CS109A

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Key Word(s): Linearity, Normality, Constant Variance, Independence, Bootstrapping, Confidence Intervals, Standard Errors, Prediction Intervals, T-Test, P-Value

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Title :

Exercise: Beta Values for Data using Bootstrapping

Description :

Solve the previous exercise by building your own bootstrap function.

Data Description:

Instructions:

• Define a function bootstrap that takes a dataframe as the input. Use NumPy's random.randint() function to generate random integers in the range of the length of the dataset. These integers will be used as the indices to access the rows of the dataset.
• Similar to the previous exercise, compute the $\beta_0$ and $\beta_1$ values for each instance of the dataframe.
• Plot the $\beta_0$, $\beta_1$ histograms.

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.random.randint() Returns list of integers as per mentioned size

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

df.iloc[] Purely integer-location based indexing for selection by position

ax.hist() Plots a histogram

ax.set_xlabel() Sets label for x-axthe is

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

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

# Import necessary libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from randomuniverse import RandomUniverse
%matplotlib inline

# Read the file "Advertising_csv"
df = pd.read_csv('Advertising_adj.csv')

# Take a quick look at the data
df.head()

	tv	sales
0	230.1	465.26
1	44.5	218.95
2	17.2	195.79
3	151.5	389.47
4	180.8	271.58

# Define a bootstrap function, which takes as input a dataframe 
# It must output a bootstrapped version of the input dataframe
def bootstrap(df):
    selectionIndex = np.random.randint(0, df.shape[0], size = df.__len__())
    new_df = df.iloc[selectionIndex]
    return new_df

Alternate approach to $\beta$ computation

# Initialize two empty lists to store the beta values
beta0_list, beta1_list = [],[]

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

# Loop through the number of bootstraps
for i in range(number_of_bootstraps):

    # Call the bootstrap function to get a bootstrapped version of the data
    df_new = bootstrap(df)

    # Find the mean of the predictor values i.e. tv
    xmean = df_new.tv.mean()

    # Find the mean of the response values i.e. sales
    ymean = df_new.sales.mean()

    #'X' is the predictor variable given by df_new.tv values 
    X = df_new.tv
    
    #'y' is the reponse variable given by df_new.sales values
    y = df_new.sales
    
    # Compute the analytical values of beta0 and beta1 using the 
    # equation given in the hints
    beta1 = (((df_new.tv - xmean)*(df_new.sales - ymean)).sum())/(((df_new.tv - xmean)**2).sum())
    beta0 = ymean - beta1*xmean

    # Append the calculated values of beta1 and beta0 to the appropriate lists
    beta0_list.append(beta1)
    beta1_list.append(beta0)

### edTest(test_beta) ###

# Compute the mean of the beta values
beta0_mean = np.mean(beta0_list)
beta1_mean = np.mean(beta1_list)

# Plot histograms of beta_0 and beta_1 using lists created above 

fig, ax = plt.subplots(1,2, figsize=(18,8))
ax[0].hist(beta0_list)
ax[1].hist(beta1_list)
ax[0].set_xlabel('beta 0')
ax[1].set_xlabel('beta 1')
ax[0].set_ylabel('Frequency')
plt.show();

Compare the plots with the results from the RandomUniverse() function

# Helper code to visualise the similarity between the bootstrap
# function here & the RandomUniverse() function from last exercise
beta0_randUni, beta1_randUni = [],[]

parallelUniverses = 1000

for i in range(parallelUniverses):
    df_new = RandomUniverse(df)
    
    xmean = df_new.tv.mean()
    ymean = df_new.sales.mean()

    # Using linear algebra result as discussed in lecture
    beta1 = (((df_new.tv - xmean)*(df_new.sales - ymean)).sum())/(((df_new.tv - xmean)**2).sum())
    beta0 = ymean - beta1*xmean

    beta0_randUni.append(beta0)
    beta1_randUni.append(beta1)

# Helper code to plot the bootstrapped beta values & the ones from random universe
def plotmulti(list1, list2):
    fig, axes = plt.subplots(1,2, figsize = (10,4), sharey = 'row')
    axes[0].hist(list1);
    axes[0].set_xlabel('Beta Distribution')
    axes[0].set_ylabel('Frequency')
    axes[0].set_title('Bootstrap')
    axes[1].hist(list2);
    axes[1].set_xlabel('Beta Distribution')
    axes[1].set_title('Random Universe')
    plt.show();

# Call the 'plotmulti' function above to compare the two histograms for beta0
plotmulti(beta0_list, beta0_randUni)

# Call the 'plotmulti' function above to compare the two histograms for beta1
plotmulti(beta1_list, beta1_randUni)

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