k 2ds00 statistics 1 for chemical engineering lecture 5
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2DS00
Statistics 1 for Chemical
Engineering
lecture 5
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Week schedule
Week 1: Measurement and statistics
Week 2: Error propagation
Week 3: Simple linear regression analysis
Week 4: Multiple linear regression analysis
Week 5: Non-linear regression analysis
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Detailed contents of week 5
• intrinsically linear models
• well-known non-linear models
• non-linear regression
– model choice
– start values
– Marquardt and Gauss-Newton algorithm
– confidence intervals
– hypothesis testing
– residual plots
– overfitting
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Approaches to non-linear models
1. transformation to linear model
2. approximation of non-linear model by linear model (linearization
through Taylor approximation)
3. non-linear regression analysis (numerically find parameters for
which sum of squares is minimal)
Remark: although 2) is often applied in the chemical literature, we
strongly recommend against this procedure because there is no
guarantee that it yields accurate results.
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Intrinsically linear models
Some non-linear models may be transformed into linear models
10
xy e
transformed model must fulfil usual
assumptions!
0 1
1 0 1
ln( ) ln( )
y x
y b b x
10 .xy e e
0 1ln( ) ln( ) y x
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Examples of non-linear models
exponential growth model
Mitscherlich model
inverse polynomial model
logistic growth model
Gompertz growth model
Von Bertalanffy model
Michaelis-Menten model
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Exponential growth model
101
teYYt
Y
i
t
iieY
10i
t
iieY 1
0
non-linearadditive error term
intrinsically linearmultiplicative error term
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Mitscherlich model
Yx
Y
01
ix
iieY 110
horizontalasymptote
determines speed of growth
ifmonomolecular model
12
e
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Inverse polynomial model
201 Y
x
Y
ii
ii x
xY
10
Slow convergence to asymptote 1/ 1
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Logistic (autoclytic) growth model
0
01
YY
t
Y
iti ieY
12
0
1
2
0
1)0(
Y 0 is horizontal asymptote
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Gompertz growth model
YY
x
Y 01 ln
ie
i
it
eY
12
0
Logarithm of Gompertz curve is monomolecular curve
horizontalasymptote
determines growth speed
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Von Bertalanffy model
i
mtmi
ieY )1/(11
01
Special cases of this general model are:• m=0 en : monomolecular model• m=2 en : logistic model• m1: Gompertz model
10
e
02 /
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Michaelis-Menten model
itt ii eeY 21 11 21
This model is often used to describe diffusion kinetics
Watch out for overfitting in model with many parameters.
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Marquardt algorithm
Non-linear regression requires numerical search for parameter values
that minimise error sum of squares.
Most important algorithms:
1. Gauss-Newton algorithm (uses 1st-order approximation; may
overshoot minimum)
2. steepest descent algorithm (searches for direction with largest
downhill slope; may be slow)
3. Marquardt algorithm (switches according to situation between
above mentioned algorithm)
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Gauss algorithm
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Marquardt algorithm
Choice between both methods is determined by Marquardt
parameter :
0 algorithm approaches to Gauss-Newton
algorithm approaches to steepest descent
The Marquardt algorithm is (deservedly) the most used method in
practice.
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Numerical search for minimum of error sum of squares
local minimumtrue minimum
Where should we start the numerical search?
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Choice of start values
• inspect data and use interpretation of parameters in model
– parameter is related to value of asymptote
– model value at certain setting
• use linear regression to obtain approximations to parameter values
– transform model to linear model
– approximate model by linear model
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Possible causes for non-convergence
• model does not match data
• badly determined numerical derivatives
• overfitting:
– model has too many parameters
– some model parameters have almost same function
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Important issues in non-linear regression analysis
• carefully consider choice of model
• choose starting values that relate to the model at hand
• experiment with different starting values to prevent convergence to
local minimum
• watch out for overfitting
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Fritz and Schluender equation: start values for a and b
For C2=0, this reduces to
Use first 10 measurements
(i.e., those with C2=0) to
obtain start values for a and b.
1
31
11
1 2 2
b X
XX
aCq
C X C
1 1bq aC
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Fritz and Schluender equation: other initial values
1
3
1
3
1
3
Xb1
X22
Xb1
X22
b11
Xb1
X22
b11
aC
CXy
aC
CX
aC
1
q
1
aC
CX
aC
1
q
1
)Cln()Cln()yln(
)Cln()Xb()Cln(X)a
Xln()yln(
)Cln()Xb()aln()Cln(X)Xln()yln(
12210
11232
11232
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Examination• bring your notebook
• Monday October 6, 14.00 – 17.00 in Paviljoen J17 and L10 (not
Auditorium)
• clean copy of Statistisch Compendium is allowed
• contents:
– one exercise on error propagation
– three statistical analyses to be performed on your notebook