Section 1
General Linear Models
1. Section Overview
(Preview)
2. The Foundations of ML - Curve Fitting
21:39 (Preview)
3. Regression Walkthrough
23:47 (Preview)
4. Underfitting and Overfitting
21:50
5. Controlling ML Models - Regularisation in Practise
16:28
6. Exploring Simple Relationships - Correlations
11:28
7. Finding Nonlinear Relations - Predictive Power Score
11:09
8. Correlation & PPS Walkthrough 📂
24:27
9. From Regression to Classification - Logistic Regression
20:27
10. Logistic Regression in Wild
15:48
11. Looking through the Right Lens - Principle Component Analysis
16:46
12. Looking in the Right Direction - Exploring PCA
21:23
13. Conclusion, Certificate, and What Next?
3. Regression Walkthrough
General Linear Models

In this walkthrough we will go through he basics of fitting linear models with sklearn using LinearRegression(). We will look at a simple polynomial model to get started, and focus on understanding how we can use PolynomialFeature() to build a general function on which to build a basic model.

Adding Libraries

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

A Fishy Data set!

In this data set we are going to look at some fish data of 7 different species, where we will try and predict weight from simple measurements. Important stuff!

In total the data set contains 7 measurements of 159 fish.

df = pd.read_csv('https://raw.githubusercontent.com/satishgunjal/datasets/master/Fish.csv')
df = df.drop(['Length1', 'Length2', 'Length3'], axis =1) # Can also use axis = 'columns'
df.sample(5) # Display random 5 records
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Ok Let us first look at the correlations in the data, for this we calculate and plot a correlation matrix.

df.corr()

plt.rcParams["figure.figsize"] = (6,4) # Custom figure size in inches
sns.heatmap(df.corr(), annot =True)
plt.title('Correlation Matrix')
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sns.pairplot(df, kind = 'scatter', hue = 'Species')
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If we want to predict weight, using height looks complex. There looks like two clear responses based on the fish type. Yet making predictions using width, looks like a much more unified relationship, much less depedent on fish type.

So let's do that.

X = df['Width']
y = df['Weight']

X_train, X_test, y_train, y_test = train_test_split(X,y, test_size = 0.2)

plt.scatter(X_train,y_train, alpha = 0.4,color='lightcoral',label='Training Data')
plt.scatter(X_test,y_test, alpha = 0.8,color='lightblue',label='Testing Data')
plt.xlabel('Width')
plt.xlabel('Weight')
plt.legend()
plt.show()
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There are clear outliers in this data set, yet since we have sufficient data their influence will be minimal. We could remove them, but we wont here.

Building a Linear Model

Ok so the first simple model we are going to build is a polynomial model, which looks like this

y=w0+w1X+w2X2=[w0,w1,w2]T[1XX2]=wTϕy = w_0 + w_1X + w_2X^2 = \begin{bmatrix} w_0, w_1, w_2 \end{bmatrix}^T\begin{bmatrix} 1 \\ X \\ X^2 \end{bmatrix} = {\bf w}^T \boldsymbol{\phi}

So here we talk about ϕ\boldsymbol{\phi} being the feature vectors, in this case they a polynomial features; really easy to do in ``sklearn`.

from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures

polynomial_features= PolynomialFeatures(degree=2)

phi_train = polynomial_features.fit_transform(np.array(X_train).reshape(-1,1))

lr = LinearRegression().fit(phi_train, y_train)

Ok so now let us plot out the function and compare it to the data.

X_all = np.linspace(1.0, 8.0, 1000).reshape(-1,1)
X_plot_poly = polynomial_features.fit_transform(X_all)
y_pred = lr.predict(X_plot_poly)
plt.scatter(X_train,y_train, alpha = 0.4,color='lightcoral',label='Training Data')
plt.scatter(X_test,y_test, alpha = 0.8,color='lightblue',label='Testing Data')
plt.plot(X_all,y_pred,'-k',alpha = 0.8,label='Linear Model')
plt.xlabel('Width')
plt.ylabel('Weight')
plt.legend()
plt.show()
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It isn't a good model. Ok it fits on average ok, but we know that weight can't be negative. So we need to impose constraints.

sklearn gives us some options.

lr_positive = LinearRegression(fit_intercept = False, positive = True).fit(phi_train, y_train)
y_pred_positive = lr_positive.predict(X_plot_poly)

plt.scatter(X_train,y_train, alpha = 0.4,color='lightcoral',label='Training Data')
plt.scatter(X_test,y_test, alpha = 0.8,color='lightblue',label='Testing Data')
plt.plot(X_all,y_pred,'-k',alpha = 0.8,label='Linear Model')
plt.plot(X_all,y_pred_positive,'-g',alpha = 0.8,label='Linear Model w. +ve constraint')
plt.xlabel('Width')
plt.ylabel('Weight')
plt.legend()
plt.show()
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Our model now at least make physically meaning predictions, although we see the performance of the model is reduce, particular at smalle width values.

How might we do better?

  • Get rid of outliers?
  • Try a different form of linear model?
  • Apply a transform of the data first? If so, then what might you try?
Next Lesson
4. Underfitting and Overfitting