Homework 4

Iteration in Python (3 ways)

In HW 1, you modified code that calculated \(\sqrt{x}\) for each \(x = 0, 0.1, 0.2,...,1\). The for loop version of the code was

x <- seq(from=0, to=1, by=0.1)
sqrt_x <- rep(0, length(x))
for(i in 1:length(x)){
  sqrt_x[i] <- sqrt(x[i])
}
sqrt_x

and the vectorized version of the code was

x <- seq(from=0, to=1, by=0.1)
sqrt_x <- sqrt(x)

To help us learn some Python, let’s see how to replicate this code in Python. Like R, Python is a modular language (we can get lots of extra functionality by loading extra libraries). Some of the most common modules (particularly for statistics, data science, and machine learning) are numpy, scipy, pandas, and scikit-learn.

Here we will reproduce the above R code in Python, with help from the numpy module. Run the following Python code (e.g.: in your Quarto document, create a Python chunk, copy the code, run the Python chunk):

import numpy as np

# create x
x = np.linspace(0, 1, 11)

# for loop version
# create sqrt_x for output
sqrt_x = np.zeros(11)
for i in range(11):
  sqrt_x[i] = np.sqrt(x[i])

sqrt_x

# "vectorized" version
sqrt_x = np.sqrt(x)
sqrt_x

# list comprehension version
sqrt_x = [np.sqrt(a) for a in x]
sqrt_x

Some notes:

• We have loaded the numpy module, and called it np for short. Now when we want to use the numpy module in our code, we will reference np
• linspace, zeros, and sqrt are functions in the numpy library, so we call them using np.linspace, np.zeros, and np.sqrt
• np.linspace works pretty similarly to the seq function in R. We specify the range (0 to 1) and the number of steps (11); the result is 0, 0.1, …, 0.9, 1
• np.zeros(11) is equivalent to rep(0, 11) in R
• for i in range(11) is essentially equivalent to for(i in 1:11) in R. The only difference is that Python is 0-indexed, so range(11) is really 0, 1, 2, …, 10
• In R, the basic object is a vector. The closest object to R’s vector is possibly a NumPy array. This allows operations like “vectorized” functions, matrix multiplication, etc.
• A list comprehension is another way of iterating in Python (and in some sense is the most “Pythonic” approach). The code [np.sqrt(a) for a in x] means “create a list, where each entry of the list is the square root of the corresponding entry in x”.
• You will notice that, for both the for loop version and the “vectorized” versions of the code, sqrt_x is a numpy array. However, the list comprehension version produces (as we would expect!) a list instead

Some simulations in Python

In the following questions, you will practice working in Python by re-implementing some of questions from previous assignments in Python. The following table shows you corresponding Python code for several R operations.

Functions in Python

Here’s a simple example of a function in R which takes two numbers and adds them together (this is literally what + does, so the function is pointless, but bear with me for the purposes of demonstration):

my_sum <- function(a, b){
  return(a + b)
}

Here is the corresponding function definition in Python:

def my_sum(a, b):
  return a + b

Let’s practice writing a short function in Python by re-writing the Huber loss function from HW 2.

def huber(x, a):

Probability simulation: the Monty Hall problem

Suppose you are on a game show, with three closed doors. Behind two of the doors is a goat, and behind one of the doors is a car. You randomly select one of the three doors. The host then opens one of the remaining doors to reveal a goat, and gives you the option of sticking with your original choice, or switching to the other unopened door. For contestants who choose to switch doors, what is the probability of winning the car?

Reproducibility

When we do research, we want our work to be reproducible: other researchers should be able to reproduce our results, given the same data. Unfortunately, published research is not always reproducible. In this part of the assignment, you will read about why reproducibility is important, and some good practices for making your own work more reproducible.

To begin, read the following articles at the links provided:

• Peng, R. D. (2011). Reproducible research in computational science. Science, 334(6060), 1226-1227.
• Broman, K., Cetinkaya-Rundel, M., Nussbaum, A., Paciorek, C., Peng, R., Turek, D., & Wickham, H. (2017). Recommendations to funding agencies for supporting reproducible research. American Statistical Association.

Then answer the following questions.