 Yeah, in the last class of the lecture series on pattern recognition, we have discussed about normal distributions which is a very important probability function used for probabilistic modeling in the field of statistical signal processing, pattern recognition image, video processing and many other applications. Towards the end of that lecture, we had also seen the scope of a distance based on the covariance matrix of the multivariate Gaussian distribution. Okay, we had seen that and now we will revisit that once more. Look at that expression and try to derive what we will call as decision boundaries to be used for the task of classification or pattern recognition specifically. So let us look back into the slide. So this was the multivariate Gaussian density function. This is the normalizing term and the expression here within the exponent contains the mean of the data as well as the inverse of the covariance matrix here. This dictates the scatter of the data okay in several dimensions put together. Now we had seen at the end of the last class of what are called contues of constant density due to this non-diagonal elements of the covariance matrix which are of constant Mahanabeg distance will define this very soon distance to the mean of the data and that is given by this function d of x. So this is where we almost stopped although we would not have talked explicitly about what are called hyper ellipsoids. We will do that which is dictated by the non-diagonal terms of the covariance matrix but let us know for the time being that the expression within the exponent term here is responsible for the distance okay and then we will see how this distance can be exploited for the task of pattern classification or pattern recognition okay. To wind up that slide this is how the expression of the covariance matrix would look like. Here in the covariance matrix you have the diagonal terms which are the individual variances along the particular direction and the off diagonal terms again it is a symmetric matrix they represent the corresponding covariance between two dimensions i and j. So that means sigma ij represents the covariance between dimensions i and j. So using this idea of this distance from the mean as given by the term called Mahanabeg's distance to the mirror mean of the data we will proceed and see now how this distance could be exploited to do certain task of pattern classification or pattern recognition okay. Let us look at some examples of how the diagonal covariance term in two dimension this is a slide which shows a data distribution in 2dx and y very simply and the covariance matrix has the diagonal terms which are equal that is why you have sigma x equals sigma y and the off diagonal terms are 0. So the covariance is here strictly a diagonal okay you have diagonal terms which reflect the corresponding variances and that too they are same and the off diagonal terms is 0 strictly diagonal matrix go back and that particular case this is what you will have as the data distribution or you can say if the data distribution is of nature like this then you are bound to have equivalent to equal terms along the diagonal sigma x square sigma y square and so on and the off diagonal terms are 0. This is also another example of a diagonal covariance term where the off diagonal term is 0 alright but the variance or the standard deviation is given here along one particular direction is more than the other one okay the variance along the x direction is more than that along y you can see that the scatter or spread along y is much less than the scatter along x direction which is the vertical one and that is much much more larger than the horizontal one. So these are two examples of diagonal covariance what if non-diagonal terms are present what if non-diagonal terms are present this is an example where sigma x equals sigma y that means the variances of standard deviations along x and y are same however you have a non-zero off diagonal term as given here and that dictates that this character of the data will not now have a sort of a inclination along a particular direction it will have a trend which shows that as x increases the data in y also increases. The reverse can happen this is a case when the off diagonal term is negative that means when you can say when x is increasing y is decreasing or goes in the negative direction or vice versa okay so this is an effect of the off diagonal term if the off diagonal term the rho x y as given here is 0 then of course you will have situations as given these are just examples to show and the gaussians which you will get remember what we are showing are basically isocontours isocontour lines which are at a certain distance d given by that expression which we just saw in the last class or strictly the Marlin Abbey's distance from the mean those are the isocontour lines which are shown there around the mean however the gaussian function may be symmetric may not be symmetric. So we can also get asymmetric or oriented gaussian functions and for those cases the isocontour lines are as given in the figure here the bottom of your screen the two cases of non-diagonal covariances also and the electrons the diagonal covariance here is an example of asymmetric or these two can be considered as oriented gaussians these are just examples in 2D to illustrate the effect of the covariance matrix remember in general it is a we deal with large dimension data now before we go and use this corresponding expression of the Marlin Abbey's distance and the probability of multivariate gaussian density for the task of pattern classification or pattern recognition we will coin certain terms which you are very important as shown in the slide now we are going to introduce 2 terms called the decision region and decision boundary we will actually call them in short a DR and DB very soon remember the goal of a pattern recognition algorithm is to actually find an optimal decision rule to categorize the incoming data into their respective categories this is a very generic statement I am making about pattern recognition remember one of the decision rules which you are going to exploit is going to be the base decision rule based on Bayes theorem which we have already heard from Professor Murthy so based on that Bayes decision rule when we categorize samples into different classes or categories how does this distance based on the probability density function of a gaussian or a multivariate gaussian distribution play an important role and from that point of view water decision regions and decision boundaries we will define those terms first and then we will see the mathematical expressions which will actually clarify your doubt about these 2 techniques so let us look back the decision boundary are the ones which separate points belonging to one class from points belonging to the other so it basically separates 2 sets of points let us say you are discriminating 2 classes of flowers white flower and a black flower very simple or certain fruits from vegetables and you have taken certain features to discriminate between these 2 categories or classes okay and we have seen remember these examples of what we have seen during the process of classification versus clustering that each instance or a sample is a point in a high dimensional feature space so you can think of a boundary in between which sits in between 2 regions in the feature space discriminating between 2 classes okay that region that that boundary is called a decision boundary very simply so if that is decision boundary with separate points separate points then of course decision regions are the ones which are formed by partitioning the feature space so a decision boundary partitions the feature space into decision regions so think of 2 regions you can take a simple example of 2 different neighboring countries or 2 different states from the field of geography okay you can have a partition or a line between 2 states 2 districts or 2 countries thinking that to be a decision boundary then what are the decision regions the regions are those geographical maps belonging to 2 neighboring countries states or districts very simple example the nature of the decision boundary is decided by what is called a discriminant function this is where the multivariate probabilistic function will come we have to form a discriminant function which is to be used for obtaining a decision what is the decision the decision of assigning a sample to a particular class based on certain decision rule it could be the base rule we can keep the base rule in our mind of course there could be other methods of classification statistical neural methods whatever it may be but for the time being let us say we have the simple base rule we will revisit that for a moment okay and based on that base decision criteria or decision rule will form a discriminant function what is that discriminant function it decides the nature of the decision boundary okay so the discriminant function which is used for deciding what is the category of the incoming data that basically forms the decision boundary it is a function of the feature vector which represents the sample of the instance. So we have introduced 3 terms remember decision region is dr decision boundary is dv and discriminant function we may not call it df but remember we will henceforth interchangeably interchangeably use dr with decision region db meaning decision boundary. Decision boundaries create hyper planes or hyper surfaces that means if you take an example of 2 categories this is a simple example we already take let us say the number of classes is 2 a positive value of the discriminant function decides that the sample belongs to class 1 and a negative value decides the other one in the last slide we said discriminant function is a function of the feature vector correct you think of any arbitrary function for the time being although we will formalize and take some examples of discriminant function later on. So what do you do you try to compute the value of the discriminant function based on a certain value of a feature vector which belongs to a particular test sample which you want to categorize the discriminant function will give you a value typically the decision of the discriminant function will tell you whether the sample belongs to class 1 or class 2 class a or class b very simple we are talking about binary classification as it is called or the number of classes is equal to 2. So you have just 2 classes it is like a white car versus a black car ok or apples versus oranges very simply ok that is what you want to discriminate class a class b class 1 class 2 ok. So there is something like a decision boundary in between these 2 classes on one side you have a decision region for class 1 you have another decision region for class 2 with a db or decision region in between we are going to have a diagram on this very soon do not worry ok. So what will the discriminant function do when it will evaluate samples for a particular region say class 1 it will give positive values here for the other class it will give negative values here. So just look at the sign of the discriminant function you do not have to worry even of the absolute value the sign of the discriminant function will tell you to which class it belongs. Now you should be able to extrapolate and tell me what happens if the what happens to the value of the discriminant function if this sample of the instance fall on the decision boundary if the sample of the instance falls on the decision boundary the value of the discriminant function will be 0 fine very easy good. So once we know this if the number of dimension is 3 then the decision boundary will be a plane or a 3D surface well in this case we are talking about a linear decision boundary remember in one of the earlier slides we are talking about hyper ellipsoids and things like that some non-linearity in a decision boundary typically if you take lines in a geographical map which partitions two regions of states districts and countries very rarely you will find a linear boundary very rarely in a rare case you will get that mostly it is non-linear regular I should say okay. So whether you will get a linear non-linear depends upon several factors in fact the covariance matrix will decide we will talk about that later on for the timing will assume that is a linear decision boundary and if the decision boundary if the dimensionality of the problem is 2 then we are talking of a line in 3D it is a plane and of course in higher dimension we talk of this as hyper planes and the decision regions then become what are called semi-infinite volumes this is a term which actually which is borrowed sometimes in the field of geometry or computer graphics why it is a semi-infinite volume because one side you have the decision boundary the other side is it is it stretches to infinity goes to infinity okay. So it is finite on one side infinite on the other so it is a semi-infinite volume it is just a terminology trying to give you an idea. So if the number of dimension increases to more than 3 then the decision boundary of the DB becomes a hyper plane or a hyper surface then the decision regions become semi-infinite hyper surfaces remember plane and surface plane is a special case of a surface is a planar surface we call it as a plane but a surface can be non-linear also so the linearity or non-linearity is a separate issue linearity or non-linearity is a separate issue we will talk about that later on because I already mentioned it depends on the properties of the covariance matrix it is off diagonal terms will be mainly responsible for this okay diagonal terms also have a role to play but off diagonal terms strictly will have a role here okay and also the covariance matrix between classes so all these will dictate but if the decision boundary is linear then it is a line in 2D we will have a slide coming up a plane in 3D and a hyper plane in higher dimension if it is non-linear it will be a curve a conic a quadratic curve let us say in 2D it will be an arbitrary conical surface in 3D and hyper surface is in higher dimension so just keep so these are something to do with decision boundaries so the terms which are given very closely associated with the process of pattern classification pattern recognition is the process of learning because when you are going to design an algorithm for pattern classification you have to learn something from the data samples and there are 2 methods of learning but what is this process of learning remember a classifier is to be designed is built using input samples which is a mixture of all the classes so the classifier is made to learn from different sample this is the process of learning this was talked about in the introduction part also in the lecture that means if you want to say discriminate between apples and oranges and let us say you want to use color as a feature which is possible because apple will have a set is reddish tinge of the color the orange of course will have its obvious color and if you use color as a feature in 3D dimension and you want to learn these colors from samples it is possible to do with apples and oranges and what you want to do in this process of learning we will see here that the essential process of learning involves learning or designing the classifier I repeat again designing the classifier involves learning the discriminant function or parameters of the discriminant function which in turn dictates what are going to be a decision boundaries or decision regions decision regions and decision boundaries are the dual of each other once you found a decision boundary you get the regions or one you establish the regions you get boundaries we will see that with examples once you form 2 neighboring states you get a boundary or you draw a boundary and create 2 regions and that decision boundary and or regions are dictated by the discriminant function so the process of learning the discriminant function is a process which you learn the parameters of that function from the input samples and it says that sometimes you need the samples from all the classes or categories to learn they could be given together or one after the other that is okay it does not matter much as but as long as you know that the sample belongs to a particular class you are fine so the classifier learns how to discriminate between samples of different classes if the learning is offline if the learning is offline typically it is called a supervised method well that is not the main criteria what I must say is if the classifier is first given a set of training samples and the optimal decision boundary or the DB is found and then the classification task is performed this method is called supervised learning why it is called supervised learning well there is a teacher the teacher tells you that these are the samples which belong to class 1 these are the other set of samples which belongs to class 2 please learn the set of features which discriminate one from the other find out what is the discriminant function and the decision boundary so it is like almost a teacher teaching you how to work out the process of classification samples are given to you like somebody the teacher in the kindergarten teaches you how to write the character a how to write the character b that means samples are given many many times in the book and in the board for students to write and practice that is the supervised method where there is a teacher explaining and telling you these are samples of particular class category if you take a b c d then you are talking about words characters to be very precise okay and that is the supervised method of learning so the classifier learns from samples given a priory finds out the optimal decision boundary and then performs classification task based on a certain set of test samples using a set of test samples you perform the task of classification after you have learned in a supervised manner the optimal decision boundary what is the other method of learning unsupervised there is no teacher there are no training samples given the process is generally considered online and there is no teacher there are no training samples this unsupervised method of learning uses the input samples as the test samples themselves the classifier learns and classifies at the same time so in this particular case of unsupervised method there is no teacher telling you this is how you should write character a this is how you should write character b all the characters are given to you together to learn all the 26 a to z okay nobody tells you that this is a this is b this is c and so on you have a set of characters you should have to now discriminate yourself between a versus b versus c versus d and so on there are sufficient samples available for that but you are doing it learning and classifying simultaneously you do it together at the same time and that is the process of unsupervised classification no teacher we talked about this feature space sometime back so I am not going to introduce it in a big way again but we all know that the samples of the input when represented by their features are represented as points in feature space a single feature is a one-dimensional example this is a simple example but points represent samples in feature space if I remember correctly we had used this slide earlier in a classification versus clustering example as well okay of course you could ask a question right away where is the relation boundary here where are the relation regions we will come to that in a moment but of course the number of features could increase when it is number 2 you get points in 2d space which you will see in the next slide you can have the number of features as 3 then we go to 3 dimension and of course higher dimensions if the number of features are more than 3 or more in a typical n dimensional feature space sometime called as a d dimensional feature space we can have decision boundaries decision regions and discriminant functions which are formed this is a typical example that is a small point which is actually shown by a line here this is a point here which indicates a decision boundary in a one dimensional case of a 2 class problem you have a set of samples belonging to class 1 as marked as red points class 2 as blue points anywhere in between you could have actually put the decision boundary the point in between okay it depends upon the algorithm let us look at this example which has been borrowed okay from a document the decision boundary in 2 or 3 dimension now this example shows 3 classes that means 3 classes 3 set of instances marked in 3 different colors magenta yellow and blue I hope the color is clear I repeat again magenta is one color say class 1 yellow in class 2 and cyan if not blue cyan color is the class 3 okay now these points could be scattered in 2 dimension XY or you can visualize that you are sitting in a room this is one corner of the room and these are points in 3D space that is also possible it is like a projection of a 3 dimensional world onto 2 dimensional space but the points may be actually in 3D okay so in that case when you see these 3 lines I have attempted to draw 3 decision boundaries or DBs separating one class to the other that means this decision boundary separates this pair of classes this decision boundary separates these 2 this decision boundary separates these 2 these could be lines in 2D these lines that could be lines in 2D or they could be planes in 3D either way you visualize it is okay that is why you look back on the slide I wrote that you can visualize this problem in either 2 or 3 dimension this is a hand drawn example let us see another example of sample points in 2 dimensional feature space this is very easy to visualize I have just 2 sets of sample points belonging to class 1 as given here you can see I have used another mark of the different color to indicate it is class 2 okay so 2 dimensional space instead of X and Y I have used F1, F2 at the 2 feature components of the feature which I have extracted from 2 sets of categories of sample if you look back into the slide this is an easy problem to categorize why I can actually draw a line through the middle which I am not drawing here but I am asking you to visualize that if I draw a line through the middle then that will be the decision boundary to discriminate between these 2 classes and I can actually use any sort of simple classifier to find out this particular discriminant function to discriminate between these 2 classes and obtain this DB that is an easy problem but is it always the case that I can separate 2 set of points using what I call as a linear decision boundary which is a line in 2D or a plane in 3D or a hyper plane in higher dimension not always possible in this case yes but what if you look back into the slide I am introducing certain sample points belonging to class 3 third category of a class and the samples have been marked by a different symbol with a different color as well now look if you want to discriminate class 1 versus class 3 or class 2 versus class 3 you will not be able to draw a line anywhere in this space which discriminates and create 2 DRs and creates 2 DRs or 2 decision regions discriminating one class versus the other look back remember class 1 versus class 2 is still it is still possible to draw a DB between the 2 which is linear but for class 1 versus class 3 you need to probably draw a non-linear elliptical boundary here the same to do with the class 2 versus class 3 it is possible that you may need a elliptical contour to discriminate between class 2 versus class 3 that means samples belonging to class 3 if you want to discriminate with respect to the other 2 classes you require a non-linear decision boundary linearity linear decision boundaries will not solve the problems in such cases these examples form the class of problems which are called non-linearly separable problems that means you cannot solve the problem of classification using a linear separable boundary or a linear decision boundary not a linearly separable or non-linearly separable these terms are sometimes used interchangeably by people. So class 2 versus class 3 look back again class 2 versus class 3 or class 1 versus class 3 or examples hand drawn of non-linearly separable or not linearly separable problems class 1 versus class 2 is a problem which is linearly separable these are points in 2 dimensional space I leave it for you to visualize the same in 3D or higher dimensions these are some nice synthetic examples which are typically available in many literature and books which shows examples of problems which you need a non-linear decision boundary these examples show dense distribution of instances belonging to 2 different classes using 2 different colors red indicates one color the blue indicates one color so this is one problem so what is the boundary you will need here sort of an elliptical boundary here okay you can see that in a non-linear boundary here between these 2 classes to separate one and ellipse will do here okay you need a very complicated boundary to discriminate between these 2 situations can be even worse than this in all these examples which I have drawn and shown so far these samples have actually not overlapped the samples have not overlapped means there has not been any amount of mixing between samples belonging to 2 different classes 1 and 2 assuming in one of these cases that the samples overlap then what amount of sophisticated classifier and feature space you use of course given a feature space you are using or designing a classifier your classifier is bound to make some errors and mistakes however if the samples do not overlap it should be possible for you to design a sophisticated classifier which will yield an optimal decision boundary linear non-linear whatever the case may be I repeat a good optimal classifier based on a good optimal design should be able to come up with a good linear or non-linear decision boundary such that you can separate between 2 categories of classes okay as long as the samples do not overlap like as given in the figure here although you do not have a linearly separable boundary to draw but you should be able to actually obtain good amount of classification. So continuing with discussion on DRs and DBs a classifier is expected to partition the feature space into class level DRs or decision regions and if decision regions are used for a possible unique class assignment which it will be then the regions must cover the entire d dimensional space and must be disjoint or non-overlapping these are certain desirable properties the border of each decision region is termed as a decision boundary in fact it is the border between a pair of adjacent decision regions to be very precise the border between 2 adjacent decision regions corresponding to 2 different classes or a pair of classes is called a decision boundary very rarely you will have a boundary encapsulating one class on the other side you do not have a class no you have another class obviously which is at least which does not belong to this particular class which is encapsulated by the decision boundary. So typical classification approach is as follows I am not talking of any particular algorithm right now for classification yet I hope you understand that I am not entering into a particular algorithm we will pick up the best decision rule when it comes and try to link it with the multivariate Gaussian density but I am just getting into the problem of classification. So what I am saying looking back the classification approach it is expected to yield a good discriminant function obtain good decision boundaries between 2 decision regions and it should determine a good DR in d dimensional space on to which a particular sample X or an instance X falls and you assign X on to that particular class the strategy is very simple this sort of approach for classification is simple the challenge is actually to determine the decision regions and sometimes it may not be possible to visualize all the DR's and the DB's in a high dimensional space for a general classification task with a very large number of classes in a high dimensional feature space often you will find that my diagrams will be concentrating on samples in 2D or 3D for a 2 class or a 3 class problem so far also you have seen and you are going to see that in this class today and in the next class following this that samples will be drawn in 2 dimensional or 3 dimensional space at the most even when we had that animation trying to describe you the difference between clustering and classification we showed you an animation in 3 dimension with a 2 class problem okay but of course it is left to always the to the imagination of the pattern recognition community to visualize what is going to happen in higher dimension with large number of classes okay there are lots of other issues which sometimes it becomes difficult to visualize in lower dimension what can happen in higher dimension so classifiers or the design of a classifier is basically involves designing the discriminant function because the discriminant function will give you the decision boundaries or DB's as well as the DR's or decision regions so designing a classifier again I repeat is building discriminant function or learning the parameters of a particular discriminant function okay remembering that if you have a C class situation that means if the number of categories are C's typically we have been taking the number of classes equal to 2 or 3 you will have as many discriminant functions as many as the number of classes are equal to the number of classes look back into the slide so classifiers are based on discriminant functions and in a C class case discriminant functions are donated by from here onwards we will introduce notation G of I of X want to see capital C indicating the number of classes and this partitions the D dimensional space into C distinct disjoint regions and the process of classification is implemented using a simple decision rule which says this is nothing new to you it has been done along the lines of along the lines of base decision rule also this can be put assign sample X to a class M or a region M or the Mth class or the Mth region if the corresponding value of G is more than all the other values of G for I not equal to M that means what I need to do after the classifier has learned the decision boundaries or the discriminant functions I substitute into C different what is the what is C? C is equal to what is capital C number of classes okay so I will have say G I is the is the discriminant function for the Ith class so if there are C classes there will be C such capital C is at G I so for all those C different G is I substitute the feature value X for that particular I mean for the particular test sample which I want to classify or discriminate or categorize or recognize I substitute the value of find out which value of G I is the highest is the peak or the maximum assign the sample X to that particular class this is the simple policy but of course who will give you the G I will get into the mathematical details of how to build the G I slowly what about a decision boundary so it seems the decision rules are giving me the DRs and the DBs are also given by this discriminant functions for any pair K and L not equal to the other okay decision boundary is defined by the locus of points where the corresponding G is at the same this we talked about already earlier at the hyper plane at the hyper plane the value of G will neither be positive or negative it is a binary classification problem you remember sometime back we talked about this the value of the G will be equal to 0 you will not have a positive or negative sign in this case of course in a multiclass classification problem at the C value of C is more than 2 then of course you need to have C different G I C different decision regions in some sense you can think of n choose to our NC 2 and the n is the number of classes those many decision boundaries and at each of the decision boundaries a pair of G I will be equal at each such point in the decision boundary appear so if you choose the boundary between two classes K and L as given here the corresponding values of G is will be the same GK will be equal to GL for that sample X a very simple classifier called a minimum distance classifier also called NN this is called the nearest neighbor is not written in the slide but please make a note it is a nearest neighbor classifier okay minimum distance classifier discriminant function is based on the distance to the class mean very simply what I am computing here look at the expression of G's the first expression of G which I am putting very simple distance of the sample from the mean for class 1 for class 2 distance of the sample from the mean of the second class okay so let us say in a two dimensional space these are the two corresponding class means mu 1 mu 2 and this is the decision boundary which I get for G 1 is equal to G 2 based on the expression which we had in the previous slide so this is the region R 1 decision region this is the R 2 the decision region 2 around the mean mu 2 for the corresponding class and the DB is where we are sitting this is the most simplest example of a decision boundary and a decision region for what is called as the minimum distance classifier or a nearest neighbor classifier the expression of G is very simple the declutent distance between the sample X and the mean how far you are from the sample nearest sample nearest mean is what you will get as the class assignment now of course you must keep in mind one particular thing that the corresponding value of G in this particular is not going to maximize in that case you have to take 1 by G if you want really maximize it okay so you can think of a minimum G for class assignment but it is okay we are interested in decision regions and decision boundaries the only thing what we have done this particular case is we have not taken into account any information about classes except the means it is like you are distinguishing apples from oranges you have a set of apples you have a set of oranges that is alright you have the two class means at these two points in feature space let us say but you do not know whether the samples of the apples are scattered around in a larger domain compared to the scatter of the oranges how far is the distance between the two means are known but the individual scatter or what are called in statistical term if you go back to the slide this method of G is do not take into account class probability density functions or PDFs which can be modeled by a multivariate Gaussian density function and the example of and the class priors so without that if you do not want to use it minimum distance in a book classifier will work correctly but it is dangerous to sometimes use this minimum distance classifier without using class priors into account you may get long wrong classification errors in classification if you do not take class priors or class distribution functions into account this has been discussed already earlier as introduction to classification task by professor Murthy and myself earlier so to incorporate that we have to bring in the class distribution function bring in the probabilities and to do that let us do the classification properly using a short of a best classifier almost available at the beginning of any book of pattern recognition or pattern classification which is the base decision rule or base theorem and we will discuss this in the next class how incorporation of the probability density functions using class prior conditional density functions etc within the base theorem gives a much more better accurate process of classification and a better estimate of the decision boundaries and formation of the discriminant functions and hence the decision regions for the purpose of classification we will end the class here today with the short note or the idea which we have discussed today or to summarize the process of classification involves designing of discriminant functions or learning the parameters of discriminant functions the discriminant functions will give the decision regions a boundary between two regions is called a decision boundary these decision boundaries could be linear or linear depending upon the individual scatter or spread of the class samples or data which will be reflected by the covariance matrix okay so we will see in the next class how the multivariate Gaussian distribution function incorporated using the base decision rule or the base theorem brings in the concept of decision boundaries by exploiting the distance of a sample from the class that particular distance term will come back again which we had in the beginning of the class today it will come back in the next class and the ideas whenever we talk about db is dr disturbance functions hyper planes hyper surfaces etc the concepts will be much clear thank you very much