Problem 4

Question

a. Find $a, b,$ and $c$ so that $y=a+b x+c x^{2}$ is the least squares approximation to data, $\left.\left(x_{1}, y_{1}\right), x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right) .$ To do so you will need to minimize $$S S=\sum_{k=1}^{n}\left(y_{k}-\left(a+b x_{k}+c x_{k}^{2}\right)\right)^{2}$$ This is a three-variable minimization problem. The solution will be similar to the least squares line approximation to data of Example 13.2.2. b. In Exercise Table 13.2 .4 are data showing the height of a ball falling in air above a Texas Instruments CBL motion detector. Find the parabola that is the least squares fit to the data. c. Check your answer on a computer or calculator.

Step-by-Step Solution

Verified

Answer

To find the least squares fit, solve the normal equations derived from minimizing the sum of squares. Use software for verification.

1Step 1: Define the Objective Function

We are given a function $ y = a + bx + cx^2 $, and we need to find the coefficients $ a, b, $ and $ c $ such that this function minimizes the sum of squared differences from the provided data points $ (x_k, y_k) $. This sum of squares (SS) is given by: \[ SS = \sum_{k=1}^{n} \left(y_k - (a + b x_k + c x_k^2) \right)^2 \].

2Step 2: Set Up the Normal Equations

To minimize $ SS $, we take the partial derivatives with respect to $ a, b, $ and $ c $, set each to zero, and solve the resulting system of equations. The normal equations are:\[ \frac{\partial SS}{\partial a} = -2 \sum_{k=1}^{n} (y_k - a - bx_k - cx_k^2) = 0 \] \[ \frac{\partial SS}{\partial b} = -2 \sum_{k=1}^{n} x_k (y_k - a - bx_k - cx_k^2) = 0 \] \[ \frac{\partial SS}{\partial c} = -2 \sum_{k=1}^{n} x_k^2 (y_k - a - bx_k - cx_k^2) = 0 \]

3Step 3: Simplify the Equations

Simplify the equations obtained from Step 2 to obtain a system of three linear equations in $ a, b, $ and $ c $. These equations are: \[ \sum y_k = na + b\sum x_k + c\sum x_k^2 \] \[ \sum x_k y_k = a \sum x_k + b \sum x_k^2 + c \sum x_k^3 \] \[ \sum x_k^2 y_k = a \sum x_k^2 + b \sum x_k^3 + c \sum x_k^4 \].

4Step 4: Solve the System of Equations

Solve the system of three linear equations for $ a, b, $ and $ c $. This can be done manually if the dataset is small (by substitution or elimination methods) or by using matrix methods such as Gaussian elimination or calculating the inverse of the coefficient matrix.

5Step 5: Apply to the Given Data

Using the data from Table 13.2.4, substitute the appropriate values into your equations from Step 3 and calculate $ a, b, $ and $ c $.

6Step 6: Verify Using a Computer or Calculator

Using a computer software (like MATLAB, Python with NumPy) or graphing calculator, verify the calculated coefficients $ a, b, $ and $ c $ by performing regression analysis to ensure they are correct.

Key Concepts

Polynomial RegressionSum of SquaresNormal EquationsParabolic Fit

Polynomial Regression

Polynomial regression is an extension of simple linear regression. Here, the goal is to model the relationship between a dependent variable and one or more independent variables by fitting a non-linear equation. Instead of a straight line, polynomial regression fits a curve, such as a parabola for quadratic relationships.

The general form of a polynomial regression is as follows:

For a linear fit: $y = a + bx$
For a quadratic (or parabolic) fit: $y = a + bx + cx^2$
For a cubic fit: $y = a + bx + cx^2 + dx^3$, and so on.

Polynomial regression is particularly useful when data shows a curvilinear trend, allowing for a more accurate representation of complex relationships. This technique involves finding the most suitable coefficients $a, b, c,$ etc., which minimize errors between predicted and observed values.

Sum of Squares

The sum of squares is a measure used to determine how well a regression function matches the data points. It quantifies the total deviation of the observed data from the fitted values derived from the model.

The formula for calculating the sum of squares in the context of polynomial regression is:\[ SS = \sum_{k=1}^{n} \left(y_k - (a + b x_k + c x_k^2) \right)^2 \]Here:

$y_k$ is the observed value.
$a + b x_k + c x_k^2$ represents the predicted value from the polynomial model.

The objective in least squares approximation is to find the coefficients $a, b, c$ that minimize this sum of squares, thus ensuring our model is the closest fit to the actual data points. A smaller sum of squares indicates a better fit.

Normal Equations

Normal equations are derived from the method of least squares, providing a systematic way to find the optimal coefficients in polynomial regression. To minimize the sum of squares, we take partial derivatives with respect to each coefficient, equate them to zero, and solve the resulting system.

The normal equations for a quadratic fit $y = a + bx + cx^2$ become:

$\sum y_k = na + b\sum x_k + c\sum x_k^2$
$\sum x_k y_k = a \sum x_k + b \sum x_k^2 + c \sum x_k^3$
$\sum x_k^2 y_k = a \sum x_k^2 + b \sum x_k^3 + c \sum x_k^4$

Solving these equations, which can be done manually for small datasets or through matrix methods for larger ones, gives us the least squares estimate of the coefficients $a, b,$ and $c$. This process ensures that the polynomial curve we derive fits our data as closely as possible.

Parabolic Fit

A parabolic fit is a specific case of polynomial regression where we fit a quadratic polynomial, $y = a + bx + cx^2$, to our data. This type of fit is useful when the data shows a parabolic trend, which is common in various fields such as physics, engineering, and economics.

To obtain the best parabolic fit, we calculate the coefficients $a, b,$ and $c$ using the steps outlined in polynomial regression and solving normal equations. The resulting parabola will minimize the sum of squared deviations from the data points to the fitted curve.

A parabolic fit accounts for the curvature in the data, unlike linear models which can only fit straight lines.
This fit is particularly well-suited for motions governed by physics, such as the trajectory of a projectile.

By providing a more accurate representation of the underlying process, parabolic fits allow for better predictions and insights from the data.

Problem 4

Problem 5

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