Assignment 01 (450 Points)

Note: Written problems must be typed and submitted as a Portable Document Format (PDF) file. All major word processors provide equation editing capabilities; therefore, all mathematics must also be typed. Here is a small list of equation editing software (there are many others):

• Apache OpenOffice Math
• Word for Mac 2011
• Office 2007
• Office 2010
• Office 2013
• MathType
• LaTeX

• Study the following presentations:
  • Introduction to the Mathematics of Regression, Part 1:
    • Presentation:
      • PDF: MATHEMATICS_OF_REGRESSION_1.PDF
      • PPTX: MATHEMATICS_OF_REGRESSION_1.PPTX
    • Video:
      • MP4:
        • 1D Data (Mean, Variance, & Standard Deviation): MATHEMATICS_OF_REGRESSION_1_1.MP4
        • 2D Data (Covariance): MATHEMATICS_OF_REGRESSION_1_2.MP4
        • Linear Regression: MATHEMATICS_OF_REGRESSION_1_3.MP4
      • WMV:
        • 1D Data (Mean, Variance, & Standard Deviation): MATHEMATICS_OF_REGRESSION_1_1.WMV
        • 2D Data (Covariance): MATHEMATICS_OF_REGRESSION_1_2.WMV
        • Linear Regression: MATHEMATICS_OF_REGRESSION_1_3.WMV
  • Introduction to the Mathematics of Classification, Part 1:
    • Presentation:
      • PDF: MATHEMATICS_OF_CLASSIFICATION_1.PDF
      • PPTX: MATHEMATICS_OF_CLASSIFICATION_1.PPTX
    • Video:
      • MP4:
        • Classification using Level Set & Normal Vector: MATHEMATICS_OF_CLASSIFICATION_1_1.MP4
        • Rotation: MATHEMATICS_OF_CLASSIFICATION_1_2.MP4
        • Translation: MATHEMATICS_OF_CLASSIFICATION_1_3.MP4
      • WMV:
        • Classification using Level Set & Normal Vector: MATHEMATICS_OF_CLASSIFICATION_1_1.WMV
        • Rotation: MATHEMATICS_OF_CLASSIFICATION_1_2.WMV
        • Translation: MATHEMATICS_OF_CLASSIFICATION_1_3.WMV
  
  
• Complete the following written problems:
  1. Terminology (50 Points):
     
     Given the following task environments, enter their properties/characteristics. The properties/characteristics of the Checkers task environment is provided as an example.

     Task Environment	Fully Observable or Partially Observable	Single Agent or Multiagent	Deterministic or Stochastic	Episodic or Sequential	Static or Dynamic	Discrete or Continuous	Known or Unknown
     Checkers	Fully Observable	Multiagent	Deterministic	Sequential	Static	Discrete	Known
     Chess (without a clock)
     Poker
     Human Driving
     Robotic Car (Machine Driving)
     Medical Diagnosis
     Image Analysis
     Newborn Baby

     
     
  2. Mean and Standard Deviation (20 Points): Given the following test scores, calculate the Sample Mean, the Biased Sample Standard Deviation, and the Unbiased Sample Standard Deviation.
     
     Scores: { 60, 70, 80, 90, 100, 70, 80, 90, 75, 85, 80 }
     
     Show all work and give exact answers (i.e. roots, fractions, etc.), not decimal approximations.
     
     
  3. Regression (30 Points): Given the following partial radar output where the first parameter is bearing in degrees and the second parameter is range in nautical miles, convert from Polar coordinates to Cartesian coordinates and perform a linear regression on this data.
     
     Radar Data:
     
     Show all work and give exact answers (i.e. roots, fractions, etc.), not decimal approximations.
     
     

     Name	Value
     Mean:
     (Slope):
     (Intercept):

     
     
  4. Level Set/Normal Vector (50 Points): Given the following two classes each with three points, find the mean of each class and then find the equation of the plane that separates those classes halfway between their respective means. In other words, find the separating plane that could be used for classification with the maximum margin between the classes.
     
     Class 1 Points: { { 0, 1 }, { 1, 0 }, { 0, 0 } }
     
     Class 2 Points: { { 10, 10 }, { 10, 9 }, { 9, 10 } }
     
     Show all work and give exact answers (i.e. roots, fractions, etc.), not decimal approximations.
     
     

     Name	Value
     Mean of Class 1:
     Mean of Class 2:
     Midpoint:
     Normalized Normal Vector:
     Equation of the Separating Plane:

     
     
  5. Level Set (20 Points): Given the equation (a circle with a radius of one) and using a level set, design a single if statement that classifies points into two classes: (1) those points on or in the circle and (2) those points outside the circle. Show that { 0, 0 } is in that circle, but that { 1, 1 } is not.
     
     Show all work and give exact answers (i.e. roots, fractions, etc.), not decimal approximations.
     
     
• Complete the following programming problems on linprog4.cs.fsu.edu:
  
  Download the ZIP file containing the directory structure and files for these programming problems: assignment_01.zip
  
  
  1. Calculate the linear regression for a known amount of two dimensional data in an array (20 Points).
     
     Use the following files:
     • print.hpp: The file containing operator << to print arrays.
     • data.cpp: The file containing the constants dimension and data for this programming problem.
     • main.cpp: The file for editing.
     • makefile: The makefile for linprog4.cs.fsu.edu.
     
     Do not make changes to print.hpp, data.cpp, and makefile. Only make changes to main.cpp.
     
     Use std::cout to output information exactly in the following format (replace numbers below with your program's calculated values):
     
     Sample Mean x: 1.111111
Sample Mean y: 2.222222
Biased Sample Covariance xy: 3.333333
Biased Sample Variance xx: 4.444444
Weight w: 5.555555
Slope m: 6.666666
Intercept b: 7.777777
Linear Regression: 8.888888 * x + 9.999999

     where:
     • 1.111111 represents the mean of the x values,
     • 2.222222 represents the mean of the y values,
     • 3.333333 represents the biased sample covariance (the scaled sum of the product of differences for the x and y values),
     • 4.444444 represents the biased sample variance for the x values (the scaled sum of squared differences for the x values),
     • 5.555555 represents the value of the weight (w),
     • 6.666666 represents the value of the slope (m),
     • 7.777777 represents the value of the intercept (b),
     • 8.888888 and 9.999999 represent the value of the slope and the intercept, respectively (m * x + b)
     
     Note: The Weight and the Slope are the same values (i.e. w = m).
     
     
     Note: The plot of data in data.cpp is as follows:
     
     
     
  2. Calculate the linear regression for a known amount of two dimensional data in an array (20 Points).
     
     Use the following files:
     • print.hpp: The file containing operator << to print arrays.
     • data.cpp: The file containing the constants dimension and data for this programming problem.
     • main.cpp: The file for editing.
     • makefile: The makefile for linprog4.cs.fsu.edu.
     
     Do not make changes to print.hpp, data.cpp, and makefile. Only make changes to main.cpp.
     
     Use std::cout to output information exactly in the following format (replace numbers below with your program's calculated values):
     
     Sample Mean x: 1.111111
Sample Mean y: 2.222222
Biased Sample Covariance xy: 3.333333
Biased Sample Variance xx: 4.444444
Weight w: 5.555555
Slope m: 6.666666
Intercept b: 7.777777
Linear Regression: 8.888888 * x + 9.999999

     where:
     • 1.111111 represents the mean of the x values,
     • 2.222222 represents the mean of the y values,
     • 3.333333 represents the biased sample covariance (the scaled sum of the product of differences for the x and y values),
     • 4.444444 represents the biased sample variance for the x values (the scaled sum of squared differences for the x values),
     • 5.555555 represents the value of the weight (w),
     • 6.666666 represents the value of the slope (m),
     • 7.777777 represents the value of the intercept (b),
     • 8.888888 and 9.999999 represent the value of the slope and the intercept, respectively (m * x + b)
     
     Note: The Weight and the Slope are the same values (i.e. w = m).
     
     
     Note: The plot of data in data.cpp is as follows:
     
     
     
  3. Calculate the linear regression for an unknown amount of two dimensional data (100 Points).
     
     Use the following files:
     • data.txt: The file containing 2D data that your program will read in from Standard Input.
     • print.hpp: The file containing operator << to print arrays and std::vector.
     • main.cpp: The file for editing.
     • makefile: The makefile for linprog4.cs.fsu.edu.
     
     Do not make changes to print.hpp and makefile. Only make changes to main.cpp. You should make changes to data.txt to test your work; however, a different data.txt (not provided) will be used to grade this problem.
     
     Use std::cin to read the x and y values in data.txt from Standard Input. data.txt will be formatted as follows:
     
     sample 1's x value
sample 1's y value
sample 2's x value
sample 2's y value
.
.
.
sample n's x value
sample n's y value

     
     Use std::cout to output information exactly in the following format (replace numbers below with your program's calculated values):
     
     Sample Mean x: 1.111111
Sample Mean y: 2.222222
Biased Sample Covariance xy: 3.333333
Biased Sample Variance xx: 4.444444
Weight w: 5.555555
Slope m: 6.666666
Intercept b: 7.777777
Linear Regression: 8.888888 * x + 9.999999

     where:
     • 1.111111 represents the mean of the x values,
     • 2.222222 represents the mean of the y values,
     • 3.333333 represents the biased sample covariance (the scaled sum of the product of differences for the x and y values),
     • 4.444444 represents the biased sample variance for the x values (the scaled sum of squared differences for the x values),
     • 5.555555 represents the value of the weight (w),
     • 6.666666 represents the value of the slope (m),
     • 7.777777 represents the value of the intercept (b),
     • 8.888888 and 9.999999 represent the value of the slope and the intercept, respectively (m * x + b)
     
     Note: The Weight and the Slope are the same values (i.e. w = m).
     
     An online version of the Sample Mean is provided in the Introduction to the Mathematics of Regression, Part 1 presentation:
     • PDF: MATHEMATICS_OF_REGRESSION_1.PDF
     • PPTX: MATHEMATICS_OF_REGRESSION_1.PPTX
     
     There are also online versions of the Unbiased Sample Variance (a.k.a. Running or Moving Variance) and Unbiased Sample Covariance (a.k.a. Running or Moving Covariance) scattered around the Internet. For instance, here are a variety of algorithms for calculating variance that also include unbiased online versions of both sample variance and covariance: http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance. Note: We want online versions that calculate the biased sample variance and covariance, so we might want to multiply by a ( n - 1 ) / n somewhere.
     
     If your program does not run out of memory when trying to calculate linear regressions during grading, then these online, running, or moving versions are not required.
     
     If your program runs out of memory when trying to calculate linear regressions during grading, then half the value of this question will immediately be deducted.
     
     
  4. Classify a known amount of two dimensional data in an array (20 Points).
     
     Given the data in data.cpp below, create two planes: (1) one that passes through { -0.75, 0 } and { 0, 0.75 } and (2) another the passes through { 0, -0.75 } and { 0.75, 0 }
     
     For plane (1), ensure that it is oriented such that { 0, 1 } results in a positive value and { 0, 0 } results in a negative value.
     
     For plane (2), ensure that it is oriented such that { 1, 0 } results in a positive value and { 1, 1 } results in a negative value.
     
     For each datum, classify that datum based on the first plane ( plane (1) ) into two classes: (1) those whose f( x, y ) values in the plane are positive or zero or (2) those whose f( x, y ) values in the plane are negative. Collect each class in separate containers. One container for data that produced positive or zero values ( i.e. container (1) ). Another container for data that produced negative values ( i.e. container (2) ).
     
     For each datum in container (2), classify that datum based on the second plane ( plane (2) ) into two classes: (1) those whose f( x, y ) values in the plane are positive or zero or (2) those whose f( x, y ) values in the plane are negative. For data that produced positive or zero values, collect that data in container (1). For the data that again produced negative values, collect that data in container (3) ... or just keep it in container (2).
     
     Use std::sort from <algorithm> and a lambda or functor to sort both of these "classification" containers in ascending order based on x values first, then y values. Example: Sorting the list { { 1, 3 }, { 2, 1 }, { 1, 2 } } would result in the list { { 1, 2 }, { 1, 3 }, { 2, 1 } }.
     
     Use the following files:
     • print.hpp: The file containing operator << to print arrays and std::vector.
     • data.cpp: The file containing the constants dimension and data for this programming problem.
     • main.cpp: The file for editing.
     • makefile: The makefile for linprog4.cs.fsu.edu.
     
     Do not make changes to print.hpp, data.cpp, and makefile. Only make changes to main.cpp.
     
     Use std::cout to output information in the following format (replace the data below with data classified by your program):
     
     CLASS 1 (POSITIVE OR ZERO):
{ 1.111111, 2.222222 }
{ 3.333333, 4.444444 }
.
.
.

CLASS 2 (NEGATIVE):
{ 5.555555, 6.666666 }
{ 7.777777, 8.888888 }
.
.
.

     Note: The ellipses above represent the rest of the data for each classification and should not be included in your output.
     
     
     
     
  5. Classify a known amount of two dimensional data in an array (20 Points).
     
     Take a look at the picture of the data below. The Upper Left-Hand-Side Point Cloud contains 1600 points. The Lower Right-Hand-Side Point Cloud contains 1000 points. Create a classifier that separates the data such that 90% or more of the Upper Left-Hand-Side Point Cloud is in one class, while the Lower Right-Hand-Side is in another class.
     
     Use the following files:
     • print.hpp: The file containing operator << to print arrays and std::vector.
     • data.cpp: The file containing the constants dimension and data for this programming problem.
     • main.cpp: The file for editing.
     • makefile: The makefile for linprog4.cs.fsu.edu.
     
     Do not make changes to print.hpp, data.cpp, and makefile. Only make changes to main.cpp.
     
     Use std::cout to output information in the following format (replace the data below with data classified by your program):
     
     CLASS 1 (UPPER LEFT-HAND-SIDE POINT CLOUD):
{ 1.111111, 2.222222 }
{ 3.333333, 4.444444 }
.
.
.

CLASS 2 (LOWER RIGHT-HAND-SIDE POINT CLOUD):
{ 5.555555, 6.666666 }
{ 7.777777, 8.888888 }
.
.
.

     Note: The ellipses above represent the rest of the data for each classification and should not be included in your output.
     
     
     
     
  6. Classify an unknown amount of two dimensional data (100 Points).
     
     Create a program that takes in information from the command line to make a linear classifier and then classifies data it receives from Standard Input using that classifier.
     
     Use the following normalized normal vector from the presentation: such that when the angle from the command line is zero degrees, the normal vector will be { 1, 0 }
     
     The program must have the following usage:
     
     main.exe -x0 number -y0 number -angle number
     
     where the number following -x0 and -y0 is the origin point of the linear classifier (i.e. the plane) and the number following -angle is the angle in degrees.
     
     For each datum received from Standard Input, classify that datum into two classes: (1) those whose f( x, y ) values in the plane are positive or zero or (2) those whose f( x, y ) values in the plane are negative. Collect each class in separate containers. One container for data that produced positive or zero values ( i.e. container (1) ). Another container for data that produced negative values ( i.e. container (2) ).
     
     Use the following files:
     • data.txt: The file containing 2D data that your program will read in from Standard Input.
     • print.hpp: The file containing operator << to print arrays and std::vector.
     • main.cpp: The file for editing.
     • makefile: The makefile for linprog4.cs.fsu.edu.
     
     Do not make changes to print.hpp and makefile. Only make changes to main.cpp. You should make changes to data.txt to test your work; however, a different data.txt (not provided) will be used to grade this problem.
     
     Use std::cin to read the x and y values in data.txt from Standard Input. data.txt will be formatted as follows:
     
     sample 1's x value
sample 1's y value
sample 2's x value
sample 2's y value
.
.
.
sample n's x value
sample n's y value

     Use std::cout to output information in the following format (replace the data below with data classified by your program):
     
     CLASS 1 (POSITIVE OR ZERO):
{ 1.111111, 2.222222 }
{ 3.333333, 4.444444 }
.
.
.

CLASS 2 (NEGATIVE):
{ 5.555555, 6.666666 }
{ 7.777777, 8.888888 }
.
.
.

     Note: The ellipses above represent the rest of the data for each classification and should not be included in your output.
     
     
  
  
• After completing Assignment 01, create an assignment_01_lastname.pdf file for your written assignment and an assignment_01_lastname.zip file for your programming assignment (where lastname is your last name). Ensure that your assignment_01_lastname.zip retains the directory structure of the original zip file. In other words, ensure your zip file has the following directory structure:
  • /
    • classification_question_1/
      • data.cpp
      • data.png
      • main.cpp
      • makefile
    • classification_question_2/
      • data.cpp
      • data.png
      • main.cpp
      • makefile
    • classification_question_3/
      • data.txt
      • main.cpp
      • makefile
    • include/
      • print.hpp
    • regression_question_1/
      • data.cpp
      • data.png
      • main.cpp
      • makefile
    • regression_question_2/
      • data.cpp
      • data.png
      • main.cpp
      • makefile
    • regression_question_3/
      • data.txt
      • main.cpp
      • makefile
    • makefile
  
  Upload both your assignment_01_lastname.pdf file for your written assignment and your assignment_01_lastname.zip file for your programming assignment to the Assignment 01 location on the BlackBoard site: https://campus.fsu.edu.
  
  
• Note: Questions derived from the following sources:
  • Artificial Intelligence: A Modern Approach, Third Edition by Stuart Russell and Peter Norvig.
  • Udacity's Introduction to Artificial Intelligence course by Peter Norvig and Sebastian Thrun.
  • Coursera's Machine Learning course by Andrew Ng.
  • Coursera's Neural Networks for Machine Learning course by Geoffrey Hinton.
  • Coursera's Machine Learning course by Pedro Domingos.
  • edX's CS188.1x Artificial Intelligence course from BerkeleyX by Dan Klein.
  • Coursera's Introduction to Logic course by Michael Genesereth.
  • NPTEL's Artificial Intelligence course by Pallab Dasgupta.
  • NPTEL's Artificial Intelligence course by Anupam Basu and Sudeshna Sarkar.