Problem 4 - A First Course in Mathematical Modeling

Question

In the following data, \(X\) represents the diameter of a ponderosa pine measured at breast height, and \(Y\) is a measure of volume-number of board feet divided by \(10 .\) Make a scatterplot of the data. Discuss the appropriateness of using a 13 th-degree polynomial that passes through the data points as an empirical model. If you have a computer available, fit a polynomial to the data and graph the results. \begin{tabular}{c|cccccccccccccc} \(X\) & 17 & 19 & 20 & 22 & 23 & 25 & 31 & 32 & 33 & 36 & 37 & 38 & 39 & 41 \\ \hline\(Y\) & 19 & 25 & 32 & 51 & 57 & 71 & 141 & 123 & 187 & 192 & 205 & 252 & 248 & 294 \end{tabular}

Step-by-Step Solution

Answer

A 13-th degree polynomial may overfit; it is generally not appropriate.

Step 1: Plot the Data Points

First, plot the given data points on a scatterplot. Use the diameters (\(X\)) as the x-values and the respective volumes (\(Y\)) as the y-values. Each data point should be represented as a distinct mark on the graph.

Step 2: Analyze the Scatterplot

Examine the scatterplot to understand the distribution of data points. Look for any apparent trends or patterns, such as linearity or curvature. It is crucial to identify if the data follows a specific trend that can be captured by a polynomial.

Step 3: Consider Polynomial Fitting

Think about the prospect of fitting a 13-th degree polynomial to the data. High-degree polynomials may fit data points exactly, especially when the number of data points is small relative to the polynomial degree. However, they tend to overfit and can produce erratic behavior outside the data range.

Step 4: Pros and Cons of a 13-th Degree Polynomial

Evaluate the pros and cons of fitting a 13-th degree polynomial. While it may pass through all points, such high-degree polynomials typically result in overfitting. They can oscillate wildly between data points, losing the general trend, especially for prediction purposes.

Step 5: Fit a Polynomial Using Software

If available, use software (e.g., Python, R, Excel) to numerically fit a polynomial to the data. Input the data points and specify the desired polynomial degree. The software will calculate the polynomial coefficients and provide a fitted curve, which can be graphed together with the scatterplot.

Step 6: Graph the Fitted Polynomial

Overlay the polynomial curve on the scatterplot. Compare how well it fits the data. Check if it interpolates between points smoothly or if it displays excessive oscillation or peculiar shapes, which often happen with high-degree fits.

Step 7: Conclusion on Polynomial Appropriateness

Based on the fitted curve and its behavior, decide on the appropriateness of using a 13-th degree polynomial. If the fit is poor and oscillatory, alternative models such as lower-degree polynomials or other forms like linear or exponential should be considered.

Key Concepts

Scatterplot Analysis

A scatterplot is a useful tool that allows us to visually examine the relationship between two variables. In this context, we're looking at the diameter of ponderosa pine trees (\(X\)) and their corresponding volume in board feet (\(Y\)). To create a scatterplot, plot each tree's diameter as an \(x\)-coordinate and the volume as a \(y\)-coordinate.

For example, if a tree has a diameter of 17 and a volume of 19, it would be represented as the point (17, 19) on the graph. Repeat this for each data point given, which helps in visualizing potential trends.

• Scatterplots help identify whether there is a linear, exponential, or polynomial relationship among the variables.
• They also reveal the spread and clustering of data points, allowing for initial judgments on the model fit type.

Look for patterns like whether the data clusters or forms a particular shape. In the case of the pine trees, you might check if there seems to be a curve that could suggest a polynomial fit. This visualization is crucial before considering more complex models.

Overfitting

Overfitting happens when a statistical model describes random error or noise instead of the underlying relationship. Using a 13-th degree polynomial to fit the data could lead to this issue.

Polynomials of such high degree are flexible enough to pass through all the data points, but this flexibility can be misleading due to:

• Excessive waviness between points, which may not represent any real-world trend.
• Lack of generalization, as the model becomes very specific to the sample data and performs poorly on new data.

Given the complexity of a 13-th degree polynomial, it fits the provided data exactly but can introduce oscillations that don't exist naturally. Figuring out the right degree of a polynomial is essential to balance fit accuracy and generalization. It is often better to use simpler models that capture the main trend without becoming overly complex.

Empirical Modeling

Empirical modeling involves creating mathematical representations of observed data without necessarily understanding the underlying processes in detail.

When modeling the relationship between tree diameter and volume, one might use empirical models like polynomial regression to capture this data pattern
- especially if there's no theoretical understanding of how these variables interact.

• A 13-th degree polynomial is an empirical approach because it was chosen based on data, not any underlying scientific mechanism.
• The purpose is to model the observed data closely to make predictions or understand the pattern.

While empirical models can be extremely useful, it is vital to ensure that they don't overfit the data to maintain predictive accuracy. Alternative models with lower degree or linear fits should be considered if overfitting is observed. Always remember, empirical models should be assessed on their predictive performance, not just their fit to existing data.