Principal Component Analysis

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Principal Component Analysis

• Adam Day

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Introduction

• Principal Component Analysis (PCA) is a
  statistical technique, it can be used for many
  things including data compression and shape
  recognition.
• A fluorescence spectrum has a shape which PCA can
  recognise.

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Method of PCA part 1.The Covariance Matrix

• It is important to understand a set of data in
  terms of its dimensions and how they vary
  together.
• For a set of 2-dimensional data it is important
  to know if high values in one dimension cause
  values in the other to be high, or the opposite
  effect, where high values in one dimension cause
  there to be low values in another dimension.

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Method of PCA part 1.The Covariance Matrix

• One way of telling if one dimension of data
  affects another is to find the covariance between
  the two dimensions.
• Covariance can be derived from the formula for
  standard deviation, where X and Y are data
  dimensions

It should be noted here that the first step in
finding the covariance is to subtract the mean of
data in a dimension from each value in that
dimension. This data set, called row data
adjust, will be used again later.
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Method of PCA part 1.The Covariance Matrix

• When covariance between two dimensions is
  positive the values in those two dimensions
  increase together, when it is negative they
  decrease together.
• When a data set comprises many dimensions, it is
  important to lay out the covariance values in a
  matrix. That is, the covariance matrix.

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Method of PCA part 1.The Covariance Matrix

• From the definition of covariance, it should be
  obvious that
• A consequence of this is that (in the case of
  3dimensional data) the covariance matrix is
  always a symmetric square matrix with the
  variances down the main diagonal.

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Method of PCA part 2.Finding Principal Components

• The principal components of the data set are,
  simply, the eigenvectors of the covariance
  matrix.
• There are as many eigenvectors as there are
  dimensions in the data set.
• The eigenvector with the highest corresponding
  eigenvalue contains the most information about
  the original data set. This information will
  correspond to large features in the shape of the
  original data set.

First Principal Component
Original Data
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Method of PCA part 2.Finding Principal Components

• As the value of the corresponding eigenvalue
  falls, so too does the size of the features
  represented by the eigenvector. In this specimen
  set of results, principal components 1-5 are
  shown of a set of data comprising 4 peaks of
  random height added to a set of 200 points of
  random noise. Notice that the peak heights fall
  as PC number rises, in fact all of the principal
  components above PC4 contain only noise.

Original Data
PC1
PC2
PC3
PC4
PC5
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Method of PCA part 3Data Reconstruction

• If the matrix dot product of this principal
  component and the row data adjust data set were
  taken, then the output data set (called final
  data) would have a similar shape to that of the
  original data however it is likely that it
  would be a poor representation of the original
  data.

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Method of PCA part 3Data Reconstruction

• If the dot product of all the principal
  components lined up in a matrix (called a row
  feature vector) and the row data adjust data
  set was taken it would be exactly the original
  data set returned.
• So, if we could choose which principal components
  were put into the row feature vector, then it
  would be possible to control the detail of the
  data returned.
• In the previous example, it was clear that there
  was no significant data relating to the peaks in
  the input graph in any principal component above
  number 4.

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PCA in Fluorescence Spectroscopy

• In the study of drug fluorescence, it is
  necessary to study spectra similar to the
  patterns of data shown in previous examples.
• These are created by shining a light from a
  source onto a drug sample, a spectrum is then
  reflected and received by a computer as data.
• In these spectra there is likely to be erroneous
  noise and large shapes caused by unwanted signals
  such as reflection of light from a source, and
  noise caused by a number of things such as
  electrical interference and fluctuations in the
  output of the source.
• Principal Component Analysis will be able to
  remove these unwanted shapes from spectra by
  careful creation of the row feature vector
  mentioned earlier.