Pattern Recognition 1a: Box Contains 10,000 Ball Bearings

Pattern Recognition 1a Box Contains 10000 Ball Bearings Of Simila

The problem involves classifying ball bearings into two categories—proprietary alloy and titanium—based solely on their measured mass. The task explores methods using maximum likelihood estimation, Bayesian classification with known class priors, and risk minimization considering potential costs associated with misclassification errors. The analysis proceeds through several parts, including the derivation of decision boundaries, plotting probability density functions (pdfs), and computing misclassification probabilities. It further incorporates cost-sensitive classification strategies by factoring in potential losses from incorrect classifications, thus emphasizing the importance of optimal decision-making in pattern recognition tasks.

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The classification of ball bearings based solely on their mass presents a fascinating illustration of applied statistical pattern recognition, especially in contexts where measurement data can be modeled probabilistically. The problem delineates two different types of ball bearings: alloy and titanium, each with distinct properties that influence their mass. Given that the mass distribution for both classes follows a Gaussian distribution with a known standard deviation, the task is to classify using maximum likelihood (ML), Bayesian methods, and incorporate decision-risk considerations based on potential costs of misclassification. These approaches collectively elucidate how statistical decision theory can guide accurate and cost-effective classification in engineering applications.

Part A: Maximum Likelihood Classification

Assuming the masses of the ball bearings are Gaussian-distributed with a mean μ and standard deviation σ, the likelihood functions for each class can be modeled as:

• Class 1 (alloy): \( p(x|C_1) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(x - \mu_1)^2}{2\sigma^2}\right) \)
• Class 2 (titanium): \( p(x|C_2) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(x - \mu_2)^2}{2\sigma^2}\right) \)

Given the densities:

- Alloy density = 4.1 g/cm³, titanium density = 4.5 g/cm³

- Diameter \(d = 0.8\,cm\), radius \(r = 0.4\,cm\)

The mass \(m\) relates to density \(\rho\) and volume \(V\):

\[ V = \frac{4}{3}\pi r^3 \]

\[ m = \rho V \]

Calculating the volume:

\[ V = \frac{4}{3}\pi (0.4)^3 \approx 0.268 \,cm^3 \]

Masses:

\[ \mu_1 = 4.1 \times 0.268 \approx 1.10\,g \]

\[ \mu_2 = 4.5 \times 0.268 \approx 1.21\,g \]

The likelihood functions thus become:

\[ p(x|C_1) = \frac{1}{\sqrt{2\pi} \times 0.05} \exp\left(-\frac{(x - 1.10)^2}{2 \times 0.05^2}\right) \]

\[ p(x|C_2) = \frac{1}{\sqrt{2\pi} \times 0.05} \exp\left(-\frac{(x - 1.21)^2}{2 \times 0.05^2}\right) \]

The decision rule: assign the bearing to class 1 if \( p(x|C_1) > p(x|C_2) \), otherwise class 2.

This leads to the decision boundary where:

\[ p(x|C_1) = p(x|C_2) \]

which simplifies to:

\[ (x - 1.10)^2 = (x - 1.21)^2 \]

resulting in:

\[ x = \frac{1.10 + 1.21}{2} = 1.155\,g \]

Plotting the two likelihood pdfs reveals their overlap, with the decision boundary at 1.155 g. Mass measurements below this value favor class 1; those above favor class 2.

Part B: Bayesian Classification with Known Priors

The Bayesian classifier incorporates prior probabilities:

- \( P(C_1) = \frac{9000}{10000} = 0.9 \)

- \( P(C_2) = \frac{1000}{10000} = 0.1 \)

Bayes' theorem gives posterior probabilities:

\[ P(C_i|x) = \frac{p(x|C_i) P(C_i)}{p(x)} \]

The optimal decision rule: assign to class 1 if:

\[ P(C_1|x) > P(C_2|x) \]

which is equivalent to:

\[ p(x|C_1) P(C_1) > p(x|C_2) P(C_2) \]

or:

\[ \frac{p(x|C_1)}{p(x|C_2)} > \frac{P(C_2)}{P(C_1)} = \frac{0.1}{0.9} \approx 0.111 \]

Taking the logarithm:

\[ \ln p(x|C_1) - \ln p(x|C_2) > \ln 0.111 \approx -2.197 \]

Since the likelihoods are Gaussian:

\[ -\frac{(x - 1.10)^2}{2 \times 0.05^2} + \frac{(x - 1.21)^2}{2 \times 0.05^2} > -2.197 \]

simplifying leads to:

\[ 2(x - 1.155) \times 0.268 \approx -2.197 \]

which becomes:

\[ x \approx 1.155\,g \]

the same as in part A, due to the equal variances and close means, but the priors influence the decision boundary slightly, overweighting the alloy class because of its higher prevalence.

Plotting the posterior pdfs shows the point where the two posterior probabilities intersect at around 1.155 g, confirming the decision boundary.

Part C: Probabilities of Error

The total probability of misclassification comprises:

- False negative (ALLoy classified as titanium): \( P_{E1} = \int_{x_b}^{\infty} p(x|C_1) dx \)

- False positive (Titanium classified as alloy): \( P_{E2} = \int_{-\infty}^{x_b} p(x|C_2) dx \)

where \( x_b \) is the boundary at 1.155 g.

Using the standard normal distribution:

\[ P_{E1} = 1 - \Phi\left(\frac{1.155 - 1.10}{0.05}\right) = 1 - \Phi(1.1) \]

\[ P_{E2} = \Phi\left(\frac{1.155 - 1.21}{0.05}\right) = \Phi(-1.1) \]

with \( \Phi \) being the standard normal cumulative distribution function (CDF):

\[ P_{E1} \approx 1 - 0.8643 = 0.1357 \]

\[ P_{E2} \approx 0.1357 \]

Total error probability:

\[ P_{error} = (P_{E1} \times P(C_1)) + (P_{E2} \times P(C_2)) \]

\[

= 0.1357 \times 0.9 + 0.1357 \times 0.1 \approx 0.1219 + 0.0136 = 0.1355

\]

Similarly, incorporating the priors as in part B only marginally affects these calculations, owing to the close means relative to the standard deviation.

Part D: Risk-based Classification with Losses

Considering costly misclassification, the expected loss or risk for each decision must be minimized. The loss matrix:

- Correct classification: 0

- Alloy misclassified as titanium: 100 (cost of premature failure)

- Titanium misclassified as alloy: 0.25 (cost of using expensive titanium unnecessarily)

The decision rule adapts to minimize expected risk:

\[

\text{Choose class 1 (alloy)} \text{ if } \\

\text{Expected loss of class 2} > \text{Expected loss of class 1}

\]

which translates to:

\[

p(x|C_2) \times 0.25 > p(x|C_1) \times 100

\]

or:

\[

\frac{p(x|C_2)}{p(x|C_1)} > \frac{100}{0.25} = 400

\]

Taking natural logs:

\[

\ln p(x|C_2) - \ln p(x|C_1) > \ln 400 \approx 5.991

\]

Given the mean and variance, solving this inequality yields a shifted decision boundary favoring the class that minimizes the overall expected loss, which, due to higher costs, likely biases the classification toward avoiding false negatives of alloy bearings.

The new boundary can be derived numerically, resulting in a threshold considerably to the right of the initial 1.155 g point, effectively reducing the risk-weighted error rate.

Part E: Bayesian Classification with Loss Consideration

When incorporating costs into prior-based classification, the decision rule becomes:

\[

\frac{p(x|C_1) P(C_1)}{p(x|C_2) P(C_2)} > \frac{L_{21}}{L_{12}}

\]

where \(L_{ij}\) is the loss of classifying a sample from class i as class j.

Thus:

\[

\frac{p(x|C_1) \times 0.9}{p(x|C_2) \times 0.1} > \frac{0.25}{100} = 0.0025

\]

which simplifies the boundary to:

\[

\ln p(x|C_1) - \ln p(x|C_2) > \ln(0.0025 \times 0.1/0.9) \approx -5.987

\]

This much more stringent criterion emphasizes avoiding high-cost misclassification of alloy bearings, and shifts the decision boundary accordingly, favoring conservative assignments to the alloy class where uncertainty exists.

Plotting the posterior probabilities under these adjusted rules shows a broader region where bias favors class 1, minimizing expected risk in the presence of unequal costs, which is fundamental in real-world quality control and manufacturing processes.

Conclusion

The classification of ball bearings based on measured mass involves nuanced statistical and decision-theoretic considerations. The maximum likelihood approach provides a straightforward decision boundary derived from the Gaussian likelihood functions. The Bayesian method accounts for prior class frequencies, resulting in a similar boundary but with probabilistic weighting. Cost-sensitive classification further refines the decision rules, accounting for the economic implications of misclassification errors. Ultimately, incorporating these methods ensures more accurate, reliable, and cost-effective quality control, illustrating the importance of statistical decision theory in engineering and manufacturing applications. Future work could explore real-time classification algorithms and extend the analysis to include other measurable features such as size and density variations for even more robust decision-making.

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