Matti HotokkaPhysical chemistryÅbo Akademi University Chemometrics

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Chemometrics

Matti Hotokka Physical chemistry Åbo Akademi University

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. The mean can be understood as the value ì which gives least deviations.

Linear regression

Mean

x₁-ì

ì = E(x)

x₂-ì

x

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. The mean can be understood as the value ì which gives least deviations.

Linear regression

Mean

x₁-ì

ì = E(x)

x₂-ì

x

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Linear regression

Some math: mean

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Linear regression

x y

y^calc(x) = a + bx

y₁ - y^calc

y₂ - y^calc

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Linear regression

x y

y^calc(x) = a + bx

y₁ - y^calc

y₂ - y^calc

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Linear regression

Formulas

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Linear regression

Formulas

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Linear regression

Formulas

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Linear regression

Formulas

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Linear regression

Formulas

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. Homogenous variance:The distribution of y is the same for each x.

Linear regression

Assumptions

x y

P(x|y)

x₁ x₂ x₃

ó

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. The population behaves linearily: The true regression obeys the rule á+âx

Linear regression

Assumptions

x y

P(x|y)

x₁ x₂ x₃

á+âx (true)

a+bx (estimate)

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. The random variables y₁, y₂, y₃, ... are statistically independent.

Linear regression

Assumptions

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. The residual variance is

Linear regression

Quality: the basic quantity

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. The estimated standard error of the slope is

Linear regression

Goodness of the slope

. Confidence interval (95 %)

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Small range of x gives poor b because Gx² is small.

Linear regression

How to improve a?

A large range gives smaller SE.

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Linear regression

How good is y^calc?

y^calc = a + b@x₀

ì₀ = true estimate = á + âx₀

x₀

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Linear regression

How good is y^calc?

y^calc = a + b@x₀

ì₀ = true estimate = á + âx₀

x₀

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Linear regression

How good is y^calc?

ì₀ = true estimate = á + âx₀ y^calc = a + b@x₀

x₀

y₀ = true value in population

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Confidence intervals for ì₀ and y₀.

Linear regression

How good is the regression

ì₀ y₀

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Slope of regression

Correlation

Regression vs correlation

Correlation

Relation

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Correlation

Different cases

r = -1.0 r = -0.8

r = 0.0

r = 0.6 r = 1.0

r = 0.0

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Correlation

Explained SSQ

Total SSQ

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Correlation

Explained SSQ

Total SSQ

Explained SSQ

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Correlation

Explained SSQ

Total SSQ

Explained SSQ

Unexplained SSQ

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Correlation

Coefficient of determination

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Multivariate regression

Why?

y

x

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A second regressor explains the spread.

Multivariate regression

Why?

y

x₁ x₂=3

x₂=2 x₂=1

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Multiple regression considers all regressors.

Multivariate regression

What?

x₁

x₂ y

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Multivariate regression

Indirect effects

x₁=Conc

y=Absorbance

x₂=pH

a₁= 0.059

a₂= -0.16 a= -0.032

Total effect = 0.059 + 0.005 = 0.064

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Multivariate regression

Indirect effects

x₁=Conc

y=Absorbance

x₂=pH

a₁= 0.059

a₂= -0.16 a= -0.032

Total effect = 0.059 + 0.005 = 0.064

Linear regression: y = a x₁ + b: a = 0.064 Gives the total effect!

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Change notations for simplicity

Multivariate regression

Math

Similarly

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A linear system of equations for the unknowns a, b and c can be written in matrix form as

Multivariate regression

Math

The unity is included to show clearly the structure of the equation.

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Rename variables

Nonlinear regression

Make it linear

Take logarithm

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Methods still exist. Just minimize the spread.

Nonlinear regression

If you cannot make it linear

However, the system of equations is not linear anymore. Therefore an iterative process is needed to solve it.

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