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

4. Two Gaussian distributions have the following characteristics:

P₁ = P₂ = 0.5, μ₁ = [-1₀ ], μ₂ = [+1₀ ], Σ₁ = Σ₁ = [ 1 0.5_0.5 1 ]

• a) (10%) Sketch the distributions;
• b) (10%) Design the Bayes classifier for minimum error for classification of the data;
• c) (10%) What is the Bayes error (compute it exactly)?
• d) (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.