15 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Linear and nonlinear data fitting

PRIMER D5D7D10

Pod 36.7 - 19.6 - 11.5 - 2.6 - 7.6 15.4 26.1 42.2 60.6 80.1 1510204060100200400760 . . . . . . . . . . 20202020202020202020410 mat transpose :

Vapor pressure = f(Temperature)

points Pod 3 col :

size Pod 4 col :

Temperature [C]

Pressure [atm]

t Pod 1 col :

p Pod 2 col :

p 10 b 1 el b 2 el T / + b 3 el T 10 log * + b 4 el T 2^* +^≡

b - unknown parameters T - temperature [K] p - vapor pressure [atm]

Proposed function for fitting the data:

Linear fitting

p 10 log b 1 el b 2 el T / + b 3 el T 10 log * + b 4 el T 2^* + ≡

Transforming the function in linear by parameters:

n k x 1 φ Xmat X lokal : j 1 k range i 1 n range X i j el x i el φ j el : for for X 31 line :

x φ 11 x / x 10 log x 2^41 mat :

Temperature [K]

log-pressure

i 1 p length range y i el p i el 10 log : for

T t 273 + :

n T length :

n 10

k x φ length :

k 4

X n k T T# φ Xmat :

Φ X transpose X * :

d X transpose y * :

b Φ 1 -^d * :

b 216.206 9295.488 - 75.568 - 4.438 105 -^* 41 mat

Parameters found

T b f b j el T φ j el * (j 1 k sum :

i 1 p length range logp.rac i el T i el b f : prac i el 10 logp.rac i el^: 21 line for

p 10 log T b f ≡

p T b f ≡

T y points size augment T logp.rac augment 2 1 sys

T 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 101 mat

p 1510204060100200400760101 mat

prac 0.9756 5.2795 10.2643 19.6362 37.7503 58.8879 101.5091 205.2313 404.3035 750.6837 101 mat

T p points size augment T prac augment 2 1 sys

Variance of parameters

s2 prac i el p i el - (2^(i 1 n sum n k - / :

s2 23.6

How to minimize this function (NLS problem)??

T b f1 10 T b f^:

b S p i el T i el b f1 - (2^(i 1 n sum :

b S 141.5368

Sum of squared residuals obtained by linear fitting Could that be reduced by NL fitting?

Levenberg-Marquardt based procedure

x b f x y b ν maxiter LMA T# b# f T# b# J m b# length : n T# length : i 1 m range j 1 n range Jac j i el T# j el b# f (b# i el diff : for for Jac 41 line : n x length : m b length : S0 y i el x i el b f (- (2^(i 1 n sum : I m identity : λ 1 : k 1 maxiter range J x# b# f x b J : j 1 n range f j el x j el b f : for δ J transpose J * λ I * + (1 -^J transpose * y f - (* eval : b b δ + : S1 y i el x i el b f (- (2^(i 1 n sum : S1 S0 < λ λ ν / : S0 S1 : 21 line λ λ ν * : if 61 line for S1 b 21 mat 81 line :

Guess values - from linear fitting

b 216.2057 9295.4881 - 75.5676 - 4.4384 105 -^* 41 mat

Damping changing parameter, ν, maximal number of iteration, maxiter (change this if needed)

ν 10 :

maxiter 5 :

res T# b# f1 T p b ν maxiter LMA :

res 7.8173 221.9812 9292.0229 - 78.1071 - 4.9921 105 -^* 41 mat 21 mat

S res 1 el :

S 7.8173

b res 2 el :

b 221.9812 9292.0229 - 78.1071 - 4.9921 105 -^* 41 mat

NLS values

Sum of squared residuals

S 7.8173

b S 7.8173

Reduced !

i 1 p length range prac i el T i el b f1 (: for

exp T p points size augment :

cal T prac augment :

exp cal 2 1 sys

T 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 101 mat

p 1510204060100200400760101 mat

prac 1.1517 5.7941 10.957 20.4307 38.3675 59.0297 100.4038 201.2966 399.2296 760.134 101 mat

s2 prac i el p i el - (2^(i 1 n sum n k - / :

s2 1.3

The fourth parameter is rather small compared to the other three, we will reject it and use the three parameters function:

p 10 b 1 el b 2 el T / + b 3 el T 10 log * +^≡

Linear fitting

p 10 log b 1 el b 2 el T / + b 3 el T 10 log * + ≡

Transforming the function in linear by parameters:

x φ 11 x / x 10 log 31 mat :

T t 273 + :

i 1 p length range y i el p i el 10 log : for

n T length :

n 10

k x φ length :

k 3

X n k T T# φ Xmat :

Φ X transpose X * :

d X transpose y * :

b Φ 1 -^d * :

b 78.0124 5018.6253 - 23.9183 - 31 mat

Parameters found

T b f b j el T φ j el * (j 1 k sum :

i 1 p length range logp.rac i el T i el b f : prac i el 10 logp.rac i el^: 21 line for

exp T y points size augment :

cal T logp.rac augment :

exp cal 2 1 sys

T 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 101 mat

p 1510204060100200400760101 mat

prac 1.0113 5.1564 9.9762 19.1864 37.4139 59.149 103.7667 213.0579 414.3331 721.1814 101 mat

T p points size augment T prac augment 2 1 sys

s2 prac i el p i el - (2^(i 1 n sum n k - / :

s2 272.2

How to minimize this function (NLS problem)??

T b f1 10 T b f^:

b S p i el 10 T i el b f^- (2^(i 1 n sum :

b S 1905.1179

Sum of quared residuals obtained by linear fitting Could that be reduced by NL fitting?

Guess values

b 78.0124 5018.6253 - 23.9183 - 31 mat

ν 10 :

maxiter 10 :

result T# b# f1 T p b ν maxiter LMA :

result 20.8884 28.4502 2720.1232 - 7.0121 - 31 mat 21 mat

S result 1 el :

S 20.8884

b S 20.8884

Reduced !

b result 2 el :

b 28.4502 2720.1232 - 7.0121 - 31 mat

NLS values

i 1 p length range prac i el T i el b f1 (: for

exp T p points size augment :

cal T prac augment :

exp cal 2 1 sys

T 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 101 mat

p 1510204060100200400760101 mat

prac 1.977 7.2452 12.4948 21.7362 38.9132 58.7193 98.9132 199.5809 401.1704 759.6735 101 mat

s2 prac i el p i el - (2^(i 1 n sum n k - / :

s2 3