PATTERN RECOGNITION : PRINCIPAL COMPONENTS ANALYSIS

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PATTERN RECOGNITION PRINCIPAL COMPONENTS
ANALYSIS Richard Brereton r.g.brereton_at_bris.ac.
uk
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• NEED FOR PATTERN RECOGNITION
• Exploratory data analysis
• e.g. PCA
• Unsupervised pattern recognition
• e.g. Cluster analysis
• Supervised pattern recognition
• e.g. Classification

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Case study Coupled chromatography in HPLC
profile
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MULTIVARIATE DATA

Wavelength columns

Time rows

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• DATA MATRICES
• The rows do not need to correspond to elution
  times in chromatography they can be any type of
  sample
• Blood sample
• Wood
• Chromatograms
• Samples from a reaction mixture
• Chromatographic columns

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• The loadings do not need to correspond to
  spectral wavelengths they can be any type of
  sample
• NMR peak heights
• Atomic spectroscopy measurements of elements
• Chromatographic intensities
• Concentrations of compounds in a mixture
• Results of chromatographic tests

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Return to example of chromatography. Rows
elution times Columns wavelengths
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Chemical factors X C.S E
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It would be nice to look at the chemical factors
underlying the chromatogram. We can use
mathematical methods to do this.
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ABSTRACT FACTORS PRINCIPAL COMPONENTS
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X T . P E C . S E
T are called scores these correspond to elution
profile P are called loadings these correspond
to spectra Ideally the size of T and P equals
the number of compounds in the mixture. This
size equals the number of principal components,
e.g. 1, 2, 3 etc. Each PC has an associated
scores vector (column of T), and loadings vector
(column of P).
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• Hence if the original data matrix is dimensions
  30 ? 28 (or I ? J) ( 30 elution times and 28
  wavelengths - or 30 blood samples and 28 compound
  concentrations - or 30 chromatographic columns
  and 28 tests) and if the number of PCs is denoted
  by A, then
• the dimensions of T will be 30 ? A, and
• the dimensions of P will be A ? 28.

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A major reason for performing PCA is data
simplification. Often datasets are very complex,
it is possible to make many measurements, but
only a few underlying factors. See the wood from
the trees. Will look at this in more detail
later.
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• SCORES AND LOADINGS HAVE SPECIAL MATHEMATICAL
  PROPERTIES
• Scores and loadings are orthogonal.
• What does this mean?
• Loadings are normalised.
• What does this mean?

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PCA is an abstract concept. Theory.
Non-mathematical Spectrum recorded at different
concentrations and several wavelengths
wavelength 6 versus 9 six spectra.
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• Each spectrum becomes ONE POINT IN 2 DIMENSIONAL
  SPACE
• (2D 2 wavelengths)
• Spectra
• Fall on a straight line which is the FIRST
  PRINCIPAL COMPONENT
• The line has a DIRECTION often called the
  LOADINGS corresponding to the SPECTRAL
  CHARACTERISTICS
• Each spectrum has a DISTANCE along the line often
  called the SCORES corresponding to CONCENTRATION

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• EXTENSIONS TO THE IDEA
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• Measurement error
• Several wavelengths
• Several compounds

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Measurement error

• Best fit straight line - statistics
• Two PCs - the second relates to the error around
  the straight

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• Several wavelengths
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• Now no longer a point in 2 dimensional space.
• Typical spectrum. Several thousand wavelengths
•  The number of dimensions equals the number of
  wavelengths.
• The spectra still fall (roughly) on a straight
  line.
• A point in 1000 dimensional space.

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Several compounds Two compounds, two wavelengths.
A
B
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• RANK AND EIGENVALUE
• How many PCs describe a dataset?
• Often unknown
• How many compounds in a series of mixtures?
• How many sources of pollution?
• How many compounds in a reaction mixture?
• Sometimes just statistical concept.
• Sometimes mixture of physical and chemical
  factors, e.g. a reaction mixture compounds,
  temperature etc.

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• EVERY PRINCIPAL COMPONENT HAS A CORRESPONDING
  EIGENVALUE
• The eigenvalue equals the sum of squares of the
  scores vector for each PC.
• The more important the PC the bigger the
  eigenvalue.
• The sum of squares of the eigenvalues of a matrix
  should never exceed that of the original matrix.
• The sum of squares of all significant PCs should
  approximate to that of the original matrix.

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RESIDUAL SUM OF SQUARES decreases as the number
of eigenvalues increases. Log eigenvalue versus
component number. Cut off?
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SEVERAL OTHER APPROACHES FOR THE DETERMINATION OF
NUMBER OF EIGENVALUES.
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• SUMMARY SO FAR
• PCA
• Principal components how many?
• Scores
• Loadings
• Eigenvalues

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GRAPHIC DISPLAY OF PCS SCORES PLOT PC2 VERSUS PC1

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SCORES AGAINST TIME PC1 AND PC2 VERSUS TIME
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LOADINGS PLOT PC2 VERSUS PC1
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FOR REFERENCE pure spectra
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LOADINGS AGAINST WAVELENGTH PC1 AND PC2 VERSUS
WAVELENGTH
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BIPLOTS SUPERIMPOSING SCORES AND LOADINGS PLOTS
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• MANY OTHER PLOTS
• Not only PC2 versus 1, also PC3 versus 1, PC3
  versus 2 etc.
• 3D PC plots, 3 axes, rotation etc.
• Loadings and scores sometimes presented as bar
  graphs, not always a sequential meaning.
• Plots of eigenvalues against component number

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• DATA SCALING AND PREPROCESSING
• Influences appearance of plots
• Column centring common in traditional
  statistics
• Standardisation of columns subtract mean and
  divide by standard deviation.
• If data of different types or absolute scales
  this is an essential technique
• Row scaling to constant total

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ANOTHER EXAMPLE Grouping of elements from
fundamental properties using PCA.
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Step 1 standardise the data. Why? On different
scales.
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PERFORM PCA Choose the first two PCs Scores plot
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Loadings plot
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• SUMMARY
• Many types of plot from PCA.
• Interpretation of the plots.
• Preprocessing important.