#OK to post homework #Quentin Dubroff, 2-24-19, Assignment 9 with(linalg): #ArDataMV(A,v,N) A procedure that outputs a list of random data from the n-dimensinal cube [-A,A]^n ArDataMV:=proc(A,v,N) local n,i,x,t,L,ra: n:=nops(v): ra:=rand(-A..A): L:=[]: for i from 1 to N do t:=1: x:=[seq(ra(),j=1..n)]: if dotprod(x,v)<0 then t:=0: fi: L:=[op(L),[x,t]]: od: L: end: #PAmv(L,n) inputs a list containing data of the form [VectorOfLengthn,ZeroOrOne] where VectorOfLengthn is a list of length n, #and n is the dimension and implements the perceptron algorithm PAmv:=proc(L,n) local i,j,c,m,v: v:=[0\$n]: m:=15*nops(L): for j from 1 to m do i:=1+(j mod nops(L)): if L[i][2]=1 and dotprod(L[i][1],v)<0 then #False negative then v:=v+L[i][1]: elif L[i][2]=0 and dotprod(L[i][1],v)>=0 then #False positive then v:=v-L[i][1]: fi: od: v: end: #TestPMV(L,v) generalizes our perceptron test to multiple dimensions. It inputs a list of classified data L and a vector v with n components #It outputs the tripe false negatives, false positives, length of L TestPMV:=proc(L,v) local fn,fp,i: fn:=0: fp:=0: for i from 1 to nops(L) do if dotprod(L[i][1],v)>=0 and L[i][2]=0 then fp:=fp+1: elif dotprod(L[i][1],v)<0 and L[i][2]=1 then fn:=fn+1: fi: od: [fn,fp,nops(L)]: end: #PA1corrected(L) First transfoms the list L to the format [[data,1],ZeroOrOne], and then uses procedure PAmv(L,n) with n=2 to output #two numbes a and b such that L[i][2]=1 iff ax+b>=0 PA1corrected:=proc(L) local J,i: J:=[]: for i from 1 to nops(L) do J:=[op(J),[[L[i][1],1],L[i][2]]]: od: [J,PAmv(J,2)]: end: #Summary of tests: It seems that the new perceptron algorithm PA1corrected has a lot more variance. Sometimes, it found a perfect separator #but sometimes it produced a large fraction of either false negatives exclusive or false positives. The old PA1 worked nicely for large n. #raising the number of times we iterate through the list seems to alleviate this problem. It could be related to the fact that our corrections #are very sensitive in the first variable but not so much for the second variable. ### THINGS FROM C9.txt #1D perceptron algorithm, #PA1(L): inputs a list of pairs [heightIncm,{0,1}] where 0 if she or he did not make it 1 if he or she did #outputs a numnber c a "predictor" that if x>c then they get to be admitted PA1:=proc(L) local i,c,n: c:=0: n:=nops(L): for i from 1 to n do if L[i][2]=1 and L[i][1]c then #False negative then c:=c+L[i][1]: fi: od: c: end: #inputs L1 and H1 (low and high heights in basketball players and the number N of applicants #and a cut-off c that >=c gets 1 (admitted) and 0 (rejected) ArData:=proc(L1,H1,N,c) local L,ra,i,t: ra:=rand(L1..H1): L:=[]: for i from 1 to N do t:=ra(): if t>=c then L:=[op(L),[t,1]]: else L:=[op(L),[t,0]]: fi: od: L: end: #Inputs a data set of pairs [height,1 or 0] and a cut-off c, outputs the #triple # of false negatives and # of false positive, total number of data entries TestP:=proc(L,c) local fn,fp,i: fn:=0: fp:=0: for i from 1 to nops(L) do if L[i][1]>=c and L[i][2]=0 then fp:=fp+1: elif L[i][1]