1. Chapter 3- Model Fitting

2. Three Tasks When Analyzing Data: 1.Fit a model type to the data. 2.Choose the most appropriate model from the ones that have been fitted. 3.Make predictions based on the data.

3. Project Outline: Seven day window during which I eliminated carbohydrates from my diet as much as possible and tracked resultant weight changes. Plotted the data and estimated a visual model to fit the data. Calculated a model to fit the data using the Least Squares method.

4. Sources of Error in the Modeling Process 1.Formulation Error - Assumption that certain variables are negligible or simplifying the interrelationships among variables in the submodels. 2.Truncation Error - Comes from the numerical methods used to solve mathematical problems, such as truncated a series. 3.Round-off Error – Occurs because all numbers cannot be represented exactly using only finite representations, such as 1/3 equaling 0.33. 4.Measurement Error – Caused by imprecision in data collection.

5. Data Collected and Graphed

6. Visual Model Fitting with the Original Data Fit a line to attempt to minimize the absolute deviation, often this method is compatible with the accuracy of the modeling process.

7. Goal In order to form a model that can be used to accurately make predictions based on the data, we must determine the parameters of a function y = f (x) using a collection of points (x i, y i ) that minimizes the absolute deviations (R 2 ).

8. Least Squares Method Let R i = |y i – f(x i )| 2 for i = 1,2,3,4,5,6,7. Since we are fitting a straight line a model of the form y = mx + b is expected, which requires minimizing : The two partial derivatives must equal zero:

9. Least Squares Method (Con’t.) Simplify the derivative to give the two formulas: With manipulation we can set the two equations equal to a and b respectively, making it easy to determine the slope and intercept.

10. Result Function y = f(x) that minimizes R, y = -0.429 + 151.36

11. Visual Estimate vs. Calculated Model EstimatedCalculated