Effects of Advertising on Sales¶
LVC 1 - Introduction to Supervised Learning: Regression¶
Context and Problem¶
- An interesting application of regression is to quantify the effect of advertisement on sales. Various channels of advertisement are newspaper, TV, radio, etc.
- In this case study, we will have a look at the advertising data of a company and try to see its effect on sales.
- We will also try to predict the sales given the different parameters of advertising.
Data Information¶
The data at hand has three features about the spending on advertising and the target variable is the net sales. Attributes are:
- TV - Independent variable quantifying budget for TV ads
- Radio - Independent variable quantifying budget for radio ads
- News - Independent variable quantifying budget for news ads
- Sales - Dependent variable
Let us start by importing necessary packages¶
In [ ]:
import pandas as pd
import numpy as np
from Scikit-learn import linear_model
import matplotlib.pyplot as plt
In [ ]:
# Let us import the files from our system. Note that you can also load the data from the drive.
# The below code is applicable only if you are working on Google Colab, In case you are using Jupyter Notebook, you can directly use pd.read_csv(filename) to load data into dataframe
#from google.colab import files
#uploaded = files.upload()
In [1]:
# Connect collab
from google.colab import drive
drive.mount('/content/drive')
Mounted at /content/drive
In [ ]:
# Load data from csv file
data = pd.read_csv('/content/drive/MyDrive/MIT - Data Sciences/Colab Notebooks/Week Four - Regression and Prediction/Results of Advertising on Sales/Advertising.csv')
data.head()
Out[ ]:
| Unnamed: 0 | TV | Radio | Newspaper | Sales | |
|---|---|---|---|---|---|
| 0 | 1 | 230.1 | 37.8 | 69.2 | 22.1 |
| 1 | 2 | 44.5 | 39.3 | 45.1 | 10.4 |
| 2 | 3 | 17.2 | 45.9 | 69.3 | 9.3 |
| 3 | 4 | 151.5 | 41.3 | 58.5 | 18.5 |
| 4 | 5 | 180.8 | 10.8 | 58.4 | 12.9 |
In [ ]:
## Create a copy fo the data and save it as a data frame
Ad_df = data.copy()
In [ ]:
#Ad_df = pd.read_csv('Advertising.csv')
# we have loaded the data into the Ad_df data frame. Let us now have a quick look.
Ad_df.head()
Out[ ]:
| Unnamed: 0 | TV | Radio | Newspaper | Sales | |
|---|---|---|---|---|---|
| 0 | 1 | 230.1 | 37.8 | 69.2 | 22.1 |
| 1 | 2 | 44.5 | 39.3 | 45.1 | 10.4 |
| 2 | 3 | 17.2 | 45.9 | 69.3 | 9.3 |
| 3 | 4 | 151.5 | 41.3 | 58.5 | 18.5 |
| 4 | 5 | 180.8 | 10.8 | 58.4 | 12.9 |
In [ ]:
Ad_df.info()
<class 'pandas.core.frame.DataFrame'> RangeIndex: 200 entries, 0 to 199 Data columns (total 5 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 Unnamed: 0 200 non-null int64 1 TV 200 non-null float64 2 Radio 200 non-null float64 3 Newspaper 200 non-null float64 4 Sales 200 non-null float64 dtypes: float64(4), int64(1) memory usage: 7.9 KB
In [ ]:
# we can drop the first column as it is just the index
Ad_df.drop(columns = 'Unnamed: 0', inplace=True)
In [ ]:
Ad_df
Out[ ]:
| TV | Radio | Newspaper | Sales | |
|---|---|---|---|---|
| 0 | 230.1 | 37.8 | 69.2 | 22.1 |
| 1 | 44.5 | 39.3 | 45.1 | 10.4 |
| 2 | 17.2 | 45.9 | 69.3 | 9.3 |
| 3 | 151.5 | 41.3 | 58.5 | 18.5 |
| 4 | 180.8 | 10.8 | 58.4 | 12.9 |
| ... | ... | ... | ... | ... |
| 195 | 38.2 | 3.7 | 13.8 | 7.6 |
| 196 | 94.2 | 4.9 | 8.1 | 9.7 |
| 197 | 177.0 | 9.3 | 6.4 | 12.8 |
| 198 | 283.6 | 42.0 | 66.2 | 25.5 |
| 199 | 232.1 | 8.6 | 8.7 | 13.4 |
200 rows × 4 columns
In [ ]:
Ad_df.info()
<class 'pandas.core.frame.DataFrame'> RangeIndex: 200 entries, 0 to 199 Data columns (total 4 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 TV 200 non-null float64 1 Radio 200 non-null float64 2 Newspaper 200 non-null float64 3 Sales 200 non-null float64 dtypes: float64(4) memory usage: 6.4 KB
In [ ]:
## Check for missing values
Ad_df.isnull().sum()
Out[ ]:
| 0 | |
|---|---|
| TV | 0 |
| Radio | 0 |
| Newspaper | 0 |
| Sales | 0 |
Observations: All the variables are of float data type.
Let us now start with the simple linear regression. We will use one feature at a time and have a look at the target variable.¶
In [ ]:
# Dataset is stored in a Pandas Dataframe. Let us take out all the variables in a numpy array.
Sales = Ad_df.Sales.values.reshape(len(Ad_df['Sales']), 1)
TV = Ad_df.TV.values.reshape(len(Ad_df['Sales']), 1)
Radio = Ad_df.Radio.values.reshape(len(Ad_df['Sales']), 1)
Newspaper = Ad_df.Newspaper.values.reshape(len(Ad_df['Sales']), 1)
In [ ]:
# let us fit the simple linear regression model with the TV feature
tv_model = linear_model.LinearRegression()
tv_model.fit(TV, Sales)
coeffs_tv = np.array(list(tv_model.intercept_.flatten()) + list(tv_model.coef_.flatten()))
coeffs_tv = list(coeffs_tv)
# let us fit the simple linear regression model with the Radio feature
radio_model = linear_model.LinearRegression()
radio_model.fit(Radio, Sales)
coeffs_radio = np.array(list(radio_model.intercept_.flatten()) + list(radio_model.coef_.flatten()))
coeffs_radio = list(coeffs_radio)
# let us fit the simple linear regression model with the Newspaper feature
newspaper_model = linear_model.LinearRegression()
newspaper_model.fit(Newspaper, Sales)
coeffs_newspaper = np.array(list(newspaper_model.intercept_.flatten()) + list(newspaper_model.coef_.flatten()))
coeffs_newspaper = list(coeffs_newspaper)
# let us store the above results in a dictionary and then display using a dataframe
dict_Sales = {}
dict_Sales["TV"] = coeffs_tv
dict_Sales["Radio"] = coeffs_radio
dict_Sales["Newspaper"] = coeffs_newspaper
metric_Df_SLR = pd.DataFrame(dict_Sales)
metric_Df_SLR.index = ['Intercept', 'Coefficient']
metric_Df_SLR
Out[ ]:
| TV | Radio | Newspaper | |
|---|---|---|---|
| Intercept | 7.032594 | 9.311638 | 12.351407 |
| Coefficient | 0.047537 | 0.202496 | 0.054693 |
In [ ]:
# Let us now calculate R^2
tv_rsq = tv_model.score(TV, Sales)
radio_rsq = radio_model.score(Radio, Sales)
newspaper_rsq = newspaper_model.score(Newspaper, Sales)
print("TV simple linear regression R-Square :", tv_rsq)
print("Radio simple linear regression R-Square :", radio_rsq)
print("Newspaper simple linear regression R-Square :", newspaper_rsq)
list_rsq = [tv_rsq, radio_rsq, newspaper_rsq]
list_rsq
TV simple linear regression R-Square : 0.611875050850071 Radio simple linear regression R-Square : 0.33203245544529525 Newspaper simple linear regression R-Square : 0.05212044544430516
Out[ ]:
[0.611875050850071, 0.33203245544529525, 0.05212044544430516]
In [ ]:
metric_Df_SLR.loc['R-Squared'] = list_rsq
metric_Df_SLR
Out[ ]:
| TV | Radio | Newspaper | |
|---|---|---|---|
| Intercept | 7.032594 | 9.311638 | 12.351407 |
| Coefficient | 0.047537 | 0.202496 | 0.054693 |
| R-Squared | 0.611875 | 0.332032 | 0.052120 |
Observations: We can see that TV has the highest R^2 value i.e. 61% followed by Radio and Newspaper
Let's try to visualize the best fit line using the regression plot
In [ ]:
plt.scatter(TV, Sales, color='red')
plt.xlabel('TV Ads')
plt.ylabel('Sales')
plt.plot(TV, tv_model.predict(TV), color='blue', linewidth=3)
plt.show()
plt.scatter(Radio, Sales, color='red')
plt.xlabel('Radio Ads')
plt.ylabel('Sales')
plt.plot(Radio, radio_model.predict(Radio), color='blue', linewidth=3)
plt.show()
plt.scatter(Newspaper, Sales, color='red')
plt.xlabel('Newspaper Ads')
plt.ylabel('Sales')
plt.plot(Newspaper, newspaper_model.predict(Newspaper), color='blue', linewidth=3)
plt.show()
Multiple Linear Regression¶
- Let us now build a multiple linear regression model.
In [ ]:
mlr_model = linear_model.LinearRegression()
mlr_model.fit(Ad_df[['TV', 'Radio', 'Newspaper']], Ad_df['Sales'])
Out[ ]:
LinearRegression()In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
LinearRegression()
In [ ]:
Ad_df['Sales_Predicted'] = mlr_model.predict(Ad_df[['TV', 'Radio', 'Newspaper']])
Ad_df['Error'] = (Ad_df['Sales_Predicted'] - Ad_df['Sales'])**2
MSE_MLR = Ad_df['Error'].mean()
In [ ]:
MSE_MLR
Out[ ]:
2.784126314510936
In [ ]:
mlr_model.score(Ad_df[['TV', 'Radio', 'Newspaper']], Ad_df['Sales'])
Out[ ]:
0.8972106381789522
Observations: The R^2 value for the multiple linear regression comes out to be 89.7% i.e. way better than simple linear regression
Let's now try to use statsmodel to get a more detailed model interpretation
In [ ]:
# let us get a more detailed model through statsmodel.
import statsmodels.formula.api as smf
lm1 = smf.ols(formula= 'Sales ~ TV+Radio+Newspaper', data = Ad_df).fit()
lm1.params
print(lm1.summary()) #Inferential statistics
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.897
Model: OLS Adj. R-squared: 0.896
Method: Least Squares F-statistic: 570.3
Date: Mon, 26 Aug 2024 Prob (F-statistic): 1.58e-96
Time: 19:45:46 Log-Likelihood: -386.18
No. Observations: 200 AIC: 780.4
Df Residuals: 196 BIC: 793.6
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 2.9389 0.312 9.422 0.000 2.324 3.554
TV 0.0458 0.001 32.809 0.000 0.043 0.049
Radio 0.1885 0.009 21.893 0.000 0.172 0.206
Newspaper -0.0010 0.006 -0.177 0.860 -0.013 0.011
==============================================================================
Omnibus: 60.414 Durbin-Watson: 2.084
Prob(Omnibus): 0.000 Jarque-Bera (JB): 151.241
Skew: -1.327 Prob(JB): 1.44e-33
Kurtosis: 6.332 Cond. No. 454.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [ ]:
print("*************Parameters**************")
print(lm1.params)
print("*************P-Values**************")
print(lm1.pvalues)
print("************Standard Errors***************")
print(lm1.bse)
print("*************Confidence Interval**************")
print(lm1.conf_int())
print("*************Error Covariance Matrix**************")
print(lm1.cov_params())
*************Parameters**************
Intercept 2.938889
TV 0.045765
Radio 0.188530
Newspaper -0.001037
dtype: float64
*************P-Values**************
Intercept 1.267295e-17
TV 1.509960e-81
Radio 1.505339e-54
Newspaper 8.599151e-01
dtype: float64
************Standard Errors***************
Intercept 0.311908
TV 0.001395
Radio 0.008611
Newspaper 0.005871
dtype: float64
*************Confidence Interval**************
0 1
Intercept 2.323762 3.554016
TV 0.043014 0.048516
Radio 0.171547 0.205513
Newspaper -0.012616 0.010541
*************Error Covariance Matrix**************
Intercept TV Radio Newspaper
Intercept 0.097287 -2.657273e-04 -1.115489e-03 -5.910212e-04
TV -0.000266 1.945737e-06 -4.470395e-07 -3.265950e-07
Radio -0.001115 -4.470395e-07 7.415335e-05 -1.780062e-05
Newspaper -0.000591 -3.265950e-07 -1.780062e-05 3.446875e-05
Visualizing the confidence bands in Simple linear regression¶
In [ ]:
import seaborn as sns
sns.lmplot(x = 'TV', y = 'Sales', data = Ad_df)
sns.lmplot(x = 'Radio', y = 'Sales', data = Ad_df )
sns.lmplot(x = 'Newspaper', y = 'Sales', data = Ad_df)
Out[ ]:
<seaborn.axisgrid.FacetGrid at 0x7983b1ce6530>
LVC 2 - Model Evaluation: Cross validation and Bootstrapping¶
- We realize that the newspaper can be omitted from the list of significant features owing to the p-value.
- Let us now run the regression analysis adding a multiplicative feature in it.
In [ ]:
Ad_df['TVandRadio'] = Ad_df['TV']*Ad_df['Radio']
In [ ]:
# let us remove the sales_predicted and the error column generated earlier
Ad_df.drop(columns = ["Error", "Sales_Predicted"], inplace = True)
In [ ]:
# Let us do the modelling with the new feature.
import statsmodels.formula.api as smf
lm2 = smf.ols(formula= 'Sales ~ TV+Radio+Newspaper+TVandRadio', data = Ad_df).fit()
lm2.params
print(lm2.summary()) #Inferential statistics
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.968
Model: OLS Adj. R-squared: 0.967
Method: Least Squares F-statistic: 1466.
Date: Mon, 26 Aug 2024 Prob (F-statistic): 2.92e-144
Time: 19:51:18 Log-Likelihood: -270.04
No. Observations: 200 AIC: 550.1
Df Residuals: 195 BIC: 566.6
Df Model: 4
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 6.7284 0.253 26.561 0.000 6.229 7.228
TV 0.0191 0.002 12.633 0.000 0.016 0.022
Radio 0.0280 0.009 3.062 0.003 0.010 0.046
Newspaper 0.0014 0.003 0.438 0.662 -0.005 0.008
TVandRadio 0.0011 5.26e-05 20.686 0.000 0.001 0.001
==============================================================================
Omnibus: 126.161 Durbin-Watson: 2.216
Prob(Omnibus): 0.000 Jarque-Bera (JB): 1123.463
Skew: -2.291 Prob(JB): 1.10e-244
Kurtosis: 13.669 Cond. No. 1.84e+04
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.84e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
Observations
- We see an increase in the R-square here. However, is this model useful for prediction? Does it predict well for the unseen data? Let us find out!
Performance assessment, testing and validation¶
Train, Test, and Validation set¶
- We will split data into three sets, one to train the model, one to validate the model performance (not seen during training) and make improvements, and the last to test the model.
In [ ]:
from Scikit-learn.model_selection import train_test_split
In [ ]:
features_base = [i for i in Ad_df.columns if i not in ("Sales" , "TVandRadio")]
features_added = [i for i in Ad_df.columns if i not in "Sales"]
target = 'Sales'
train, test = train_test_split(Ad_df, test_size = 0.10, train_size = 0.9)
In [ ]:
train, validation = train_test_split(train, test_size = 0.2, train_size = 0.80)
In [ ]:
train.shape, validation.shape,test.shape
Out[ ]:
((144, 5), (36, 5), (20, 5))
In [ ]:
# now let us start with the modelling
from Scikit-learn.linear_model import LinearRegression
mlr = LinearRegression()
mlr.fit(train[features_base], train[target])
print("*********Training set Metrics**************")
print("R-Squared:", mlr.score(train[features_base], train[target]))
se_train = (train[target] - mlr.predict(train[features_base]))**2
mse_train = se_train.mean()
print('MSE: ', mse_train)
print("********Validation set Metrics**************")
print("R-Squared:", mlr.score(validation[features_base], validation[target]))
se_val = (validation[target] - mlr.predict(validation[features_base]))**2
mse_val = se_val.mean()
print('MSE: ', mse_val)
*********Training set Metrics************** R-Squared: 0.9148715588726591 MSE: 2.2887134456785474 ********Validation set Metrics************** R-Squared: 0.7789766535118162 MSE: 3.3952648037509183
In [ ]:
# Can we increase the model performance by adding the new feature?
# We found that to be the case in the analysis above but let's check the same for the validation dataset
mlr_added_feature = LinearRegression()
mlr_added_feature.fit(train[features_added], train[target])
print("*********Training set Metrics**************")
print("R-Squared:", mlr_added_feature.score(train[features_added], train[target]))
se_train = (train[target] - mlr_added_feature.predict(train[features_added]))**2
mse_train = se_train.mean()
print('MSE: ', mse_train)
print("********Validation set Metrics**************")
print("R-Squared:", mlr_added_feature.score(validation[features_added], validation[target]))
se_val = (validation[target] - mlr_added_feature.predict(validation[features_added]))**2
mse_val = se_val.mean()
print('MSE: ', mse_val)
*********Training set Metrics************** R-Squared: 0.9775457308122865 MSE: 0.6036923397426106 ********Validation set Metrics************** R-Squared: 0.9631013641483601 MSE: 0.566820843155554
Observations
- We found the R-squared increased as we would expect after adding a feature. Also the error decreased. Let us now fit a regularized model.
Regularization¶
In [ ]:
features_added
Out[ ]:
['TV', 'Radio', 'Newspaper', 'TVandRadio']
In [ ]:
from Scikit-learn.linear_model import Ridge
from Scikit-learn.linear_model import Lasso
#fitting Ridge with the default features
ridge = Ridge()
ridge.fit(train[features_added], train[target])
print("*********Training set Metrics**************")
print("R-Squared:", ridge.score(train[features_added], train[target]))
se_train = (train[target] - ridge.predict(train[features_added]))**2
mse_train = se_train.mean()
print('MSE: ', mse_train)
print("********Validation set Metrics**************")
print("R-Squared:", ridge.score(validation[features_added], validation[target]))
se_val = (validation[target] - ridge.predict(validation[features_added]))**2
mse_val = se_val.mean()
print('MSE: ', mse_val)
*********Training set Metrics************** R-Squared: 0.958023517237818 MSE: 1.0016155396698094 ********Validation set Metrics************** R-Squared: 0.9795476415861633 MSE: 0.7600681866903618
In [ ]:
from Scikit-learn.linear_model import Ridge
from Scikit-learn.linear_model import Lasso
#fitting Lasso with the default features
lasso = Lasso()
lasso.fit(train[features_added], train[target])
print("*********Training set Metrics**************")
print("R-Squared:", lasso.score(train[features_added], train[target]))
se_train = (train[target] - lasso.predict(train[features_added]))**2
mse_train = se_train.mean()
print('MSE: ', mse_train)
print("********Validation set Metrics**************")
print("R-Squared:", lasso.score(validation[features_added], validation[target]))
se_val = (validation[target] - lasso.predict(validation[features_added]))**2
mse_val = se_val.mean()
print('MSE: ', mse_val)
*********Training set Metrics************** R-Squared: 0.9575230612795557 MSE: 1.0135570943626075 ********Validation set Metrics************** R-Squared: 0.9769399450467467 MSE: 0.856977655028845
In [ ]:
#Let us predict on the unseen data using Ridge
rsq_test = ridge.score(test[features_added], test[target])
se_test = (test[target] - ridge.predict(test[features_added]))**2
mse_test = se_test.mean()
print("*****************Test set Metrics******************")
print("Rsquared: ", rsq_test)
print("MSE: ", mse_test)
print("Intercept is {} and Coefficients are {}".format(ridge.intercept_, ridge.coef_))
*****************Test set Metrics****************** Rsquared: 0.9817005837522722 MSE: 0.49043304766182383 Intercept is 6.8067374349457 and Coefficients are [0.01801443 0.01332594 0.00500065 0.00118479]
- We will now evaluate the performance using the LooCV and KFold methods.
K-Fold and LooCV¶
In [ ]:
from Scikit-learn.linear_model import Ridge
from Scikit-learn.linear_model import Lasso
from Scikit-learn.model_selection import cross_val_score
In [ ]:
ridgeCV = Ridge()
cvs = cross_val_score(ridgeCV, Ad_df[features_added], Ad_df[target], cv = 10)
print("Mean Score:")
print(cvs.mean(), "\n")
print("Confidence Interval:")
cvs.mean() - cvs.std(), cvs.mean() + cvs.std()
# note that the same can be set as LooCV if cv parameter above is set to n, i.e, 200.
Mean Score: 0.9649887636257694 Confidence Interval:
Out[ ]:
(0.9430473456799696, 0.9869301815715691)
Extra: Statsmodel to fit regularized model¶
In [ ]:
# You can also use the statsmodel for the regularization using the below code
# import statsmodels.formula.api as smf
# We will use the below code to fit a regularized regression.
# Here, lasso is fit
# lm3 = smf.ols(formula= 'Sales ~ TV+Radio+Newspaper+TVandRadio', data = Ad_df).fit_regularized(method = 'elastic_net', L1_wt = 1)
# print("*************Parameters**************")
# print(lm3.params)
# Here, ridge regularization has been fit
# lm4 = smf.ols(formula= 'Sales ~ TV+Radio+Newspaper+TVandRadio', data = Ad_df).fit_regularized(method = 'elastic_net', L1_wt = 0)
# print("*************Parameters**************")
# print(lm4.params)
Bootstrapping¶
In [ ]:
# let us get a more detailed model through statsmodel.
import statsmodels.formula.api as smf
lm2 = smf.ols(formula= 'Sales ~ TV', data = Ad_df).fit()
lm2.params
print(lm2.summary()) #Inferential statistics
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.612
Model: OLS Adj. R-squared: 0.610
Method: Least Squares F-statistic: 312.1
Date: Sat, 28 Aug 2021 Prob (F-statistic): 1.47e-42
Time: 15:28:22 Log-Likelihood: -519.05
No. Observations: 200 AIC: 1042.
Df Residuals: 198 BIC: 1049.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 7.0326 0.458 15.360 0.000 6.130 7.935
TV 0.0475 0.003 17.668 0.000 0.042 0.053
==============================================================================
Omnibus: 0.531 Durbin-Watson: 1.935
Prob(Omnibus): 0.767 Jarque-Bera (JB): 0.669
Skew: -0.089 Prob(JB): 0.716
Kurtosis: 2.779 Cond. No. 338.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [ ]:
#Now, let us calculate the slopes a 1000 times using bootstrapping
import statsmodels.formula.api as smf
Slope = []
for i in range(1000):
bootstrap_df = Ad_df.sample(n = 200, replace = True )
lm3 = smf.ols(formula= 'Sales ~ TV', data = bootstrap_df).fit()
Slope.append(lm3.params.TV)
plt.xlabel('TV Ads')
plt.ylabel('Sales')
plt.plot(bootstrap_df['TV'], lm3.predict(bootstrap_df['TV']), color='green', linewidth=3)
plt.scatter(Ad_df['TV'], Ad_df['Sales'], color=(0,0,0.5))
plt.show()
In [ ]:
# Let's now find out the 2.5 and 97.5 percentile for the slopes obtained
Slope = np.array(Slope)
Sort_Slope = np.sort(Slope)
Slope_limits = np.percentile(Sort_Slope, (2.5, 97.5))
Slope_limits
Out[ ]:
array([0.04126775, 0.05293063])
In [ ]:
# Plotting the slopes and the upper and the lower limits
plt.hist(Slope, 50)
plt.axvline(Slope_limits[0], color = 'r')
plt.axvline(Slope_limits[1], color = 'r')
Out[ ]:
<matplotlib.lines.Line2D at 0x22adda6d7f0>
In [2]:
# Convert notebook to html
!jupyter nbconvert --to html "/content/drive/MyDrive/MIT - Data Sciences/Colab Notebooks/Week Four - Regression and Prediction/Results of Advertising on Sales/LVC_1_%26_2_Practical_Application_Effects_of_Advertising_on_Sales.ipynb"
[NbConvertApp] Converting notebook /content/drive/MyDrive/MIT - Data Sciences/Colab Notebooks/Week Four - Regression and Prediction/Results of Advertising on Sales/LVC_1_%26_2_Practical_Application_Effects_of_Advertising_on_Sales.ipynb to html [NbConvertApp] WARNING | Alternative text is missing on 8 image(s). [NbConvertApp] Writing 680495 bytes to /content/drive/MyDrive/MIT - Data Sciences/Colab Notebooks/Week Four - Regression and Prediction/Results of Advertising on Sales/LVC_1_%26_2_Practical_Application_Effects_of_Advertising_on_Sales.html