Intro to Machine Learning Assignment 1

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CS 484: Intro to Machine Learning
Assignment 1
Question 1 (30 points)
Write a Python program to calculate the density estimator of a histogram. Use the field x in the
NormalSample.csv file.
a) (5 points) Let a be the largest integer less than the minimum value of the field x, and b be the
smallest integer greater than the maximum value of the field x. What are the values of a and b?
The minimum value of the field x is 26.3 and maximum value is 35.4. Thus, a is 26 and b is 36.
b) (5 points) Use h = 0.25, minimum = a and maximum = b. List the coordinates of the density
estimator. Paste the histogram drawn using Python or your favorite graphing tools.
[(26.125, 0.0), (26.375, 0.003996003996003996), (26.625, 0.0), (26.875, 0.0), (27.125,
0.003996003996003996), (27.375, 0.0), (27.625, 0.007992007992007992), (27.875,
0.011988011988011988), (28.125, 0.027972027972027972), (28.375, 0.01998001998001998), (28.625,
0.05194805194805195), (28.875, 0.05194805194805195), (29.125, 0.07992007992007992), (29.375,
0.0919080919080919), (29.625, 0.16783216783216784), (29.875, 0.12387612387612387), (30.125,
0.21178821178821178), (30.375, 0.17182817182817184), (30.625, 0.27972027972027974), (30.875,
0.15984015984015984), (31.125, 0.24375624375624375), (31.375, 0.1958041958041958), (31.625,
0.3756243756243756), (31.875, 0.1878121878121878), (32.125, 0.36363636363636365), (32.375,
0.17182817182817184), (32.625, 0.32367632367632365), (32.875, 0.15184815184815184), (33.125,
0.1838161838161838), (33.375, 0.08791208791208792), (33.625, 0.0959040959040959), (33.875,
0.05194805194805195), (34.125, 0.04795204795204795), (34.375, 0.015984015984015984), (34.625,
0.01998001998001998), (34.875, 0.007992007992007992), (35.125, 0.0), (35.375,
0.007992007992007992), (35.625, 0.0), (35.875, 0.0)]
c) (5 points) Use h = 0.5, minimum = a and maximum = b. List the coordinates of the density
estimator. Paste the histogram drawn using Python or your favorite graphing tools.
[(26.25, 0.001998001998001998), (26.75, 0.0), (27.25, 0.001998001998001998), (27.75,
0.00999000999000999), (28.25, 0.023976023976023976), (28.75, 0.05194805194805195), (29.25,
0.08591408591408592), (29.75, 0.14585414585414586), (30.25, 0.1918081918081918), (30.75,
0.21978021978021978), (31.25, 0.21978021978021978), (31.75, 0.2817182817182817), (32.25,
0.2677322677322677), (32.75, 0.23776223776223776), (33.25, 0.13586413586413587), (33.75,
CS 484: Fall 2020 Assignment 1
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0.07392607392607392), (34.25, 0.03196803196803197), (34.75, 0.013986013986013986), (35.25,
0.003996003996003996), (35.75, 0.0)]
d) (5 points) Use h = 1, minimum = a and maximum = b. List the coordinates of the density
estimator. Paste the histogram drawn using Python or your favorite graphing tools.
[(26.5, 0.000999000999000999), (27.5, 0.005994005994005994), (28.5, 0.03796203796203796), (29.5,
0.11588411588411589), (30.5, 0.2057942057942058), (31.5, 0.25074925074925075), (32.5,
0.25274725274725274), (33.5, 0.1048951048951049), (34.5, 0.022977022977022976), (35.5,
0.001998001998001998)]
e) (5 points) Use h = 2, minimum = a and maximum = b. List the coordinates of the density
estimator. Paste the histogram drawn using Python or your favorite graphing tools.
[(27.0, 0.0034965034965034965), (29.0, 0.07692307692307693), (31.0, 0.22827172827172826), (33.0,
0.17882117882117882), (35.0, 0.012487512487512488)]
CS 484: Fall 2020 Assignment 1
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f) (5 points) Among the four histograms, which one, in your honest opinions, can best provide your
insights into the shape and the spread of the distribution of the field x? Please state your
arguments.
I would say h = 0.5 because it is the most uniform, can see the shape and spread of the
distribution better than other h’s, and also best fits the probability density estimation in terms
of the ‘area’ it covers.
CS 484: Fall 2020 Assignment 1
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Question 2 (10 points)
Use in the NormalSample.csv to generate box-plots for answering the following questions.
a) (5 points) What is the five-number summary of x for each category of the group? What are the
values of the 1.5 IQR whiskers for each category of the group?
All x summary
Min: 26.3
Q1: 30.4
Median: 31.5
Q3: 32.4
Max: 35.4
1.5 IQR whiskers = (27.4, 35.4)
------------------------------------
0.0 summary
Min: 26.3
Q1: 29.4
Median: 30.0
Q3: 30.6
Max: 35.4
1.5 IQR whiskers = (27.6, 32.4)
------------------------------------
1.0 summary
Min: 26.3
Q1: 31.4
Median: 32.1
Q3: 32.7
Max: 35.4
1.5 IQR whiskers = (29.45, 34.65)
b) (5 points) Draw a graph where it contains the boxplot of x, the boxplot of x for each category of
Group (i.e., three boxplots within the same graph frame). Use the 1.5 IQR whiskers, identify the
outliers of x, if any, for the entire data and for each category of the group.
All : 2 outliers = [26.3 27.2]
0 : 2 outliers = [26.3 27.2]
1 : 8 outliers = [29.1 29.3 29.3 34.7 34.8
34.9 35.3 35.4]
CS 484: Fall 2020 Assignment 1
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Question 3 (40 points)
The data, FRAUD.csv, contains results of fraud investigations of 5,960 cases. The binary variable FRAUD
indicates the result of a fraud investigation: 1 = Fraudulent, 0 = Otherwise. The other interval variables
contain information about the cases.
1. TOTAL_SPEND: Total amount of claims in dollars
2. DOCTOR_VISITS: Number of visits to a doctor
3. NUM_CLAIMS: Number of claims made recently
4. MEMBER_DURATION: Membership duration in number of months
5. OPTOM_PRESC: Number of optical examinations
6. NUM_MEMBERS: Number of members covered
You are asked to use the Nearest Neighbors algorithm to predict the likelihood of fraud.
a) (5 points) What percent of investigations are found to be fraudulent? Please give your answer
up to 4 decimal places.
Total: 5960, Fraud: 1189, Not Fraud: 4771
19.9496% of investigations are found to be fraudulent
b) (5 points) Use the BOXPLOT function to produce horizontal box-plots. For each interval variable,
one box-plot for the fraudulent observations, and another box-plot for the non-fraudulent
observations. These two box-plots must appear in the same graph for each interval variable.
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c) (10 points) Orthonormalize interval variables and use the resulting variables for the nearest
neighbor analysis. Use only the dimensions whose corresponding eigenvalues are greater than
one.
i. (5 points) How many dimensions are used?
6 dimensions are used
ii. (5 points) Please provide the transformation matrix? You must provide proof that the
resulting variables are actually orthonormal.
Transformation matrix:
[[-6.49862374e-08 -2.41194689e-07 2.69941036e-07 -2.42525871e-07
 -7.90492750e-07 5.96286732e-07]
[ 7.31656633e-05 -2.94741983e-04 9.48855536e-05 1.77761538e-03
 3.51604254e-06 2.20559915e-10]
[-1.18697179e-02 1.70828329e-03 -7.68683456e-04 2.03673350e-05
 1.76401304e-07 9.09938972e-12]
[ 1.92524315e-06 -5.37085514e-05 2.32038406e-05 -5.78327741e-05
 1.08753133e-04 4.32672436e-09]
[ 8.34989734e-04 -2.29964514e-03 -7.25509934e-03 1.11508242e-05
 2.39238772e-07 2.85768709e-11]
[ 2.10964750e-03 1.05319439e-02 -1.45669326e-03 4.85837631e-05
 6.76601477e-07 4.66565230e-11]]
Transpose of transformed x multiplied by transformation matrix should result in
identity matrix
Result is : [[ 1.00000000e+00 -6.28837260e-17 -1.69753534e-15 8.78069590e-15
 1.00267017e-15 -2.18900419e-16]
[-6.28837260e-17 1.00000000e+00 -5.58580959e-16 -1.91886437e-14
 -1.51354623e-15 -2.93168267e-16]
[-1.69753534e-15 -5.58580959e-16 1.00000000e+00 1.73819292e-15
 8.15320034e-17 -2.94902991e-17]
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[ 8.78069590e-15 -1.91886437e-14 1.73819292e-15 1.00000000e+00
 1.13286117e-14 -3.82766735e-15]
[ 1.00267017e-15 -1.51354623e-15 8.15320034e-17 1.13286117e-14
 1.00000000e+00 -6.27969898e-16]
[-2.18900419e-16 -2.93168267e-16 -2.94902991e-17 -3.82766735e-15
 -6.27969898e-16 1.00000000e+00]]
= Which is an identity matrix because the diagonal is 1, while others are basically 0
Also, orthonormalizing using the orth function from scipy and confirming:
[[ 1.00000000e+00 -7.97972799e-17 9.10729825e-17 -2.77555756e-17
 -2.42861287e-17 -4.33680869e-18]
[-7.97972799e-17 1.00000000e+00 -1.90819582e-17 2.94902991e-17
 -5.20417043e-17 -6.14742632e-17]
[ 9.10729825e-17 -1.90819582e-17 1.00000000e+00 8.45677695e-18
 8.13151629e-18 -3.16858085e-17]
[-2.77555756e-17 2.94902991e-17 8.45677695e-18 1.00000000e+00
 3.26995375e-16 -8.51098705e-18]
[-2.42861287e-17 -5.20417043e-17 8.13151629e-18 3.26995375e-16
 1.00000000e+00 3.38921599e-16]
[-4.33680869e-18 -6.14742632e-17 -3.16858085e-17 -8.51098705e-18
 3.38921599e-16 1.00000000e+00]]
You also get an identity matrix,
So the resulting variables are actually orthonormal
d) (10 points) Use the NearestNeighbors module to execute the Nearest Neighbors algorithm using
exactly five neighbors and the resulting variables you have chosen in c). The
KNeighborsClassifier module has a score function.
i. (5 points) Run the score function, provide the function return value
0.8778523489932886
ii. (5 points) Explain the meaning of the score function return value.
The value represents the accuracy of the function with the given data, meaning the
model has a 87.78% accuracy.
CS 484: Fall 2020 Assignment 1
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e) (5 points) For the observation which has these input variable values: TOTAL_SPEND = 7500,
DOCTOR_VISITS = 15, NUM_CLAIMS = 3, MEMBER_DURATION = 127, OPTOM_PRESC = 2, and
NUM_MEMBERS = 2, find its five neighbors. Please list their input variable values and the target
values. Reminder: transform the input observation using the results in c) before finding the
neighbors.
5 neighbors - Tuple of distance and element of neighbors = [(6.585445079827193e-10, 588),
(0.010457155671774961, 2897), (0.01206994606792243, 1199), (0.012186196058304651, 1246),
(0.014037896643257355, 886)]
Values of the 5 neighbors:
{'CASE_ID': 589.0, 'FRAUD': 1.0, 'TOTAL_SPEND': 7500.0, 'DOCTOR_VISITS': 15.0,
'NUM_CLAIMS': 3.0, 'MEMBER_DURATION': 127.0, 'OPTOM_PRESC': 2.0, 'NUM_MEMBERS':
2.0}
{'CASE_ID': 2898.0, 'FRAUD': 1.0, 'TOTAL_SPEND': 16000.0, 'DOCTOR_VISITS': 18.0,
'NUM_CLAIMS': 3.0, 'MEMBER_DURATION': 146.0, 'OPTOM_PRESC': 3.0, 'NUM_MEMBERS':
2.0}
{'CASE_ID': 1200.0, 'FRAUD': 1.0, 'TOTAL_SPEND': 10000.0, 'DOCTOR_VISITS': 16.0,
'NUM_CLAIMS': 3.0, 'MEMBER_DURATION': 124.0, 'OPTOM_PRESC': 2.0, 'NUM_MEMBERS':
1.0}
{'CASE_ID': 1247.0, 'FRAUD': 1.0, 'TOTAL_SPEND': 10200.0, 'DOCTOR_VISITS': 13.0,
'NUM_CLAIMS': 3.0, 'MEMBER_DURATION': 119.0, 'OPTOM_PRESC': 2.0, 'NUM_MEMBERS':
3.0}
{'CASE_ID': 887.0, 'FRAUD': 1.0, 'TOTAL_SPEND': 8900.0, 'DOCTOR_VISITS': 22.0,
'NUM_CLAIMS': 3.0, 'MEMBER_DURATION': 166.0, 'OPTOM_PRESC': 1.0, 'NUM_MEMBERS':
2.0}
f) (5 points) Follow-up with e), what is the predicted probability of fraudulent (i.e., FRAUD = 1)? If
your predicted probability is greater than or equal to your answer in a), then the observation
will be classified as fraudulent. Otherwise, non-fraudulent. Based on this criterion, will this
observation be misclassified?
When I do the function predict_proba, it results in [[0, 1.]] which means 0% chance its not
fraud, and 100% its fraud. Since the predicted probability is greater than the answer in a, 19%,
the observation is classified to be fraudulent and probably wouldn’t be misclassified as it has a
really high probability, 100%.
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Question 4 (10 points)
Try the Manhattan distance and the Chebyshev distance in analyzing the ten-observations example in
Chapter 2 of A Practitioner's Guide to Machine Learning. Does the optimal number of neighbors the
same as that found with the Euclidean distance?
Yes, optimal number of neighbors is the same as found with Euclidean, 2 neighbors
Using python, redid the steps for Manhattan and Chebyshev and got that 2 neighbors is the most
optimal.
Manhattan : Chebyshev:
Euclidean :
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Question 5 (10 points)
Include Entropy_Image in analyzing the banknote data mentioned in Chapter 2 of A Practitioner's Guide
to Machine Learning. Does orthonormalizing the four input features make a difference in determining
the number of neighbors?
The result of the original and orthonormalizing with Entropy Image:
With the original, only misclassification rate at k = 3, but
orthonormalizing, there are misclassification rates at k = 6, 8, 9.
So, orthonormalizing the four input features does make a
difference in determining the number of neighbors.