{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "###
San Jose State University
Department of Applied Data Science
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DATA 220
Mathematical Methods for Data Analysis
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Spring 2021
Instructor: Ron Mak
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Assignment #11
Matrix Operations Problem Set
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100 points total (20 points each)

Work together with your lab partner.
Write your solutions in one or more cells after each problem.
Whenever possible, use matrix operations and not loops.
You can add your own functions or other support code.
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Due May 6, 2020
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### PROBLEM 1. Amalgamated Widgets, Inc. manufactures several products using various amounts of materials per product:\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
ProductMetalWoodPlasticClothPaper
A01.30.20.80.4
B001.50.40.3
C0.25000.20.7
D000.30.70.5
E0.1500.50.40.8
\n", "\n", "#### On a certain day, the following total amounts of materials were used:\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
MetalWoodPlasticClothPaper
51.0312.0215.4373.1356.0
\n", "\n", "#### How many of each product were made that day?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "#### PROBLEM 2. Find the least-squares regression line for the points (1,1), (2,2), (3,4), (4,4), and (5,6) using the matrix form of the normal equations A.T@A@x = A.T@b." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "#### PROBLEM 3. The following table shows the mean distances x in astonomical units (multiples of the distance between the sun and the earth) and the periods y in years (how many earth-years to orbit the sun):\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
PlanetMercuryVenusEarthMarsJupiterSaturn
Distance x0.3870.7231.0001.5245.2039.537
Years y0.2410.6151.0001.88111.86229.457
\n", "\n", "#### It turns out that if you take the natural logarithm of the numbers in the table, the transformed points lie roughly in a straight line. Find the least-squares regression line using the matrix form of the normal equations A.T@A@x = A.T@b. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "#### PROBLEM 4.a. Here's more data about that certain day at Amalgamated Widgets, Inc. There are three more products and the cost to make one of each product.\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
ProductMetalWoodPlasticClothPaperCost
A01.30.20.80.421.60
B001.50.40.317.20
C0.25000.20.74.70
D000.30.70.59.60
E0.1500.50.40.815.70
F1.000.900.000.650.0026.80
G0.220.001.801.001.2525.50
H0.200.750.651.001.2522.40
\n", "\n", "#### The manufacturing cost of a product is related to the amounts of materials used to make the product. Find the least-squares regression line using the matrix form of the normal equations A.T@A@x = A.T@b that will compute an estimated manufacturing cost of a product." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "#### PROBLEM 4.b. Find the least-squares regression line for the same data using QR factorization. Verify that matrix R is upper-triangular and invertible. Verify that the columns of matrix Q are orthonormal. " ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.7" } }, "nbformat": 4, "nbformat_minor": 4 }