PCA | What Is Principal Component Analysis & How It Works? (Updated 2024)

Introduction

Too much of anything is good for nothing!

Picture this – you are working on a large-scale data science project. What happens when the given data set has too many variables? There are a few possible situations that you might come across. For instance, you find that most of the variables are correlated on analysis, and you become indecisive about what to do; hence you lose patience and decide to run a model on the whole data. This returns poor accuracy, and you feel terrible and start thinking of some strategic method to find a few important variables. That’s where Principal Component Analysis (PCA) is used.

Trust me, dealing with such situations isn’t as difficult as it sounds. This is the most common scenario in machine learning projects. Statistical techniques such as factor analysis and principal component analysis (PCA) help to overcome such difficulties. In this post, I’ve explained the concept of PCA. I’ve kept the explanation to be simple and informative. I’ve also demonstrated using this technique in R with interpretations for practical understanding.

Learning Objectives

• Step by step, you will learn the widely used technique of dimension reduction, which is Principal Component Analysis (PCA).
• You will also learn how to extract the important factors from the data with the help of PCA.
• At last, you will learn the Implementation of PCA in both R and Python.

What Is Principal Component Analysis?

Principal Component Analysis (PCA) is a powerful technique used in data analysis, particularly for reducing the dimensionality of datasets while preserving crucial information. It does this by transforming the original variables into a set of new, uncorrelated variables called principal components. Here’s a breakdown of PCA’s key aspects:

• Dimensionality Reduction: PCA helps manage high-dimensional datasets by extracting essential information and discarding less relevant features, simplifying analysis.
• Data Exploration and Visualization: It plays a significant role in data exploration and visualization, aiding in uncovering hidden patterns and insights.
• Linear Transformation: PCA performs a linear transformation of data, seeking directions of maximum variance.
• Feature Selection: Principal components are ranked by the variance they explain, allowing for effective feature selection.
• Data Compression: PCA can compress data while preserving most of the original information.
• Clustering and Classification: It finds applications in clustering and classification tasks by reducing noise and highlighting underlying structure.
• Advantages: PCA offers linearity, computational efficiency, and scalability for large datasets.
• Limitations: It assumes data normality and linearity and may lead to information loss.
• Matrix Requirements: PCA works with symmetric correlation or covariance matrices and requires numeric, standardized data.
• Eigenvalues and Eigenvectors: Eigenvalues represent variance magnitude, and eigenvectors indicate variance direction.
• Number of Components: The number of principal components chosen determines the number of eigenvectors computed.

PCA Example

Let’s say we have a data set of dimension 300 (n) × 50 (p). n represents the number of observations, and p represents the number of predictors. Since we have a large p = 50, there can be p(p-1)/2 scatter plots, i.e., more than 1000 plots possible to analyze the variable relationship. Wouldn’t it be a tedious job to perform exploratory analysis on this data?

In this case, it would be a lucid approach to select a subset of p (p << 50) predictor which captures so much information, followed by plotting the observation in the resultant low-dimensional space.

The image below shows the transformation of high-dimensional data (3 dimension) to low-dimensional data (2 dimension) using PCA. Not to forget, each resultant dimension is a linear combination of p features

What Are Principal Components?

A principal component (PCA) is a normalized linear combination of the original features in a data set. In the image above, PC1 and PC2 are the principal components. Let’s say we have a set of predictors as X¹, X²...,Xp

The principal component can be written as:

Z¹ = Φ¹¹X¹ + Φ²¹X² + Φ³¹X³ + .... +Φ^p¹Xp

where,

• Z¹ is the first principal component
• Φp¹ is the loading vector comprising loadings (Φ¹, Φ²..) of the first principal component. The loadings are constrained to a sum of squares equals to 1. This is because a large magnitude of loadings may lead to a large variance. It also defines the direction of the principal component (Z¹), along which data varies the most. It results in a line in p dimensional space, which is closest to the n observations. Closeness is measured using average squared euclidean distance.
• X¹..Xp are normalized predictors. Normalized predictors have mean values equal to zero and standard deviations equal to one.

First Principal Component

The first principal component is a linear combination of original predictor variables that captures the data set’s maximum variance. It determines the direction of highest variability in the data. Larger the variability captured in the first component, larger the information captured by component. No other component can have variability higher than first principal component.

The first principal component results in a line that is closest to the data, i.e., it minimizes the sum of squared distance between a data point and the line.

Similarly, we can compute the second principal component also.

Second Principal Component (Z²)

The second principal component is also a linear combination of original predictors, which captures the remaining variance in the data set and is uncorrelated with Z¹. In other words, the correlation between first and second components should be zero. It can be represented as:

Z² = Φ¹²X¹ + Φ²²X² + Φ³²X³ + .... + Φp2Xp

If the two components are uncorrelated, their directions should be orthogonal (image below). This image is based on simulated data with 2 predictors. Notice the direction of the components; as expected, they are orthogonal. This suggests the correlation b/w these components is zero.

All succeeding principal component follows a similar concept, i.e., they capture the remaining variation without being correlated with the previous component. In general, for n × p dimensional data, min(n-1, p) principal component can be constructed.

The directions of these components are identified unsupervised; i.e., the response variable(Y) is not used to determine the component direction. Therefore, it is an unsupervised approach.

Note: Partial least square (PLS) is a supervised alternative to PCA. PLS assigns a higher weight to variables that are strongly related to response variable to determine principal components.

How Principal Component Analysis (PCA) Work ?

1. Standardize the Data:

  – If the features of your dataset are on different scales, it’s essential to standardize them (subtract the mean and divide by the standard deviation).

2. Compute the Covariance Matrix:

  – Calculate the covariance matrix for the standardized dataset.

3. Compute Eigenvectors and Eigenvalues:

  – Find the eigenvectors and eigenvalues of the covariance matrix. The eigenvectors represent the directions of maximum variance, and the corresponding eigenvalues indicate the magnitude of variance along those directions.

4. Sort Eigenvectors by Eigenvalues:

  – Sort the eigenvectors based on their corresponding eigenvalues in descending order.

5. Choose Principal Components:

  – Select the top k eigenvectors (principal components) where k is the desired dimensionality of the reduced dataset.

6. Transform the Data:

  – Multiply the original standardized data by the selected principal components to obtain the new, lower-dimensional representation of the data.

PCA Implementation in Python

import numpy as np
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import matplotlib.pyplot as plt

# Example Data
np.random.seed(42)
X = np.random.rand(100, 3)  # 100 samples with 3 features

# Step 1: Standardize the Data
scaler = StandardScaler()
X_std = scaler.fit_transform(X)

# Step 2-5: PCA
pca = PCA()
X_pca = pca.fit_transform(X_std)

# Plot Explained Variance Ratio
explained_var_ratio = pca.explained_variance_ratio_
cumulative_var_ratio = np.cumsum(explained_var_ratio)

plt.plot(range(1, len(cumulative_var_ratio) + 1), cumulative_var_ratio, marker='o')
plt.xlabel('Number of Principal Components')
plt.ylabel('Cumulative Explained Variance Ratio')
plt.title('Explained Variance Ratio vs. Number of Principal Components')
plt.show()

Why Is Normalization of Variables Necessary in Principal Component Analysis (PCA)?

The principal components are supplied with a normalized version of the original predictors. This is because the original predictors may have different scales. For example: Imagine a data set with variables measuring units as gallons, kilometers, light years, etc. The scale of variances in these variables will obviously be large.

Performing PCA on un-normalized variables will lead to exponentially large loadings for variables with high variance. In turn, this will lead to the dependence of a principal component on the variable with high variance. This is undesirable.

As shown in the image below, PCA was run on a data set twice (with unscaled and scaled predictors). This data set has ~40 variables. You can see a variable Item_MRP dominates first principal component and a variable Item_Weight dominates the second principal component. This domination prevails due to high value of variance associated with a variable. When the variables are scaled, we get a much better representation of variables in 2D space.

Implement PCA in R & Python (With Interpretation)

How many principal components to choose from the original dataset? I could dive deep into theory, but it would be better to answer these questions practically.

For this demonstration, I’ll be using the data set from Big Mart Prediction Challenge III.

Remember, Principal Component Analysis can be applied only to numerical data. Therefore, if the data have categorical variables, they must be converted to numerical ones. Also, make sure you have done the basic data cleaning prior to implementing this technique.

Data Loading and Cleaning

Let’s quickly finish with initial data loading and cleaning steps:

#directory path
 > path <- ".../Data/Big_Mart_Sales"
#set working directory
 > setwd(path)
#load train and test file
 > train <- read.csv("train_Big.csv")
 > test <- read.csv("test_Big.csv")
#add a column
 > test$Item_Outlet_Sales <- 1
#combine the data set
 > combi <- rbind(train, test)
#impute missing values with median
 > combi$Item_Weight[is.na(combi$Item_Weight)] <- median(combi$Item_Weight, na.rm = TRUE)
#impute 0 with median
 > combi$Item_Visibility <- ifelse(combi$Item_Visibility == 0, median(combi$Item_Visibility),                                   combi$Item_Visibility)
#find mode and impute
 > table(combi$Outlet_Size, combi$Outlet_Type)
 > levels(combi$Outlet_Size)[1] <- "Other"

Removing the Dependent and Other Identifier Variables

Till here, we’ve imputed missing values. Now we are left with removing the dependent (response) variable and other identifier variables( if any). As we said above, we are practicing an unsupervised learning technique; hence response variable must be removed.

#remove the dependent and identifier variables

 > my_data <- subset(combi, select = -c(Item_Outlet_Sales, Item_Identifier,                                       Outlet_Identifier))

Let’s check the available variables ( a.k.a predictors) in the data set.

#check available variables

 > colnames(my_data)

Since PCA works on numeric variables, let’s see if we have any variables other than numeric.

#check variable class

 > str(my_data)

'data.frame': 14204 obs. of 9 variables:

 $ Item_Weight : num 9.3 5.92 17.5 19.2 8.93 ...

 $ Item_Fat_Content : Factor w/ 5 levels "LF","low fat",..: 3 5 3 5 3 5 5 3 5 5 ...

 $ Item_Visibility : num 0.016 0.0193 0.0168 0.054 0.054 ...

 $ Item_Type : Factor w/ 16 levels "Baking Goods",..: 5 15 11 7 10 1 14 14 6 6 ...

 $ Item_MRP : num 249.8 48.3 141.6 182.1 53.9 ...

 $ Outlet_Establishment_Year: int 1999 2009 1999 1998 1987 2009 1987 1985 2002 2007 ...

 $ Outlet_Size : Factor w/ 4 levels "Other","High",..: 3 3 3 1 2 3 2 3 1 1 ...

 $ Outlet_Location_Type : Factor w/ 3 levels "Tier 1","Tier 2",..: 1 3 1 3 3 3 3 3 2 2 ...

 $ Outlet_Type : Factor w/ 4 levels "Grocery Store",..: 2 3 2 1 2 3 2 4 2 2 ...

Hot Encoding

Sadly, 6 out of 9 variables are categorical in nature. We have some additional work to do now. We’ll convert these categorical variables into numeric ones using one hot encoding.

#load library

> library(dummies)

#create a dummy data frame

> new_my_data <- dummy.data.frame(my_data, names = c("Item_Fat_Content","Item_Type",

                                "Outlet_Establishment_Year","Outlet_Size",

                                "Outlet_Location_Type","Outlet_Type"))

To check if we now have a data set of integer values, simply write:

#check the data set
> str(new_my_data)

Divide Data in Test and Train

And we now have all the numerical values. Let’s divide the data into test and train.

#divide the new data

> pca.train <- new_my_data[1:nrow(train),]

> pca.test <- new_my_data[-(1:nrow(train)),]

We can now go ahead with PCA.

The base R function prcomp() is used to perform PCA. By default, it centers the variable to have a mean equal to zero. With parameter scale. = T, we normalize the variables to have a standard deviation equal to 1.

#principal component analysis

 > prin_comp <- prcomp(pca.train, scale. = T)

 > names(prin_comp)

 [1] "sdev"     "rotation" "center"   "scale"    "x"

The prcomp() Function Results in 5 Useful Measures

1. center and scale

These refers to respective mean and standard deviation of the variables that are used for normalization prior to implementing PCA

#outputs the mean of variables

 prin_comp$center

#outputs the standard deviation of variables

 prin_comp$scale

2. rocation

The rotation measure provides the principal component loading. Each column of rotation matrix contains the principal component loading vector. This is the most important measure we should be interested in.

> prin_comp$rotation

This returns 44 principal component loadings. Is that correct? Absolutely. The maximum number of principal component loadings in a data set is a minimum of (n-1, p). Let’s look at the first 4 principal components and first 5 rows.

> prin_comp$rotation[1:5,1:4]

                                PC1            PC2            PC3             PC4

Item_Weight                0.0054429225   -0.001285666   0.011246194   0.011887106

Item_Fat_ContentLF        -0.0021983314    0.003768557  -0.009790094  -0.016789483

Item_Fat_Contentlow fat   -0.0019042710    0.001866905  -0.003066415  -0.018396143

Item_Fat_ContentLow Fat    0.0027936467   -0.002234328   0.028309811   0.056822747

Item_Fat_Contentreg        0.0002936319    0.001120931   0.009033254  -0.001026615

3. Principal Component Score

To compute the principal component score vector, we don’t need to multiply the loading with data. Rather, the matrix x has the principal component score vectors in an 8523 × 44 dimension.

> dim(prin_comp$x) 
[1] 8523    44

Let’s plot the resultant principal components.

The parameter scale = 0ensures that arrows are scaled to represent the loadings. To infer from the image above, focus on this graph’s extreme ends (top, bottom, left, right).

We infer that the first principal component corresponds to Outlet_TypeSupermarket, Outlet_Establishment_Year 2007. Similarly, it can be said that the second component corresponds to a measure of Outlet_Location_TypeTier1, Outlet_Sizeother. For the exact measure of a variable in a component, you should look at rotation matrix(above) again.

4. prcomp()

This function also provides the facility to compute standard deviation of each principal component. sdev refers to the standard deviation of principal components.

#compute standard deviation of each principal component
> std_dev <- prin_comp$sdev
#compute variance
> pr_var <- std_dev^2
#check variance of first 10 components
> pr_var[1:10]
[1] 4.563615 3.217702 2.744726 2.541091 2.198152 2.015320 1.932076 1.256831
[9] 1.203791 1.168101

We aim to find the components which explain the maximum variance. This is because, we want to retain as much information as possible using these components. So, higher is the explained variance, higher will be the information contained in those components.

5. Sum total of Variance

To compute the proportion of variance explained by each component, we simply divide the variance by sum of total variance. This results in:

#proportion of variance explained

 > prop_varex <- pr_var/sum(pr_var)

 > prop_varex[1:20]

 [1] 0.10371853 0.07312958 0.06238014 0.05775207 0.04995800 0.04580274

 [7] 0.04391081 0.02856433 0.02735888 0.02654774 0.02559876 0.02556797

 [13] 0.02549516 0.02508831 0.02493932 0.02490938 0.02468313 0.02446016

 [19] 0.02390367 0.02371118

This shows that first principal component explains 10.3% variance. Second component explains 7.3% variance. Third component explains 6.2% variance and so on. So, how do we decide how many components should we select for modeling stage ?

Screen Plot

The answer to this question is provided by a scree plot. A scree plot is used to access components or factors which explains the most of variability in the data. It represents values in descending order.

#scree plot
 
> plot(prop_varex, xlab = "Principal Component",

             ylab = "Proportion of Variance Explained",

             type = "b")

The plot above shows that ~ 30 components explains around 98.4% variance in the data set. In order words, using PCA we have reduced 44 predictors to 30 without compromising on explained variance. This is the power of PCA> Let’s do a confirmation check, by plotting a cumulative variance plot. This will give us a clear picture of number of components.

#cumulative scree plot
> plot(cumsum(prop_varex), xlab = "Principal Component",
              ylab = "Cumulative Proportion of Variance Explained",
              type = "b")

This plot shows that 30 components result in a variance close to ~ 98%. Therefore, in this case, we’ll select the number of components as 30 [PC1 to PC30] and proceed to the modeling stage. This completes the steps to implement PCA on train data. For modeling, we’ll use these 30 components as predictor variables and follow the normal procedures.

This plot shows that 30 components result in a variance close to ~ 98%. Therefore, in this case, we’ll select the number of components as 30 [PC1 to PC30] and proceed to the modeling stage. This completes the steps to implement PCA on train data. For modeling, we’ll use these 30 components as predictor variables and follow the normal procedures.

Predictive Modeling With PCA Components

Principal Component Analysis (PCA) is used along with machine learning algorithms for predictive modeling.

After we’ve performed PCA on the training set, let’s now understand the process of predicting test data using these components. The process is simple. Just like we’ve obtained PCA components on the training set, we’ll get another bunch of components on the testing set. Finally, we train the model.

But, few important points to understand:

1. We should not combine the train and test set to obtain PCA components of the whole data at once, as this would violate the assumption of generalization since the test data would get ‘leaked’ into the training set. In other words, the test data set would no longer remain ‘unseen’. Eventually, this will hammer down the generalization capability of the model.
2. We should not perform PCA on test and train data sets separately because the resultant vectors from the train and test PCAs will have different directions (due to unequal variance). Due to this, we’ll end up comparing data registered on different axes. Therefore, the resulting train and test data vectors should have same axes.

So, what should we do?

We should do exactly the same transformation to the test set as we did to the training set, including the center and scaling feature. Let’s do it in R:

#add a training set with principal components
> train.data <- data.frame(Item_Outlet_Sales = train$Item_Outlet_Sales, prin_comp$x)
#we are interested in first 30 PCAs
> train.data <- train.data[,1:31]
#run a decision tree
> install.packages("rpart")
> library(rpart)
> rpart.model <- rpart(Item_Outlet_Sales ~ .,data = train.data, method = "anova")
> rpart.model
#transform test into PCA
> test.data <- predict(prin_comp, newdata = pca.test)
> test.data <- as.data.frame(test.data)
#select the first 30 components
> test.data <- test.data[,1:30]
#make prediction on test data
> rpart.prediction <- predict(rpart.model, test.data)
#For fun, finally check your score of leaderboard
> sample <- read.csv("SampleSubmission_TmnO39y.csv")
> final.sub <- data.frame(Item_Identifier = sample$Item_Identifier, Outlet_Identifier = sample$Outlet_Identifier, Item_Outlet_Sales = rpart.prediction)
> write.csv(final.sub, "pca.csv",row.names = F)

That’s the complete modeling process after PCA extraction. I’m sure you wouldn’t be happy with your leaderboard rank after you upload the solution. Try using random forest!

For Python Users:
To implement PCA in python, import PCA from sklearn library. The interpretation remains same as explained for R users above. Of course, the result is some as derived after using R. The data set used for Python is a cleaned version where missing values have been imputed, and categorical variables are converted into numeric. The modeling process remains same, as explained for R users above.

import numpy as np
 from sklearn.decomposition import PCA
 import pandas as pd
 import matplotlib.pyplot as plt
 from sklearn.preprocessing import scale
 %matplotlib inline
#Load data set
 data = pd.read_csv('Big_Mart_PCA.csv')
#convert it to numpy arrays
 X=data.values
#Scaling the values
 X = scale(X)
pca = PCA(n_components=44)
pca.fit(X)
#The amount of variance that each PC explains
 var= pca.explained_variance_ratio_
#Cumulative Variance explains
 var1=np.cumsum(np.round(pca.explained_variance_ratio_, decimals=4)*100)
print var1
 [  10.37   17.68   23.92   29.7    34.7    39.28   43.67   46.53   49.27
 51.92   54.48   57.04   59.59   62.1    64.59   67.08   69.55   72.
 74.39   76.76   79.1    81.44   83.77   86.06   88.33   90.59   92.7
 94.76   96.78   98.44  100.01  100.01  100.01  100.01  100.01  100.01
 100.01  100.01  100.01  100.01  100.01  100.01  100.01  100.01]
plt.plot(var1)

#Looking at above plot I'm taking 30 variables

 pca = PCA(n_components=30)

 pca.fit(X)

 X1=pca.fit_transform(X)

print X1

For more information on PCA in python, visit scikit learn documentation.

Conclusion

This brings me to the end of this Principal Component Analysis tutorial. Without delving deep into mathematics, I’ve tried to familiarize you with the most important concepts required to use this technique. It’s simple but needs special attention when deciding the number of components. Practically, we should strive to retain only the first few k components. The idea behind PCA is to construct some principal components (Z << Xp) that satisfactorily explain most of the data’s variability and relationship with the response variable.

This brings me to the end of this Principal Component Analysis tutorial. Without delving deep into mathematics, I’ve tried to make you familiar with the most important concepts required to use this technique. It’s simple but needs special attention when deciding the number of components. Practically, we should strive to retain only the first few k components. The idea behind PCA is to construct some principal components (Z << Xp) which satisfactorily explain most of the data’s variability and relationship with the response variable.

Key Takeaways

• Principal Component Analysis (PCA) is used to overcome feature redundancy in a data set. These features are low dimensional in nature. These features, a.k.a components, are a result of normalized linear combinations of original predictor variables.
• The first component has the highest variance, followed by second, third, and so on. The components must be uncorrelated (remember orthogonal direction ?). See above.
• Normalizing data becomes extremely important when the predictors are measured in different units.PCA works best on data sets having 3 or higher dimensions. Because, with higher dimensions, it becomes increasingly difficult to make interpretations from the resultant data cloud.

Frequently Asked Questions