questions.txt

Coding Exercises
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Please create a public git repo in github/gitlab/bitbucket for your solutions and share the link. Please also include
your code, solutions, and any README, solution explanations etc. that you may have within this
repo.

1) Unsupervised + supervised learning.
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Attached is a data file dataClustering.csv which contains a data set of 2500 samples
with 8 features. 
i) Perform any clustering of your choice to determine the optimal # of clusters in the data
ii) Using the result of i) assign clusters labels to each sample, so each sample's label is the
cluster to which it belongs. Using these labels as the exact labels, you now have a labeled dataset.
Build a classification model that classifies a sample with its corresponding label. Use multinomial 
regression as a benchmark model, and any ML model (trees, forests, SVM, NN etc.) as a comparison model.
Comment on which does better and why.

2) Prediction + filtering
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Attached are 3 files: xvalsSine.csv, cleanSine.csv and noisySine.csv. xvalsSine.csv contains
1000 x-values in the interval -pi/2 to pi/2. cleanSine.csv is a pure sine(x) function for the
x values mentioned earlier. noisySine.csv contains sine(x) corrupted by noise. 
i) Using xvalsSine.csv and cleanSine.csv as a labeled dataset (x,sine(x)) being (value,label) with a
random train/test split of 0.7/0.3, build an OLS regression model (you may want to use polynomial
basis of a sufficiently large order). 
(bonus) If you used the normal equations to solve the OLS problem, can you redo it with stochastic
gradient descent?
ii) Now, assume you are given the noisySine.csv as a time series with the values of 
xvalsSine.csv being the time variable. Filter the noisySine.csv data with any filter of your choice
and compare against cleanSine.csv to report the error.
(bonus) Can you code a Kalman filter to predict out 10 samples from the noisySine.csv data?

3) Time series with pi
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Attached is a function genPiAppxDigits(numdigits,appxAcc) which returns an approximate value of pi
to numdigits digits of accuracy. appxAcc is an integer that controls the approximation accuracy, with
a larger number for appxAcc leading to a better approximation.
i) Fix numdigits and appxAcc to be sufficiently large, say 1000 and 100000 respectively.
Treat each of the 1000 resulting digits of pi as the value of a time series. Thus x[n]=nth digit
of pi for n=1,1000. Build a simple time series forecasting model (any model of your choice)
that predicts the next 50 digits of pi. Report your accuracy. Using your results, can
you conclude that pi is irrational? If so, how?
(bonus) Now let's vary appxAcc to be 1000,5000,10000,50000,100000 with fixed numdigits=1000. You thus
have 5 time series, each corresponding to a value of appxAcc. Can you find the pairwise correlation
between each of the time series?

#
def genPiAppxDigits(numdigits,appxAcc):
	import numpy as np
	from decimal import getcontext, Decimal
	getcontext().prec = numdigits
	mypi = (Decimal(4) * sum(-Decimal(k%4 - 2) / k for k in range(1, 2*appxAcc+1, 2)))
	return mypi