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This problem asks you to plot data points, then apply linear regression and quadratic regression to predict the engin efficiency in the requested years

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

"""

This problem asks you to plot data points, then apply linear regression and quadratic regression to predict the engin efficiency in the requested years

"""

 

import numpy as np

import scipy.linalg as linalg import matplotlib.pyplot as plt

 

 

#---------------------------------------------------------------------

# note: LS is essentially solving Ax=b. all you need are A and b from the input data.

# for this problem the LS solution x will be the coefficients of polynomials

#

 

 

def eigen_linear(xData, yData, years): #

# add code to genetate A and b for linear-LS, then solve it to get the solution

# x = [c0, c1].

# here x contains only two components, from which you can construct the LS line

# (the line equation should be a function of year, you can use it to generate

# prediction for any input years, from which you can do plotting etc.

#

# for this one: solve the LS problem via solving the normal equation (A.T*A)*x = A.T*b

# you first form the A.T*A and A.T*b (note: * do not work for numpy array multiplication),

# after that you can call linalg.solve() to get the solution #

# this problem asks you to return the linear function eveluated on the input years,

# i.e. c0 + c1* years #

L=len(xData) G=np.ones((L,2))

 

G[:,1]=xData

z,_,_,_ = linalg.lstsq (G, yData)

 

return (z[0]+z[1]*years)

 

 

 

 

def eigen_quadratic(xData, yData, years): #

# add code to genetate A and b for quadratic-LS, then solve it to get the solution

# x = [c0, c1, c2].

# here x contains only three components, from which you can construct the LS curve

# (the curve should be a function of year, you can use it to generate

# prediction for any input years, from which you can do plotting etc.

#

# for this one: solve the LS problem by calling the scipy linalg.lstsq() function

# note this function has 4 outputs, you only need the 1st one to get the solution,

# a call like x, _, _, _ = linalg.lstsq(A, b) should get the LS solution for Ax=b

# if your A and b are correct #

# return the quadratic function eveluated on the input years, # i.e. c0 + c1* years + c2* years**2

#

n=len(xData) A=np.ones((n,3))

A[:,1]=xData

for i in range(1,3): A[:,i] = A[:, i-1] *xData c,_, _ , _ = linalg.lstsq (A, yData)

return c[0]+c[1]*years +c[2]*years**2

 

 

 

 

def engin_eff(xData, yData, years):

 

# add code to plot your data points,

# plot also the linear LS line, the quadratic LS curve, evaluated on the input years

# (the data from plotting should be by calling your defined functions above)

# set the labels and lengends etc. save your plot to an image file for upload

 

 

 

 

 

 

 

2022

 

 

#

# find the predicted values that are for years 2005, 2015, and

 

# you can simply pass predic_years=np.array([2005, 2015, 2022])

 

to your above functions to get

# the predictions for all these three years.

# (round your results to exactly 5 decimal points, including possible ending zeros,

# for canvas input) #

 

 

 

 

 

#=====================================================================

===================

# if your code above is correct, you do not need to modify anything below

#

if  name ==' main ':

 

#

# input the data from the table, make them to be np.array so that they can be operated componentwisely

#

xData = np.array([1718, 1767, 1774, 1775, 1792, 1816, 1828, 1834,

1878, 1906, 2000 ])

yData = np.array([0.005, 0.008, 0.014, 0.027, 0.045, 0.075, 0.12,

0.17, 0.172, 0.23, 0.46 ])

 

years = np.arange(1718, 2030, 1) #step=1 should make all integer years be included

 

engin_eff(xData, yData, years)

 

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