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Solutions to Exercises from STA663-2015
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‎PublishedSolutions/Exercise05-Solutions.ipynb‎

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‎PublishedSolutions/Exercise06-Solutions.ipynb‎

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‎PublishedSolutions/Exercises01Solutions.ipynb‎

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{
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"metadata": {
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"name": "",
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"signature": "sha256:1f652cb57d42a081aa2ae79bae5664160b451147c9a2113e053bb812c7c2ae68"
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},
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"nbformat": 3,
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"nbformat_minor": 0,
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"worksheets": [
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{
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"cells": [
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"import string\n",
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"import numpy as np\n",
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"from collections import Counter\n",
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"import urllib2"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 1
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Exercise 1"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"# Possible solution\n",
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"\n",
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"x3 = [3*i for i in range(1, (1000+2)/3)]\n",
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"x5 = [5*i for i in range(1, (1000+4)/5)]\n",
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"x35 = [15*i for i in range(1, (1000+14)/15)]\n",
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"print sum(x3) + sum(x5) - sum(x35)"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"233168\n"
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]
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}
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],
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"prompt_number": 2
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"# Alternative solution\n",
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"\n",
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"s = 0\n",
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"for i in range(1,1000):\n",
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" if i % 3 == 0 or i % 5 == 0:\n",
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" s += i\n",
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"print s"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"233168\n"
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]
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}
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],
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"prompt_number": 3
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Exercise 2"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def sample_mean(xs):\n",
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" \"\"\"Samplele mean.\"\"\"\n",
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" return sum(xs)/float(len(xs))\n",
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"\n",
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"def sample_std(xs):\n",
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" \"\"\"Sample standard deviaiton.\"\"\"\n",
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" n = len(xs)\n",
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" xbar = sample_mean(xs)\n",
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" s = 0.0\n",
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" for x in xs:\n",
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" s += (x - xbar)**2\n",
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" return (s/(n-1))**0.5\n",
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"\n",
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"def sample_correlation(xs, ys):\n",
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" \"\"\"Sample correlation coefficient.\"\"\"\n",
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" n = len(xs)\n",
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" xbar = sample_mean(xs)\n",
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" ybar = sample_mean(ys)\n",
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" sx = sample_std(xs)\n",
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" sy = sample_std(ys)\n",
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" r = 0.0\n",
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" for x, y in zip(xs, ys):\n",
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" r += ((x - xbar)/sx) * ((y - ybar)/sy)\n",
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" return r/(n-1)\n",
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"\n",
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"x = [10.0, 8.0, 13.0, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0]\n",
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"y = [8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68]\n",
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"print sample_correlation(x, y)"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"0.816420516345\n"
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]
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}
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],
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"prompt_number": 4
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Exercise 3"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def hailstone(n):\n",
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" \"\"\"Given a positive integer n, return the series of hailstone numbers.\"\"\"\n",
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" acc = []\n",
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" while n != 1:\n",
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" acc.append(n)\n",
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" if n%2 == 0:\n",
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" n /= 2\n",
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" else:\n",
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" n = n*3 + 1\n",
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" acc.append(1)\n",
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" return acc\n",
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"\n",
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"seq = hailstone(23)\n",
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"print seq\n",
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"print len(seq)"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"[23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1]\n",
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"16\n"
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]
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}
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],
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"prompt_number": 5
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Exercise 4"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def let2num(c):\n",
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" \"\"\"Convert lowercase character to number.\"\"\"\n",
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" return ord(c) - ord('a')\n",
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"\n",
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"def num2let(n):\n",
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" \"\"\"Convert number to lowercase character.\"\"\"\n",
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" return chr(ord('a') + n)\n",
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"\n",
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"def encode(cs, n):\n",
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" \"\"\"Caesar cipher with offset n.\"\"\"\n",
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" return ''.join([num2let((let2num(c) + n) % 26) if c.islower() else c for c in cs])"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 6
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def chisq(os, es):\n",
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" \"\"\"Retruns chi-square score given observed os and expected es frequencies.\"\"\"\n",
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" os = np.array(os)\n",
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" es = np.array(es)\n",
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" return np.sum((os - es)**2.0/es)"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 7
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def freqs(text):\n",
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" \"\"\"Returns relative frequenies of lowercase letter in text.\"\"\"\n",
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" ctr = Counter(text)\n",
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" counts = np.array([ctr[c] for c in string.lowercase], dtype='float')\n",
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" return counts/counts.sum()"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 8
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"# Use Pride and Prejudice to build up base frequencies\n",
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"text = urllib2.urlopen('http://www.gutenberg.org/ebooks/1342.txt.utf-8').read()\n",
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"ref_freqs = freqs(text)"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 9
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def crack(code):\n",
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" \"\"\"Find the original text by identifying the most likely encoding offset.\"\"\"\n",
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" shift = np.argmin([chisq(freqs(encode(code, n)), ref_freqs) for n in range(26)])\n",
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" return encode(code, shift)"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 10
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"cs = 'Statistical computing and computation is fun!'\n",
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"code = encode(cs, np.random.randint(1, 26))\n",
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"print code\n",
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"crack(code)"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"Szgzoyzoigr iusvazotm gtj iusvazgzout oy lat!\n"
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]
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},
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{
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"metadata": {},
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"output_type": "pyout",
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"prompt_number": 11,
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"text": [
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"'Statistical computing and computation is fun!'"
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]
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}
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],
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"prompt_number": 11
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 11
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}
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],
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"metadata": {}
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}
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]
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}

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