Regression using Anscombe’s Quartet Dataset
let’s delve into the topic with practical implementation.
Import the necessary libraries
Python3
# Import the necessary librariesimport numpy as npimport pandas as pdimport matplotlib.pyplot as plt |
Load the dataset
let’s import Anscombe’s quartet dataset.
Python3
df = pd.read_csv('https://query.data.world/s/6p2ntncvkzj5mnvbpkaswfilryvnrk')print(df) |
output:
x1 x2 x3 x4 y1 y2 y3 y4
0 10 10 10 8 8.04 9.14 7.46 6.58
1 8 8 8 8 6.95 8.14 6.77 5.76
2 13 13 13 8 7.58 8.74 12.74 7.71
3 9 9 9 8 8.81 8.77 7.11 8.84
4 11 11 11 8 8.33 9.26 7.81 8.47
5 14 14 14 8 9.96 8.10 8.84 7.04
6 6 6 6 8 7.24 6.13 6.08 5.25
7 4 4 4 19 4.26 3.10 5.39 12.50
8 12 12 12 8 10.84 9.13 8.15 5.56
9 7 7 7 8 4.82 7.26 6.42 7.91
10 5 5 5 8 5.68 4.74 5.73 6.89
Find the Descriptive Statistical Properties for the all four Dataset
- Find mean for x and y for all four datasets.
- Find standard deviations for x and y for all four datasets.
- Find correlations with their corresponding pair of each datasets.
- Find slope and intercept for each datasets.
- Find R-square for each datasets.
- To find R-square first find residual sum of square error and Total sum of square error
- Create a statistical summary by using all these data and print it.
Python3
# mean values (x-bar)x1_mean = df['x1'].mean()x2_mean = df['x2'].mean()x3_mean = df['x3'].mean()x4_mean = df['x4'].mean()# y-bary1_mean = df['y1'].mean()y2_mean = df['y2'].mean()y3_mean = df['y3'].mean()y4_mean = df['y4'].mean()# Standard deviation values (x-bar)x1_std = df['x1'].std()x2_std = df['x2'].std()x3_std = df['x3'].std()x4_std = df['x4'].std()# Standard deviation values (y-bar)y1_std = df['y1'].std()y2_std = df['y2'].std()y3_std = df['y3'].std()y4_std = df['y4'].std()# Correlationcorrelation_x1y1 = np.corrcoef(df['x1'],df['y1'])[0,1]correlation_x2y2 = np.corrcoef(df['x2'],df['y2'])[0,1]correlation_x3y3 = np.corrcoef(df['x3'],df['y3'])[0,1]correlation_x4y4 = np.corrcoef(df['x4'],df['y4'])[0,1]# Linear Regression slope and interceptm1,c1 = np.polyfit(df['x1'], df['y1'], 1)m2,c2 = np.polyfit(df['x2'], df['y2'], 1)m3,c3 = np.polyfit(df['x3'], df['y3'], 1)m4,c4 = np.polyfit(df['x4'], df['y4'], 1)# Residual sum of squares errorRSSY_1 = ((df['y1'] - (m1*df['x1']+c1))**2).sum()RSSY_2 = ((df['y2'] - (m2*df['x2']+c2))**2).sum()RSSY_3 = ((df['y3'] - (m3*df['x3']+c3))**2).sum()RSSY_4 = ((df['y4'] - (m4*df['x4']+c4))**2).sum()# Total sum of squares TSS_1 = ((df['y1'] - y1_mean)**2).sum()TSS_2 = ((df['y2'] - y2_mean)**2).sum()TSS_3 = ((df['y3'] - y3_mean)**2).sum()TSS_4 = ((df['y4'] - y4_mean)**2).sum()# R squared (coefficient of determination)R2_1 = 1 - (RSSY_1 / TSS_1)R2_2 = 1 - (RSSY_2 / TSS_2)R2_3 = 1 - (RSSY_3 / TSS_3)R2_4 = 1 - (RSSY_4 / TSS_4)# Create a pandas dataframe to represent the summary statisticssummary_stats = pd.DataFrame({'Mean_x': [x1_mean, x2_mean, x3_mean, x4_mean], 'Variance_x': [x1_std**2, x2_std**2, x3_std**2, x4_std**2], 'Mean_y': [y1_mean, y2_mean, y3_mean, y4_mean], 'Variance_y': [y1_std**2, y2_std**2, y3_std**2, y4_std**2], 'Correlation': [correlation_x1y1, correlation_x2y2, correlation_x3y3, correlation_x4y4], 'Linear Regression slope': [m1, m2, m3, m4], 'Linear Regression intercept': [c1, c2, c3, c4]},index = ['I', 'II', 'III', 'IV'])print(summary_stats.T) |
Output:
I II III IV
Mean_x 9.000000 9.000000 9.000000 9.000000
Variance_x 11.000000 11.000000 11.000000 11.000000
Mean_y 7.500909 7.500909 7.500000 7.500909
Variance_y 4.127269 4.127629 4.122620 4.123249
Correlation 0.816421 0.816237 0.816287 0.816521
Linear Regression slope 0.500091 0.500000 0.499727 0.499909
Linear Regression intercept 3.000091 3.000909 3.002455 3.001727
Clearly, we can see identical descriptive statistics summary, this uniformity in summary statistics might lead one to believe that the datasets are essentially the same.
However, when examining the scatter plots of these datasets, we’ll observe the inherent differences.
Plot the scatter plot and linear regression line for each datasets
Python3
# plot all four plotsfig, axs = plt.subplots(2, 2, figsize=(18,12), dpi=500)axs[0, 0].set_title('Dataset I', fontsize=20)axs[0, 0].set_xlabel('X', fontsize=13)axs[0, 0].set_ylabel('Y', fontsize=13)axs[0, 0].plot(df['x1'], df['y1'], 'go')axs[0, 0].plot(df['x1'], m1*df['x1']+c1,'r',label='Y='+str(round(m1,2))+'x +'+str(round(c1,2)))axs[0, 0].legend(loc='best',fontsize=16)axs[0, 1].set_title('Dataset II',fontsize=20)axs[0, 1].set_xlabel('X', fontsize=13)axs[0, 1].set_ylabel('Y', fontsize=13)axs[0, 1].plot(df['x2'], df['y2'], 'go')axs[0, 1].plot(df['x2'], m2*df['x2']+c2,'r',label='Y='+str(round(m2,2))+'x +'+str(round(c2,2)))axs[0, 1].legend(loc='best',fontsize=16)axs[1, 0].set_title('Dataset III',fontsize=20)axs[1, 0].set_xlabel('X', fontsize=13)axs[1, 0].set_ylabel('Y', fontsize=13)axs[1, 0].plot(df['x3'], df['y3'], 'go')axs[1, 0].plot(df['x3'], m1*df['x3']+c1,'r',label='Y='+str(round(m3,2))+'x +'+str(round(c3,2)))axs[1, 0].legend(loc='best',fontsize=16)axs[1, 1].set_title('Dataset IV',fontsize=20)axs[1, 1].set_xlabel('X', fontsize=13)axs[1, 1].set_ylabel('Y', fontsize=13)axs[1, 1].plot(df['x4'], df['y4'], 'go')axs[1, 1].plot(df['x4'], m4*df['x4']+c4,'r',label='Y='+str(round(m4,2))+'x +'+str(round(c4,2)))axs[1, 1].legend(loc='best',fontsize=16) plt.show() |
Output:
Anscombe’s quartet Plot
Note: It is mentioned in the definition that Anscombe’s quartet comprises four datasets that have nearly identical simple statistical properties, yet appear very different when graphed.
Explanation of this output:
- In the first one(top left) if you look at the scatter plot you will see that there seems to be a linear relationship between x and y.
- In the second one(top right) if you look at this figure you can conclude that there is a non-linear relationship between x and y.
- In the third one(bottom left) you can say when there is a perfect linear relationship for all the data points except one which seems to be an outlier which is indicated be far away from that line.
- Finally, the fourth one(bottom right) shows an example when one high-leverage point is enough to produce a high correlation coefficient.
Anscombe’s quartet
Anscombe’s Quartet, comprising four datasets with nearly identical summary statistics, underscores the limitations of relying solely on numerical metrics.
This article explores the quartet’s datasets, emphasizing the importance of visualizing data for a comprehensive understanding.
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