Title¶

Exercise: A.1 - Beta values for data from Random Universe

Description¶

Given a RandomUniverse(dataframe)->dataframe function that gives a new dataset from a "parallel" universe, calculate the $\beta_0$ 's and $\beta_1$ 's and plot a histogram like the one below.

Roadmap¶

Get a new dataframe using the RandomUniverse function already provided in the exercise
Calculate $\beta_0$, $\beta_1$ for that particular dataframe
Add the calculated $\beta_0$ and $\beta_1$ values to a python list
Plot a histogram using the lists calculated above

Change the number of parallelUniverses and comment on what you observe. Discuss within the group why you see this behavior. Did you expect the spread to change? Why or why not?

Hints¶

To compute the beta values use the following equations:

$\beta_{0}=\bar{y}-\left(b_{1} * \bar{x}\right)$

$\beta_{1}=\frac{\sum(x-\bar{x}) *(y-\bar{y})}{\sum(x-\bar{x})^{2}}$

where $\bar{x}$ is the mean of $x$ and $\bar{y}$ is the mean of $y$

np.dot() : Computes the dot product of two arrays

ax.hist() : Plots a histogram

ax.set_xlabel() : Sets label for x-axis

ax.set_ylabel() : Sets label for the y-axis

Note: This exercise is auto-graded and you can try multiple attempts.

In [1]:

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from randomuniverse import RandomUniverse
%matplotlib inline

Reading the standard Advertising dataset¶

In [2]:

df = pd.read_csv('Advertising_adj.csv')

In [2]:

df.head()

In [1]:

#Create two empty lists that will store the beta values
beta0_list, beta1_list = [],[]


#Choose the number of "parallel" Universes to generate the new dataset
parallelUniverses = 1000

for i in range(parallelUniverses):
    df_new = RandomUniverse(df)

# x is the predictor variable given by 'tv' values 
# y is the reponse variable given by 'sales' values
    x = ___
    y = ___

#Find the mean of the x values
    xmean = x.___

#Find the mean of the y values
    ymean = y.___

# Using Linear Algebra as discussed in lecture for beta0 and beta1

    beta1 = ___
    beta0 = ___

# Append the calculated values of beta1 and beta0
    beta0_list.___
    beta1_list.___

In [3]:

### edTest(test_beta) ###
beta0_mean = np.mean(beta0_list)
beta1_mean = np.mean(beta1_list)

Now we plot the histograms¶

Returns a plot for a histogram

In [4]:

# plot histogram of 
fig, ax = plt.subplots(1,2, figsize=(18,8))
ax[0].___
ax[1].___
ax[0].set_xlabel('Beta 0')
ax[1].set_xlabel('Beta 1')
ax[0].set_ylabel('Frequency');

Discussion¶

Change the number of parallelUniverses and comment on what you observe. Discuss within the group why you see this behavior. Did you expect the spread to change? Why or why not?

Fin¶

Fin