10.34, Numerical Methods Applied to Chemical Engineering
Professor William H. Green
Lecture #25: Conclude Models vs. Data
Parameter Estimation
1) Model definition/Formulation:
choosing θ
2) Compile/Assess what you already knew before adjusting θ
a. estimate parameters, error bars
b. initial guess θ
3) Adjust θ
a. Determine if Model is Consistent with Data: if inconsistent, you have learned
something important
b. θbestfit at θlocalminima of χ2(θ)
4) Refine p(θ): narrow range for the parameters
a. summarize what we have learned
4-STEP PROCESS IS ALSO CALLED: “LEAST-SQUARES FITTING”
1) Repeat measurement “i” Nreplicates times Yi(j) j = 1,Nreplicates
N rep
Yi data =
χ =
2
∑ Yi
j =1
N rep
N data ⎛
∑
i =1
N rep
( j)
Y data − Yi model (θ ) ⎞
⎜ i
⎟
⎜
⎟
σ
i
⎝
⎠
σi =
∑ (Yi( j ) − Yi data )
2
j =1
N rep − 1
2
Quantitative Definition of “Consistent”
calc χ2(θ; Ydata)
Prob(measure Ydata with χ2 ≥ χ2expt)
Î
∫
∞
2
xexpt
2
⎛ v xexpt
Γ⎜ ,
⎜2 2
t e
dt = ⎝
N
⎛v⎞
⎛v⎞
2 2 Γ⎜ ⎟
Γ⎜ ⎟
⎝2⎠
⎝2⎠
v
t
−1 −
2
2
{if small, unlikely that our model is right}
⎞
⎟
⎟
⎠
ν = Ndata - Nparams_adjusted
GAMMA FUNCTION
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
⎛v χ2
Γ⎜⎜ ,
⎝2 2
If
⎛v⎞
Γ⎜ ⎟
⎝2⎠
⎞
⎟⎟
⎠ < 0.01 Æ very unlikely model is consistent with data
χ2(θ) < χ2max for consistency
If you have more data, you have more confidence. Need lots more data than number of
θ’s.
If
>>
: inconsistent
If
=
: consistent
2
P(χ )
χ2expt
Ndata
[χ2]
Figure 1. Chi-squared distribution tests.
MatLab: chi2cdf.m
inconsistent
∆H1
⎛ ⎛ k1 ⎞
⎞
⎟⎟, Tdata ⎟⎟
⎝ ⎝ ΔH 1 ⎠
⎠
x
2 ⎜
χ exp
⎜ ⎜⎜
∆H
consistent
k1
k
θthat is inconsistent with data
neglect
Figure 2. An example of two parameter fitting.
χ2(θ) = χ2(θbestfit) + ½(θ – θbest)T*H(θbest)*(θ – θbest) + O(∆θ3)
H ≈ JTJ
Jin =
∂Yi model
∂θ n
θ best
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 25
Page 2 of 3
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
χ2 = χ2best + 2
χ2 = χ2best + 1
∆H
δ
k
χ2(θ) = χ2m=x
\Figure 3. Contours around best fit.
People want these contours to be circles
Range of parameters that are acceptable: ∆H ± δ(∆H)
k = kbest + δk
Covariance Matrix
Equilibrium
Concentration
kinetics experiments
give the ellipses
∆Hdist.
P(θ)
Experiment
∆H
gives the
values of
k
the
horizontal
Figure 4. Each experiment tells you about different “cuts” or ellipses and
cuts.
where they all intersect is the answer.
BAYESIAN: store p(θ)
STORE ALL THE DATA: Thermochemistry Active Tables, PrIMe
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 25
Page 3 of 3
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].