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In [5]:

    
import pandas as pd
#%matplotlib inline
import matplotlib.pyplot as plt



In [6]:

    
vis = pd.DataFrame({
    'cells': [116,117,120,1,52,79,109,27,85,51,78,55,26,39,107],
    'photo': [60,67,64,8,13,63,63,2,46,27,43,24,10,28,56]})



In [7]:

    
graph = []
def graph_all(graph): [g() for g in graph]



In [8]:

    
def scatter(): return plt.scatter(x=vis.cells, y=vis.photo, color='g')
graph = [scatter]
graph_all(graph)



In [9]:

    
x_range = vis.cells.min(), vis.cells.max()
y_mean = vis.photo.mean(), vis.photo.mean()
def mean(): return plt.plot(x_range, y_mean, color="brown")
graph.append(mean)
graph_all(graph)

The equations for the slope of the regression line can be constructed as:

$$ \hat\beta = \frac{ \operatorname{Cov}[x,y] }{ \operatorname{Var}[x] } $$



In [10]:

    
slope = vis.cells.cov(vis.photo)/ vis.cells.var()
slope









    Out[10]:





0.57225117604575371

The equation for the intercept of the regression line can be construted as:

$$ \hat\alpha = \bar{y} - \hat\beta\,\bar{x} $$



In [11]:

    
intercept = vis.photo.mean() - (slope * vis.cells.mean())
intercept









    Out[11]:





-2.2487165973726917

So we can construction the regression line by solving for our expected y:

$$ \hat{y} = (slope * x) + intercept $$



In [12]:

    
vis.expected = (vis.cells * slope) + intercept
vis.expected









    Out[12]:





0     64.132420
1     64.704671
2     66.421425
3     -1.676465
4     27.508345
5     42.959126
6     60.126662
7     13.202065
8     46.392633
9     26.936093
10    42.386875
11    29.225098
12    12.629814
13    20.069079
14    58.982159
Name: cells, dtype: float64



In [13]:

    
def regression(): return plt.plot(vis.cells, vis.expected)
graph.append(regression)
graph_all(graph)

SStotal:



In [14]:

    
def SStotal(): return [plt.plot((vis.cells[i], vis.cells[i]), (vis.photo.mean(), vis.photo[i]), color='black') for i in range(len(vis))]
graph.append(SStotal)



In [15]:

    
graph_all(graph)

SSregrssion



In [16]:

    
def SSreg(): return [plt.plot((vis.cells[i], vis.cells[i]), (vis.expected[i], vis.photo[i]), color='red') for i in range(len(vis))]
graph.append(SSreg)
graph_all(graph)

recall that to calculate SStotal we use the formula:

$$ SS_{total} = \sum\limits_{x=i}^n(y_{i} - \hat{y})^{2} $$



In [17]:

    
sum_of_squares = {'total' : ((vis.photo - vis.photo.mean())**2).sum()}
sum_of_squares['total']









    Out[17]:





7684.9333333333334



In [18]:

    
sum_of_squares['regression'] = ((vis.photo - vis.expected)**2).sum()
sum_of_squares['regression']









    Out[18]:





974.25824207999085

now r squared is just..

$$ R^{2} = 1 - \frac{SS_{regression}}{SS_{total}} $$



In [19]:

    
r_squared = 1 - (sum_of_squares['regression']/sum_of_squares['total'])



In [20]:

    
r_squared









    Out[20]:





0.8732248934607989



In [1]:

    
%load_ext vimception



In [5]:



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