these square variance and the result of that like any regression model is that it is a smooth model that also means that there are certain statistics and this moves model that are not reproduced and that the data is much more varying than the smooth model basically because it can be considered in many cases as a conditional expectation so if we would calculate the variance of the smooth map and the variable of the data or verum the reference we notice you will notice two differences the first is that the behavior at the origin will be very different because we have a smooth map so we get this Gaussian type behavior and secondly is that we get less various and indeed it would calculate the variance of this map and compare with the variance of the data you'll see that that's quite different so we say why is that now relevant well it's relevant in many applications certainly nap locations for example where are we going to use our map for certain purpose for example we'd like to use our map for building run water models for flow simulation and so what matters in flow simulation is the spatial nature and heterogeneity of the hydraulic connectivity and so if we use a smooth map we get completely wrong predictions for example of the drawdown if I would be pumping in the world etc all over I would be do tracer injection the tracer survivor times would be wrong and I only get one answer and so it would not be able to quantify uncertainty so that is the limitation of spatial regression here's just another example of the same problem so we have a very complex phenomena we extract limited amount of data and we do regression and we get this map which has looks like there's almost very little resemblance to reality so what is it that we'd like to achieve so what do we like to achieve is now it's not so much at least square type optimal map means that the deviation between the map and the true map is as small as possible but to have maps or a map that reflect certain statistics of reality and the minimum amount of statistics that we can extract or should extract from some data is first of all it's the univariate distribution and secondly it will be the covariance function or Vera Graham function since the covariance function of Aragon function are already summary statistics of some by variant distribution so indeed if I have this data and calculate the histogram and I model the Vera Graham that would I'd like to have our maps that first of all interpolate still the data so we'd like this idea of greeting to be maintained but secondly that these maps now also reproduce certain statistics in particularly calculate the histogram of this map it should be similar to this histogram calculate the variance of this map should be very similar to this variable it's not going to be exactly the same but it should be statistically similar and so then we put the goal of stochastic simulation the one thing you have to realize is that such map is no unique I can create many maps that essentially have the same histogram and say very Ram statistics and so I'll no longer try to find them how that is as close as possible to do it choose but that reflects certain statistics of the truth so here's the idea we have some unknown reference truth and we from that we have limited sample data so from that we can calculate variance let's just focus here on that red and blue Vera Graham so so imagine you have that red and blue Vera Graham could be very good model could be an experimental Vera Graham what we'd like to do is create maps such that if I calculate the very room on that map you'll get a similar Vera Graham as this paradigm here in addition you also reproduce the histogram so let's discuss a little bit some equivalences and differences with what you already know which is univariate statistics so in the univariate statistics we we gather samples for example I could look at the weight of each person in the classroom and what I do I now ask is if I get a new person what is the your best guess for the weight of that new person well if you're thinking about these squares then the best guess is the mean would be the best custom the least square sense so the best case of a new sample would simply be this so if you say well I'd like to know an uncertainty distribution on the next person then seemingly what you do is you can fit this model here which just be fitting a distribution model with some parameters so you can fit these parameters and then in order to we can then calculate directly probabilities or quantiles from this fitted distribution but we can also say well we can also do multi Carlos so that means that we can represent the uncertainty on the weight of that next person simply by drawing many many numbers and say well it could be any of these weights and that would be done your answer in terms of an empirical distribution the special case is somewhat similar but are also differences first of all we get samples at spatial locations so x1 to xn our spatial locations anyway I have now a new unobserved location where I have no observation and I'd like to make a best prediction then I can use kriging and so greeting is again one answer and gives you these weights in this and this particular linear model surrogates weights to each of the points of the data so now your question could be well what is the uncertainty on in that particular location and so then we would have to do the same as we do in a univariate case which is to fit some specific form of this region and once we have that distribution we can create many samples now because they're in a special case we have essentially many unknown locations and so we'll have to simulate all these locations together which means creating samples of at many locations and that's what we showed in that in the case with Walker Lake is that we can generate many images that are constrained to this data and reflect on certainty so the question you probably have now is what are these distributions and so the problem which is in reality is that there very few analytical forms available as compared to say the univariate case we have many forms available local normal distribution exponential distribution etc so now there are very few forms available in particular will mostly be focusing on the multivariate normal distribution so more specifically in spatial context or the temporal context will be and now defining what's called a Gaussian process model so early in the course we already talked about long process model which is the Poisson process model for some point process model so now a Gaussian process model we be looking at a spatial process where we extend the notion of random variable which say the random variable Z which was univariate case 2 now a variable Z that's function of X so one way of representing that is with a district red to say that we have a grid on which we have all the unknown basically the unobserved locations and so in that case this process reduces to simply a multivariate problem we need all these variables at those locations where we don't have servation is become random variables but that doesn't have to be the case throughout the process suggest that this could be just a continuous function of what the discrete grid is this is a very special case so in do statistics we often call this a special process also a random function and so that has to do simply with the function is now a random variable that's not a function of space or space-time but a mental to the Gaussian process is the multivariate normal distribution and we discussed that earlier in the course where the multivariate normal distribution is uniquely defined if I know the mean and I know the covariance between my various so this is then used is reflected in the covariance matrix which is given over here so this is just for the multivariate case so what we'll do now for the special case is replace these X 1 to X n right now 0 X 1 to X n so we'll put in the unobserved locations as random variables in this and so but do mind that this could be an infinite set but of course in reality when we execute it as in computer programs this will be a finite set so once we specify to multivariate normal distributions you can easily draw values from from that multivariate distribution so here's this simply the case where the mean is 0 and so what we do to drop on the multivariate normal distribution is simply take the covariance matrix do an Lu decomposition so you decompose it into a lower triangular and upper triangular matrix and then you generate a set of Gaussian independent Gaussian random deviates so these have a standard normal distribution you multiply that with this matrix so you make many combinations of those in the middle variance which you gain then gives you dependent variables and a simple proof will show you that these X have the resulting excess covariance matrix C so this is just throwing for the multivariate normal distribution so imagine now that we are working on a grid and red cells then this covariance matrix will be n by n so let's do an example of that so here's the simple simple examples where our covariance matrix is the identity matrix so that means that there is no correlation and we will be sampling on a 25 by 25 grid so we have essentially 625 unknowns means unknown is a value at each individual grid cell and so my covariance matrix will be 625 by 625 so obviously a divergence matrix it's simply the same as drawing independent values from the standard normal distribution at each of these locations and then just gives you prove noise so now we imagine that I have a special covariance function specified so for example I've done some of the Aragon modeling covariance modelling and I find that this is the model that I have and it's isotropic and that's the case then we notice now that we're getting off diagonal elements and that makes sense right because now I look at any two locations so for example the first two and here they have a distance of 1 and so forth a distance of 1 I have a certain correlation which is about 0.8 so that means and this off diagonal here that all distances 1 apart and we'll get is 0.8 and so on until I reach 3 then I get again 0 so if I invert this matrix then doing that gives me go to Lu decomposition on that matrix gives me these samples and so now I see that correlations are starting to happen so now let's look at this right one if I increase the range is you get more off diagonal elements and so your random view realizations of the multivariate normal distribution start now obviously show considerable correlation which is due to the fact that I have enforce this correlation in my models my samples so this is a as you know this is already a very simple way of generating models that have a given covariance or variable function now if you have an isotropy or you have nesting or zonal anisotropy it's exactly the same thing you would just have to input into the covariance matrix just different values of the correlation vary in various direction and so we have to just keep track of where you put those values and no longer it will no longer look like this will look somewhat different of course so that takes care of create examples in a certain variant but what about the histogram so the problem here is that the Gaussian distribution the multivariate normal distribution has univariate distribution is also Gaussian and so the problem is that the previous samples that we generated all have standard Gaussian distributions of course in reality we never have Gaussian the serrations is what the question is evaluate my distribution looks something like that skew how do i generate samples that have that distribution so to do that we simply rely on a simple method called the rank transformation it's a transformation which is rank preserving and that transform any instagramming end to end in the auto histogram so here is a very simple example of how you transform data into uniform histogram or the uniform score transformation so here we have original data and the first thing we do always in a right transformation is writing this data and then we looked at standardizing the ranks we get this here and so as a result these V values now have a uniform distribution so what that basically is is and that is the equivalent is if you have any kind of distribution whether empirical or theoretical then you can simply transform the values of that distribution into uniform values by taking the CDF of that value so that in other words the CDF of Z is a probability and therefore is uniformly distributed on the interval zero-one so that's kind of the inverse of Montecarlo right the Montecarlo we give ourselves a V and we find using the Colton function what the value of C is so that's already nice that also means that you can start understand that I can now transform any distribution into other just by going by out a uniform so that this is shown here here's a theoretical explanation but it's actually a very simple idea I have for example Instagram of values imagine this is the empirical distribution and I'd like to transform all these values into a new histogram which is this histogram so this is useful for example if my original values are normal score and what I want my new values to have a distribution like this so the way it's done is very simple you take your value z you map it on to the uniform distribution then you take the uniform distribution and you map it back and you get a value of y and so you can do that for all Z and you get a little Y so this is called the rank preserving because if I have a value of Z here then logically because this type of distribution because the the CDF distribution is non decreasing I will find this Y here which is on the left side if Darcy is on the left side of that Z so that makes sense so let's do that for this let's say this image is this but I don't like the sandwich I'd like it to have a uniform distribution so then I can just use the previous method and create a uniform score transformation so here we see that the result so this is the histogram of this new transform data and so what I have is a uniform distribution but you also notice here that I have the spike at origin so why is that well our rank transformation does require at least in in terms of creating a new histogram that we don't have ties so for example in this in this particular case I have a whole valley here where the elevation is exactly the same so that means that all these values were transformed the same value which in this case is zero so these ties can be broken and so they they will remain as files to basically have a mixture distribution how which is the distribution with a bunch of zeros and then all the rest which is then out of this uniform distribution so this could be useful for many applications for example for any kind of any square application particularly for example calculating variables which are highly affected by high values so very dense over worked necessarily very well with very skewed distributions and therefore taking uniform score transformation makes it a whole lot more easier to do because it just removes all the extreme values of course then you have to be mindful interpreting the variable because that would be didn't Vera Graham of the uniform score transform well in what we typically see is that ranges and types and so on they're not that much affected by these transformations so how is this practically done in its Jennison to that in the YouTube video on conditional simulation so in essence for example we specify our date of which for example in the case of copper case we had some great data and so for example if I generate now a model of the copper deposit I'd liked it to have some similar history I must histogram of my data so imagine here I have some copper percentages and so what I like to have his histograms that looked like this is Sir for doing that I have to first model in in order for me to make transformation for example transfer moment the Gaussian deviates into these values I'll have to model the history first and so we want to avoid any kind of parametric modeling because that remains so in essence we do a very simple nonparametric fitting here where we take the six values and we make an empirical distribution by taking steps of 1 over 7 what that does is that it leaves me a little bit of space here because of course the value twenty point six is not the highest value that I can ever reach in this deposit this is the highest value my data would like to extrapolate here and so the way essence does that is essentially it does this piecewise linear interpolations and then you also have to specify the smallest value and the largest value you can expect and so in s Jemez those those are specified over here so this is the smallest value and waters its value and of course then we can also debate a little bit about how this extrapolation works is it linear is it more hyperbolic parabolic and hence Jemez has some really good suggestions there and I suggest not to do alt Radice for most of our distributions we have skewed distributions to the right and then these settings are quite adequate ok so now I've shown essentially that I have can create models that have very many sort of histogram but of course I also like to produce the information that's there that was really nice about kriging is that at those location it will interpolate that data so I'd like to do the same i'd like to create maps that reflect this information for example here I see that I have this red so I should have more red in this corner here I've got all the blue so that means I have more blue in this area so I need to be able to do that to account for those information okay so how does that work well we already have an idea of how to intercalate that was our greeting and we have now an idea of how to create models that have certain verbum and history so that also means that if I take a first idea namely do what's called unconditional simulation so that will exalt rate in terms of fare of M and histogram but at a location of the data you make an error so the idea in part two is that including we don't make an error at the location at those locations better we have a problem with the reproduction of the statistics so what we would like to do is is use creaking to correct essentially the error you make on the conditional simulation in terms of reproducing the data and so the assumption here is that that error can be interpolated with greeting again and so the theory shows that indeed if you do that then you'll find conditional realizations that have the same histogram in Feridun will not be talking about the theory essentially if how why that is it's it's actually not that difficult of a development but just tell you a little bit about how that works so the way that works is we say I have a conditional simulation so I write it as screaming but greeting has not enough variance we saw that when we calculated the averag I'm on the greeting map it has another my variance so we'd like to add some variance to that so that variance is over here so we're going to essentially estimate or simulate that error and add them to the cribbing map such that we have essentially no a conditional simulation so here is essentially how that is simply done it's just a different notation here but bear with me here is again that's the Z of X now it's called X of s so basically what we have is we have some data which is these stars here we have this data which are these stars here so if I do creaking I calculate the variable Luke regain I've got a nice interpolation that then I do this but you do it the difference interpolate this error okay this and so you find many names for this which is point eight so remember that the random function is not defined on a grid but of course in reality we'll have to go and simulate every turn on our grid and some there there are certain complications that happen in particularly we have to be mindful about what it is that we're simulating namely the volume which is also called support and geostatistics and the definition of the grid so the reason for that is is a little bit it's very confusing in the sense that your sample could have a very very small volume for example you could go in the field and take a bucket of soil and there lies that and a bucket of soil is mapped in an area say of 10 kilometers by 10 kilometers so then ready to generate models you're not going to create models with billions of cells at the side the bucket of soil you will just do that at a much sparser grit but when we make essentially the map at the end whether it's a creaking map or a simulation map the the pixel plot essentially upscales these individual simulations to a larger grid cell which could be easily say 100 meters by 100 meters and so that's not really what's happening into you statistics and so I'd like to spend a little bit more time in explaining that because it's a sustainer confusion so what I was what I'm getting at here is that if you want to simulate or estimate you have to define a grid and a grid basically means that you have to find a mesh which says that my sample my that the values I'd like to simulate or estimate or on a certain distance from each other so it should be dy here but also that they have a certain volume for example if I use this square here calculate the volume of that square or the area of that square but I can also take a different volume over here I can take a circle and I won't do an ellipse and one of the requirements is that all these volumes and these shapes are exactly the same means they are all centered around this location so I don't have a bucket you can imagine that this would be a circle instead of a square and so basically I have buckets at each individual rid point that maybe these buckets may be very small but the distance between the buckets may be may be very large so then I have then I have obtained data so I have buckets that I sampled over here over there over there and so these are called hard data why because they are the same size of this essentially this information you also notice that this data does not have to lie on the grid and so for example here's the data belies excitement the location of course I will never happen in reality or rarely happens unless you use a very dense grid so most of your data lies off the grid and so in that case it's not a problem because in creating the only thing we need to know is the distance between the unknown and the data location and so that's all obtained from the very room which as you know is a smoothly interpolated so the takeaway here is that random functions are just functions and continuous functions of X but and the computer implementations have discrete implementations and so we have to be mindful not just just if this value becomes say 5 what we do is so America picks up like this we take the entire zone here we call it 5 so it's kind of like we do it and implicit upscaling and that's not really there so to be mindful when you do creation when you create simulation agreeing that you are mindful of the fact that you are simulating or creating very small volumes and that's why we call this sometimes point data or hard data in the sense that it doesn't have an error but then again this idea of hard data it's a bit of a misnomer because it doesn't really exist in reality so the interesting idea is actually at this conditioning with kriging it's a it's a nice way to start thinking about stochastic simulation I like it like that because it shows you the nature of stochastic simulation that we'd like to import certain properties on on these stochastic models namely that they reproduce certain statistics but also that they reproduce some deal in reality however we'll this is kind of conditioning with creaking is rarely used and one of the reasons is that theoretically you will have to use global simple kriging the second one of course is that you have to do this Lu decomposition which has some additional issues when you go to march or it's a billion so I'll grade you you can imagine doing that and also we'd like to work on more local problems and so because global simple kriging is not going to cut it first that secondly is that this Lu decomposition will only simulate cultural processes and so later in the course you'll see that there are some significant shortcomings of these processes that they can't model more say organized materials particular geology that is something we notice is that the world is simply not Gaussian so we'd like to extend on that I will be doing that now what I'll be doing now is looking at some alternative method for simulating the Gaussian process and that method can then be extended to two other non Gaussian processes so the method is called sequential simulation and simply because what we rely on it's will rely on thicker decomposition of the multivariate distribution because after all our it is a set of unobserved locations that are unknown so it's a multi-tier distribution just as before but we can decompose this multivariate distribution into a series of individual distributions and that starts shown here namely the multivariate distribution can be written as for example the pointer the multivariate distribution can be written as the probability of the first one and then the prevail the first one given the second one given the first one and so on down the down the chain and the end one is the probability of the last one given all the previous one so this is general so this could be just random variables could be a discrete random variables or could be again this is before it could be unobserved locations and variables and unobserved locations and so then we have essentially a special problem so this decomposition is completely general in the order of average to do this composition does not matter in other words I can write that for an odd order and it be exactly the same result so then the jodan simulate all these events it becomes now a matter of simulating from univariate distributions and that's something we know very well so you simulate a from the key of a a one from P of a then you once you have that value so now we have a simulated value you calculate which be the value of a 1 here you calculate the condition or estimate the condition distribution of aid to given us this and you simulate from that now you have two values a 1 and a 2 you calculate the conditional distribution of a 3 given these two and so on and you get at the end you simulate from this big one and so we end up essentially have a simulation from that multivariate distribution okay so now it'll get probably that we want to apply this to the multivariate Gaussian distribution that's correct so here we have again the multivariate Gaussian distribution and so some important properties that we'll be using is that any subset you've got to get any subset of these random variables is still a multivariate Gaussian distribution it also means that if I want to condition any set on any other set or any subset on an intercept set for example I want to know x1 given all the rests that is again a multivariate Gaussian distribution of course the mean and the covariance will change but the form of the distribution does not change also very important is indeed if I look at a univariate condition and univariate distribution conditional distribution for example f of x1 given the rest then I have a conditional mean would be an expected value of x1 given the rest or it could be for any X N and what's really nice is that that's simply a linear combination of these conditioning values so now you start to see okay I've seen that before and then need this is linear regression right so this says that my conditional expectation is a linear combination of my data values and so we'll see in the next slide that we abuse in creaming to determine these weights so okay so here we have some more properties in particularly we'd like to estimate the conditional distribution of a Gaussian process so remember that imagine that I would like to estimate something at an unobserved location and I have some observations so what essentially if I would like now to look at the conditional distribution then essentially what I'm asking is what's the conditional distribution of this value given these values so what will be assuming is that the Joint Distribution of all these is multivariate Gaussian it also means that the conditional distribution of this given thesis its unification and have a certain mean and certain variance so let's again look at the multivariate distribution of this well what it has it has a certain covariance matrix remember that the multi-bear distribution as it certain covariance matrix assume for now the mean is 0 then that then that multi-layer distribution is uniquely defined by this covariance matrix and the covariance matrix is written over here so we notice here we have a number of parts in this covariance matrix the first is the covariance between any two data locations so this is is a subside n times n that's the same as the covariance matrix that I've seen before then we have also the covariance between these values and the unknown so that's this case here and then we have the variance of your predictions of W to be directives for example could be standard Gaussian then this variance here would be simply 1 if that is the case then I can now and then shown it's an easy proof you can show that the conditional mean and the conditional variance can be written in this form so you have this matrix K etc and so what we know this would be a here not an X and so what we notice is that this is for him in this form is something that you've seen before namely that is simply der the answer of simple kriging remember you have to in vertical variance matrix and multiply that with this value of K to get the weights in the system multiply that with the outcomes namely the values of Z and so here we have the conditional variance that was simply our treating variance remember this was the variance - a part I belong to the tricking areas okay so now let's put everything together so we saw two things about the sequential simulation we saw that we can write a multivariate distribution as a product of the univariate conditional distribution and secondly when we assume that the multivariate Gaussian distribution is a Gaussian sorry if the multi-layer distribution is Gaussian or a Gaussian process then we know that every single conditional distribution is a universal distribution and we also know that the conditional mean is obtained used in creating and the conditional variance is the creating barriers so how does it done unconditional simulation work in practice and it's called sequential Gaussian simulation some people call it SGS and other people call it as G city so it s Jam it's called as Jason and here we first talked about the unconditional form so this form is exactly equivalent to this Lu decomposition so what we doing here is we we sample first from an unconditional the first value we sample could just be the first value in the grade so the bottom left corner of the critics that say and then we remember that the the order does not matter and so we sample value we get a some kind of value say minus 2 so then we would like to estimate using simple kriging we have now the second one say that the value to the left of it we estimate the conditional mean which is obtained with stringing and the conditional variance and we draw that value from a normal distribution which is now not longer standard Gaussian but it has a mean which is the simple greedy mean and variance integrating variance and so on so we go and we repeat the screen so what it does is is you have to repeat the screening every time it so this creating system becomes bigger and bigger and bigger and so so that's what we get so the order does not matter as long as we use all of the previous simulated values and you say yeah but that doesn't solve my problem because my problem before was essentially that I had a big grid and so you had a bigger covariance matrix so the covariance matrix is essentially growing with every single value that I simulated before answering that how we address that issue let's first look at at the well let's first look at what we have observations so in that case it doesn't change it's just that we the first step we don't know withdraw from the universe on but we already constrained the observations that we have so we just do the first one is just a creaming estimate we get a conditional expectation we get a conditional variance and we draw from this distribution and then we go on so as I said in reality we're not going to do it that way first of all it will not be any better than their conditioning risk reading and secondly we're going to use just as we're creating a limited search neighborhood the advantage of that is that limits search neighborhood will allow me to do other forms of craving for ordinary craving or creaking with limited search neighborhood and we saw increasing that has many advantages and secondly it will mitigate the the problem of the computational issue however if use a limited search neighborhood then you you no longer having this this conditional decomposition or this sequential decomposition is no longer fully valid it now becomes an approximation because the decomposition that I had required that I conditioned to all of the the way that is mitigated in practice is that we don't use sort of a deterministic path or a constant path but each for each simulation we choose a different path in other words we choose a different where to start what's the next one and so on so we no longer always start with them with the same one and follow it in a deterministic path and so that is then called that is an intended path that changes for each new simulation and so now we can also started replacing simple kriging by any other form of riggings which is universal creating ordinary meaning etc so here's a simple example and we'll talk more about that in the estrogen tutorial on SG sin so another in reference truth we have some data extracted from it we may have some external drift so I could do University reading so now I do sequential guard simulation it means that I visit a location I searched locally the neighborhood I do a universal kriging at that location then I have a creeping mean and a craving variance and then instead of putting estimate there now I construct a Gaussian distribution with that mean and variance and I draw from it and then I moved to an honor point until the grid is entirely full and then I have a single simulation do you get a lot of simulation you repeat the exact same way exactly you start somewhere else and then you have a different random path which gives you different simulations so that's now sequential goals of simulation and that already has a lot of practical applications simply because now as you could notice you are producing a lot of interesting things you're producing a drift or a trend you're producing conditioning to data you're producing the vrm in a histogram and so these have really nice properties that you can put in flow simulators or other kind of applications and start quantifying the impact of these applications in terms of spatial uncertainty