logo Use CA10RAM to get 10%* Discount.
Order Nowlogo
(5/5)

Let p(x|wi) ∼ N (µi, σ2) for a two-category one-dimensional problem with P (ω1) = P (ω2) = 2.

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Problem 1. Let p(x|wi) ∼ N (µi, σ2) for a two-category one-dimensional problem with P (ω1) = P (ω2) = 2

Show that the minimum probability of error is given by where a = |µ1 µ2 | .

  • Use the inequality

Pe = √2π

eu /2du

Pe = √2πet /2    1dt ≤ √2πaea /2

show that Pe goes to zero as |µ1 µ2 | goes to infinity.

Problem 2. Consider a two-category classification problem in two dimensions with(x|ω1) ∼ N (0, I), p(x|ω2) ∼ N (.1Σ , I)and P (ω1) = P (ω2) = 1 ,

  • Calculate the Bayes decision
  • Calculate the Bhattacharyya error
  • Repeat the above for the same prior probabilities, p(x|ω1) ∼ N (0, . 2         0.5Σ), p(x|ω2) ∼ N (.1Σ , .5    4Σ)

Problem 3.        Suppose that we have three categories in two dimensions with the following underlying distributions:

.  Σ•      |           ∼ N2

 
  • p(x|ω1) ∼ N (0, I)

p(x ω )           ( 1  , I) 1

  • p(x|ω3) ∼ 1N (.0., I) + 1 N (.−0.5Σ , I)with P (ωi) = 1 , i = 1, 2, 3.
  • By explicit calculation of posterior probabilities, classify the point x = 0.3    for minimum probability

0.3

of error.

  • Suppose that for a particular test point the first feature is That is, classify x = .0∗.3Σ.
  • Suppose that for a particular test point the second feature is That is, classify x = .0.3Σ.
  • Repeat all of the above for x = 0.2 .

Problem 4. Consider two normal distributions with arbitrary but equal covariances. Prove that the Fisher linear discriminant, for suitable threshold, can be derived from the negative of the log-likelihood ratio

Problem 5. It is easy to see that the nearest-neighbor error rate P can equal the Bayes rate P if P = 0 (the best possibility) or if P = c1 (the worst possibility). One might ask whether or not there are problems for which P = P when P is between these extremes.

  • Show that the Bayes rate for the one-dimensional case where P (ωi) = 1 and
  • Show that for this case the nearest-neighbor rate is P = P .

Problem 6. Prove that the computational complexity of the basic nearest-neighbor editing algorithm for

n points in d dimension is O(d3n| d ∫ ln n).

Nearest-Neighbor Editing Algorithm

 1:  begin initialize j            0,           data set, n          #prototypes

2: construct the full Voronoi diagram of 3:   do j    j + 1, for each prototype x³ j 4:      find the Voronoi neighbors of x³ j

5:           if any neighbor is not from the same class as x³ j , then mark x³ j

6: until j = n

7:       discard all points that are not marked

8:       construct the Voronoi diagram of the remaining (marked) prototypes

9: end

Problem 7. Consider classifiers based on samples with priors P (ω1) = P (ω2) = 0.5 and the distributions

p(x|ω  ) = .2x,    0 ≤ x ≤ 1

(2)0, elsewhere

p(x|ω  ) = .2 − 2x,    0 ≤ x ≤ 1

  elsewhere

  • What is the Bayes decision rule and the Bayes classification error?
  • Suppose we randomly select a single point ω1 and a single point from ω2, and create a nearest-neighbor classifier. Suppose too we select a test point from one of the categories (ω1 for definiteness). Integrate to find the expected error rate P2(e).
  • Repeat with two training samples from each category and a single test point in order to find P2(e).
  • Generalize to show that in general,

Pn(e) = 3 + (n + 1)(n + 3) + 2(n + 2)(n + 3)

Confirm this formula makes sense in the n = 1 case.

  • Compare limn→∞ Pn(e) with the Bayes
(5/5)
Attachments:

Related Questions

. Introgramming & Unix Fall 2018, CRN 44882, Oakland University Homework Assignment 6 - Using Arrays and Functions in C

DescriptionIn this final assignment, the students will demonstrate their ability to apply two ma

. The standard path finding involves finding the (shortest) path from an origin to a destination, typically on a map. This is an

Path finding involves finding a path from A to B. Typically we want the path to have certain properties,such as being the shortest or to avoid going t

. Develop a program to emulate a purchase transaction at a retail store. This program will have two classes, a LineItem class and a Transaction class. The LineItem class will represent an individual

Develop a program to emulate a purchase transaction at a retail store. Thisprogram will have two classes, a LineItem class and a Transaction class. Th

. SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their instance variables we wish to define:

1 Project 1 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of

. Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their instance variables we wish to define:

1 Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of

Ask This Question To Be Solved By Our ExpertsGet A+ Grade Solution Guaranteed

expert
Um e HaniScience

734 Answers

Hire Me
expert
Muhammad Ali HaiderFinance

738 Answers

Hire Me
expert
Husnain SaeedComputer science

887 Answers

Hire Me
expert
Atharva PatilComputer science

787 Answers

Hire Me

Get Free Quote!

444 Experts Online