Calculating slope and intercept for best fit line

SWBAT

• Calculate the slope of a line using standard slope formula
• Calculate the y-intercept using the slope value
• Draw a regression line based on calculated slope and intercept
• Predict the label of a previously unseen data element

Introduction

Regression analysis forms the basis of machine learning experiments. Understanding regression will help you to get the foundations of most machine learning algorithms. Ever wondered what's at the heart of an artificial neural network processing unstructured data like music and graphics? It can be linear regression!

A first step towards understanding regression is getting a clear idea about "linear" regression and basic linear algebra.

The calculation for the best-fit line's slope, m is calculated as :

As in our previous lesson, let's break down the formula into its parts. First we shall import the required libraries and define some data points to work with.

# import necessary libraries

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import style
style.use('ggplot')

We shall first create some dummy data as numpy arrays and plot it to view the variability of data. Use following values for this example:

X = [1,2,3,4,5,6,8,8,9,10]

Y = [7,7,8,9,9,10,10,11,11,12]

# Initialize vectors X and Y with given values and create a scatter plot
X = None
Y = None

In a data analysis context, we can think of these points as two vectors:

• vector X: the features of our model
• vector Y: the labels for given features

def calc_slope(xs,ys):

    # Use the slope formula above and calculate the slope
    m = None
    return m

calc_slope(X,Y)

# 0.5393518518518512

As a reminder, the calculation for the best-fit line's y-intercept is:

def best_fit(xs,ys):

    # use the slope function with intercept formula to return calculate slop and intercept from data points
    m = None
    b = None

    return m, b

m, b = best_fit(X,Y)
m,b
# (0.5393518518518512, 6.379629629629633)

(None, None)

We now have a working model with m and b as model parameters. We can create a line for the data points using the calculated slope and intercept:

• Recall that y=mx+b. We can now use slope and intercept values along with X data points (features) to calculate the Y data points (labels) of the regression line.

def reg_line (m, b, xs):

    return None

regression_line = reg_line(m,b,X)

Now that we have calculated the regression line, we can plot the data points and regression line for visual inspection.

Predicting label for new data

So, how might you go about actually making a prediction based on this model you just made?

Now that we have a working model with m and b as model parameters. We can fill in a value of x with these parameters to identify a corresponding value of y according to our model.

Let's try to find a y prediction for a new value of x = 7 and unknown y, and plot the new prediction with existing data

x_new = 7
y_predicted = None
y_predicted

# 10.155092592592592

# Plot the regression line and y prediction for new data element

We now know how to create our own models, which is great, but we're stilling missing something integral: how accurate is our model? This is the topic for discussion in the next lab.

Summary

In this lesson, we learnt how we can draw a best fit line for given data labels and features, by first calculating the slope and intercept. The calculated regression line was then used to predict the label (y-value) of a previously unseen feature (x-value). The lesson uses a simple set of data points for demonstration. Students should be able to plug in other datasets and practice with predictions for accuracy.