Case study Galactose diffusion in silica mesopore

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Cleopatra Lindsey
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1 Liear regressio

2 Case study Galactose diffusio i silica mesopore

3 Cotrolled drug release systems

4 MSD cagig regime diffusive regime balistic regime

5 MSD

6 MSD <r 2 > T 2 T T 0.5 MSD 6Dt log MSD log t + log(6d)

7 1. How to check if the molecule is i the diffusive regime? calculate slope of log(msd) vs. log(t)

8 1. How to check if the molecule is i the diffusive regime? calculate slope of log(msd) vs. log(t) 2. How to calculate self-diffusio coefficiet? calculate slope of MSD vs. T ad divide it by 6. y = ax + b D = a 6

9 How to aalyse data?

10 How to aalyse data? Plot!

11 How to aalyse data? Plot! Huma brai is oe the most powerfull computatioall tools Works differetly tha a computer

12 Simple example fidig maximum y(x max ) Computer x 1 x 2 x 3

13 Simple example fidig maximum y(x max ) Computer 1. Set y(x max ) = y(x 1 ) x 1 x 2 x 3

14 Simple example fidig maximum y(x max ) Computer 3 1. Set y(x max ) = y(x 1 ). 2. Go to the ext poit x 2 : 2 1 x 1 x 2 x 3

15 Simple example fidig maximum y(x max ) Computer Set y(x max ) = y(x 1 ). 2. Go to the ext poit x 2 : 1. If y(x 2 ) > y(x max ) the x max = x 2 2. Else do othig. 1 x 1 x 2 x 3

16 Simple example fidig maximum y(x max ) Computer Set y(x max ) = y(x 1 ). 2. Go to the ext poit x 2 : 1. If y(x 2 ) > y(x max ) the x max = x 2 2. Else do othig. 3. Repeat this procedure util you reach the ed. x 1 x 2 x 3

17 Simple example fidig maximum y(x max ) Huma brai x 1 x 2 x 3

18 Simple example fidig maximum y(x max ) Huma brai Here! x 1 x 2 x 3

19 Simple example fidig maximum y(x max ) Huma brai Here! x 1 x 2 x 3 With icreasig umber of poits quicker aswer

20 How to aalyse data? Plot x agaist y Observe tred - correlatio

21 How to measure liearity? Geometry a b

22 How to measure agle betwee two vectors? Scalar product a α b

23 How to measure agle betwee two vectors? Scalar product a = (a 1, a 2 ), b = (b 1, b 2 ) a α b

24 How to measure agle betwee two vectors? Scalar product a = (a 1, a 2 ), b = (b 1, b 2 ) a α a o b = a 1 b 1 + a 2 b 2 = 2 a i b i b

25 How to measure agle betwee two vectors? Scalar product a = (a 1, a 2 ), b = (b 1, b 2 ) a α b a o b = a 1 b 1 + a 2 b 2 a o a = a a 2 2

26 How to measure agle betwee two vectors? Scalar product a = (a 1, a 2 ), b = (b 1, b 2 ) a α b a o b = a 1 b 1 + a 2 b 2 a o a = a a 2 2 cos α = a o b a b

27 Example z x y

28 Example How to do it? z x y

29 Example z How to do it? We choose two vectors b a x y

30 Example z b y a x How to do it? We choose two vectors a = (1, 0, 1), b = (0, 1, 1)

31 Example How to do it? z We choose two vectors a = (1, 0, 1), b = (0, 1, 1) b a x cos α = a o b a b y α = 60 o

32 Example z How to do it? We choose two vectors a = (1, 0, 1), b = (0, 1, 1) b a x cos α = a o b a b y cos α = 1, 0, 1 o 0, 1, 1 (1, 0, 1) 0, 1, 1

33 Example z How to do it? We choose two vectors a = (1, 0, 1), b = (0, 1, 1) b a x cos α = a o b a b y cos α = 1, 0, 1 o 0, 1, 1 (1, 0, 1) 0, 1, 1 = = 1 2

34 Example z How to do it? We choose two vectors a = (1, 0, 1), b = (0, 1, 1) b a x cos α = a o b a b y cos α = 1, 0, 1 o 0, 1, 1 (1, 0, 1) 0, 1, 1 = = 1 2 α = 60 o

35 What s the relevace? X Y Two sets of data y 4 y 3 y 2 x = (1, 2, 3, 4) y = (2, 4. 1, 5. 4, 8. 3) y 1 x 1 x 2 x 3 x 4 Data are vectors!

36 What s the relevace? X Y Two sets of data y 4 y 3 y = a x y 2 Liear relatioship y 1 x 1 x 2 x 3 x 4 x y parallel

37 How to measure parallelism betwee two vectors? Liear relatioship y 4 y 3 y 2 x y parallel = zero agle y 1 x 1 x 2 x 3 x 4 α 0 cos α 1

38 How to calculute the agle? Scalar product! X Y Two sets of data y 4 y 3 y 2 cos α = xo y x y = x i y i 2 x i y i 2 y 1 x 1 x 2 x 3 x 4

39 How to calculute the agle? Scalar product! Chagig origi (0,0) x, y y R 2 = cos α = (x i x)(y i y) (x i x) 2 y i y 2 x x = 1 x i, y = 1 y i

40 Our case X Y Two sets of data y 4 y 3 y 2 y 1 x = = y = = x x y y (xi x)(y i y) = ( ) + ( ) + ( ) + ( ) = x 1 x 2 x 3 x 4 2 ( (x i x) = 5 y i y 2 = R 2 = = 0. 98

41 What is the best positio of the lie? X Y The best = smallest error Error = data value estimated value

42 What is the best positio of the lie? X Y The best = smallest error y 2 f x 2 E 2 E 1 = y 1 f(x 1 ) SSE = E i 2 = y i f x i 2 f x 1 y 1 E 1 E 2 = y 2 f x 2 f x = ax + b E i = y i f x i SSE = y i ax i b 2

43 How to adjust a ad b so SSE is the smallest? SSE(a, b) = y i ax i b 2 How to calculate miimum of the SSE(a,b) fuctio? SSE a, b a = 0 SSE a, b b = 0

44 How to adjust a ad b so SSE is the smallest? SSE(a, b) = y i ax i b 2 SSE a, b a = a y i ax i b 2 = a y i ax i b 2 = x i 2 y i ax i b = 2 x i y i ax i b SSE a, b b = b y i ax i b 2 = b y i ax i b 2 = 2 y i ax i b = 2 y i ax i b

45 How to adjust a ad b so SSE is the smallest? SSE(a, b) = y i ax i b 2 SSE a, b a = 0 2 x i y i ax i b = 0 SSE a, b b = 0 2 y i ax i b = 0

46 We obtai a set of liear equatios of two variables a ad b x i y i ax i b = 0 (x i y i ax i 2 bx i ) = 0 a x i 2 + b x i x i y i = 0 y i ax i b = 0 (y i ax i b) = 0 a x i + b 1 y i = 0

47 Fially Set of liear equatios a x i 2 + b x i x i y i = 0 a x i 2 + b x i = x i y i a x i + b 1 y i = 0 a x i + b = y i 2 x i x i x i x i y i a b = y i

48 How to solve it? Set of liear equatios. 2 x i x i x i x i y i a b = y i Ax = b

49 Has solutio if det A 0 2 x i x i x i = x 2 i x i x i 0 1 x i 2 1 x i 1 x i 0 x 2 x x 2 x 2 0 Cov X, X 0

50 Liear regressio procedure 1. Plot data make observatio, decide which model fits best. 2. If you decide to use liear regressio compute R Solve liear regressio problem.

Nonlinear regression

Nonlinear regression oliear regressio How to aalyse data? How to aalyse data? Plot! How to aalyse data? Plot! Huma brai is oe the most powerfull computatioall tools Works differetly tha a computer What if data have o liear