Name:
Student ID #:
Please read the instructions carefully:
import numpy as np
import matplotlib.pyplot as plt
ver = "B"
Let
data = np.genfromtxt("p1-%s.csv"%ver, delimiter=",")
If you project the points (rows) in data onto its 0-th and 1-st principal components, then you will see a four-letter word in upper case.
What is the word?
(Note the word might be upside down or left/right mirrored.)
### your code to reach the answer
data = np.genfromtxt("p1-%s.csv"%ver, delimiter=",")
from sklearn.decomposition import PCA
model = PCA(2)
X_new = model.fit_transform(data)
plt.axis("equal")
plt.scatter(*X_new.T)
<matplotlib.collections.PathCollection at 0x7fa12ba81180>
Your answer: LIFE
Let
dist = np.genfromtxt("p2-%s.csv"%ver, delimiter=",")
be the distance matrix of a dataset.
Use MDS to reconstruct the dataset in $\mathbb{R}^2$.
Then draw the scatter plot of them.
You are supposed to see one of the four suit symbols:
What do you see?
### your code to reach the answer
dist = np.genfromtxt("p2-%s.csv"%ver, delimiter=",")
from sklearn.manifold import MDS
model = MDS(dissimilarity="precomputed")
X_new = model.fit_transform(dist)
plt.axis("equal")
plt.scatter(*X_new.T)
<matplotlib.collections.PathCollection at 0x7fa12b99cc70>
Your answer: diamond
Let
X = np.genfromtxt("p3-%s.csv"%ver, delimiter=",")
y = np.array([0]*50 + [1]*50 + [2]*50 + [3]*50)
w = np.array([0,0,0,0])
Suppose after training a $k$-means model with the data X ,
the output label array is y .
What is the predicted label of w under this setting?
### your code to reach the answer
X = np.genfromtxt("p3-%s.csv"%ver, delimiter=",")
y = np.array([0]*50 + [1]*50 + [2]*50 + [3]*50)
w = np.array([0,0,0,0])
centers = np.zeros((4,X.shape[1]))
for i in range(4):
centers[i] = X[y == i].mean(axis=0)
dist = np.sqrt(np.sum((centers - w) ** 2, axis=1))
dist.argmin()
1
Your answer: 1
Let
X = np.genfromtxt("p4-%s.csv"%ver, delimiter=",")
One of the data points (rows) in X is an obvious outlier.
Use DBSCAN to find its index.
### your code to reach the answer
X = np.genfromtxt("p4-%s.csv"%ver, delimiter=",")
from sklearn.cluster import DBSCAN
model = DBSCAN()
y_new = model.fit_predict(X)
np.where(y_new == -1)
(array([648]),)
Your answer: 648
Let
data = np.genfromtxt("p5-%s.csv"%ver, delimiter=",")
x,y = data.T
Let $x_i$ and $y_i$ be the entries in x and y , respectively.
Let $f(x) = 3 + 4x$.
Find the value $\sum_{i} (f(x_i) - y_i)^2$.
### your code to reach the answer
data = np.genfromtxt("p5-%s.csv"%ver, delimiter=",")
x,y = data.T
np.sum((3 + 4*x - y)**2)
0.6588026544945301
Your answer: ~0.65
Let
data = np.genfromtxt("p6-%s.csv"%ver, delimiter=",")
x,y = data.T
Let $x_i$ and $y_i$ be the entries in x and y , respectively.
Let $f(x) = c_2x^2 + c_4x^4$.
Find $c_2, c_4$ such that $\sum_{i} (f(x_i) - y_i)^2$ is minimized.
### your code to reach the answer
data = np.genfromtxt("p6-%s.csv"%ver, delimiter=",")
x,y = data.T
X = x[:,np.newaxis] ** np.array([2,4])
from sklearn.linear_model import LinearRegression
model = LinearRegression(fit_intercept=False)
model.fit(X, y)
model.coef_
array([ 4.00031613, -1.0000116 ])
Your answer: $c_2 \sim 4.00$, $c_4 \sim -1.00$
Let
path = "p7-%s"%ver
There are 100 pictures in the path folder,
where digits00 ~ digits49 are pictures of hand-written digit 0,
while digits50 ~ digits99 are pictures of hand-written digit 1.
Train an $k$-nearest neighbor classification model with k = 5 .
Then predict what is the hand-written digit (0 or 1) in blur.png .
### your code to reach the answer
path = "p7-%s"%ver
import os
from PIL import Image
X = np.zeros((100, 28, 28), dtype=int)
for j in range(100):
img = Image.open(os.path.join(path, "digit%02d.png"%j))
X[j] = np.array(img)
X_flat = X.reshape(100, 28*28)
y = np.array([0]*50 + [1]*50)
img = Image.open(os.path.join(path, "blur.png"))
blur = np.array(img)
blur_flat = blur.reshape(1, 28*28)
from sklearn.neighbors import KNeighborsClassifier
model = KNeighborsClassifier(5)
model.fit(X_flat, y)
y_new = model.predict(blur_flat)
y_new
array([0])
Your answer: 0
Let
y = np.genfromtxt("p8-%s.csv"%ver, delimiter=",")
The array y contains 1000 labels,
representing the three categories 0, 1, 2 .
Find the Gini impurity of y .
### your code to reach the answer
y = np.genfromtxt("p8-%s.csv"%ver, delimiter=",")
dtrib = np.unique(y, return_counts=True)[1]
prob = dtrib / dtrib.sum()
np.sum(prob * (1 - prob))
0.56
Your answer: ~0.56
Let
X = np.genfromtxt("p9-%s.csv"%ver, delimiter=",")
Train $k$-means models with k = 1,2,...,10
and find the corresponding inertias.
Use the elbow method to suggest the number of clusters in X .
### your code to reach the answer
X = np.genfromtxt("p9-%s.csv"%ver, delimiter=",")
from sklearn.cluster import KMeans
inertias = []
for k in range(1,11):
model = KMeans(k)
model.fit(X)
inertias.append(model.inertia_)
plt.plot(np.arange(1,11), inertias)
[<matplotlib.lines.Line2D at 0x7fa12b4baa40>]
Your answer: 4
Let
X = np.genfromtxt("p10-%s.csv"%ver, delimiter=",")
It is known that the points (rows) in X look like three layers of spheres centered at the origin.
Find their radii.
Recall that the radius of a sphere centered at the origin is $\sqrt{x^2 + y^2 + z^2}$
for any point $(x,y,z)$ on the sphere.
### your code to reach the answer
X = np.genfromtxt("p10-%s.csv"%ver, delimiter=",")
radii = np.sqrt(np.sum(X**2, axis=1))[:,np.newaxis]
from sklearn.cluster import KMeans
model = KMeans(3)
model.fit(radii)
model.cluster_centers_
array([[2. ],
[1.41421356],
[2.44948974]])
Your answer: ~1.41, ~2, ~2.44
Suppose your have 1000 pictures of pictures and
they are labeled by either 0 for dogs or 1 for cats.
If you want to train a model to make a prediction of other pictures
on whether they are pictures of dogs or cats,
which model you would choose.
Choose from one of the following:
The answer might not be unique.
Add one or two sentences to justify your choice.
Your answer: Either 7 or 8 is okay, since it is a classification problem.
Exam ends here.
Total point = 20 (+2)
### points for each problem
pts = [0,0,0,0,0,
0,0,0,0,0,
0]
total = sum(pts)
print("Your total score =", total)