1. Linear regression

2. Case study Galactose diffusion in silica mesopore

3. Controlled drug release systems

4. MSD balistic regime caging regime diffusive regime

5. MSD

6. T

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

8. 2. How to calculate self-diffusion coefficient?  calculate slope of MSD vs. T and divide it by 6.

9. How to analyse data?

10. Plot!

11. Plot! Human brain is one the most powerfull computationall tools Works differently than a computer…

12. Simple example – finding maximum y(x max ) Computer 1 2 3 x1x1 x2x2 x3x3

13. Computer 1 2 3 x1x1 1.Set y(x max ) = y(x 1 ). x2x2 x3x3

14. Simple example – finding maximum y(x max ) Computer 1 2 3 x1x1 1.Set y(x max ) = y(x 1 ). 2.Go to the next point x 2 : x2x2 x3x3

15. Simple example – finding maximum y(x max ) Computer 1 2 3 x1x1 1.Set y(x max ) = y(x 1 ). 2.Go to the next point x 2 : 1.If y(x 2 ) > y(x max ) then x max = x 2 2. Else do nothing. x2x2 x3x3

16. Simple example – finding maximum y(x max ) Computer 1 2 3 x1x1 1.Set y(x max ) = y(x 1 ). 2.Go to the next point x 2 : 1.If y(x 2 ) > y(x max ) then x max = x 2 2. Else do nothing. 3. Repeat this procedure until you reach the end. x2x2 x3x3

17. Simple example – finding maximum y(x max ) Human brain 1 2 3 x1x1 x2x2 x3x3

18. Simple example – finding maximum y(x max ) Human brain 1 2 3 x1x1 x2x2 x3x3 Here!

19. Simple example – finding maximum y(x max ) Human brain 1 2 3 x1x1 x2x2 x3x3 Here! With increasing number of points quicker answer

20. How to analyse data? Plot x against y Observe trend - correlation

21. How to „measure” linearity? Geometry

22. How to measure angle between two vectors? Scalar product

23. How to measure angle between two vectors? Scalar product

24. How to measure angle between two vectors? Scalar product

25. How to measure angle between two vectors? Scalar product

26. How to measure angle between two vectors? Scalar product

27. Example

28. Example How to do it?

29. Example We choose two vectors

30. Example How to do it? We choose two vectors

31. Example How to do it? We choose two vectors

32. Example How to do it? We choose two vectors

33. Example How to do it? We choose two vectors

34. Example How to do it? We choose two vectors

35. What’s the relevance? y1y1 x1x1 x2x2 x3x3 x4x4 y2y2 y3y3 y4y4 X 1 2 3 4 Y 2 4.1 5.4 8.3 Two sets of data Data are vectors!

36. What’s the relevance? y1y1 x1x1 x2x2 x3x3 x4x4 y2y2 y3y3 y4y4 X 1 2 3 4 Y 2 4.1 5.4 8.3 Two sets of data Linear relationship parallel

37. How to measure parallelism between two vectors? y1y1 x1x1 x2x2 x3x3 x4x4 y2y2 y3y3 y4y4 Linear relationship parallel = zero angle

38. How to calculute the angle? Scalar product! y1y1 x1x1 x2x2 x3x3 x4x4 y2y2 y3y3 y4y4 X 1 2 3 4 Y 2 4.1 5.4 8.3 Two sets of data

39. How to calculute the angle? Scalar product!

40. Our case y1y1 x1x1 x2x2 x3x3 x4x4 y2y2 y3y3 y4y4 X 1 2 3 4 Y 2 4.1 5.4 8.3 Two sets of data

41. X 1 2 3 4 Y 2 4.1 5.4 8.3 The best = smallest error What is the best position of the line? Error = data value – estimated value

42. X 1 2 3 4 Y 2 4.1 5.4 8.3 The best = smallest error What is the best position of the line?

43. How to adjust a and b so SSE is the smallest? How to calculate minimum of the SSE(a,b) function?

44. How to adjust a and b so SSE is the smallest?

46. We obtain a set of linear equations of two variables a and b

47. Finally… Set of linear equations

48. How to solve it? Set of linear equations.

50. Linear regression procedure