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SGN-41007 Pattern Recognition and Machine Learning
Exam 9.10.2019
Heikki Huttunen

» Use of calculator is allowed.

» Use of other materials is not allowed.

b The exam guestions need not be returned after the exam.
b You may answer in English or Finnish.

1. Are the following statements true or false? No need to justify your answer, just T or F.
Correct answer: 1 pts, wrong answer: < pts, no answer 0 pts.

(a) Maximum likelihood estimators are unbiased.

(b) The Receiver Operating Characteristics curve plots the probability of detection versus
the probability of false alarm for all thresholds.

(c) Least sguares estimator minimizes the sguared distance between the data and the
model.

(d) The number of support vectors of a support vector machine eguals the total number
of samples.

(e) The LDA maximizes the variance of samples in each classes.

(f) Cross-validation is used for model accuracy evaluation.

2. The Poisson distribution is a discrete probability distribution that expresses the probability
of a number of events x > 0 occurring in a fixed period of time:

 

Ax
e A
PSN) =
We measure N samples: xo, X1,...,Xn-1 and assume they are Poisson distributed and in-

dependent of each other.

(a) Compute the probability p(x;A) of observing the samples x = (X0, X1,...,XN-1). (1p)
(b) Compute the natural logarithm of p, i.e., log p(x;N). (1p)

(c) Differentiate the result with respect to A. (2p)

(d) Find the maximum of the function, i.e., the value where z logp(x;N) = 0. (2p)

 
Prediction Truelabel

   

Sample 1
Sample 2 05
Sample 3 0.6
Sample 4 0.1

oO 0

Table 1: Results on test data for guestion 5a.

3. Two measurements x(n) and y(n) depend on each other in a linear manner, and there are
the following measurements available:

 

 

 

 

n 0 1 2
x(n) | 7 9 4
y(m) | 12 15 4

We want to model the relationship between the two variables using the model:
yln) = ax(n) + b.

Find the L,-regularized least sguares estimates & and 6 that minimize the sguared error
using penalty X = 10!

4. (6 pts) Consider the Keras model defined in Listing 1. Inputs are 64 x 64 color images from
10 categories.

(a) Draw a diagram of the network.

(b) Compute the number of parameters for each layer, and their total number over all
layers.

5. (a) (4p) A random forest classifier is trained on training data set and the
predict proba method is applied on the test data of Table 1. Draw the receiver
operating characteristic curve. What is the Area Under Curve (AUC) score?

(b) (2p) A binary classifier is trained with 1 million samples from two classes. The AUC
of the classifier on test data with another 1 million samples, is 0.768. We choose one
sample from the positive class at random and another three samples from the negative
class at random. What is the probability that the sample from the positive class has
highest score of the four samples [hint: study the literature on the last page]?

1 Alternatively, the unregularized solution will give you max. 4 points.
 

Listing 1: ACNN model defined in Keras

 

model = Seguential()

model.add(Convolution2D (32, w, h, input shape=sh, border mode="sams' ))
model .add(MaxPooling2D (pool size=(2, 2)) )
model.add(Activation('relu'))

model.add(Convolution2D (32 w, h, border mode="same”))
model.add (MaxPooling2D (pool size=(2, 2)))
model.add(Activation('relu'))

model.add(Convolution2D (48, w, h, border mode="same'))
model.add (MaxPooling2D (pool size=(2, 2)))
model.add(Activation('relu'))

model.add(Convolution2D (48, w, h, border mode="same'))
model.add(MaxPooling2D (pool size=(2, 2)))
model.add(Activation(/relu'))

model.add(Flatten())
model.add(Dense (128) )

model.add(Activation('relu'))

model.add(Dense (10, activation = /softmax” ))