4  Python Refreshment

You have programmed in Python. Regardless of your skill level, let us do some refreshing.

4.1 The Python World

  • Function: a block of organized, reusable code to complete certain task.
  • Module: a file containing a collection of functions, variables, and statements.
  • Package: a structured directory containing collections of modules and an __init.py__ file by which the directory is interpreted as a package.
  • Library: a collection of related functionality of codes. It is a reusable chunk of code that we can use by importing it in our program, we can just use it by importing that library and calling the method of that library with period(.).

See, for example, how to build a Python libratry.

4.2 Standard Library

Python’s has an extensive standard library that offers a wide range of facilities as indicated by the long table of contents listed below. See documentation online.

The library contains built-in modules (written in C) that provide access to system functionality such as file I/O that would otherwise be inaccessible to Python programmers, as well as modules written in Python that provide standardized solutions for many problems that occur in everyday programming. Some of these modules are explicitly designed to encourage and enhance the portability of Python programs by abstracting away platform-specifics into platform-neutral APIs.

Question: How to get the constant \(e\) to an arbitary precision?

The constant is only represented by a given double precision.

import math
print("%0.20f" % math.e)
print("%0.80f" % math.e)
2.71828182845904509080
2.71828182845904509079559829842764884233474731445312500000000000000000000000000000

Now use package decimal to export with an arbitary precision.

import decimal  # for what?

## set the required number digits to 150
decimal.getcontext().prec = 150
decimal.Decimal(1).exp().to_eng_string()
decimal.Decimal(1).exp().to_eng_string()[2:]
'71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642742746639193200305992181741359662904357290033429526'

4.3 Important Libraries

  • NumPy
  • pandas
  • matplotlib
  • IPython/Jupyter
  • SciPy
  • scikit-learn
  • statsmodels

Question: how to draw a random sample from a normal distribution and evaluate the density and distributions at these points?

from scipy.stats import norm

mu, sigma = 2, 4
mean, var, skew, kurt = norm.stats(mu, sigma, moments='mvsk')
print(mean, var, skew, kurt)
x = norm.rvs(loc = mu, scale = sigma, size = 10)
x
2.0 16.0 0.0 0.0
array([-7.61542223,  4.04410857, -0.66018039, -3.22051506,  3.46732088,
        8.19518069, -0.249417  , -1.29855302,  4.58570583,  4.19481766])

The pdf and cdf can be evaluated:

norm.pdf(x, loc = mu, scale = sigma)
array([0.00554702, 0.08752706, 0.07994828, 0.04255689, 0.09324591,
       0.03005831, 0.08514888, 0.07098782, 0.08093034, 0.08579701])

4.4 Writing a Function

Consider the Fibonacci Sequence \(1, 1, 2, 3, 5, 8, 13, 21, 34, ...\). The next number is found by adding up the two numbers before it. We are going to use 3 ways to solve the problems.

The first is a recursive solution.

def fib_rs(n):
    if (n==1 or n==2):
        return 1
    else:
        return fib_rs(n - 1) + fib_rs(n - 2)

%timeit fib_rs(10)
8.83 μs ± 51 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)

The second uses dynamic programming memoization.

def fib_dm_helper(n, mem):
    if mem[n] is not None:
        return mem[n]
    elif (n == 1 or n == 2):
        result = 1
    else:
        result = fib_dm_helper(n - 1, mem) + fib_dm_helper(n - 2, mem)
    mem[n] = result
    return result

def fib_dm(n):
    mem = [None] * (n + 1)
    return fib_dm_helper(n, mem)

%timeit fib_dm(10)
1.93 μs ± 9.57 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)

The third is still dynamic programming but bottom-up.

def fib_dbu(n):
    mem = [None] * (n + 1)
    mem[1] = 1;
    mem[2] = 1;
    for i in range(3, n + 1):
        mem[i] = mem[i - 1] + mem[i - 2]
    return mem[n]


%timeit fib_dbu(500)
66.4 μs ± 918 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

Apparently, the three solutions have very different performance for larger n.

4.4.1 Monty Hall

Here is a function that performs the Monty Hall experiments. In this version, the host opens only one empty door.

import numpy as np

def montyhall(ndoors, ntrials):
    doors = np.arange(1, ndoors + 1) / 10
    prize = np.random.choice(doors, size=ntrials)
    player = np.random.choice(doors, size=ntrials)
    host = np.array([np.random.choice([d for d in doors
                                       if d not in [player[x], prize[x]]])
                     for x in range(ntrials)])
    player2 = np.array([np.random.choice([d for d in doors
                                          if d not in [player[x], host[x]]])
                        for x in range(ntrials)])
    return {'noswitch': np.sum(prize == player), 'switch': np.sum(prize == player2)}

Test it out:

montyhall(3, 1000)
montyhall(4, 1000)
{'noswitch': np.int64(242), 'switch': np.int64(381)}

The true value for the two strategies with \(n\) doors are, respectively, \(1 / n\) and \(\frac{n - 1}{n (n - 2)}\).

In the homework exercise, the host opens \(m\) doors that are empty. An argument nempty could be added to the function.

4.5 Variables versus Objects

In Python, variables and the objects they point to actually live in two different places in the computer memory. Think of variables as pointers to the objects they’re associated with, rather than being those objects. This matters when multiple variables point to the same object.

x = [1, 2, 3]  # create a list; x points to the list
y = x          # y also points to the same list in the memory
y.append(4)    # append to y
x              # x changed!
[1, 2, 3, 4]

Now check their addresses

print(id(x))   # address of x
print(id(y))   # address of y
4696292544
4696292544

Nonetheless, some data types in Python are “immutable”, meaning that their values cannot be changed in place. One such example is strings.

x = "abc"
y = x
y = "xyz"
x
'abc'

Now check their addresses

print(id(x))   # address of x
print(id(y))   # address of y
4526902032
4610332432

Question: What’s mutable and what’s immutable?

Anything that is a collection of other objects is mutable, except tuples.

Not all manipulations of mutable objects change the object rather than create a new object. Sometimes when you do something to a mutable object, you get back a new object. Manipulations that change an existing object, rather than create a new one, are referred to as “in-place mutations” or just “mutations.” So:

  • All manipulations of immutable types create new objects.
  • Some manipulations of mutable types create new objects.

Different variables may all be pointing at the same object is preserved through function calls (a behavior known as “pass by object-reference”). So if you pass a list to a function, and that function manipulates that list using an in-place mutation, that change will affect any variable that was pointing to that same object outside the function.

x = [1, 2, 3]
y = x

def append_42(input_list):
    input_list.append(42)
    return input_list

append_42(x)
[1, 2, 3, 42]

Note that both x and y have been appended by \(42\).

4.6 Number Representation

Numers in a computer’s memory are represented by binary styles (on and off of bits).

4.6.1 Integers

If not careful, It is easy to be bitten by overflow with integers when using Numpy and Pandas in Python.

import numpy as np

x = np.array(2 ** 63 - 1 , dtype = 'int')
x
# This should be the largest number numpy can display, with
# the default int8 type (64 bits)
array(9223372036854775807)

Note: on Windows and other platforms, dtype = 'int' may have to be changed to dtype = np.int64 for the code to execute. Source: Stackoverflow

What if we increment it by 1?

y = np.array(x + 1, dtype = 'int')
y
# Because of the overflow, it becomes negative!
array(-9223372036854775808)

For vanilla Python, the overflow errors are checked and more digits are allocated when needed, at the cost of being slow.

2 ** 63 * 1000
9223372036854775808000

This number is 1000 times larger than the prior number, but still displayed perfectly without any overflows

4.6.2 Floating Number

Standard double-precision floating point number uses 64 bits. Among them, 1 is for sign, 11 is for exponent, and 52 are fraction significand, See https://en.wikipedia.org/wiki/Double-precision_floating-point_format. The bottom line is that, of course, not every real number is exactly representable.

If you have played the Game 24, here is a tricky one:

8 / (3 - 8 / 3) == 24
False

Surprise?

There are more.

0.1 + 0.1 + 0.1 == 0.3
False
0.3 - 0.2 == 0.1
False

What is really going on?

import decimal
decimal.Decimal(0.1)
Decimal('0.1000000000000000055511151231257827021181583404541015625')
decimal.Decimal(8 / (3 - 8 / 3))
Decimal('23.999999999999989341858963598497211933135986328125')

Because the mantissa bits are limited, it can not represent a floating point that’s both very big and very precise. Most computers can represent all integers up to \(2^{53}\), after that it starts skipping numbers.

2.1 ** 53 + 1 == 2.1 ** 53

# Find a number larger than 2 to the 53rd
True
x = 2.1 ** 53
for i in range(1000000):
    x = x + 1
x == 2.1 ** 53
True

We add 1 to x by 1000000 times, but it still equal to its initial value, 2.1 ** 53. This is because this number is too big that computer can’t handle it with precision like add 1.

Machine epsilon is the smallest positive floating-point number x such that 1 + x != 1.

print(np.finfo(float).eps)
print(np.finfo(np.float32).eps)
2.220446049250313e-16
1.1920929e-07

4.7 Virtual Environment

Virtual environments in Python are essential tools for managing dependencies and ensuring consistency across projects. They allow you to create isolated environments for each project, with its own set of installed packages, separate from the global Python installation. This isolation prevents conflicts between project dependencies and versions, making your projects more reliable and easier to manage. It’s particularly useful when working on multiple projects with differing requirements, or when collaborating with others who may have different setups.

To set up a virtual environment, you first need to ensure that Python is installed on your system. Most modern Python installations come with the venv module, which is used to create virtual environments. Here’s how to set one up:

  • Open your command line interface.
  • Navigate to your project directory.
  • Run python3 -m venv myenv, where myenv is the name of the virtual environment to be created. Choose an informative name.

This command creates a new directory named myenv (or your chosen name) in your project directory, containing the virtual environment.

To start using this environment, you need to activate it. The activation command varies depending on your operating system:

  • On Windows, run myenv\Scripts\activate.
  • On Linux or MacOS, use source myenv/bin/activate or . myenv/bin/activate.

Once activated, your command line will typically show the name of the virtual environment, and you can then install and use packages within this isolated environment without affecting your global Python setup.

To exit the virtual environment, simply type deactivate in your command line. This will return you to your system’s global Python environment.

As an example, let’s install a package, like numpy, in this newly created virtual environment:

  • Ensure your virtual environment is activated.
  • Run pip install numpy.

This command installs the requests library in your virtual environment. You can verify the installation by running pip list, which should show requests along with its version.

4.8 Numpy