6  Normalizing your data and PCA

6.1 Introduction

This chapter demonstrates basic unsupervised machine learning concepts using Python.

TipLearning Objectives
  • Understand the difference between supervised and unsupervised learning.
  • Apply PCA and clustering to example data.
  • Visualize results.

6.2 Normalization (Z-score Standardization)

Normalization, specifically Z-score standardization, is a data scaling technique that transforms your data to have a mean of 0 and a standard deviation of 1. This is useful for many machine learning algorithms that are sensitive to the scale of input features.

Intuition: the value represents the number of standard deviations away from the mean for that variable. For example an 80-year-old person might be 3 standard deviations above the mean age.

The formula for Z-score is:

\[ z = \frac{x - \mu}{\sigma} \]

Where: - \(x\) is the original data point. - \(\mu\) is the mean of the data. - \(\sigma\) is the standard deviation of the data.

For example, say you have two variables or features on very different scales.

Age Weight (grams)
25 65000
30 70000
35 75000
40 80000
45 85000
50 90000
55 95000
60 100000
65 105000
70 110000
75 115000
80 120000

If these are not brought on similar scales, weight will have a dispproportionate influence on whatever machine learning model we build.

Hence we normalize each of the features separately, i.e. age is normalized relative to age and weight is normalized relative to weight.

Original data:
Age: mean=43.6, std=13.1
Weight: mean=69.8, std=9.8

  • In an ideal scenario a feature/variable such as weight might be transformed in the following way after normalization:

  • And here is what it might look like for a feature such as age.
Z-scored mean: -0.00, std: 1.00

Tip

NOTE (IMPORTANT CONCEPT):

  • After normalization, the normalized features are on comparable scales. The features (such as weight and age) no longer have so much variation. They can be used as input to machine learning algorithms.

  • The rule of thumb is to (almost) always normalize your data before you use it in a machine learning algorithm. (There are a few exceptions and we will point this out in due course).

6.2.1 Data visualization before doing PCA

ExerciseExercise 1 - exercise_data_visualization

Level:

You should always visualize your data before trying any algorithms on it.

Discuss in a group. What is wrong with the following plot?

Looking at your data

Always look at your data before you try and machine learning technique on it. There is a 150 year old person in your data!

Tip

NOTE (IMPORTANT CONCEPT):

  • Visualize your data before you do any normalization. If there is anything odd about your data, discuss this with the person who gave you the data or did the experiment. This could be an error in the machine that generated the data or a data entry error. If there is justification, you can remove the data point.

  • Then perform normalization and apply a machine learning technique.

6.3 Setup

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.cluster import KMeans

6.4 Example Data

6.5 PCA Example

pca = PCA(n_components=2)
X_pca = pca.fit_transform(X)
plt.scatter(X_pca[:, 0], X_pca[:, 1])
plt.title("PCA Projection")
plt.show()

A simple PCA plot

6.6 Scree plot

A scree plot is a simple graph that shows how much variance (information) each principal component explains in your data after running PCA. The x-axis shows the principal components (PC1, PC2, etc.), and the y-axis shows the proportion of variance explained by each one.

You can use a scree plot to decide how many principal components to keep: look for the point where the plot levels off (the elbow): this tells you that adding more components doesn’t explain much more variance.

# Scree plot: variance explained by each component
plt.plot(range(1, len(pca.explained_variance_ratio_) + 1), pca.explained_variance_ratio_, marker='o')
plt.title("Scree Plot")
plt.xlabel("Principal Component")
plt.ylabel("Variance Explained Ratio")
plt.show()

A scree plot may have an elbow like the plot below.

6.6.1 Hands-on coding

  • Perform PCA on a dataset of US Arrests

  • Simple method first

!pip install pandas numpy scikit-learn seaborn matplotlib
  • Load libraries and data
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import matplotlib.pyplot as plt
import pandas as pd

# Load the US Arrests data
# Read the USArrests data directly from the GitHub raw URL
url = "https://raw.githubusercontent.com/cambiotraining/ml-unsupervised/main/course_files/data/USArrests.csv"
X = pd.read_csv(url, index_col=0)

# alternatively if you have downloaded the data folder 
#   on your computer try the following
# import os
# os.getcwd()
# os.chdir("data")
# X = pd.read_csv("USArrests.csv", index_col=0)

# what is in the data?
X.head()
Murder Assault UrbanPop Rape
State
Alabama 13.2 236 58 21.2
Alaska 10.0 263 48 44.5
Arizona 8.1 294 80 31.0
Arkansas 8.8 190 50 19.5
California 9.0 276 91 40.6
  • Normalize the data
scaler_standard = StandardScaler()
X_scaled = scaler_standard.fit_transform(X)
  • Perform PCA
pca_fn = PCA()
X_pca = pca_fn.fit_transform(X_scaled)
  • Plotting
plt.figure()
plt.scatter(X_pca[:,0], X_pca[:,1])
plt.xlabel("PC1")
plt.ylabel("PC2")
plt.title("PCA on crime data")
plt.show()

  • Label the plot by US State
# States come from the index
X.index

states = X.index # fetch states and assign it to a variable

# map each state to a code
colour_codes_states = pd.Categorical(states).codes

# pd.Categorical(states): Converts the sequence states (e.g., a list/Index of state names) into a categorical type. It internally builds:
# categories: the unique labels (e.g., all distinct state names)
# codes: integer labels pointing to those categories

plt.figure()
plt.scatter(X_pca[:,0], X_pca[:,1], c = colour_codes_states)
plt.xlabel("PC1")
plt.ylabel("PC2")
plt.title("PCA on crime data (coloured by US state)")
plt.show()

  • A method to have text labels in the PCA plot

We need slightly more complex code to do this.

#loadings = pca.components_.T * np.sqrt(pca.explained_variance_)  # variable vectors

# Plot
fig, ax = plt.subplots()

# Scatter of states
ax.scatter(X_pca[:, 0], X_pca[:, 1], alpha=0.7)

# Label each state
# X.index has the state names
# go through each point (which is each row in the table)
for i, state in enumerate(X.index):
    ax.text(X_pca[i, 0], X_pca[i, 1], state, fontsize=8, va="center", ha="left")

ax.set_xlabel("PC1")
ax.set_ylabel("PC2")
ax.set_title("PCA Biplot: US Arrests")

plt.tight_layout()
plt.show()

A simple PCA plot with text labels
  • Get the loadings
# get the loadings
# pca.components_ contains the principal component vectors
# transpose them using T
loadings = pca_fn.components_.T

# create a data frame
df_loadings = pd.DataFrame(loadings,
                            index=X.columns
)

# the first column is PC1, then PC2, and so on ...
print(df_loadings)
                 0         1         2         3
Murder    0.533785 -0.428765 -0.331927 -0.648891
Assault   0.583489 -0.190485 -0.267593  0.742732
UrbanPop  0.284213  0.865950 -0.386784 -0.140542
Rape      0.542068  0.173225  0.817690 -0.086823

6.6.2 Interpreting the US Crime PCA biplot

Here is an intutive explanation of the PCA biplot.

Tip

NOTE (IMPORTANT CONCEPT):

  • Distances matter: Points that are far apart represent states with more dissimilar overall crime/urbanization profiles. For example, Vermont being far from California indicates very different feature patterns in the variables used (e.g., assault, murder, urban population).

  • PC1 (horizontal) ≈ Crime level/severity: Higher values indicate greater overall crime intensity (e.g., higher assault/murder rates), lower values indicate lower crime intensity.

  • PC2 (vertical) ≈ Urbanization: States to the right tend to have higher urban population and associated traits; those to the left are more rural.

  • You can inspect the loadings to understand what each principal component represents.

  • We will have an exercise on this later.

  • Reading clusters: States that cluster together have similar profiles. States on opposite sides of the plot (e.g., Vermont vs. California) differ substantially along the dominant patterns captured by PC1 and PC2.

  • Interpretation of loadings:

  • PC1 (Urbanization axis): All crime variables (Murder, Assault, Rape) load positively, while UrbanPop has a smaller positive loading. This suggests PC1 captures overall crime levels.

Tip

NOTE (IMPORTANT CONCEPT):

  • Notice, that we have not told PCA anything about the US states

  • Yet it is still able to find some interesting patterns in the data

  • This is the strength of unsupervised machine learning

  • Another method using the pca package; prettier plots

Install the pca Python package

!pip install pca
  • Load data
from pca import pca
import pandas as pd

# Load the US Arrests data
# Read the USArrests data directly from the GitHub raw URL
url = "https://raw.githubusercontent.com/cambiotraining/ml-unsupervised/main/course_files/data/USArrests.csv"
df = pd.read_csv(url, index_col=0)

print("US Arrests Data (first 5 rows):")
print(df.head())
print("\nData shape:", df.shape)
US Arrests Data (first 5 rows):
            Murder  Assault  UrbanPop  Rape
State                                      
Alabama       13.2      236        58  21.2
Alaska        10.0      263        48  44.5
Arizona        8.1      294        80  31.0
Arkansas       8.8      190        50  19.5
California     9.0      276        91  40.6

Data shape: (48, 4)
  • Normalize the data and perform PCA
model = pca(normalize=True)
out = model.fit_transform(df)
ax = model.biplot()

  • Variance explained plots
model.plot()
(<Figure size 1440x960 with 1 Axes>,
 <Axes: title={'center': 'Cumulative explained variance\n 3 Principal Components explain [100.0%] of the variance.'}, xlabel='Principal Component', ylabel='Percentage explained variance'>)

  • 3D PCA biplots
model.biplot3d()
(<Figure size 3000x2500 with 1 Axes>,
 <Axes3D: title={'center': '3 Principal Components explain [100.0%] of the variance'}, xlabel='PC1 (61.6% expl.var)', ylabel='PC2 (24.7% expl.var)', zlabel='PC3 (9.14% expl.var)'>)

  • Loadings

Recall

What is being plotted on the axes (PC1 and PC2) are the scores.

The scores for each principal component are calculated as follows:

\[ PC_{1} = \alpha X + \beta Y + \gamma Z + .... \]

where \(X\), \(Y\) and \(Z\) are the normalized features.

The constants \(\alpha\), \(\beta\), \(\gamma\) are determined by the PCA algorithm. They are called the loadings.

print(model.results)
{'loadings':        Murder   Assault  UrbanPop      Rape
PC1  0.533785  0.583489  0.284213  0.542068
PC2 -0.428765 -0.190485  0.865950  0.173225
PC3 -0.331927 -0.267593 -0.386784  0.817690, 'PC':                      PC1       PC2       PC3
Alabama         0.923886 -1.127792 -0.437720
Alaska          1.884005 -1.032585  2.032973
Arizona         1.705462  0.730059  0.043498
Arkansas       -0.198714 -1.092074  0.111217
California      2.462479  1.513698  0.585558
Colorado        1.453427  0.982671  1.080932
Connecticut    -1.406810  1.081895 -0.661238
Delaware       -0.003621  0.319738 -0.730442
Florida         2.947649 -0.070435 -0.569823
Georgia         1.571384 -1.281416 -0.326932
Hawaii         -0.966398  1.557165  0.034386
Idaho          -1.689257 -0.178154  0.241665
Illinois        1.320695  0.653978 -0.681444
Indiana        -0.561650  0.161720  0.218372
Iowa           -2.302281  0.133259  0.145716
Kansas         -0.850716  0.279295  0.013602
Kentucky       -0.808869 -0.934920 -0.029023
Louisiana       1.500981 -0.882536 -0.772483
Maine          -2.444195 -0.340245 -0.083049
Maryland        1.702710 -0.431039 -0.158134
Massachusetts  -0.536401  1.454143 -0.626920
Michigan        2.044350  0.144860  0.383014
Minnesota      -1.742422  0.647555  0.133541
Mississippi     0.932617 -2.374555 -0.724196
Missouri        0.637255  0.263934  0.369919
Montana        -1.239466 -0.507562  0.236769
Nebraska       -1.317489  0.212450  0.160150
Nevada          2.806905  0.760007  1.157898
New Hampshire  -2.431886  0.048021  0.018380
New Jersey      0.127587  1.417883 -0.775421
New Mexico      1.917815 -0.148279  0.181459
New York        1.623118  0.790157 -0.646164
North Carolina  1.064086 -2.207350 -0.854340
North Dakota   -3.038797 -0.548177  0.281399
Ohio           -0.281823  0.736114 -0.041732
Oklahoma       -0.366423  0.292555 -0.026415
Oregon          0.003276  0.556212  0.921912
Pennsylvania   -0.941353  0.568486 -0.411608
Rhode Island   -0.909909  1.464948 -1.387731
South Carolina  1.257310 -1.914756 -0.290121
South Dakota   -2.038884 -0.778125  0.375435
Tennessee       0.935690 -0.851392  0.192734
Texas           1.293269  0.387317 -0.490484
Utah           -0.602262  1.466342  0.271830
Vermont        -2.851337 -1.332665  0.825094
Virginia       -0.153441 -0.190521  0.005751
Washington     -0.270617  0.975724  0.604878
West Virginia  -2.160933 -1.375609  0.097337, 'explained_var': array([0.61629429, 0.86387677, 0.95532444, 1.        ]), 'variance_ratio': array([0.61629429, 0.24758248, 0.09144767, 0.04467556]), 'model': PCA(n_components=np.int64(3)), 'scaler': StandardScaler(), 'pcp': np.float64(1.0000000000000002), 'topfeat':     PC   feature   loading  type
0  PC1   Assault  0.583489  best
1  PC2  UrbanPop  0.865950  best
2  PC3      Rape  0.817690  best
3  PC1    Murder  0.533785  weak, 'outliers':                  y_proba     p_raw    y_score  y_bool  y_bool_spe  y_score_spe
Alabama         0.883607  0.572223   4.780770   False       False     1.457903
Alaska          0.771864  0.061516  12.020355   False       False     2.148419
Arizona         0.883607  0.487777   5.447889   False       False     1.855152
Arkansas        0.994964  0.849865   2.662427   False       False     1.110005
California      0.771864  0.073767  11.512662   False       False     2.890516
Colorado        0.883607  0.291940   7.323773   False       False     1.754449
Connecticut     0.883607  0.376298   6.434671   False       False     1.774715
Delaware        0.997194  0.934870   1.827387   False       False     0.319759
Florida         0.827974  0.105185  10.498050   False       False     2.948491
Georgia         0.883607  0.334106   6.858767   False       False     2.027628
Hawaii          0.883607  0.482129   5.494442   False       False     1.832672
Idaho           0.883607  0.589071   4.652640   False       False     1.698626
Illinois        0.883607  0.518417   5.200102   False       False     1.473744
Indiana         0.997522  0.957121   1.535211   False       False     0.584469
Iowa            0.883607  0.341171   6.785183   False       False     2.306135
Kansas          0.997194  0.915334   2.046914   False       False     0.895390
Kentucky        0.942972  0.766164   3.332046   False       False     1.236262
Louisiana       0.883607  0.372576   6.470687   False       False     1.741210
Maine           0.883607  0.264681   7.652531   False       False     2.467763
Maryland        0.883607  0.543590   5.001739   False       False     1.756421
Massachusetts   0.883607  0.512543   5.247023   False       False     1.549922
Michigan        0.883607  0.406682   6.149239   False       False     2.049476
Minnesota       0.883607  0.477475   5.533020   False       False     1.858861
Mississippi     0.827974  0.134373   9.776761   False       False     2.551134
Missouri        0.997194  0.908446   2.118885   False       False     0.689750
Montana         0.942972  0.701130   3.819185   False       False     1.339364
Nebraska        0.942972  0.758317   3.391711   False       False     1.334509
Nevada          0.771864  0.052403  12.462954   False       False     2.907976
New Hampshire   0.883607  0.317979   7.031119   False       False     2.432360
New Jersey      0.883607  0.577856   4.737795   False       False     1.423612
New Mexico      0.883607  0.502690   5.326337   False       False     1.923539
New York        0.883607  0.382425   6.375898   False       False     1.805231
North Carolina  0.827974  0.137996   9.697217   False       False     2.450444
North Dakota    0.771864  0.080403  11.269266   False       False     3.087845
Ohio            0.997194  0.934716   1.829224   False       False     0.788218
Oklahoma        0.997522  0.982272   1.082928   False       False     0.468886
Oregon          0.994964  0.843369   2.717557   False       False     0.556221
Pennsylvania    0.942972  0.749878   3.455517   False       False     1.099691
Rhode Island    0.883607  0.234460   8.050041   False       False     1.724530
South Carolina  0.883607  0.243404   7.928295   False       False     2.290659
South Dakota    0.883607  0.291991   7.323178   False       False     2.182321
Tennessee       0.942972  0.719456   3.683208   False       False     1.265063
Texas           0.942972  0.649265   4.202711   False       False     1.350022
Utah            0.883607  0.576700   4.746601   False       False     1.585206
Vermont         0.771864  0.038043  13.332939   False       False     3.147398
Virginia        0.997522  0.997522   0.524914   False       False     0.244627
Washington      0.942972  0.761722   3.365861   False       False     1.012557
West Virginia   0.883607  0.177782   8.926009   False       False     2.561626, 'outliers_params': {'paramT2': (np.float64(-2.4671622769447922e-17), np.float64(1.273765915366402)), 'paramSPE': (array([-9.25185854e-17, -1.38777878e-17]), array([[2.51762774e+00, 6.29946348e-17],
       [6.29946348e-17, 1.01140077e+00]]))}}

6.7 Exercise for normalization in PCA

ExerciseExercise 2 - exercise_pca_normalization

Level:

Work in a group.

  • Try the same code above but now without normalisation.

  • What differences do you observe in PCA with and without normalization?

6.8 Exercise (theoretical)

ExerciseExercise 3 - exercise_theoretical

Level:

Break up into groups and discuss the following problem:

  1. Shown are biological samples with scores

  2. The features are genes

  • Why are Sample 33 and Sample 24 separated from the rest? What can we say about Gene1, Gene 2, Gene 3 and Gene 4?

  • Why is Sample 2 separated from the rest? What can we say about Gene1, Gene 2, Gene 3 and Gene 4?

  • Can we treat Sample 2 as an outlier? Why or why not? Argue your case.

The PCA biplot is shown below:

The table of loadings is shown below:

            PC1       PC2       PC3       PC4
Gene1 -0.535899  0.418181 -0.341233  0.649228
Gene2 -0.583184  0.187986 -0.268148 -0.743075
Gene3 -0.278191 -0.872806 -0.378016  0.133877
Gene4 -0.543432 -0.167319  0.817778  0.089024

6.9 🧠 PCA vs. Other Techniques

  • PCA is unsupervised (no labels used)

  • Works best for linear relationships

  • Alternatives:

    • t-SNE for nonlinear structures (we will encounter this in the next session)

6.10 🧬 In Practice: Tips for Biologists

  • Always standardize data before PCA
  • Be cautious interpreting PCs biologically: PCs are mathematical constructs
TipSummary
  • Need to normalize data before doing dimensionality reduction
  • PCA reduces dimensionality for visualization.
  • Clustering algorithms finds clusters in unlabeled data.
  • The goal of unsupervised learning is to find patterns and form hypotheses.

6.11 Resources

[1] Article on normalization on Wikipedia

[2] ISLP book

[3] Video lectures by the authors of the book Introduction to Statistical Learning in Python

[4] Visual explanations of machine learning algorithms