What's up everybody, JZ over at jazzyoverjazzbeastcasebreaks.com. We just sold out 2020 Panini Prism Draft Hybrid, one box break, number three, two random packs each. So again guys, this is coming from a sealed box of hybrid. And again, we randomize the customer names. You get two times your names and the 12 pack numbers, one through 12, and then whatever number you match up with is the number you get. So let's double up your names here. And then of course, the numbers are one through 12. Let's quickly number the packs. All right, good luck guys. So let's go pack one, pack two, pack three, four, seven, eight, nine, 11. All right, also one through 12. What up, 11, how's it going guys? All right, let's roll it. 11 times six and a five, one, two, three, four, five, six, seven, eight, nine, 10, 11. Brian, down to Michael Colucci. 11 times here, now on the numbers, one, two, three, four, five, six, seven, eight, nine, 10, and 11. 11 times 11, five down to four. All right, Brian, you have five, Chris with eight, Brian you have 10, Zach with 11, seven and six, Brian with 12 and two, Michael Colucci with one, Zach with three, Chris with nine, and Michael Colucci with four. All right, so let's alphabetize them. We'll go by customer name first, print this out. That's really quick. All right, Brian, you're up first. And you got a couple of packs here. Five, 10, 12, two. Five, 10, 12, so two, 12, 10, and five. So let's actually put these together for you, put them right here. You'll go first, I just wanna put them all together though. Chris, you have eight and nine, eight, nine, put that here for you as well. Michael Colucci, one and four, one, four. I have that there. And then I think these are all yours, Zach. 11, seven, six and three. 11, seven, six. All right, you have first, Brian, good luck, man. Nick Bosa, you got Jacob Eason, silver. Nice pack, Tom Brady Crusade, Javon Kinlaw. Next one, pack 12. You got Chris McCaffrey, and a silver Terrell Lewis. I'm gonna last pack two. Two's your number, you got a George Kittle, green, and Colin Johnson. No autograph in your packs at least, but some good color, as you can see. And some good players too, a little Tom Brady in there as well. All right, man, let's put this away for the shipping team. All right, Chris, you're up next with eight and nine, eight and nine, Pay Manning, and Deandre Swift. Lamar Jackson, and Derek Brown. Nice, Derek, yeah, I mean, I think you guys are all gonna sell it so many breaks at one time. Just remember, guys, if you want those mixtures to break tonight, you gotta sell those up quick, because I may have to push one back. Like I said, I'm here till 12, but I won't stay here till after 12. Get in on it, guys. I think, honestly, guys, the basketball mixer is the closest. I think football mixture we're gonna need are two more pack breaks, plus sell out three or four straight up spots. Football basketball mixture I think is down like two, and then just the Chronicles pack break, if it hasn't sold out just yet. Tom Brady, Crusade. Not sure, Jason, I mean, all someone has to do is start taking spots in those random number blocks, and then we can break them. Like I said, we took out those, like, however many teams are remaining, and put them in random number blocks, so we can sell them out. We can break it tonight. Barry Sanders, and a nice Justin Herbert, purple. I think I'll make a little itchy, very nice. And Zach, you must have the autograph hiding here, man, good luck. Aaron Jones, the work. Alabama, oh, it's Terrell Lewis. There you go, there's the auto. McChubb, and Jordan loves silver, nice. All right, two more packs, Zach. Derek Henry, and a Justin Jefferson. No, James, I'm about to take my lunch break right now, and when I get back, I'm gonna go through orders, and then we'll create a new break schedule, and a Justin Herbert green. NT should be on that break schedule, and other breaks to have sold out. All right, there you go, guys, that was the break. Appreciate it, everybody. This is hybrid number three, jaspyscasebreak.com.


In the last session, we were discussing the concept of supervised method of pattern classification which is based on the Fisher's linear discriminant analysis or LDA. It is sometimes called FLDA as well in the pattern recognition literature. And we have just introduced 2 different matrices for the one is the within class scatter matrix and there is the between class scatter matrix. We will have a look at that expressions one more time and then look ahead and see some important properties and expressions of LDA and we will wind up this class with a few examples which can be handworked out with you okay by you. So let us look back what we are saying is LDA is a method of supervised learning it is one of the classifiers which needs set of training samples for learning a set of parameters. In this case the parameters are SW and SB the within class scatter and the between class scatter respectively. The learning set is labeled unlike the PCA where we had unlabeled data and that was also called unsupervised learning. So it is a class specific method in the sense that it tries to shape the scatter in order to make it more reliable for classification. Remember in the case of PCA we were interested in trying to find out the directions in which the maximum scatter of the entire data exists that is what principal components analysis or PCA does. In case of LDA we want to maximize class separability okay that means in some sense the between class scatter if you take a 2 class problems we want to find a certain direction of the data within the data certain dimension in which the distance between the 2 class means or the 2 clusters of the 2 classes become large or become more and in the same direction the within class scatter becomes less okay. If we recollect the animation slides which we had long ago in an earlier class when we were trying to distinguish between classification vis-a-vis clustering we always said that it is better and easier for a classifier to perform better if the between class distance or the distance between the 2 cluster centers or the 2 clusters themselves is very large and the clusters are very compact. So if you can find such a dimension or set of dimensions what can be considered also as a subspace okay with respect to the original higher dimension of the data if we can find a subspace where the directions point out that we are going to have or is satisfied satisfies a constraint that the inter class distance is very large and the inter cluster distance is very small that is what is LDA trying to achieve okay. So that is what is the meaning of this sentence which you see now that the class specific method it tries to shape the scatter in order to make it more reliable or better for classification okay. So this is accomplished with the help of trying to find out a weight matrix W which maximizes the ratio of between class scatter HB and within class scatter SW where the terms will define them again for your ease of understanding that this is the between class scatter HB if you look at the expression here the N i is the number of samples for a particular class X i let us say the class label is X i, mu Y is the mean for a particular class X i and the mu is the overall mean of the data okay. So it is something if you look at P Y minus B mu it is basically that the individual cluster centers are normalized with respect to the mean okay of the entire data set and summation over N i will actually give the number of samples but a particular class a certain weight age capital C or small c here whatever you see is the number of overall number of classes okay. So you need to sum this over all classes that is the between class scatter let us look at the expression once again for SW which is the within class scatter matrix you need to sum it over all classes alright but you sum it over now this expression is similar to the PCA it is an outer product of the samples with respect to mu I that in case of PCA you had the overall data mean here which was mu the same mu which you see here on the left hand side would have occurred here in the case of PCA but in case of LDA you have the mean subtracted from the data which is the individual class means okay the mu i is the mean of the class X i which you have to take that and take the outer product you sum it over all the samples for the particular class in fact X k belongs to the samples at X i that means basically n i the summation will go over n i number of times and then over c number of classes okay so the total number of summation terms which you will be having is basically n multiplied by c SW and SB have the same dimension as the scatter matrix we had for PCA the dimension is the same but the matrices themselves are little bit different okay if you look at the expressions both are in some sense the outer product but one of them is computed with respect to the means only the other is computed with respect to the samples means subtracted with respect to the class specific means okay the PCA we subtracted the overall data mean I am repeating again here you are subtracting the class means okay so SW and SB is what you have okay and let us so what we are trying to do is find out an optimal W and we will just talk a little bit later on that what happens if SW is singular but assuming that the within class scatter matrix is non-singular you try to find out an optimal value of W which maximizes this expression okay so you try to find out a W which maximizes expression and in the process of doing so you get an optimal W which can be written in terms of a set of eigenvectors as given here instead of eigenvectors and they are the eigenvectors of this particular matrix which is SW inverse multiplied by SB in fact what you are doing here is trying to find out the m largest eigenvectors of this characteristic equation which is if you think of SW inverse multiplied by SB as an overall scatter matrix S then you are actually trying to find out the eigenvectors and eigenvalues lambda is the corresponding eigenvalues of this particular matrix okay and this is the reason why it is essential that the within class scatter matrix is non-singular because you need to obtain its inverse of that matrix then multiplied with SB and then find the corresponding eigenvalues and eigenvectors this is the basic approach for FLDA or LDA correct and do that you need to find the inverse of the matrix SW the question comes is SW always singular we will have a look at it very soon I will just give you a key point with respect to some properties of within class scatter matrix SW. So there are actually at the most C plus 1 minus nonzero eigenvalues in the lambda so if we look at what is called the eigen spectrum of this particular matrix the upper bound of m here is basically number of classes minus 1 so the restriction on the number of nonzero eigenvalues and the corresponding eigenvectors for the W will actually depend on the characteristics or properties of SW whether it is singular or not these are some of the main criteria which we need to follow SW is singular if the total number of samples n is less than the dimension d remember this is total number of samples not the so you can say that this is the average number of samples per class multiplied by the total number of classes okay that will give you the total number of samples n its rank is at most n minus c okay where c is the number of classes so you can see that the number of samples here is going to dictate has a very important say on the singularity or rank of SW typically if you have sufficient number of samples you do not need to worry about the singularity of SW if you have sufficient number of samples very very large okay but there are certain applications where there are a dearth of the number of samples both for a particular class and for all the data samples put together or all the classes put together a typical example is the case of face recognition problem in the case of face recognition the dimensionally typically may go to a few millions okay it is very very large okay in fact the resolution of size of the image n square if you take as not this n but if you take this 640 by 480 image 640 by multiplied by 480 will be the dimension of the problem whether you are doing PC or LDA now you will not have those many samples per individual or total number of faces available in a database if you take let us say an example of a very large database which may have even a few thousand individuals you may have around 10 let us say okay 10 to 100 at the most samples per particular class in fact there are certain situations where you have just a few samples may be just about half a dozen or 10 samples per particular for a particular class in this case the class is a particular subject so in such a case definitely what will happen is the number of samples n will be less than the dimension if you look at the slide this constraint will not be satisfied in all applications or situations of pattern recognition problems all problem this may not satisfy in certain situations if the number of samples is less than the dimension or the dimension is much larger than the number of samples this becomes singular okay and its rank is at the most n minus c we will move at with this assumption that the rank is n minus c for the time being we are looking into this problem however if the number of samples are very large and it exceeds the number of dimensions in certain cases which it is possible yes in such cases you do not need to worry about the singularity of SW but there are certain situations where you need to worry the number of samples are less typically those are the cases which are called the SSS problem which is called a small sample size problem and the number of samples being very large per class or for all the classes put together that again is less than number of dimensions in such a case you cannot invert SW okay it is not a full rank matrix it ranks restricted by the number of class and number of samples solution to the such problems if you have singularity in SW is the following project the samples to a low dimensional space and do that we use our method of PCA which we have studied in the last class to reduce the dimensions from the feature space from the original dimension D to a dimension n cross c and once that is possible to apply the standard Fisher linear discriminant criteria FLD to the reduced dimension c minus 1 okay. So in such a case when you are applying PCA to reduce the dimension the W optimal value of W can be visualized as that you have done a PCA earlier and then the corresponding W which you would have obtained by the FLD criteria and you the PCA criteria for getting the W for the PCA is given by this where ST is the origin scatter matrix remember we had an expression earlier which said that the scatter matrix ST is the sum of SW plus SB. So ST is SW plus SB and W FLD can be written as an expression you can see that this is the similar expression to whatever we got earlier except that you have the PCA done before the LDA. So that corresponding matrix comes and sits as a pre and a post multiplication with respect to SW. So if you do this method under the condition that n is less than the SW singular you want to reduce this dimensionality of the space go to a subspace where you can reduce the dimension corresponding to the expected rank of the matrix then you do a PCA first and then do an FLD to reduce the dimension and people typically take C minus 1 dimensions for the FLD. So let us look at some hand worked out example this is a very synthetic diagram drawn with respect to 2 data sample points and we had this similar diagram in the case of PCA where the PCA would have given a direction along the maximum scatter or variance of the data samples which have been along the direction orthogonal to this vector. So that along this line along the data samples as this arrow indicates that would have been what the PCA directs but the LDA will give you a direction in which you are expected to have the maximum scatter that means you know if you project the samples point you expect the red points to be on the left hand side the blue points on the right hand side. So you may not have separability in this case because the data samples are very near by alright but this is the direction in which you will have maximum separability or the condition of SW inverse multiplied by SB is maximized correspondingly whatever with the way W transpose. Let us take this hand worked out example it is easy to do this calculations in a sheet of paper but of course you can keep your calculators ready if you want. So let us take this example of data sample points in 2D we are taking an example in 2D because it is easy to visualize. So these are set of XY points on the top you have set of X coordinates on the bottom you have set of Y coordinates you have them for 2 different classes. So the class 1 this is the class level at the bottom so for the class 1 you have 7 points for the class 2 you also have 7 points and the data samples in the XY 2 dimensional space will look something like this where you will have the data samples to the left given by these blue points here and the data samples to the right as identified by class number 2 will be given by the set of green points.  So forget the color of this code color code for these two different classes let us take all of these as a set of points as set of black markers let us say forget the color and let us try PC that means we are just taking the data samples and ignoring the class level and we do that this is the overall data mean for all the samples which can be averaged out by taking the average of all X values which will get as this average of all the Y values is what you will get as 5 and this will help you to compute the overall covariance of the mean subtracted data from this that means what you need to do is subtract from each of the sample points the mean and then do an X transpose to actually obtain the covariance matrix which will be 2 cross 2 matrix as given here because it is a two-dimensional data so that is what is the size or dimension of your ST or covariance matrix or scatter matrix. The corresponding eigenvalues obtained after singular value decomposition of this matrix is basically this which basically shows that along the first dimension you have quite a bit of a scatter along the second dimension you have you know maybe much less than the first dimension what are the corresponding eigenvectors for the first dimension this is the dimension of the eigenvector we will draw that in this diagram very soon you will have it up and this is the second eigenvector what do you expect along the first eigenvector you should have expected to have a large class scatter because the eigenvalue is very large along the second dimension you will also have a scatter but it would not be that scattered or that stretch as with respect to the first eigenvalue okay so if you look at the diagrams here the eigenvector shown here is given by the same color along this particular direction so it basically says that along this direction you have the maximum scatter of the data which is quite obvious if you look into this the overall data seems to form an electrical pattern in terms of a scatter and this seems to be the major axis of that particular scatter what is the other dimension which is given here this is given by also the blue color corresponding the same color code I have used for displaying the values of the components of the eigenvector as well as the vector shown in 2D so you can see that along this dimension you have very less scatter of the data samples which is probably almost one third or even less than the scatter which you have along the so what PCA is giving you are 2 different directions okay as we have discussed in the earlier class along the first direction you have a very large scatter along the second direction dimension given by the PCA you also have a scatter but it would not be as large as the first dimension how much of the scatter exists in the second or even third if it exists in this case of course we have just discussed a hand worked out example of two dimension but if you take an n dimensional problem you will get a scatter matrix of dimension n corresponding n eigenvalues and n set of eigenvectors so along the first eigenvector second eigenvector third and vector and so on you will have this diminishing scatter this scatter will be going less how much is it tapers down will actually depend on what is called the eigen spectrum or the distribution of the eigenvalues along the diagonal of that diagonal matrix okay so in this case is a two dimensional problem if you look here that the scatter along the first is much larger than the second which is expected and the ratio will probably tell us how much more is a scatter along the first dimension here or the first principle component that is the way one should say precisely this is the first principle component is the second principle component you could have a third if the dimension was 3 and the corresponding eigenvectors are given here and the corresponding eigenvalues are given here this is what PCA will do but as you can see here that if you project the data samples along the first eigen dimension you will not have separability in fact is the second dimension of the PCA the second principle eigenvector which will give you separability so PCA should not be used for classification in general we talked about this earlier it is typically used for dimension reduction for data representation okay are trying to find out a subspace where you can have maximum scatter up to that point it is okay as for example you have just seen in today's class that PCA supersedes LDA comes before LDA because we want to reduce the dimension in certain cases when the SW the within class scatter matrix is singular okay so PCA can come before the LDA to reduce the dimension that is one of the applications if you want to actually obtain class separability and if of course class labels are available you better try an LDA rather than a PCA so let us use this example in the next slide and move ahead and try to use this class information which is now labeled with two different colors to show that the data on the left side belongs to class 1 the data on the right side belongs to class 2 this is the distribution which we had for class 1 and class 2 in the previous slide as well where we use the samples for PCA without the class information now using LDA now using class information we will perform LDA on this dimension on this data look at this W okay so we had given the expression earlier but I am just we are giving you the values so you can use the entire data set to class problem two dimension to compute within class scatter 2 cross 2 within class sorry between class scatter matrix as given here the inverse of SW in this case it is non-singular okay because you have enough number of data samples with respect to dimension of the problem so it would not be singular here so inverse of this matrix I leave it as an exercise for you to do that it is simple 2 cross 2 matrix multiplied with SB okay so if you look at the inverse of SW which is non-singular matrix in this case as the number of data samples is large compared to the dimension this is what you get so what is the corresponding eigen decomposition of this okay eigen values of SW inverse SW is given by this okay this what does it basically tell you that along the first eigen vector you will have a large scatter but along the second eigen vector given by LDA you will the scatter is 0 the separability is 0 of the data and the corresponding eigen vectors are given here let us draw them on the data so this is the first corresponding eigen vector given by LDA remember PCA gave us direction which was along the maximum direction of the scatter LDA is given along the direction of maximum separability this is the direction as given in red the corresponding color has been used also to show you the direction what about the I have not shown you the other direction but let me tell you this direction will be orthogonal to this so it will be along this direction in which which is almost similar to the direction given by the first principal component of PCA let us go back to have a look if you look at the PCA eigen vector here and we invert the sign it is 0.8 and 0.6 which will give you the direction towards the upper right and it is same as the second eigen vector direction almost similar almost similar on the it is not the same it is not identical but along the similar direction you have a scatter which is 0 so I am not drawing the second eigen vector the main reason being that you are not expected to have any separability or the criteria is null so in this particular case we will take only the first dimension and project the data to have the maximum separability okay let us look at the effect of inter cluster distance on the eigen vectors and the eigen values as an example. Now let us assume that these are on the left hand side you have the within class scatter and between class scatter matrix of the data which you have just discussed in the previous example okay these are the same values okay you can check your notes that these will be the same values obtained from the data as done in the previous study if you go back you can see here this is what SW and SB is for this particular data so we have kept that on the left hand side on the right hand side I give you the same SW but a different SB okay and I am telling you the mechanism by which I obtained this what I did was or what you can do is you need to actually separate the two data out okay the data sample points which you have seen in the previous slide you need to bring in more separability I will show that with the diagram what I mean by separability separate the one cluster with respect to the other so if you do that basically give it a shift along the direction of not the maximum scatter by the maximum direction of separability okay so that can be done and I will show you what I mean by that when you do this keeping the within class separability same within class scatter same you can see that this is change or increase in the corresponding values of the between class scatter the between class scatter values have changed but the within class scatter values have remain unchanged what this basically means I have not change the distribution of the individual data samples but taken the two class means and separated them apart okay I repeat again kept the intra class that means within class scatter for each particular set of class samples I have not change this scatter of the data points but I have change the class means let us say one of the class means if you change to bring in more separability this is the what is the result which is going to happen that means we will have a larger between class scatter matrix and the same SW we will go ahead and do again the compositions for both the net result is the following look at the Eigen spectrum of the original data and compare this with what you have now okay so I have done judiciously such that I do not make the second component of the Eigen value non zero still keep it zero but there is an increase in the lambda one which is basically the corresponding first Eigen value along the first principle component obtained by LDA let us look at the corresponding Eigen vectors well there is not much of a change okay the change is what you will see in the second or third decimal places of the corresponding Eigen vectors so their directions more or less and the magnitudes have remain the same let us look at the data okay this on the left hand side what you see is at the data points which were given to you earlier in the previous slide as the previous example and that is the corresponding component given by the first Eigen vector we will never consider the second Eigen vector because the corresponding Eigen value zero so this is the first Eigen vector this is the same case here minor change in the value with respect to the Eigen vector here so it is almost the same but look at the separability this is the separately which I separability which I brought into the data to actually have a large value for the both the diagonal terms for the between class catamatics compared to the data which we had earlier and the net result is a larger again we can see that the separability of the data between classes is getting reflected in the corresponding Eigen value so this is a lesson so in some sense you have maximum separability along the first dimension the less you go to the second and so on and after some point of time you may not have any separability on certain dimensions which are you know beyond C minus 1 the number of classes equal to C here is equal to 2 the number of classes equal to 2 so C minus 1 equals 1 so you will have only one dimension in which you will have separability and you will not have after LDA performing LDA you will not have any separability after C minus 1 Eigen vectors that why lambda 1 C minus 1 equal to 1 lambda 1 is non-zero and lambda 2 is equal to 0 but if you bring in more separability of the data it gets reflected in the corresponding Eigen values the Eigen values will shoot up if the class separability is more or the inter-class within class scatter also gets reduced this example shows that between class if the spread or the distance is more the Eigen values are going higher the same will happen if the within class scattered matrix value goes down that means the data samples are much close to the mean themselves for a particular class that also can happen and that also can actually change the value of lambda 1 okay so this is an example which shows that if I take the data points and project it along this direction well there is a little bit of a scale problem here which the direction is not shown appropriately but anyway it is more or less accurate here this is generated by a small program so if you do this you can see that the separability is here that means all these blue points are projected to the left hand side so what you are seeing as the x axis here is this axis along the direction of the principal Eigen vector given by LDA so all these blue points will project on the left hand side as is here all these green points are projected here and there is some degree of separability between these two data samples okay the cluster means would be somewhere here look at this particular data and if I project now this on to the corresponding Eigen vectors you see the separability now you can see the separability in 2D the same gets reflected in the corresponding principal Eigen vector first principal Eigen vector given by LDA so if you project these blue points they all form a very strong cluster here the corresponding green points form another cluster here you can when you compare the separability on the right hand side with respect to this you can see the not only the separability is larger but the within class scatter you look at the scatter of the blue samples here compare them with the scattered here look at the scatter of the green points look at the scattered here so both this example show that the between class scatter has increased and the within class scatter has come down the scatter within a particular class remember of course one must be careful when comparing these two plots remember one thing here that they are not to the same scale now this is of range minus 3 to plus 3 so it is basically in a range of 6 pixels or 6 units in dimension here the dimension is 10 so in a scale of 10 the separability here if you say will be about 5 units in terms of length the huge separation and this separation is about 2 units just more than 1 so the separation is also increased and the within class scatter separability has also come down these are the 2 lessons and that gets reflected where in the Eigen values not in the corresponding Eigen vectors to wind up let us take an example where I increase the number of classes to 3 so what I have done the distribution of the points is same you have set of points in blue corresponding to class 1 set of points in green corresponding to class 2 and 4 points in red corresponding to class 3 will quickly go through the SW and SB since the number of classes is 3 you can expect that you will have separability in 2 dimensions C minus 1 will equal to 2 when you perform Eigen decomposition on SW inverse multiplied by SB you can see that you have both the Eigen values which are non-zero now and the corresponding Eigen vectors are given here okay so you have some small amount of separability in the other dimension which is the second dimension but look at the first dimension here you have a high degree of separability which is reflected in the corresponding Eigen value this is the first degree of separation for the class so you can see this is the class 1 this is class 2 and class 3 so of course this is the best possible dimension in which you can project the samples to have a fair degree of separability you may not have a separability along the other dimension so that is why you have a less value of the discriminant Eigen value when you project these samples here in the first Eigen vector this is what you have okay that means again I have kept the inclination of this direction same as this that means to take this blue point project it here take the green point project from this take the red point project it here this is what you have okay if you tilt it and show it here this is the separability you have a good amount of separability with respect to the blue class compared to the green and the red points there is some degree of overlap here between the red and green points which you cannot probably overcome so the red points here are here the blue points projected here original point data points here in two-dimension space they all project here so there is a fair degree of separability in the first Eigen vector obtained by LDA the same may not be true in the second Eigen vector you can see a huge degree of overlap this is along the second Eigen vector as given by the green arrow along the second Eigen vector when you overlap you can see a scatter which is overlapping in all three different classes so this is another example is possible to hand work out this using a calculator where you can find that the Eigen spectrum or the set of Eigen values obtained by SVD is going to give you the degree of separability between classes okay so in such cases one need to do well there are other methods of supervised learning which is called independent component analysis one can even try that to separate the data to wind up this class I have put forward a list of some may not be exhaustive but some advances or the technology over the last decade or some advances in the field of pattern recognition which has taken place over the last decade and I am just going to name them may not discuss at all any one of them in detail as talked about by Professor Murthy and myself that this is a course for beginners in the field of pattern recognition and we hope that you get encouraged in this field of study from these lectures and read books and read much more advanced topics which have come out in several other books as well as advanced literature getting published in rich conferences and journals so we look at them but let us look at the list of techniques which we may not have covered and they are some recent advances people adopts of computing methods based on neuro fuzzy techniques we talked a little bit of perceptron and neural networks you combine that with fuzzy reasoning to actually get a neuro fuzzy architecture for class discriminability rich in fact these are methods which are used often to discriminate between classes which are overlapping some of them do not function for Gaussian distribution and so on and so forth. Multi-classifier ensemble or combination which involves both decision and feature fusion this actually talks about trying to use all set of different classifiers together to form a unified decision the classifiers themselves will be different in terms of architecture or they could be trained with different number of samples there are lots of theories on this people also work on reinforcement and probabilistic learning there are works which try to handle small data size problem or what is called as a small sample size problem the triple S in the field of neural networks and pattern recognition people work on generalization capabilities of neural networks and pattern recognition algorithms people also work on evolutionary computations which is very near is a sub-paradigm under soft computing there are methods based on decision trees multi-objective clustering and the most of the clustering algorithms algorithms which we have discussed tries to minimize one criteria if you let us say take K means it tries to reduce some sort of an Euclidean distance with respect to the mean but there are better clustering algorithms which actually tries to minimize one criteria or try to minimize one criteria with by keeping some other constraint in mind so those are some examples of multi-objective clustering very recently people also working on manifold based learning and optimization where people take ideas from the areas of differential geometry and topology people work on genetic algorithms pervasive computing neural dynamics yes support vector machines and kernel methods modern machine learning methods have contributed a lot in the field of pattern recognition in fact there are certain areas in which machine learning and pattern recognition overlap quite a lot very hard to discriminate whether you are exclusively talking about machine learning as by itself or pattern recognition some of the terms and methods are semi-supervised learning transfer learning deep learning dominant application etc these situations have become very important because it is possible that you may not have lot number of samples in a certain data set but there is an auxiliary or an ancillary data set which has huge number of data samples so you could train your classifier with such samples and try to see if it performs well in another domain well well it does not you cannot train directly on other data set and train on another data set and test it another one so there are methods in which you try to transfer the information from one domain to the other there are methods on other methods of transfer learning and semi-supervised in which actually you have the human intervention in between which tells you that some of the classifier decisions made during testing are correct or not and you try to readjust the weights of a particular classifier carrying on there are methods on random forest independent component analysis well I will put a one minute sentence here remember PCA LDN then you have ICA people talk about these three things almost together PCA works without class levels it gives you this scatter along the maximum direction okay LDA supervised learning which tries to give you the scatter along the direction which you have maximum separability between classes the difference is PCA takes the entire data into account and gives you maximum scatter whereas LDA gives you maximum separability we know that criteria SW inverse multiplied by HB that is the criteria which tries to which it tries to maximum and maximize and it you need class levels for that and hence LDA is supervised PCA is unsupervised ICA relaxes one constraint for LDA it needs class levels it is a supervised method of learning but it relaxes the constraint that the second eigenvector is normal to the first third is normal to the first and second and so on if you rely because there are certain data sets in which after you obtain the first principle eigenvector let us say in which you have the maximum separability it is not necessarily true always in all data sets that the second separability will be orthogonal to the first that may not be the case so in such cases you need to find out a mechanism where you get the second component of the second maximum degree of separation which should be in a certain direction may not be necessarily orthogonal to the first one but basically means if you are looking at some ith principle component in LDA that may not be orthogonal to the rest previous I minus one principle eigenvectors that is what ICA tries to do to overcome the restriction on LDA people have worked on pulse spike and probabilistic neural networks and there are methods of nonlinear and convex optimization related to both manifold based learning related to model methods of ML which people are trying out to solve complex problems in the field of pattern recognition graph based kernel and embedding are also some of the methods in structural pattern recognition and syntactic pattern recognition which we have not covered in this course thank you very much.


In this module, we will discuss about type specimen. What is type specimen? Type specimen is a specimen which we use as a reference material in the museum. Type specimen is when a researcher is doing his research, he is writing the description of his specimen, so the specimens available in front of the researchers, whose base is the original description, one of them selects a specimen and submits this specimen as a reference material in the museum. Type specimen is a reference material for the researchers who are coming later. If any researcher wants to compare his specimens with this reference sample, he can easily find this specimen based on the museum ID of this specimen and study its characters and compare his specimens with this sample. Now, when a reference specimen is kept in the museum, we have to consider some of the characters and some of the things. The most important thing is that we have to mention the precise type locality of the specimen. What is the locality of the specimen? The locality of the specimen is the place where the specimen is collected. So precise type locality and other important notes should be present. Along with that, the specimen which we kept as a type specimen, in the museum with a specific museum ID, what is its sex? It is a male specimen or a female specimen. It should be clearly mentioned. Along with that, the specimen belongs to any developmental stage. It is a larvae, it is an adult, it is in any form. It should be clearly mentioned. Similarly, when a specimen is collected from a type locality, then who collected it? What was the name of the collector? It should be clearly mentioned. Similarly, this specimen is kept as a proper ID number so that the researchers who come later, when they use it as a reference, they can easily identify this specimen on the basis of this specimen. Similarly, when a type locality is mentioned, then its altitudes are mentioned that how many altitudes are collected. Its proper coordinates are mentioned. Apart from that, if it is collected from a depth, then how many meters of the animal is collected from a depth, then it should be clearly mentioned. Apart from that, if this specimen is a fossil, then its geographical horizon should have detailed information about it. Now, whenever a publication is published, in which a type specimen is given reference, then it is necessary that this type specimen has been submitted to the museum. Apart from that, its designation should be clear and there should be no ambiguity in it. Apart from that, whenever a type specimen is kept in the museum, then all the requirements for it, such as we said that what was its type locality, after that, what date it was collected, what was the name of the collector, what was its altitude, apart from that, what were its coordinates, what was its sex, all this information should be present so that any researcher can trace this specimen later.


Hello, good morning everyone. I think the two sessions were very insightful, the case study, which Apurva talked about and the Janai Flipkart, so a lot of Janai. Now let's come to the bread and butter, the basic things which you do. So this is a point of view which we are talking about with a lot of our customers and I thought, you know, let's share that what, how do we design for future or how do we cater the various expectations of the next generation user group which mostly, you know, includes the Gen Z's or Millenials or the Zenilials, I would say. But we are not saying anything to, you know, specific to Gen Z, but today every user is our next generation, whether it's a, you know, a 70-year-old senior citizen or a teenager or maybe an adult. What we are looking in the financial services, like if we go almost two decade or a half decade before, most of the, you know, digital transformation were targeted on the manual process to be digitized. The context here is that when in late 90s or early 2000 before iPhone came, a lot of focus was on how the things can be, you know, improve the things can be more digitized. But the user experience or I would say the customer experience was not targeted. People were only thinking that how I can move from mainframe to a web-based application or maybe when iPhone came, how some of the journeys can be on mobile as a responsive. We were not even in the mobile app journey or a mobile-first journey in the late 2010 or something. Now that's the context, you know, and a lot of people, I think, the senior people who have contributed to create that digital journey or they were the initial, you know, debutant of, you know, creating the digital world. Let's again go back, as I said, the most of the focus on the manual journeys to be digitized. They were not thinking that how it's going to impact the user groups, how it's going to impact to the legacy applications. They were focused only on, you know, how the things can be more on computers or more on a better technology. But I think this is the time where we have to reimagine the entire user experience or a customer experience for our next generation user group. So let's have some key stats. We did a research now. This is a research based out of North America, Europe, done by our research institute in Capgemnai that almost 70% people wanted to be connected.  They want everything on their mobile. They want all notifications to be come out.  They want to be connected to every, whether they're family, friends, their offices and all those stuffs. People want personalized data. They want that, you know, whatever information I'm getting that should be around my ecosystem. They want tailored services. They want, you know, even the offers to align to be their expectation.  Now, what point of view we brought, and this is very much, you know, realistic in today's world, that the next generation is basically digitally native. The people who are using all kind of devices, they are digitally native. They are using mobile phone. They are using watches. They are using, you know, OTT platforms and, you know, different variety of computers and laptops. And what they exactly want, they want a personalized data.  They are influenced by social media. For an example, people don't book tickets or go for a vacation from a website. They follow their influencer. People don't, you know, invest in stocks just looking to the market.  And also, they are very much socially responsible. They look for, you know, investment in the green companies which are, you know, more sustainable. They are more ethical. Plus, they want seamless experience.  Lastly, they are looking for, you know, multi-channel. They want anything everywhere. They want a seamless experience. Some of, you know, more details on that. As I said, personalized, they want AI-based materials. They want, you know, tailored services to them. In banking industry, mostly people use a lot of services because of the rewards. They want to leverage. They want to reimburse those into their various activity. They are, as I said, they are influenced by social media. They have a lot of followers. They look for those kind of, you know, suggestions from their followers, their friends and family. They don't take decisions just based on the product which they see. And as I said, they are socially responsible. They do ethical banking. They look for products which are more, you know, sustainable. Their ASG scores are good. People are also, these days, we have observed that Gen Zs are also looking for, you know, green investment. The investment that they are investing in are stocks which are more eco-friendly, who give, you know, green loans. For example, loans for e-vehicles. The interest rates are more, you know, lower as compared to the other vehicles. So these are some, you know, more parameters like they want intuitive UX. They want only channel experience. Finally, they want everywhere, anytime. They want the accessible. All these, you know, devices, there's a mobile watch. The accessibility should be at par. And finally, their expectation, the customer expectation or satisfaction is very high. Now, how do we, you know, create this next generation or customer experience? Because the digital economy is driven by these people, the next generation user group. So to do this, what we have done, this is a life cycle of all of us in entire region where, you know, we start. The first thing is that we open our WhatsApp or we go to Facebook or we see Insta Reels.  Then probably we will look at what is the traffic. For example, today's, you know, bundling Bangalore so people were looking, you know, the traffic is very less. Same way, you know, people will look when I will reach office or when should I book my cab, if they are, or when should I, you know, hire or, you know, co-do with some of my friends and colleagues. They are more on, the generation are more on dating apps. They are, you know, using various dating apps like Tinder, Bumble's. In throughout the journey, they also, you know, go and watch their favorite web series or OTT if they are free. And finally, you know, in between while doing office, if they are stuck in the traffic, they are on their listing music. So these are some trends which a person do or every next generation is doing. Now, what we thought the customer experience is very high of these applications. These are some of the applications. How we can adopt these into the financial application? What happens? What do we observe when these people, when they are using all this application, whether it's a Tinder where, you know, they do right swipe or left swipe or, you know, they are using Google products, people use WhatsApp, not only for text. They are not doing text. They are actually sending audio message. So people are more lazy, they don't want to type. They will just send audio notes. People are, you know, they are looking for music apps also which throws playlists based on their listening, based on their artistic, the artists which they follow, what kind of music they want. A lot of people don't directly book the tables through restaurants.  And also, we have seen, like, you know, people are using cloud. They store their data on a variety of application which provides, you know, documentation. Obviously, all of us are on OTT. We've watched a lot of movies, a lot of series. So these are some trends which people are adopted and the customer experience is very fantastic. But what happens when they do their banking or when they go to their insurance application? However, a person goes only twice or thrice in a year to an insurance website either to buy a policy or to renew or unfortunately, if he wants to claim. But still, they do banking day by day. But when they open their banking app in any region, I'm not talking only about India. It's more on North America, European. The customer experience goes down because the interactions, the content which they get, that is not personalized, that is not, you know, it's a very old school application, I would say. Because as I said, the background, the insurance and banking industry was the first to adopt the computers and first to adopt the digitization. But that was the focus only on manual to computerization. And that is continuous. But the change happened when iPhone was launched. Lot of mobile phone, people are mobile first everywhere. No one opens even their laptops to do basic things. Everything is available on phone. That's why we thought, why can't we use these interactions in some of the basic journeys or some crucial journeys of banking and insurance application? Not only the front-facing application, also to the enterprise application. And that's why we thought, let's create some of the concepts and we did some consideration on that, how the next generation user experience or customer experience is going to be. So the first one is lifestyle insurance application. Basically, we will, you know, onboard the person, understand their lifestyle. A lifestyle can be, you know, going to a gym. A lifestyle can be going to vacation or it can be, you know, to do a yoga or to take care of their mental health. A lifestyle can be going to an office also. And based on that data points, the insurance company will check whether they are insured. For example, a person, a lot of people in Europe, you know, goes to office with cycle. But are they insured when they, you know, when they are riding cycle, if something happens, some accident happens, they are not. Not every insurance companies provide that thing. So we thought of, you know, defining all those lifestyles and based on their lifestyles, the insurance company can, you know, provide that byproduct. Second, Super App is a, you know, Super App was a lot of talk of the town last year or years back. It is heavily declined in North America, but it is very good in China and Japan. But what we think that let's create next generation Super App. And the last one we thought, which is no one has talked about that how we can do banking on drive. Like we know Tesla, a good, very nice, you know, dashboard, lot of Indian cars also having that thing.   Do you want you to pay? So all the securities are taken care and then while driving, you can do. Here are some examples. For example, we took, again, these are some concepts I cannot share, the actual, because of the idea, but what if I use the left and right sweep of Tinder in a banking application to do, if I do a right, I can do a credit card payment immediately. If I swipe left, it can be a scheduler. I can pay, you know, on certain dates. The another example of life tracking versus status. People once basically a life tracking. For example, we order food or we order, you know, grocery. We get the actual status when the order is picked, where he's, you know, the delivery boy is stuck. So how that can be used? That may use in a policy claim that your claim status is this or your loan status is approved or declined or dispending for documentation. And all this notification should come with your watches or from your mobile app. Another example would be two-way authentication, which we see in Google or in all of our, you know, mailing or with collaboration apps. If I want to log into my website, I need to provide my customer ID, my user ID and password. Why can't we do something that if I open my website or net banking, immediately, you know, a two-way authentication should come on my watch and I will say yes, the way we do in Google or in YouTube. Automatically, I will land it to my dashboard. Another example can be, which we have actually applied to a lot of our customers, the Spotify model. The Spotify, which actually while onboarding, you know, they ask who are your favorite artists and then based on that and based on our behavior, based on your history of listening those songs, it's throw you a playlist. In same way, you know, you can be onboarded to a insurance application. You don't need to tell that, you know, you have a baby at your home. They will understand your life events and then they will suggest you various products.  Another one more example, which is actually very good. Like, you know, I don't need, I should not, you know, open my mobile app or my net banking to see my account status or do some transaction. I can, you know, the way people are using WhatsApp audio, the same can be used like, hey Siri or hey XYZ bank, what is this my score or can you please transfer XYZ dollars to XYZ person. So instead of, you know, clicking instead of going with traditional, all the trends which is into the non-financial world where people are very much, you know, satisfied we can leverage here. Another, there are a couple of examples, but what we are trying to say here that if we leverage those interactions, those gestures, those patterns, which are good in the all, you know, hundreds of apps into financial and insurance application, not only in the front facing the enterprise application, the customer experience is going to increase, which is not, which is very little bit of an average or okay kind of. So that was a summary, we see a lot of scope in the, you know, fintech industry where we can, you know, improve the overall experience. And I was just, you know, wondering if I can share this point of view where you can, you know, also see where you are interacting. I'm not sure if everyone is from financial, but all the strengths can be leveraged in any of the model. So that was a very small, you know, point of view. Unfortunately, I got only 10 minutes. My time is already up, but I'm here to interact more. I have a detail. This point of view is also available on Capgemini's website. You can, you know, interact open to public, but I'm here, we can have interaction in any questions, any solutions. Thank you very much.


They were in the concentration camp in Germany at that time, World War II. They come and call numbers. Those numbers are going to the gas chamber. There is a young man, his number is called. The guy is terrified. A rabbi came up to him and said, son, don't, don't fear this, like this. A few days later the war ended and this man lived. It's a very beautiful incident that happened. It's a guy who, you know, they were in the concentration camp, Germany at that time, World War II. Every day they come out, they come and call numbers. Those numbers are going to the gas chamber. So when they call out numbers, people hide under the this, that, they'll be dragged to beat. It's a horrible thing. So you don't know tomorrow your numbers may be called. Every day they live in terror. Tomorrow it could be my number. Because they have a certain capacity for the gas chamber, they can only take that many numbers per day. And also for the world they don't want to look like a whole concentration camp died in one day. They died slowly out of sickness and illness and this and that. So every day fifteen, twenty numbers will be called and they have to go. So one day there is a young man, his number is called. The guy is terrified. He doesn't want to go. He's shivering and crying and this. A rabbi feared this, like this, took off his shirt. Give it to him, you wear my shirt with my number. I'll wear yours. In that fear and that overwhelmed with the possibility of death, he took out his shirt and gave it to him. The rabbi wore the shirt and went into the gas chambers. A few days later the war ended and this man lived. Somebody actually wore your shirt and went. He could have lived and this fellow, his whole life was transformed because of this. That this rabbi actually walked to death. Knowing fully well where he's going, no chance. When a man is like this, for him attaining is no problem.  But just being like this, that I'm willing to keep myself aside and do what's needed right now. Just very casually, no big fanfare about it. Now for this man, knowing the ultimate is not far away, it is very simple. Gautama said, when you're really hungry, when you badly need food, if you give away your food to somebody else, you will become stronger, not weaker. No logical human being would accept such nonsense, but it's hundred percent truth. If you know how to give away that which matters most to you at a certain moment, it leaves you enormously capable, enormously strong, not weak. To have that strength to give away what matters to you is not a small strength, it's a tremendous strength, isn't it? To give away things which don't matter to you, that means nothing. To give away that which matters most to you is everything, isn't it? Right now, what matters most to you is yourself, of course, isn't it? You say, no, no, no, my child, my husband, my wife, it's not true. What matters most to you is yourself, isn't it? If you can give yourself up, if you can just throw yourself into something, not because it is going to buy you a ticket to heaven, not because you will realize mukti out of it, not because of anything, simply to give away. You know Ramana said, atma gnanam ati solobham. Knowing the ultimate self-knowledge is the easiest thing, it is. If you know how to keep yourself aside, make an attempt to break the threshold of where you are, it can lead to lots of confusion. For those aspects for which you have no answers, I will answer. Just deepen the question, just make the question very intense within you, I will take care of the answer.


We're here on the 20th day of January in the year 2024 and we're diving into the practical application of the lessons of A Course in Miracles, the workbook lessons. And it's so freeing because when you really look at it, it's going to be the practical application in the transfer of training of the ideas to everyone and everything in every situation that actually brings in the joy. The more you transfer the ideas means the more you are unwilling to make exceptions to the ideas. You are actually opening your mind and opening your heart to the direct experience that the lessons point to. And that is an experience of love and peace and joy, serenity, tranquility. That is the peace of God. It's no small thing. And we're opening to, yeah, the purity of heart that Jesus talked about, blessed are the pure of heart for they shall see God. That's it. That's what we're into. So today we're going to look at lesson 28. Above all else, I want to see things differently. Today we are really giving specific application to the idea for yesterday. In these practice periods, you will be making a series of definite commitments. The question of whether you will keep them in the future is not our concern here. If you're willing, at least to make them now, you have started on the way to keeping them and we are still at the beginning. You may wonder why it is important to say, for example, above all else, I want to see this table differently. In itself, it is not important at all. Yet what is by itself and what does in itself mean? You see a lot of separate things about you, which really means you are not seeing it all. You either see or not, and you have seen one thing differently. You will see all things differently. The light you will see in any one of them is the same light you will see in them all. When you say above all else, I want to see this table differently, you're making a commitment to withdraw your preconceived ideas about the table and open your mind to what it is and what it is for. You are not defining it in past terms. You are asking what it is rather than telling it what it is. You are not binding its meaning to your tiny experience of tables, nor are you limiting its purpose to your little personal thoughts. You will not question what you have already defined, and the purpose of these exercises is to ask questions and receive the answers. In saying above all else, I want to see this table differently, you are committing yourself to seeing. It is not an exclusive commitment. It is a commitment that applies to the table just as much to anything else, neither more nor less. You could in fact gain vision from just that table if you would withdraw all your own ideas from it and look upon it with a complete open mind. It has something to show you, something beautiful and clean and of infinite value full of happiness and hope, hidden under all your ideas about its real purpose, the purpose it shares with all the universe is there. In using the table as a subject for applying the idea for today, you are therefore really asking to see the purpose of the universe. You will be making the same request of each subject that you use in the practice periods, and you are making a commitment to each of them to let its purpose be revealed to you, instead of placing your own judgment upon it. We will have six two-minute practice periods today in which the idea for today is stated first and then applied to whatever you see about you. Not only should the subjects be chosen randomly, but each one should be accorded equal sincerely, as today's idea is applied to it, in an attempt to acknowledge the equal value of them all in their contribution to your seeing. As usual, the applications should include the name of the subject your eyes happen to light on, and you should rest your eyes on it while saying, above all else I want to see this differently. Each application should be made quite slowly and as thoughtfully as possible. There is no hurry. So today Jesus tells us we will be making a series of definite commitments and that word he repeats in the lesson. It's going to take a commitment to see this world differently. It's going to take a commitment to train the mind. It will take a commitment to see with the vision of Christ. The commitment is in our mind and it's to wake up and to remember God, to know the creator. And we're applying this commitment now into shifting our perception to line up with the Holy Spirit and line up with Jesus. Jesus and the Holy Spirit see the world in a very holistic way, not in a fragmented way which has many parts and pieces. It's the perception that has been the problem, not the things that seem to be inside of the perception. For example, it's not really all the problems that are defined by the ego that are the problem. It's the ego itself and it's distorted perceptual world that is the problem. The belief in separation from God is the problem and the atonement is the solution. So today we're putting vision and seeing above everything else. That's why we're saying above all else I want to see things differently. And so our focus is the shift of perception away from the fragmented way that we have been accustomed to, perceiving linear time and subjects and objects and everything holding a space by itself and Jesus is reminding us in this lesson that nothing exists in and of itself. It's part of a tapestry, every image that is perceived as part of a tapestry and the lesson is saying we first must come to see the whole tapestry before we see the light that comes shining through the tapestry. So our goal is to see the whole tapestry. So in talking about commitment, I thought I would use some of the text from the course and let's go to chapter three, the innocent perception and jump down to section two, miracles as true perception. I have stated that the basic concepts referred to in this course are not matters of degree. Fundamental concepts cannot be understood in terms of opposites. It is impossible to conceive of light and darkness or anything and nothing as joint possibilities. They are all true or false. Either all true or they're all false. It is essential that you realize your thinking will be erratic until a firm commitment to one or the other is made. A firm commitment to darkness or nothingness, however, is impossible. No one has ever lived who has not experienced some light and some thing. No one therefore is able to deny truth totally, even if he thinks he can. Innocence is not a partial attribute. It is not real until it is total. The partly innocent are apt to be quite foolish at times. It is not until their innocence becomes a viewpoint with universal application that it becomes wisdom. Innocent or true perception means that you never misperceive and always see truly. Or simply, it means that you never see what does not exist and always see what does exist. How beautiful. Let's continue talking about the commitment that is required for this seeing, this vision of Christ. Let's go to chapter 7 and drop down to the section 6 and we're in the eighth paragraph here starting with the seventh sentence. If truth is total, the untrue cannot exist. Commitment to either must be total. They cannot coexist in your mind without splitting it. If they cannot coexist in peace, and if you want peace, you must give up the idea of conflict entirely and for all time. This requires vigilance, only as long as you do not recognize what is true. How you believe that two totally contradictory thought systems share truth. Your need for vigilance is apparent. That's very strong. So yeah, if truth is total, the untrue cannot exist. So commitment to either must be total and he's just calling us to make our commitment to the truth. He will later tell us the truth is true and only the truth is true. So he's really focusing on opening to what is real and what is true. Her grip nine, your mind is dividing its allegiance between two kingdoms and you are totally committed to neither. Your identification with the kingdom is totally beyond question except by you when you are thinking insanely. What you are is not established by your perception and is not influenced by it at all. Perceived problems and identification at any level are not problems of fact. They're problems of understanding since their presence implies a belief that what you are is up to you to decide. The ego believes this totally being fully committed to it. It is not true. The ego therefore is totally committed to untruth, proceeding in total contradiction to the Holy Spirit and to the knowledge of God. And then we turn ourselves to the total commitment that every commitment that we make in terms of our spiritual awakening. You might say your commitment to a course in miracles as a path or your commitment to doing the workbook or your commitment to practice the lessons throughout the day or any other commitment in terms of relationship commitments, job commitments, different family commitments in terms of responsibility and so on and so forth. These are all temporary commitments by definition and they just give us a sense of confidence and trust and expanding faith as we move deeper and deeper inward toward the light. But the temporary commitments will fade away and give way to the total commitment and that is basically what our lesson is today. You know, again, we're saying that I am determined above all else I want to see things differently and that commitment to that seeing is the commitment to accept the Atonement. So, let's go to chapter 2 and to section 2, the Atonement as Defense. Jesus says, you can do anything I ask. I have asked you to perform miracles and it made clear that miracles are natural, corrective, healing and universal. There is nothing they cannot do but they cannot be performed in the spirit of doubt or fear. When you are afraid of anything, you are acknowledging its power to hurt you. Remember that where your heart is, there is your treasure also. You believe in what you value. If you are afraid, you are valuing wrongly. Your understanding will then inevitably value wrongly and by endowing all thoughts with equal power will inevitably destroy peace. That is why the Bible speaks of the peace that patheth the understanding, the peace of God which patheth understanding. This peace is totally incapable of being shaken by errors of any kind. It denies the ability of anything not of God to affect you. This is the proper use of denial. It is not used to hide anything but to correct error. It brings all error into the light and since error and darkness are the same it corrects error automatically. True denial is a powerful protective device. You can and should deny any belief that error can hurt you. This kind of denial is not a concealment but a correction. Your right mind depends on it. Denial of error is a strong defense of truth but denial of truth results in miscreation, the projections of the ego. In the service of the right mind, the denial of error frees the mind and reestablishes the freedom of the will. And the will is really free. It cannot miscreate because it recognizes only truth. And then we jump down to paragraph four in this section two. The atonement is the only defense that cannot be used destructively because it is not a device you made. The atonement principle was in effect long before the atonement began. The principle was love and the atonement was an act of love. Acts were not necessary before the separation because belief in space and time did not exist. It was only after the separation that the atonement and the conditions necessary for its fulfillment were planned. Then a defense so splendid was needed that it could not be misused, although it could be refused. The atonement could not, however, turn it into a weapon of attack, which is the inherent characteristic of other defenses. The atonement thus becomes the only defense that is not a two-edged sword. It can only heal. The atonement was built into the time space believed to set a limit on the need for the belief itself and ultimately to make learning complete. The atonement is the final lesson. Learning itself, like the classrooms in which it occurs, is temporary. The ability to learn has no value when change is no longer necessary. The eternally created have nothing to learn. You can learn to improve your perceptions and can become a better and better learner. This will bring you closer and closer to chord with the sonship. As the sonship itself is perfect creation and perfection is not a matter of degree. Only while there is the belief in differences is learning meaningful. Evolution is a process in which you seem to proceed from one degree to the next. You correct your previous missteps by stepping forward. This process is actually incomprehensible in temporal terms because you return as you go forward. The atonement is the device by which you can free yourself from the past as you go ahead. It undoes your past errors, thus making it unnecessary for you to keep retracing your steps without advancing to your return. In this sense, the atonement saves time, but like the miracle it serves does not abolish it. As long as there is need for atonement, there is need for time. But the atonement as a completed plan has a unique relationship to time. Until the atonement is complete in its various phases, it will proceed in time. But the whole atonement stands at time's end. At that point, the bridge of return has been built. And here's the important part. The atonement is a total commitment. You may still think this is associated with loss, a mistake all the separated sons of God make in one way or another. It is hard to believe a defense that cannot attack is the best defense. This is what is meant by the meek shall inherit the earth. They will literally take it over because of their strength. A two-way defense is inherently weak precisely because it has two edges and can be turned against you very unexpectedly. This possibility cannot be controlled except by miracles. The miracle turns the defense of atonement to your real protection. And as you become more and more secure, you assume your natural talent of protecting others, knowing yourself as both a brother and a son. So that is a call to deep, deep commitment, deep devotion.  A devoted life is a life that is dedicated to one purpose. And today we're practicing that purpose is the lesson above all else. I want to see things differently. And this is the devotion, the dedication to be lighthearted, to be happy, to be kind, to be gentle, to let the spirit flow through you to you and be perceived in everything as a softness. It's like a soft joining in which you don't separate out, you don't focus on differences. You focus on the sameness. You focus that we are the same mind, we are the same one, we are the same Christ. And the Christ is very, very loving and very, very loving. So thank you so much for joining me today. We are floating in grace today as we are really wanting to see things differently. Above all else, I want to see things differently. Amen. Blessings.