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Introduction to Linear Discriminant Analysis

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Jaclyn Kokx

An introduction to using linear discriminant analysis as a dimensionality reduction technique.

Data & Analytics
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Introduction to Linear Discriminant Analysis Jaclyn Kokx
1. Who am I? 2. What is LDA? 3. Why use LDA? 4. How do I use LDA? https://memegenerator.net/instance/55587306/winter-is-coming-brace-yourself-powerpoint-slides-are-coming
Who Am I?
Jackie Kokx MS Mechanical Engineering BS Materials Science & Engineering Biotech Industry Algebra Instructor Self-Taught Data Enthusiast Ultimate Frisbee, Running, Mom www.jaclynkokx.com
What is LDA?
Linear Discriminant Analysis A supervised dimensionality reduction technique to be used with continuous independent variables and a categorical dependent variables A linear combination of features separates two or more classes Because it works with numbers and sounds science-y
More LDA Ronald Fisher Fisher’s Linear Discriminant (2-Class Method) - 1936 C R Rao Multiple Discriminant Analysis - 1948 Also used As a data classification technique http://www.swlearning.com/quant/kohler/stat/biographical_sketches/Fisher_3.jpeg http://www.buffalo.edu/ubreporter/archive/2011_07_21/rao_guy_medal.html
Why Would I Want to Use LDA?
Because real life data is ugly. Iris Dataset Your (My) Dataset
The curse of dimensionality http://www.visiondummy.com/2014/04/curse-dimensionality-affect-classification/ Reducing the number of your features might improve your classifier
How much can I reduce my number of features? One less than the number of labeled classes you have My waterpoint dataset 1 2 3 4 5 6 87 Functional Functional Repair NonFuctional LDA2 My waterpoint dataset after LDA LDA1 Functional Functional Repair NonFuctional 3 Classes - 1 = 2 features (dimensions)
How Do I Use LDA?
Due credit to Sebastian Raschka http://sebastianraschka.com/Articles/2014_python_lda .html
https://www.researchgate.net/publication/316994943_Linear_discriminant_analysis_A_d etailed_tutorial
What you need to start: ● Labeled data ● Ordinal features (the number kind, not the category kind) ● Some idea about matrix algebra ● (Python) https://www.memecenter.com/fun/330902/matrix--expectation-vs-reality
The Big Picture Create a subspace that separates our classes really well, then we’ll project our data onto that subspace. Linear Discriminant: To find that optimal subspace we’re going to: ● Minimize within-class variance (we want tight groups) ● Maximize the between-class distance (we want our groups as far apart as possible)
Let’s visualize it Iris Dataset
Another View
Let’s start the calculations
5 Steps to LDA 1) Means 2) Scatter Matrices 3) Finding Linear Discriminants 4) Subspace 5) Project Data Iris Dataset
Step 1: Means 1. Find each class mean 1. Find the overall mean (central point)
Sepal Petal Width Width 3.504 0.248 3.408 0.213 2.975 0.327 3.225 0.289 2.648 0.222 3.346 0.203 3.198 0.255 2.899 0.301 3.045 0.297 Mean s Sepal Petal Width Width µ0 µ1 µ2
5 Steps to LDA 1) Means 2) Scatter Matrices 3) Finding Linear Discriminants 4) Subspace 5) Project Data Iris Dataset
Step 2 : Scatter Matrices 2a) Between-Class Scatter Matrix (Sb) 2b) Within-Class Scatter Matrix (Sw)
2a) Between-Class Scatter Matrix µi - CP µi - CP Ni = Sample Size of Class Squared Difference M = # of Features
Between-Class Scatter Matrix µi - CP µi - CP
2b) Within-Class Scatter Matrix µi µi
Within-Class Scatter Matrix µi µi
5 Steps to LDA 1) Means 2) Scatter Matrices 3) Finding Linear Discriminants 4) Subspace 5) Project Data Iris Dataset
Step 3: Finding Linear Discriminants Finding W using eigenvectors and eigenvalues Distance
Now, remember… Iris Dataset
The Math of Finding Linear Discriminants http://research.cs.tamu.edu/prism/lectures/pr/pr_l10.pdf Working towards: We have so far: Projected means Scatter of the projection
The Math of Linear Discriminants Fisher’s Criterion Eigenvector EigenvalueMatrix
Eigenvectors and Eigenvalues using NumPy YouTube: PatrickJMT Quadratic Equation W A A
5 Steps to LDA 1) Means 2) Scatter Matrices 3) Finding Linear Discriminants 4) Subspace 5) Project Data Iris Dataset
Step 4: Subspace ● Sort our Eigenvectors by decreasing Eigenvalue ● Choose the top Eigenvectors to make your transformation matrix used to project your data Choose top (Classes - 1) Eigenvalues
5 Steps to LDA 1) Means 2) Scatter Matrices 3) Finding Linear Discriminants 4) Subspace 5) Project Data Iris Dataset
Step 5: Project Data X_lda = X.dot(W) https://memegenerator.net/instance/48422270/willy-wonka-well-now-that-was-anticlimactic
Results and scikit-learn
Disclaimer Your (My) Dataset
Thanks! www.jaclynkokx.com jaclynkokx@gmail.com @JackieKokx

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Introduction to Linear Discriminant Analysis

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