91Ó°ÊÓ

Feature recognition from surface models of complicated parts is becoming increasingly important in the development of efficient computer-aided design (CAD) systems. The article 'A Computationally Efficient Approach to Feature Abstraction in DesignManufacturing Integration" (J. of Engr. for Industry, 1995: 16-27) contained a graph of \(\log _{\mathrm{s}}\) (total recognition time), with time in sec, versus \(\log _{10}\) (number of edges of a part), from which the following representative values were read: a. Does a scatterplot of \(\log\) (time) versus \(\log (\) edges) suggest an approximate linear relationship between these two variables? b. What probabilistic model for relating \(y=\) recognition time to \(x=\) number of edges is implied by the simple linear regression relationship between the transformed variables? c. Summary quantities calculated from the data are $$ \begin{aligned} &n=16 \quad \Sigma x_{i}^{\prime}=42.4 \quad \Sigma y_{i}^{\prime}=21.69 \\ &\Sigma\left(x_{i}^{\prime}\right)^{2}=126.34 \quad \Sigma\left(y_{i}^{\prime}\right)^{2}=38.5305 \\ &\Sigma x_{i}^{\prime} y_{i}^{\prime}=68.640 \end{aligned} $$ Calculate estimates of the parameters for the model in part (b), and then obtain a point prediction of time when the number of edges is 300 .

Step by step solution

01

Understanding the Problem

We need to check if there's a linear relationship between \(\log \text{ (time)}\) and \(\log \text{ (edges)}\). Further, we need to find a probabilistic model that relates recognition time to the number of edges and estimate the parameters to make predictions.

02

Scatterplot Analysis for Linear Relationship

A scatterplot of \(\log \text{ (time)}\) versus \(\log \text{ (edges)}\) indicates a linear relationship if the points tend to align in a straight line.

03

Recognize the Probabilistic Model

If \(\log \text{ (time)}\) and \(\log \text{ (edges)}\) have a linear relationship, the model for \(y\) (time) and \(x\) (edges) is \(\log y = \beta_0 + \beta_1 \log x + \varepsilon\), suggesting that recognition time has a power law relationship with edges.

04

Use Formulas to Calculate Parameters

Calculate \(\hat{\beta_1}\) using \( \hat{\beta_1} = \frac{\Sigma x_i'y_i' - \frac{\Sigma x_i' \Sigma y_i'}{n}}{\Sigma (x_i')^2 - \frac{(\Sigma x_i')^2}{n}}\) and \(\hat{\beta_0}\) using \( \hat{\beta_0} = \frac{\Sigma y_i' - \hat{\beta_1} \Sigma x_i'}{n}\).

05

Calculate \(\hat{\beta_1}\)

Using the given data, \(\hat{\beta_1} = \frac{68.640 - \frac{42.4 \times 21.69}{16}}{126.34 - \frac{(42.4)^2}{16}}\). Calculate this to get \(\hat{\beta_1} = 0.5974\).

06

Calculate \(\hat{\beta_0}\)

Using the result from Step 5 and the given data, \(\hat{\beta_0} = \frac{21.69 - 0.5974 \times 42.4}{16}\). Calculate this to get \(\hat{\beta_0} = -1.1491\).

07

Make Predictions for 300 Edges

To predict \(y\) (time) for \(x = 300\), use the model: \(\log y = -1.1491 + 0.5974 \log_{10} 300\). First calculate \(\log_{10} 300\), approximately 2.4771. Then substitute to get \(\log y = -1.1491 + 0.5974 \times 2.4771 = 0.3283\). Convert back to \(y\) by exponentiating: \(y = 10^{0.3283} \approx 2.13\) seconds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probabilistic Model

Understanding a probabilistic model is crucial for analyzing the relationship between variables in statistical terms. When working with simple linear regression, we often aim to establish a connection between two variables using a linear equation.
This equation includes a deterministic component and a random error term to reflect real-world unpredictability.
In our context, the probabilistic model helps us understand how recognition time (y) relates to the number of part edges (x).
We express this relationship as

• \(\log y = \beta_0 + \beta_1 \log x + \varepsilon\)

Here, \(\beta_0\) and \(\beta_1\) are parameters, and \(\varepsilon\) represents a random error term.
This model helps capture the systematic trend between transformed variables along with random variability that cannot be explained by the deterministic part alone.

Scatterplot Analysis

A scatterplot is an essential tool in data analysis, providing a graphic representation of the relationship between two numerical variables.
In this case, you create a scatterplot using logs of recognition time and the number of edges.
When analyzing this scatterplot, you're checking for a potential linear relationship between these logs.

• If points closely follow a straight line, this suggests linearity in the relationship.
• The slope and direction of the line indicate the nature of this relationship.

Constructing a scatterplot allows you to visually detect trends, clusters, and outliers which might not be obvious from raw data alone.
Additionally, it sets the stage for formulating a suitable regression model.

Parameter Estimation

Parameter estimation involves calculating the coefficients of our regression model, essentially determining the equation of the line that best fits the data.
In simple linear regression, we find the slope \(\hat{\beta_1}\) and the intercept \(\hat{\beta_0}\) using formulas based on least squares estimation.
For slope \(\hat{\beta_1}\), the formula is:

• \( \hat{\beta_1} = \frac{\Sigma x_i'y_i' - \frac{\Sigma x_i' \Sigma y_i'}{n}}{\Sigma (x_i')^2 - \frac{(\Sigma x_i')^2}{n}} \)
• This represents the change in \(\log y\) for a one-unit change in \(\log x\).

The intercept \(\hat{\beta_0}\) uses the formula:

• \(\hat{\beta_0} = \frac{\Sigma y_i' - \hat{\beta_1} \Sigma x_i'}{n} \)
• This is the expected value of \(\log y\) when \(\log x\) is zero.

By calculating these estimates using our given data, you predict the average response accurately for any value of the predictor variable present in the dataset.

Power Law Relationship

The power law relationship is a distinct type of functional relationship commonly found in natural phenomena.
In our regression context, transforming both variables logarithmically can reveal a power law relationship.
This relationship is identified when a straight line in the log-log plot signifies a functional relationship between the original variables of the form:

• \(y = kx^b\)

Where:

• k is a constant and b is the "power" or "exponent"
• When \(y\) is plotted against \(x\), it may appear non-linear, but plotting \(\log y\) against \(\log x\) will linearize it due to the equation taking the form \(\log y = \log k + b\log x \)

Recognizing this relationship allows us to better understand how changes in one variable scale with the other.
In practical terms, using such a model allows us to conduct predictions and analyses on the proportional effect of edges on recognition time in design processes.