HW 2 Perceptron Training

 HW 2

1 Instructions:
You may form small groups (e.g. of up to four people) to work on this assignment, but you must write
up all solutions by yourself. List your study partners for the homework on the first page, or “none”
if you had no partners.
Keep all responses brief, a few sentences at most. Show all work for full credit.
Start each problem on a new page, and be sure to clearly label where each problem and subproblem
begins. All problems must be submitted in order (all of P1 before P2, etc.).
No late homeworks will be accepted. This is not out of a desire to be harsh, but rather out of
fairness to all students in this large course.
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CS 188, Winter 2020 HW 2 M. Sarrafzadeh
2 Perceptron Training
Assume a three input perceptron plus bias (it outputs 1 if b +
P
i wi ∗ xi 0, else 0). Assume a
learning rate c of 1 and initial weights all 1: ∆wi = c(t − z) ∗ xi
, where t is the true label and z is the
predicted label.
Show weights after each pattern in Table 1 until the result converges. Use an Excel sheet (attach
your Excel sheet to the homework). Iterate over the training samples from top to bottom.
x1 x2 x3 t
1 0 1 0
1 1 0 0
1 0 1 1
0 1 1 1
Table 1: Train Set
3 Input Validation
A SickBit health sensor produces a stream of readings from 20 different sensors (think blood pressure,
heart rate body temperature, etc.). List two techniques you could use to check whether the stream of
data coming from the sensors are valid or not. Write one or two sentences to describe each approach.
4 Distributions
Figure 1: Height Distributions
Galton measured the heights of individuals in 200 families, each of which included one mother,
one father, and a varying number of adult sons. The three histograms of heights in Figure 1 depict
the distributions for all mothers, fathers, and adult sons. All bars are 2 inches wide. All bar heights
are integers. The heights of all people in the data set are included in the histograms.
(a) Calculate each quantity described below or write Unknown if there is not enough information
above to express the quantity as a single number (not a range). Show your work!
(i) The percentage of mothers that are at least 60 inches but less than 64 inches tall.
(ii) The percentage of fathers that are at least 64 but less than 67 inches tall.
(iii) The number of sons that are at least 70 inches tall.
(iv) The number of mothers that are at least 60 inches tall.
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CS 188, Winter 2020 HW 2 M. Sarrafzadeh
(b) If the father’s histogram were redrawn, replacing the two bins from 72-to-74 and from 74-to-76
with one bin from 72-to-76, what would be the height of its bar? If it’s impossible to tell, write
Unknown.
(c) The percentage of sons that are taller than all of the mothers is between and . Fill
in the blanks in the previous sentence with the smallest range that can be determined from the
histograms, then explain your answer below.
5 Voronoi
Draw the Voronoi diagram of 10 points all on a line. Draw separately the Voronoi diagram of 10
points all on a circle. What do these two diagrams have in common?
6 Augmentation
Many methods for making predictions from data, such as linear regression, are limited in terms of the
transformations that they can apply to input data before making a prediction. As linear regression
assumes that the output is the sum of coefficients multiplied by input features, it is unable to account
for cases where the impact of two features together is greater than the sum of their parts. For example,
a house that both has 5 bedrooms and is in California may be worth four times more than would
be expected from the learned price impact of each feature on its own.
Feature Crosses are synthetic features you can form by crossing two or more features together, and
they can help to improve the predictive power of techniques such as linear regression. Expanding on
the above housing example, you could generate a new feature that indicates a combination of both a
home’s number of bedrooms and location.
(a) Describe two pairs of features from Project 2 that might be interesting to cross together, and
explain why.
(b) You have latitude and longitude for homes, and you think feature crosses may allow you to make
better predictions. However, your latitude and longitude are continuously valued. How might you
do a feature cross in this case?
(c) Think up a dataset consisting of features X and Y and associated labels Z that is shaped such
that a linear model would perform poorly without feature crosses. Provide a table with at least
7 points from your dataset.
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