1. MA3264 Mathematical Modelling Lecture 3 Model Fitting

2. Introduction Analysing a set of data may involve 3 tasks 1. Choose an appropriate model (if possible) Example: using a 2. Fit each the selected model(s) to the data 3. Make predictions from the data Example: estimate set data points Example: interpolate (extrapolate, smooth)

3. Model Fitting versus Interpolation Models explain data (observed behavior) they are theory driven, often derived from laws Example: Interpolation can be also be used to predict data and can ALSO be used to predict data EVEN in the absence of an explanatory model it is data driven, rather than theory driven and will be discussed in the next Chapter / Lecture

4. Breaking Distance Data page 71 >> [v' d'] 20.0000 20.0000 25.0000 28.0000 30.0000 40.5000 35.0000 52.5000 40.0000 72.0000 45.0000 92.5000 50.0000 118.0000 55.0000 148.5000 60.0000 182.0000 65.0000 220.5000 70.0000 266.0000 75.0000 318.0000 80.0000 376.0000 >> v = 20:5:80 v = 20 25 30 35 40 45 50 55 60 65 70 75 80 >> d = [20 28 40.5 52.5 72 92.5 118 148.5 182 220.5 266 318 376] d = Columns 1 through 7 20.0000 28.0000 40.5000 52.5000 72.0000 92.5000 118.0000 Columns 8 through 13 148.5000 182.0000 220.5000 266.0000 318.0000 376.0000

5. Plot of Data Points

6. Plot of Transformed Data Points

7. Analytic Models for Data Fitting 1. Minimize Maximum Absolute Deviation (Chebyshev): Data 1, 2 require linear programming, simplex-method 2. Minimize Sum of Absolute Deviations 3. Minimize Sum of Squared Deviations (Least-Squares) Fit

8. Parameterized Functions The function is to be chosen from a set of functions That have the same form but depend only on one or more PARAMETERS Example 1. m=1 breaking distance Example 2. m=2 total distance

9. Least-Squares Equations Example 1. If Question. If then the quantity is minimized only if Then what are the least-squared equations ?

10. Suggested Reading&Problems in Textbook Study the derivations of the Least-Squares Equations on pages 114-117 and think about Problems 3.3 Read pages 97-105 and think about Problems 3.1 Using MATLAB or an equivalent computer environment experiment with least-squares algorithms and plotting and approximating data

11. Tutorial 3 Due Week 8-12 September Page 118 Problem 2 a,b,c Page 105 Problem 3 Page 118 Problem 4 Compute and plot the transformed data and estimate a and b graphically Tutorials will NOT be collected. I suggest you to write your solutions to present to the class

12. Homework 1 Due Friday 12 September I will collect this homework during the class Page 50 Problem 6 a, b Page 124 Problem 1 Plot the data, the transformed data, and use the least-squares method on the transformed data to estimate both a and b Compute and plot the points Page 83 Problem 3