definition and overview of chemometrics
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Definition and overview of chemometrics. Paul Geladi. Head of Research NIR CE Chairperson NIR Nord Unit of Biomass Technology and Chemistry Swedish University of Agricultural Sciences Umeå Technobothnia Vasa paul.geladi @ btk.slu.se paul.geladi @ syh.fi. Project geography. - PowerPoint PPT PresentationTRANSCRIPT
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Definition and overview of chemometrics
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Paul Geladi
Head of Research NIRCEChairperson NIR Nord
Unit of Biomass Technology and ChemistrySwedish University of Agricultural SciencesUmeåTechnobothniaVasa
paul.geladi @ btk.slu.se paul.geladi @ syh.fi
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Project geography
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Chemometrics
Mathematics
Statistics
Computer Science
In Chemistry
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Similar fields
• Biometrics ±1900
• Psychometrics ±1930
• Econometrics ±1950
• Technometrics ±1960
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Chemometrics
• Design of Experiments (DOE)
• Exploratory Data Analysis
• Classification
• Regression and Calibration
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Design of Experiments
• Most important where possible
• Uses:
• ANOVA
• F-test
• t-test
• Plots
• Response Surfaces
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Design of Experiments
y = b0 + b1x1 + b2x2 +...+bKxK + b11x12 +
b22x22 +...+ bKKxK
2 + b12x1x2 +...+
Factors x1, x2,...xK changed systematically
Response y measured and modeled
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Exploratory Data Analysis
• Design not possible• Sampling situations• Find structure• Find groupings• Find outliers
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Classification
• Check for groupings = UNSUPERVISED• Existing groupings = SUPERVISED• Visualize groupings• Classify• Test
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Regression / Calibration
• Two types of variables X / y
• Relationship linear / nonlinear
• Model
• Diagnostics
• Residual
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x
y
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Multivariate Data Analysis
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Multivariate Data Analysis
• Sampled data and design with too many reponses:• Mining• Hospitals• Agriculture• Food industry• More
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Nomenclature
• Samples are objects
• What is measured on the object is a variable
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34.92 Spectrum
Samples
Vectors
1 K1
I
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123.6
11.15.9340.51.417
A vector is a collectionof numbers.
It is always a columnvector.
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The transpose of a vector is a row vector.
Symbols for transpose are’ and T. a’ or aT.
12 3.6 11.1 5.9 34 0.5 1.4 17
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0 5 10 15 20 250
2
4
6
8
10
12
14
16
18
Particle size, 1 sample
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0 5 10 15 20 25 30 35 400
2
4
6
8
10
12
Small particles, 35 samples
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The Data Matrix
A data matrix is a vector of vectors
I
K
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0 5 10 15 20 250
5
10
15
20
25
30
35
40
Size histograms, all samples
Particle area
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0 200 400 600 800 1000 12000
0.5
1
1.5
2
2.5
3
3.5
4
NIR wavelengths
Times in batch reaction
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Geometry of multivariate space
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Problem
I and K can be large
Correlation
Univariate statistics does not apply
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I patients
3 variables: blood oxygen,iron, hemoglobin
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O2
Fe
Hb
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O2
Fe
Hb
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O2
Fe
Hb
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O2
Fe
Hb
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O2
Fe
Hb
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O2
Fe
Hb
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O2
Fe
Hb
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O2
Fe
Hb
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O2
Fe
Hb
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Properties of multivariate spaceRotation
vectors unchanged / distance unchanged
Translation
vectors changed / distance unchanged
Rescaling / change units
all changes
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Consequences
• We can move the coordinate sytem around
• The relative distances between objects do not change
• We can rotate the coordinate system
• Scale changes are important
• Move coordinate system to center of data
• Scale properly
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Vectors (physics)
x = [ x1, x2, x3 ]
|| x || = ( x12 + x2
2 + x32 ) 1/2
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Geometry
a
b
cc2 = a2 + b2
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Vectors (K dimensions)
x = [ x1, x2,..., xK ]
|| x || = ( x12 + x2
2 +...+ xK2 ) 1/2
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Problem
We can not see in more than 3 dimensions
Paper, computer screen: 2-2.5 dimensions
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O2
Fe
Hb
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O2
Fe
Hb
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Projection
2D plane (screen, paper)
Many projections possible
Find a good one
Find a few good ones
What is good?