Description
1) Simulation: a) Write a function to simulate N uniformly-drawn points within bound polyhedra. The function SimPolyHedra would proceed by a 1st stage of uniformly drawing (possibly N) points within a box that is bounding the desired polyhedra, followed by a 2nd stage of filtering, so that the function outputs only the first N of those points that fall within the polyhedra. The 2nd stage would retain a point iff it is in any of p1 convex polyhedra that are provided as input. The input arguments are: N: The number of points to simulate Bounds: A real D×2 matrix, each of its rows specifying the (lower,upper) bounds of a Ddimensional box from which all points are drawn at the 1st stage Polyhedra: A cell-array of real matrices M1,…,Mp . Mi is of size fi×(D+1). It defines a convex polyhedron of fi faces, as all the vectors x in RD such that [ ] ⃗ . Each face is thus defined by a row of Mi interpreted as a hyperplane in RD . Note that Mi may be unbounded, by devfined faces, as we only consider its intersection with the Bounds box. Output: X: A real N×D matrix, each of its rows specifying a vector that is inside the Bounds box and inside at least one of the polyhedra Mi . To generate those you draw potential D dimensional vectors x whose transpose could serve as rows of X . You would then check whether to include each row, i.e. whether any for each such x there exists at least one Mi such that the [ ] ⃗ condition is satisfied. [20 points]
b) Use the above to write a function SimTanzania() that simulate points of particular colors in the Tanzanian flag (see attached Tanzania.pdf). This flag spans the axis-bounded rectangle between (0,0) and (1.5,1.0), and has 5 regions in green, yellow, black, yellow and cyan, separated by the lines Draw points inside the
planar rectangle of the flag, and save 4 text files, each of N=50 rows and D=2 columns of numbers in text, specifying 50 points of the appropriate color: Tanzania_green.txt, Tanzania_cyan.txt,Tanzania_black.txt and Tanzania_yellow.txt .
[5 points]
2) Probabilistic interpretation: a) Consider SimPoly of Assignment2, Question 1 as defining a probability space of potential outputs y. As such, it defines a probability density function f (y) over possible values of y= y . Of course, this probability space is different for each input, so f (y) depends on the inputs RealThetas, sigma and x. Denote it f θ , ,x (y) for RealThetas= θ , sigma= and x= x .Prove that the least-squares regression result θ = θ * maximizes f θ , ,x (y) for any ,x and y. [15 points] b) Consider SimLogistic of Assignment2, Question 3 , with zero noise, as defining a probability space of potential outputs y. As such, it defines a probability function P(y) over possible values of y= y . Of course, this probability space is different for each input, so P (y) depends on the inputs RealThetas and x. Denote it P θ , x (y) for RealThetas= θ and x= x . Prove that the logistic regression result θ = θ * maximizes P θ ,x (y) for any x and y. [15 points]
Guidance: Neither 2a nor 2b requires computing derivatives. Both can be solved by the definition of θ * as an ERM, so you are welcome to just use that.
3) Optional: Prepare a single sided, single page, 12-font English-letter (with potential notations in Greek) cheat sheet for the quiz scheduled for Feb 19th. This would be the only allowed material. [0 points]
