Math Problem Statement
answer
Solution
The problem involves calculating a linear discriminant function (LDF) for two datasets and , and then classifying a new observation based on this function. Let's break it down into parts.
(a) Calculate the Linear Discriminant Function:
Given two datasets:
3 & 7 \\ 2 & 1 \\ 4 & 7 \end{pmatrix}, X_2 = \begin{pmatrix} 6 & 9 \\ 5 & 4 \\ 4 & 8 \end{pmatrix}$$ The steps to calculate the linear discriminant function are as follows: 1. **Step 1: Compute the means of $$X_1$$ and $$X_2$$**: Let the mean vectors of $$X_1$$ and $$X_2$$ be $$\mu_1$$ and $$\mu_2$$ respectively. $$\mu_1 = \frac{1}{3} \sum X_1 \quad \text{and} \quad \mu_2 = \frac{1}{3} \sum X_2$$ 2. **Step 2: Compute the pooled covariance matrix**: The pooled covariance matrix $$\Sigma$$ is the weighted sum of the covariance matrices for $$X_1$$ and $$X_2$$ assuming equal priors. 3. **Step 3: Compute the linear discriminant function**: The linear discriminant function has the form: $$g(x) = w^T x + w_0$$ where $$w = \Sigma^{-1} (\mu_1 - \mu_2)$$ and $$w_0 = -\frac{1}{2} (\mu_1^T \Sigma^{-1} \mu_1 - \mu_2^T \Sigma^{-1} \mu_2)$$. --- ### (b) **Classify the Observation $$x_0^T = (2, 4)$$:** Using the linear discriminant function obtained in part (a), classify the observation $$x_0 = (2, 4)$$. If $$g(x_0) > 0$$, assign it to population $$\pi_1$$, otherwise to population $$\pi_2$$. --- Let me compute these steps explicitly for you.### (a) Linear Discriminant Function: - **Mean vector of $$X_1$$**: $$\mu_1 = (3, 5)$$ - **Mean vector of $$X_2$$**: $$\mu_2 = (5, 7)$$ - **Pooled covariance matrix**: $$\Sigma = \begin{pmatrix} 1 & 1.75 \\ 1.75 & 9.5 \end{pmatrix}$$ - **Inverse of the pooled covariance matrix** was used to compute: - **Weight vector** $$w = (-2.408, 0.233)$$ - **Constant** $$w_0 = 8.233$$ The linear discriminant function is: $$g(x) = (-2.408)x_1 + (0.233)x_2 + 8.233$$ --- ### (b) Classification of $$x_0^T = (2, 4)$$: For the observation $$x_0 = (2, 4)$$: $$g(x_0) = (-2.408)(2) + (0.233)(4) + 8.233 \approx 4.35$$ Since $$g(x_0) > 0$$, we classify $$x_0$$ as belonging to population $$\pi_1$$. --- Would you like a detailed breakdown of any of the steps? Here are 5 related questions to expand on: 1. How is the pooled covariance matrix derived? 2. What happens if the priors are unequal in this classification problem? 3. Can this method be extended to more than two populations? 4. How does the discriminant function change if we have different covariance matrices for each class? 5. What is the geometric interpretation of the linear discriminant function? **Tip**: In classification problems with equal priors and costs, the decision boundary is often linear, making computations straightforward with discriminant analysis.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Statistical Classification
Linear Discriminant Analysis
Formulas
Mean vector formula: μ = (1/n) Σ X_i
Pooled covariance matrix: Σ = (1/(n1 + n2 - 2)) * (Σ1 + Σ2)
Linear discriminant function: g(x) = w^T x + w_0
Weight vector: w = Σ^(-1) (μ1 - μ2)
Constant: w_0 = -(1/2) (μ1^T Σ^(-1) μ1 - μ2^T Σ^(-1) μ2)
Theorems
Linear Discriminant Analysis (LDA) Theorem
Bayes' Theorem for classification
Suitable Grade Level
Undergraduate
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