Chapter 5 Regression analysis

knitr::opts_chunk$set(echo = TRUE)

library(reticulate)

use_condaenv("tf")

5.1 Importing python packages

import matplotlib.pyplot as plt
import matplotlib.dates as mdates
import pandas as pd
import numpy as np
import statistics

from scipy.stats import norm
from matplotlib.ticker import EngFormatter, StrMethodFormatter

The fundamental data type of NumPy is the array type called numpy.ndarray. The rest of this article uses the term array to refer to instances of the type numpy.ndarray.

from sklearn.datasets import fetch_california_housing

california_housing = fetch_california_housing(as_frame=True)

print(california_housing.DESCR)

california_housing.data.head()

#Looks good - let's convert it into a pandas dataframe

california_housing_df = pd.DataFrame(california_housing.data)

print(california_housing_df)

california_housing.frame.hist(figsize=(12, 10), bins=30, edgecolor="black")

plt.subplots_adjust(hspace=0.7, wspace=0.4)

plt.show()

5.2 Create a model and fit it

The next step is to create the regression model as an instance of LinearRegression and fit it with .fit().

import sklearn
from sklearn.linear_model import LinearRegression
from sklearn import linear_model

# Choose our variables of interest
x = california_housing_df[['HouseAge']]
y = california_housing_df[['MedInc']]

# Make a model
model = LinearRegression().fit(x, y)

# Analyse the model fit 
r_sq = model.score(x, y)
print('coefficient of determination:', r_sq)

## coefficient of determination: 0.014169090760525749

print('intercept:', model.intercept_)

## intercept: [4.38527909]

print('slope:', model.coef_)

## slope: [[-0.01796848]]

5.3 Polynomial regression

We can fit different order polynomials by defining the relevant polynomial functions.

# Load in relevant packages
from numpy import arange
from pandas import read_csv
from scipy.optimize import curve_fit
from matplotlib import pyplot
 
# Define the true objective function for a linear estimation
def objective(x, a, b):
    return a * x + b
 
# load the dataset
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/longley.csv'
dataframe = read_csv(url, header=None)
data = dataframe.values

# choose the input and output variables
x, y = data[:, 4], data[:, -1]

# curve fit
popt, _ = curve_fit(objective, x, y)

# summarize the parameter values
a, b = popt
print('y = %.5f * x + %.5f' % (a, b))

# plot input vs output

## y = 0.48488 * x + 8.38067

plt.scatter(x, y, c ="blue")

# define a sequence of inputs between the smallest and largest known inputs

## <matplotlib.collections.PathCollection object at 0x186f613c0>

x_line = arange(min(x), max(x), 1)

# calculate the output for the range
y_line = objective(x_line, a, b)

# create a line plot for the mapping function
plt.plot(x_line, y_line,
        label='Polynomial', 
        color='purple', 
        alpha=1, 
        linewidth=1.2,
        linestyle='dashed')

## [<matplotlib.lines.Line2D object at 0x186f62050>]

plt.title('Using a linear model to approximate data', fontsize=12)

## Text(0.5, 1.0, 'Using a linear model to approximate data')

plt.xlabel('', fontsize=10)

## Text(0.5, 0, '')

plt.ylabel('', fontsize=10)

## Text(0, 0.5, '')

plt.gca().spines['top'].set_visible(False)
plt.gca().spines['bottom'].set_visible(True)
plt.gca().spines['right'].set_visible(False)
plt.gca().spines['left'].set_visible(False)

plt.tick_params(
    axis='x',          # changes apply to the x-axis
    which='both',      # both major and minor ticks are affected
    bottom=False,      # ticks along the bottom edge are off
    top=False,         # ticks along the top edge are off
    labelbottom=True)  # labels along the bottom edge are off
    
plt.tick_params(
    axis='y',          # changes apply to the y-axis
    which='both',      # both major and minor ticks are affected
    left=False,      # ticks along the bottom edge are off
    right=False,         # ticks along the top edge are off
    labelleft=True)  # labels along the bottom edge are off    
 
plt.grid(False)
plt.gca().yaxis.grid(True)

plt.legend(fancybox=False, framealpha=1, shadow=False, borderpad=1)

## <matplotlib.legend.Legend object at 0x187f5d870>

plt.savefig('linear_model_chart.png',dpi=300,bbox_inches='tight')

plt.show()

Now let’s try a polynomial model

# Fit a second degree polynomial to the economic data
from numpy import arange
from pandas import read_csv
from scipy.optimize import curve_fit
from matplotlib import pyplot
 
# Define the true objective function
def objective(x, a, b, c):
    return a * x + b * x**2 + c
 
# Load the dataset
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/longley.csv'
dataframe = read_csv(url, header=None)
data = dataframe.values

# choose the input and output variables
x, y = data[:, 4], data[:, -1]

# curve fit
popt, _ = curve_fit(objective, x, y)

# summarize the parameter values
a, b, c = popt
print('y = %.5f * x + %.5f * x^2 + %.5f' % (a, b, c))

# plot input vs output

plt.scatter(x, y, c ="blue")

# define a sequence of inputs between the smallest and largest known inputs

x_line = arange(min(x), max(x), 1)

# calculate the output for the range
y_line = objective(x_line, a, b, c)

# create a line plot for the mapping function
# create a line plot for the mapping function
plt.plot(x_line, y_line,
        label='Polynomial', 
        color='purple', 
        alpha=1, 
        linewidth=1.2,
        linestyle='dashed')

plt.title('Using a polynomial model to approximate data', fontsize=12)

plt.xlabel('', fontsize=10)

plt.ylabel('', fontsize=10)

plt.gca().spines['top'].set_visible(False)
plt.gca().spines['bottom'].set_visible(True)
plt.gca().spines['right'].set_visible(False)
plt.gca().spines['left'].set_visible(False)

plt.tick_params(
    axis='x',          # changes apply to the x-axis
    which='both',      # both major and minor ticks are affected
    bottom=False,      # ticks along the bottom edge are off
    top=False,         # ticks along the top edge are off
    labelbottom=True)  # labels along the bottom edge are off
    
plt.tick_params(
    axis='y',          # changes apply to the y-axis
    which='both',      # both major and minor ticks are affected
    left=False,      # ticks along the bottom edge are off
    right=False,         # ticks along the top edge are off
    labelleft=True)  # labels along the bottom edge are off    
 
plt.grid(False)
plt.gca().yaxis.grid(True)

plt.legend(fancybox=False, framealpha=1, shadow=False, borderpad=1)

plt.savefig('linear_model_chart.png',dpi=300,bbox_inches='tight')

plt.show()

5.4 —

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