The idea of principal coordinates analysis pca is

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The idea of Principal Coordinates Analysis (PCA) is firstly to represent the different objectsunder consideration (in this case the wines) graphically, as pointsv1,v2, . . . ,vninRpfor somesuitable choice of dimensionp. The distancekvi-vjkbetween points in the plot is chosen to reflectas closely as possible the entryTijin the dissimilarity matrixT. Our eye is much more capable ofspotting relationships and patterns in such a plot than it is spotting them in the raw data. Howeveranother problem arises in doing this: we can only visualize two or possibly three dimensions (p= 2or 3) whereas the data would most naturally be graphed in a much higher dimensional, possiblyevenn-1-dimensional, space. The second key idea of PCA is to reduce the number of dimensionsbeing considered to those in which the variation is greatest.IV.8.2. Definitions and useful propertiesA real square matrixAis calledpositive definiteifhx, Axi>0for anyx6=0.A real square matrixAis calledpositive semi-definiteifhx, Axi ≥0for anyx6=0.A real matrix of the formATAfor any real matrixAis always positive semi-definite becausehx, ATAxi=hAx, Axi=kAxk20.205
A positive semi-definite matrix has all non-negative eigenvalues. The proof is as follows. Letvbe an eigenvector ofAwith eigenvalueλsoAv=λv.BecauseAis positive semi-definite we havehv, Avi ≥0.Using the fact thatvis an eigenvector we also havehv, Avi=hv, λvi=λhv,vi.Combining these two results and remembering thathv,vi>0 for an eigenvector, we have ourresult:λ0.IV.8.3. Reconstructing points inRpWe begin with a simple example, taking our objects to be a set ofnpointsw1,w2, . . . ,wninRp.We take the dissimilarity between objectsiandjto be the usual distance between those pointsTij=kwi-wjk.Is it possible to reconstruct the relative location of the points from the data inTalone?It is important to notice that any reconstruction that we find is not unique. We are free to rotate,reflect and translate the points and they still satisfy the only requirement that we make of them,namely that the distances between the points are as specified.Reconstructing two points inRpConsider two points,w1andw2, that are a known distance, say 2, apart. A single line can alwaysbe placed between any two points and therefore we expect that the points can be represented inone dimension. If we only know the distance between the points, then a possible representation ofthem isv1=0andv2=2 .Reconstructing three points inRpNow consider three points,w1,w2andw3with dissimilarity matrix:T=035304540.206
A plane can always be placed through any three points and therefore we expect that the pointscan be represented in two dimensions. We can find such a representation using trilateration. Firstwe choose a point to representw1:1Next we draw a circle of radius 3 centred at our first point and choose a second point on thecircle to representw2:123

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Term
Fall
Professor
RICHARDFROESE
Tags
Linear Algebra, Algebra, Eigenvectors, Vectors, Matrices, Eigenvalues, Orthogonal matrix, Matlab Octave

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