Comunicazioni multimediali - le teorie e le tecniche del riconoscimento

Teoria e tecniche del riconoscimento

Allowed supporting material: Notes and Calculator

1. (5%) For a 2x2 matrix, Q, tr(Q) = 8 and |Q| = 12. One of the eigenvalues of Q is:

   • A) 0
   • B) 1
   • C) 2
   • D) 3
   • E) 4
   • F) 5

2. (5%) For a matrix Q, QΦ = ΦΛ. The eigenvector matrix of Q^-1 is:

   • A) Φ
   • B) Φ^-1
   • C) Φ^T
   • D) Something else

3. (5%) 10 samples, X₁,..., X₁₀, are drawn from a 20-dimensional random vector X, and the mean and covariance matrices are estimated by M̂ = (1/10) Σ_i=1¹⁰ X_i, and Σ̂ = (1/10) Σ_i=1¹⁰ (X_i - M)(X_i - M)^T, respectively. The rank of the covariance matrix Σ̂ is:

   • A) 9
   • B) 10
   • C) 20
   • D) Something else

Two Gaussian distributions

Two Gaussian distributions have the following characteristics: P₁ = P₂ = 0.5, μ₁ = [-1₀ ], μ₂ = [+1₀ ], Σ₁ = Σ₁ = [ 1 0.5_0.5 1 ]

1. (10%) Sketch the distributions;

2. (10%) Design the Bayes classifier for minimum error for classification of the data;

3. (10%) What is the Bayes error (compute it exactly)?

4. (10%) Design and plot the linear classifier which maximizes:

   f = P₁n₁² + P_nn₂²_{1/σ1² + P2}σ₂²

   where P₁ and P₂ are prior probabilities, n₁ and n₂ are the corresponding means and σ₁ and σ₂ are the corresponding standard deviations.

Teoria e tecniche del riconoscimento

Allowed supporting material: Notes and Calculator

1. (5%) For a 2x2 matrix, Q, tr(Q) = 8 and |Q| = 12. One of the eigenvalues of Q is:

   • A) 0
   • B) 1
   • C) 2
   • D) 3
   • E) 4
   • F) 5

2. (5%) For a matrix Q, QΦΦ = ΦΛ. The eigenvector matrix of Q^-1 is:

   • A) Φ
   • B) Φ^-1
   • C) Φ^T
   • D) Something else

3. (5%) 10 samples, X₁,...,X₁₀, are drawn from a 20-dimensional random vector X, and the mean and covariance matrices are estimated by M = ¹/₁₀ Σ_i=1¹⁰ X_i and Σ̂ = ¹/₁₀ Σ_i=1¹⁰ (X_i - M)(X_i - M)^T, respectively. The rank of the covariance matrix Σ̂ is:

   • A) 9
   • B) 10
   • C) 20
   • D) Something else

4. Two Gaussian distributions have the following characteristics: P₁ = P₂ = 0.5, μ₁ = -10, μ₂ = +10, Σ₁ = Σ₁ = 10.50.51

   • (10%) Sketch the distributions;
   • (10%) Design the Bayes classifier for minimum error for classification of the data;
   • (10%) What is the Bayes error (compute it exactly)?
   • (10%) Design and plot the linear classifier which maximizes:

   f = ^{P₁n₁2 + P_n2₂2} / _{P1σ1² + P2σ2²}

   where P₁ and P₂ are prior probabilities, n₁ and n₂ are the corresponding means and σ₁ and σ₂ are the corresponding standard deviations.