# Lesson 29: Practice with Numpy solutions¶

This document was prepared at Caltech with financial support from the Donna and Benjamin M. Rosen Bioengineering Center.

This lesson was generated from a Jupyter notebook. You can download the notebook here.

In [1]:
import numpy as np
import scipy.stats
import pandas as pd

import bokeh_catplot

import bokeh.io
import bokeh.plotting

bokeh.io.output_notebook()


Numpy arrays can take a while to get the hang of. Therefore, it's important to practice practice practice!

## Practice 1: Computing things!¶

In the last lesson, we looked at a data set from Harvey and Orbidans on the cross-sectional area of C. elegans eggs. Recall, we loaded the data and converted everything to Numpy arrays like this:

In [2]:
df = pd.read_csv('data/c_elegans_egg_xa.csv', comment='#')

xa_high = df.loc[df['food']=='high', 'area (sq. um)'].values
xa_low = df.loc[df['food']=='low', 'area (sq. um)'].values


Now we would like to compute the diameter of the egg from the cross-sectional area. Write a function that takes in an array of cross-sectional areas and returns an array of diameters. Recall that the diameter $d$ and cross-sectional area $A$ are related by $A = \pi d^2/4$. There should be no for loops in your function! The call signature is

xa_to_diameter(xa)


Use your function to compute the diameters of the eggs.

### Practice 1: solution¶

In [3]:
def xa_to_diameter(xa):
"""
Convert an array of cross-sectional areas
to diameters with commensurate units.
"""
# Compute diameter from area
diameter = 2 * np.sqrt(xa / np.pi)

return diameter

print('Diameters of eggs from well fed mothers:\n', xa_to_diameter(xa_high))
print('\nDiameters of eggs from poorly fed mothers:\n', xa_to_diameter(xa_low))

Diameters of eggs from well fed mothers:
[46.29105911 51.22642581 47.76657057 48.5596503  51.59790585 47.61973991
49.33998388 47.89966242 47.21697198 46.94654036 49.08125119 49.84064959
47.9926071  46.29105911 47.69988539 48.40207395 48.15152345 49.3141717
49.57168871 47.87307365 48.30991705 46.29105911 46.12573337 46.24978308
46.41466697 47.87307365 48.15152345 48.95137203 45.72372833 47.18999856
46.68817945 45.98750791 46.53794651 52.2111661  48.70364742 47.23045291
47.06842687 46.81073869 45.97366251 49.57168871 50.8397116  48.54653847
52.08909166 48.24398292]

Diameters of eggs from poorly fed mothers:
[48.40207395 51.58556628 52.55146594 50.31103472 53.06982074 54.57203767
50.32368681 52.24773281 53.99739399 49.44309786 53.87936676 47.9926071
52.41804019 47.87307365 52.11352942 51.21399674 52.44232467 50.47526453
50.8397116  51.56087828 49.84064959 55.96578669 50.72688754 50.58864976
52.18677405 52.44232467 51.78264653 52.57568879 51.86863366 52.67246879
49.05530287 52.67246879 50.72688754 50.07003758 52.32078957 49.18490759
53.72554372 46.67454189 49.19784929 51.88090591 51.85635852 54.8280819
52.07686848 51.22642581 51.96673046 48.29673743 53.04582353 52.07686848
52.35727972 50.57606396 51.70882946 53.54750652 52.23554675 53.54750652
53.18964437 51.96673046 55.38261517]


## Practice 2: Working with two-dimensional arrays¶

Numpy enables you do to matrix calculations on two-dimensional arrays. In exercise, you will practice doing matrix calculations on arrays. We'll start by making a matrix and a vector to practice with. You can copy and paste the code below.

In [4]:
A = np.array([[6.7, 1.3, 0.6, 0.7],
[0.1, 5.5, 0.4, 2.4],
[1.1, 0.8, 4.5, 1.7],
[0.0, 1.5, 3.4, 7.5]])

b = np.array([1.1, 2.3, 3.3, 3.9])


a) First, let's practice slicing.

1. Print row 1 (remember, indexing starts at zero) of A.
2. Print columns 1 and 3 of A.
3. Print the values of every entry in A that is greater than 2.
4. Print the diagonal of A. using the np.diag() function.

#### Solution¶

In [5]:
# 1.
print(A[1,:])

# 2.
print(A[:,[1,3]])

# 3.
print(A[A>2])

# 4.
print(np.diag(A))

[0.1 5.5 0.4 2.4]
[[1.3 0.7]
[5.5 2.4]
[0.8 1.7]
[1.5 7.5]]
[6.7 5.5 2.4 4.5 3.4 7.5]
[6.7 5.5 4.5 7.5]


b) The np.linalg module has some powerful linear algebra tools.

1. First, we'll solve the linear system $\mathsf{A}\cdot \mathbf{x} = \mathbf{b}$. Try it out: use np.linalg.solve(). Store your answer in the Numpy array x.
2. Now do np.dot(A, x) to verify that $\mathsf{A}\cdot \mathbf{x} = \mathbf{b}$.
3. Use np.transpose() to compute the transpose of A.
4. Use np.linalg.inv() to compute the inverse of A.

#### Solution¶

In [6]:
# 1.
x = np.linalg.solve(A, b)

# 2.
print('A . x:\n', np.dot(A, x))
print('\nb:\n', b)

# 3.
print('\ntranspose of A:\n', np.transpose(A))

# 4.
print('\ninverse of A:\n', np.linalg.inv(A))

A . x:
[1.1 2.3 3.3 3.9]

b:
[1.1 2.3 3.3 3.9]

transpose of A:
[[6.7 0.1 1.1 0. ]
[1.3 5.5 0.8 1.5]
[0.6 0.4 4.5 3.4]
[0.7 2.4 1.7 7.5]]

inverse of A:
[[ 0.15267508 -0.03365026 -0.01778     0.00054854]
[-0.00906001  0.19788853  0.03719385 -0.07090934]
[-0.04391535 -0.0144834   0.26880108 -0.05219479]
[ 0.02172029 -0.0330119  -0.12929526  0.17117684]]


c) Sometimes you want to convert a two-dimensional array to a one-dimensional array. This can be done with np.ravel().

1. See what happens when you do B = np.ravel(A).
2. Look of the documentation for np.reshape(). Then, reshape B to make it look like A again.

#### Solution¶

In [7]:
# 1.
B = np.ravel(A)
print(B)

# 2.
np.reshape(B, A.shape)

[6.7 1.3 0.6 0.7 0.1 5.5 0.4 2.4 1.1 0.8 4.5 1.7 0.  1.5 3.4 7.5]

Out[7]:
array([[6.7, 1.3, 0.6, 0.7],
[0.1, 5.5, 0.4, 2.4],
[1.1, 0.8, 4.5, 1.7],
[0. , 1.5, 3.4, 7.5]])

## Practice 3: Understanding and building ECDFs¶

Write a function with call signature

ecdf_vals(data)


which takes a one-dimensional NumPy array (or Pandas Series; the same construction of your function will work for both) of data and returns the x and y values for plotting a "dot-style" ECDF. That is, each dot has a y value given by the ECDF evaluated at x. As a reminder,

ECDF(x) = fraction of data points ≤ x.

When writing a function, you should have detailed doc strings and checking of input. However, here the focus is on your understanding of ECDFs and developing skills with NumPy, so you do not need to have a very descriptive doc string nor lots of input checking.

Hint: The functions np.sort() and np.arange() may be useful.

### Practice 3 solution¶

In [8]:
def ecdf_vals(data):
"""Return x and y values for an ECDF."""
return np.sort(data), np.arange(1, len(data)+1) / len(data)


## Practice 4: Are they Normally distributed?¶

We might be interested to see if the egg cross-section data follow a Normal distribution. After all, this is commonly an underlying assumption when people report data from repeated measurements in the literature.

One way to assess this is to plot the theoretical CDF with the same mean and standard deviation as the data on the same plot as the ECDFs. (There are better graphical ways to do this, but this is ok for our purposes here.) We know the cumulative distribution function for a Normal distribution with mean $\mu$ and standard deviation $\sigma$ is

\begin{align} \mathrm{cdf}(x) = \frac{1}{2}\left(1 + \mathrm{erf}\left(\frac{x - \mu}{\sqrt{2\sigma^2}}\right)\right), \end{align}

but instead of coding this up directly, we can use the scipy.stats module to do it for us! We just need to supply where we want the CDF evaluated ($x$), and the mean (the location parameter) and standard deviation (the scale parameter).

Now, let's make the plot.

1. Compute smooth curves for Normal CDFs. Use the scipy.stats module to make the curves. I am intentionally not telling you how to do this nor giving you a link to the docs. This will help you practice using Google to figure out how to use these tools.
2. Overlay the ECDFs of the cross-sectional areas to give a graphical evaluation of the Normality of the data.

### Practice 4: solution¶

In [9]:
# Make smooth x-values
x = np.linspace(1500, 2500, 400)

# Compute theoretical Normal distributions
cdf_low = scipy.stats.norm.cdf(x, loc=np.mean(xa_low), scale=np.std(xa_low))
cdf_high = scipy.stats.norm.cdf(x, loc=np.mean(xa_high), scale=np.std(xa_high))

# Color palette
palette = bokeh.palettes.d3['Category10'][10]

# Make a plot of the theoretical distributions
p = bokeh.plotting.figure(
height=250,
width=350,
x_axis_label='cross sectional area (µm²)',
y_axis_label='CDF',
)

p.line(
x=x,
y=cdf_low,
color=palette[0],
line_width=2,
)

p.line(
x=x,
y=cdf_high,
color=palette[1],
line_width=2,
)

# Overlay ECDFs
p = bokeh_catplot.ecdf(
data=df,
cats='food',
val='area (sq. um)',
order=['low', 'high'],
palette=palette,
p=p,
)

p.legend.title = 'food level'

bokeh.io.show(p)


It is important to note that I didn't just memorize how all of these functions work when I wrote these practice exercises. I looked at the online documentation. For example, I looked at the scipy.stats.norm documentation. To find those links, I just Googled "scipy.stats normal".

These packages are all very well documented, and those docs will be your guide. You don't need to memorize (though you will eventually just by accident).

## Computing environment¶

In [10]:
%load_ext watermark
%watermark -v -p numpy,scipy,pandas,bokeh,bokeh_catplot,jupyterlab

CPython 3.7.3
IPython 7.1.1

numpy 1.16.4
scipy 1.2.1
pandas 0.24.2
bokeh 1.2.0
bokeh_catplot 0.1.2
jupyterlab 0.35.5