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SUPPORT VECTOR MACHINE,[object Object],BY PARIN SHAH,[object Object]
SVM FOR LINEARLY SEPARABLE DATA,[object Object],Plot the points.,[object Object],Find the margin and support vectors.,[object Object],Find the hyperplane having maximum margin.,[object Object],Based on the computed margin value classify the new input data sets into different categories.,[object Object]
FIGURE REPRESENTING LINEARLY SEPARABLE DATA,[object Object],Figure representing the support vector and maximum margin hyper plane.,[object Object],                          (w · x) + b = +1 (positive labels),[object Object],                          (w · x) + b = -1 (negative labels),[object Object],(w · x) + b = 0 (hyperplane),[object Object], ,[object Object],Margin       ::    ,[object Object]
SVM FOR NON LINEARLY SEPARABLE DATA,[object Object]
STEPS FOR NON LINEARLY SEPARABLE DATA,[object Object],1.) Map into feature space.,[object Object],2.) Use Polynomial kernel Φ(X1) = (X1, X1^2) to ,[object Object],     map points.,[object Object],3.) Compute the positive , negative and zero ,[object Object],hyperplane.,[object Object],4.) We get the support vectors and the margin value ,[object Object],     from it. ,[object Object],5.) Classify the new input values from margin value ,[object Object]
KERNEL AND ITS TYPES.,[object Object],Computation of various points in the feature space can be very costly because feature space can be typically said to be infinite-dimensional.,[object Object],The kernel function is used for to reduce these cost because the data points appear in dot product and the kernel function are able to compute the inner products of these points. ,[object Object],By kernel function we can directly compute the data points through inner product without explicitly mapping on the feature space.,[object Object]
KERNEL AND ITS TYPES.,[object Object],1.)  Polynomial kernel with degree d.,[object Object], ,[object Object], ,[object Object],2.)  Radial basis function kernel with width s ,[object Object], ,[object Object],  ,[object Object],3.)  Sigmoid with parameter k and q ,[object Object], ,[object Object], ,[object Object], ,[object Object],4.)  Linear Kernel,[object Object], K(x,y)= x' * y,[object Object]
SPARSE MATRIX AND SPARSE DATA,[object Object],Simple data structure of 2-dimensional array storing non-zero values.,[object Object],Sparse Data iterates over non-zero values only.,[object Object],Stores the values, row number and column number of non-zero values from the matrix.,[object Object],Easy to compute the inner product of  zeroes.,[object Object],Speed of SVM algorithms increases by use of Sparse data. ,[object Object]
STORING SPARSE DATA,[object Object],Dictionary of keys (DOK),[object Object],DOK represents non-zero values as a dictionary mapping (row, column) tuples to values,[object Object], ,[object Object],List of lists (LIL),[object Object],LIL stores one list per row, where each entry stores a column index and value. Typically, these entries are kept sorted by column index for faster lookup. ,[object Object], ,[object Object],Coordinate list (COO),[object Object],COO stores a list of (row, column, value) tuples. In this the entries are sorted (row index  then column index  value) to improve random access times. ,[object Object], ,[object Object],Yale format,[object Object]
STORING SPARSE DATA,[object Object],The Yale Sparse Matrix Format stores an initial sparse m×n matrix,,[object Object],     Where M = row in three one-dimensional arrays. ,[object Object],                NNZ = number of nonzero entries of M. ,[object Object],                Array A = length= NNZ, and holds all nonzero entries. Order-top bottom right left.,[object Object],                Array IA= length is m + 1.  IA(i) contains the index in A of the first nonzero element of row i. ,[object Object],                                     Row i of the original matrix extends from A(IA(i)) to A(IA(i+1)-1), i.e. from the start  ,[object Object],                                     of one row to the last index before the start of the next. ,[object Object],                Array JA= column index of each element of A, length= NNZ.,[object Object],EXAMPLES:::,[object Object], [ 1 2 0 0 ],[object Object], [ 0 3 9 0 ],[object Object], [ 0 1 4 0 ],[object Object], ,[object Object],    So computing it we get values as,,[object Object],         A  = [ 1 2 3 9 1 4 ]  ,    IA = [ 0 2 4 6 ]      and  JA = [ 0 1 1 2 1 2 ].,[object Object], ,[object Object]
ADVANTAGES OF SVM,[object Object],In high dimensional spaces Support Vector Machines are very effective.,[object Object],When number of dimensions is greater than the number of samples in such cases also it is found to be very effective.,[object Object],Memory Efficient because it uses subset of training points(support vectors) as decisive factors for classification.,[object Object],Versatile:  For different decision function we can define different kernel as long as they provide correct result. Depending upon our requirement we can define our own kernel.,[object Object]
DISADVANTAGES OF SVM,[object Object],If the number of features is much greater than the number of samples, the method is likely to give poor performances. It is useful for small training samples.,[object Object],SVMs do not directly provide probability estimates, so these must be calculated using indirect techniques.,[object Object],We can have Non-traditional data like strings and trees as input to SVM instead of featured vectors.,[object Object],Should select appropriate kernel for their project according to requirement,[object Object]

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