Encoders and Decoders in NLP
This article explores the concepts of encoders and decoders in Natural Language Processing (NLP), particularly within the context of Transformer architectures. It explains how encoders like BERT process input sequences to create contextual embeddings, while decoders like GPT generate text by predicting the next token based on previous context. The article also compares these components, discusses their roles in various NLP tasks, and provides a detailed look at the Transformer architecture, including self-attention mechanisms and practical examples.
Encoder and Decoder in NLP (Transformer Architecture)
In the context of Transformer models, the architecture is made of two main components:
- Encoder: Converts input tokens (e.g. words, subwords) into a continuous, context-aware representation (a sequence of embeddings).
- Decoder: Takes this encoded representation and generates output tokens, often one-by-one, conditioned on both the encoder outputs and previously generated tokens.
Encoder in BERT (Bidirectional Encoder Representations from Transformers)
- Only uses the encoder stack of the Transformer.
- Processes input sequences bidirectionally—meaning each token attends to all others simultaneously, both left and right.
- Outputs a contextual embedding for each token in the input.
-
Tasks:
- Masked Language Modeling (MLM): Predict missing tokens in input.
- Next Sentence Prediction (NSP): Predict relationship between two sentences.
Key point: BERT is optimized for understanding text.
Decoder in GPT (Generative Pretrained Transformer)
- Only uses the decoder stack of the Transformer.
- Uses causal (autoregressive) attention: each token can only attend to the left (past tokens), not future ones.
- Trained to predict the next token in a sequence, given the previous ones.
- Generates coherent sequences one token at a time, ideal for completion/generation tasks.
Key point: GPT is optimized for generating text.
Comparison: Encoder vs Decoder
| Feature | Encoder (BERT) | Decoder (GPT) |
|---|---|---|
| Attention | Bidirectional | Unidirectional (causal) |
| Primary Usage | Understanding (classification, QA) | Generation (text, code, etc.) |
| Input/Output | Input → Embedding | Embedding → Output tokens |
| Architecture | Stack of encoders | Stack of decoders |
In essence:
- BERT = Input → Encode deeply → Use for tasks
- GPT = Seed → Decode token-by-token → Output
Absolutely. Let’s go into the Transformer architecture, then walk through a concrete example using pseudo-embeddings for intuitive clarity.
🔧 Transformer: Core Architecture (Vaswani et al., 2017)
The Transformer is a neural network architecture designed for handling sequential data without relying on recurrence (like RNNs or LSTMs).
It consists of:
1. Encoder-Decoder Structure
- Encoder stack: Transforms input into a contextual representation.
- Decoder stack: Consumes encoder output + prior tokens to generate output.
Each stack is made of identical layers, and each layer contains:
2. Key Components
-
Multi-Head Self-Attention:
- Allows tokens to attend to each other (within input or output).
- Multi-head: learn different attention patterns in parallel.
-
Feedforward Neural Network (FFN):
- Fully connected layers applied to each token position.
-
Add & Norm:
- Residual connection + Layer Normalization.
🧠 Self-Attention (Core Idea)
Each token is re-encoded based on a weighted sum of all other tokens, depending on their relevance.
For token T₁, it might attend 80% to T₃, 10% to T₂, 10% to itself, etc.
These attention weights come from:
Attention(Q, K, V) = softmax((Q × Kᵗ) / √dₖ) × V
Where:
- Q, K, V are linear projections of input embeddings.
- Q = Queries, K = Keys, V = Values.
📦 Pseudo Example: Transformer Encoder Layer
Let’s say you input this sentence:
"Birds fly high"
1. Token Embeddings
Assume each word is embedded into 4-dimensional space (simplified):
Birds: [0.1, 0.3, 0.0, 0.5]
Fly: [0.2, 0.1, 0.4, 0.0]
High: [0.0, 0.4, 0.2, 0.3]
2. Positional Encoding
Added to capture order:
Birds + PE: [0.1+0.01, 0.3+0.02, 0.0+0.03, 0.5+0.04] → [0.11, 0.32, 0.03, 0.54]
Fly + PE: [0.2+0.01, 0.1+0.02, 0.4+0.03, 0.0+0.04] → [0.21, 0.12, 0.43, 0.04]
High + PE: [0.0+0.01, 0.4+0.02, 0.2+0.03, 0.3+0.04] → [0.01, 0.42, 0.23, 0.34]
3. Self-Attention
Each token attends to all others. For instance:
- “Birds” might give 50% weight to “Fly”, 30% to “High”, 20% to itself.
- The output of self-attention for “Birds” becomes a weighted combination of all 3 token embeddings.
4. Feedforward + Norm
Each result from attention goes through a small MLP and is normalized. This layer stack is repeated N times.
✅ Result
After passing through several encoder layers:
-
The embedding for each word becomes contextualized:
- “Fly” now encodes information about “Birds” and “High”.
-
These embeddings can be used for classification, Q\&A, etc.
Great — let’s now walk through the Transformer Decoder, especially in the GPT-style, using the same sentence:
“Birds fly high”
We’ll simulate how GPT might generate it step-by-step, token-by-token, using a decoder-only transformer.
🔁 Transformer Decoder: Structure
Each decoder layer contains:
-
Masked Multi-Head Self-Attention
- Only allows attending to previous tokens (causal).
- Feedforward Network
- (In full encoder-decoder models: Cross-Attention with encoder output, not used in GPT)
📚 Generation Setup
Let’s say GPT is asked to generate text from a prompt:
"Birds"
We already have the embedding for “Birds”:
Birds: [0.11, 0.32, 0.03, 0.54]
We want it to generate the rest: "fly high".
🔄 Step-by-Step Generation
Step 1: Input “Birds”
- Add positional encoding.
- Apply masked self-attention (only token “Birds”, attends to itself).
- Pass through feedforward layer → output vector for “Birds”.
-
Final layer: apply softmax over vocabulary → model predicts:
- Highest probability:
"fly"
- Highest probability:
So it outputs “fly”.
Step 2: Input “Birds fly”
- Now we embed both tokens:
Birds: [0.11, 0.32, 0.03, 0.54]
Fly: [0.21, 0.12, 0.43, 0.04]
- Apply positional encoding.
- Masked attention matrix now looks like:
attends to →
B F
Input ↓
B ✔ ✖
F ✔ ✔
So:
- “Birds” attends to itself.
- “Fly” attends to “Birds” and itself.
- This allows “Fly” to learn context: “Birds fly”
Final layer predicts:
- Highest probability:
"high"
GPT outputs “high”.
⛓ Final Output
Input → "Birds"
Generated → "fly" → then "high"
Thus, model generated:
"Birds fly high"
Each step only depends on tokens before it — that’s the key of causal decoding.
Summary: GPT-Style Decoder Flow
| Step | Input Tokens | Attention Scope | Output Prediction |
|---|---|---|---|
| 1 | Birds | Birds | Fly |
| 2 | Birds, Fly | Birds, Fly | High |
| 3 | Birds, Fly, High | Birds, Fly, High |
Certainly. Let’s break down Softmax, especially in the context of NLP and Transformers.
🔢 What is Softmax?
Softmax is a mathematical function that converts a vector of raw scores (logits) into a probability distribution over a set of possible outcomes.
It ensures:
- All output values are in the range (0, 1).
- All probabilities sum to 1.
📐 Formula
For a vector $z = [z_1, z_2, …, z_n]$, the softmax of the $i^{th}$ element is:
\[\text{softmax}(z_i) = \frac{e^{z_i}}{\sum_{j=1}^n e^{z_j}}\]This does two things:
- Exponentiates the input values to ensure they’re all positive.
- Normalizes them to sum to 1.
💡 Why Use Softmax in NLP?
In NLP models (like GPT), the output of the final layer is a vector of logits — one score per vocabulary token (e.g., 50,000+ values).
Softmax turns these logits into a probability distribution:
- Highest probability → most likely next token.
- Used during training (cross-entropy loss) and inference (token sampling).
📊 Example
Imagine the model outputs raw logits for 3 tokens:
Logits: [3.0, 1.0, 0.2]
Tokens: ["cat", "dog", "fish"]
Apply softmax:
-
Exponentiate:
\[e^3 = 20.1,\quad e^1 = 2.7,\quad e^{0.2} = 1.22\] -
Sum = 20.1 + 2.7 + 1.22 ≈ 24.02
-
Probabilities:
- “cat”: 20.1 / 24.02 ≈ 0.836
- “dog”: 2.7 / 24.02 ≈ 0.112
- “fish”: 1.22 / 24.02 ≈ 0.051
Result: “cat” has the highest probability, so it’s most likely the next token.
🧠 In Transformers
At each decoding step, GPT:
- Produces a vector of logits (size = vocab size).
- Applies softmax to get token probabilities.
-
Chooses the next token:
- Greedy: pick max probability.
- Sampling: pick based on distribution.
Great question — Autoencoders are another key concept in representation learning, distinct from Transformers but with conceptual overlap, especially in how they encode information.
Let’s go over it in a precise, technical way.
🔄 What is an Autoencoder?
An Autoencoder is a type of neural network that learns to:
- Compress input into a latent representation (encoding).
- Reconstruct the original input from that encoding (decoding).
Its architecture has two parts:
- Encoder: $x \rightarrow z$
- Decoder: $z \rightarrow \hat{x}$
The goal is to minimize the difference between $x$ and $\hat{x}$ (e.g., using MSE loss).
🔧 Architecture
Input → [Encoder] → Latent Code → [Decoder] → Output
- No attention, no autoregression.
- Pure feedforward networks (or convolutional, recurrent, etc.).
- Typically used for dimensionality reduction, denoising, or unsupervised pretraining.
📚 In NLP Context
Autoencoders in NLP are used to learn dense representations of text.
🔹 Example:
You feed in a sentence like:
"birds fly high"
Encoder compresses it into a vector, say:
[0.12, -0.43, 0.88, ..., 0.06] (size = 512 or 768)
Decoder tries to reconstruct:
"birds fly high"
🔍 Variants of Autoencoders in NLP
-
Denoising Autoencoder (used in BERT)
- Input is corrupted (e.g. tokens masked).
- Model must reconstruct the clean/original input.
- BERT’s MLM (Masked Language Modeling) task is a denoising autoencoding objective.
-
Variational Autoencoder (VAE):
- Adds probabilistic encoding.
- Used in generative models (like text generation), e.g., VAE-LSTM.
⚖️ Comparison: Autoencoder vs Transformer
| Feature | Autoencoder | Transformer (e.g., GPT/BERT) |
|---|---|---|
| Structure | Encoder + Decoder | BERT: Encoder-only, GPT: Decoder-only |
| Attention | ❌ Not used | ✅ Self-attention |
| Output | Reconstruct input | Understand (BERT) or Generate (GPT) |
| Use Case | Compression, reconstruction | Language modeling, generation |
| Training Objective | Minimize reconstruction error | MLM (BERT), Next-token prediction (GPT) |
🧠 Takeaway
- Autoencoders: Learn to compress and reconstruct. Focused on representation learning.
- Transformer-based models: Use attention for context modeling. Focused on language understanding or generation.
Certainly. Here’s a concrete autoencoder example using pseudo values, similar to what we did for Transformers. We’ll keep it numeric and hands-on to make the data flow crystal clear.
🔧 Autoencoder Pseudo Example
Let’s say we have a 4-dimensional input vector representing a sentence embedding (e.g., from “birds fly high”):
Input vector (x): [0.6, 0.1, 0.3, 0.8]
🏗️ Autoencoder Structure
- Encoder: Compresses 4D → 2D
- Decoder: Reconstructs 2D → 4D
▶️ Step 1: Encoding
We use a weight matrix and bias to compress:
Encoder weight matrix (Wₑ): 2x4
[
[0.5, 0.1, 0.2, 0.0],
[0.3, 0.4, 0.1, 0.2]
]
Bias (bₑ): [0.0, 0.1]
Latent (z) = Wₑ · x + bₑ
Compute:
z[0] = 0.5*0.6 + 0.1*0.1 + 0.2*0.3 + 0.0*0.8 = 0.3 + 0.01 + 0.06 = 0.37
z[1] = 0.3*0.6 + 0.4*0.1 + 0.1*0.3 + 0.2*0.8 = 0.18 + 0.04 + 0.03 + 0.16 = 0.41
Latent representation (z): [0.37, 0.41]
◀️ Step 2: Decoding
Decoder expands 2D → 4D:
Decoder weight matrix (W_d): 4x2
[
[0.6, 0.2],
[0.1, 0.7],
[0.4, 0.1],
[0.3, 0.5]
]
Bias (b_d): [0.0, 0.0, 0.0, 0.0]
Reconstructed x̂ = W_d · z + b_d
Compute:
x̂[0] = 0.6*0.37 + 0.2*0.41 = 0.222 + 0.082 = 0.304
x̂[1] = 0.1*0.37 + 0.7*0.41 = 0.037 + 0.287 = 0.324
x̂[2] = 0.4*0.37 + 0.1*0.41 = 0.148 + 0.041 = 0.189
x̂[3] = 0.3*0.37 + 0.5*0.41 = 0.111 + 0.205 = 0.316
Reconstructed output (x̂): [0.304, 0.324, 0.189, 0.316]
📉 Step 3: Loss
Compare original vs reconstructed:
Original x : [0.6, 0.1, 0.3, 0.8]
Reconstructed x̂: [0.304, 0.324, 0.189, 0.316]
Use Mean Squared Error (MSE):
\[\text{MSE} = \frac{1}{4} \sum (x_i - \hat{x}_i)^2 = \frac{1}{4} [(0.296)^2 + (−0.224)^2 + (0.111)^2 + (0.484)^2] ≈ \frac{1}{4} (0.0876 + 0.0502 + 0.0123 + 0.2343) ≈ 0.0961\]✅ Summary
| Component | Data |
|---|---|
| Input (x) | [0.6, 0.1, 0.3, 0.8] |
| Latent (z) | [0.37, 0.41] |
| Reconstructed | [0.304, 0.324, 0.189, 0.316] |
| Loss (MSE) | ~0.0961 |
Absolutely — autoencoders are used well beyond NLP, especially for tasks like image compression, anomaly detection, and dimensionality reduction.
Let’s look at why and walk through a concrete example using pseudo values from a non-NLP domain.
🎯 Why Use Autoencoders Outside NLP?
🔍 Core Idea: Learn a compressed representation (encoding) of high-dimensional data that retains essential information.
Common Use Cases:
| Domain | Use Case |
|---|---|
| Images | Compression, denoising, inpainting |
| Cybersecurity | Anomaly detection (e.g., unusual logins) |
| Finance | Fraud detection from transaction patterns |
| IoT/Sensors | Detect malfunctioning sensors |
| Genomics | Dimensionality reduction of DNA data |
💡 Example: Autoencoder for Anomaly Detection in Sensor Data
Imagine a system monitoring temperature readings from 4 sensors:
Normal input (x): [22.0, 21.5, 22.1, 21.8]
🏗️ Autoencoder Architecture
- Input: 4 features
- Latent dimension: 2
- Decoder reconstructs to 4D
▶️ Encoding
Encoder Weights (Wₑ): 2x4
[
[0.4, 0.3, 0.2, 0.1],
[0.1, 0.2, 0.3, 0.4]
]
x = [22.0, 21.5, 22.1, 21.8]
z[0] = 0.4*22 + 0.3*21.5 + 0.2*22.1 + 0.1*21.8 ≈ 8.8 + 6.45 + 4.42 + 2.18 = **21.85**
z[1] = 0.1*22 + 0.2*21.5 + 0.3*22.1 + 0.4*21.8 ≈ 2.2 + 4.3 + 6.63 + 8.72 = **21.85**
Latent z = [21.85, 21.85]
◀️ Decoding
Decoder Weights (W_d): 4x2
[
[0.3, 0.1],
[0.1, 0.4],
[0.2, 0.2],
[0.4, 0.3]
]
Reconstructed x̂ = W_d · z
x̂[0] = 0.3*21.85 + 0.1*21.85 = 6.555
x̂[1] = 0.1*21.85 + 0.4*21.85 = 10.925
x̂[2] = 0.2*21.85 + 0.2*21.85 = 8.74
x̂[3] = 0.4*21.85 + 0.3*21.85 = 15.295
Reconstructed x̂: [6.56, 10.93, 8.74, 15.30]
Oops! This reconstruction is far from the original — which signals something is off (possibly a bad encoding setup in this toy example or abnormal input).
🔥 Now Assume an Anomalous Input
Anomalous input x: [22.0, 60.0, 22.1, 21.8] (Sensor 2 is way off)
- Encoder → latent z changes
- Decoder reconstructs poorly
- Reconstruction error:
Original x: [22.0, 60.0, 22.1, 21.8]
Reconstructed x̂: [6.56, 10.93, 8.74, 15.30]
MSE = large → flag as anomaly ✅
✅ Summary
| Purpose | Detect when input doesn’t match learned “normal” patterns |
|---|---|
| Input | Sensor values (normal or not) |
| Output | Reconstructed values |
| Detection signal | High reconstruction error → anomaly |
Perfect — let’s compare Autoencoders vs PCA (Principal Component Analysis) in the context of a practical machine learning example like food delivery analytics.
🍕 Use Case: Food Delivery Behavior Compression
Imagine you’re analyzing customer behavior using the following 5 features:
| Feature | Example Value |
|---|---|
delivery_time_minutes |
25 |
num_items_ordered |
3 |
total_cost |
18.5 |
discount_used |
1 (yes) |
distance_km |
4.2 |
You want to reduce this to 2 dimensions for clustering, visualization, or anomaly detection.
🧠 PCA vs Autoencoder
| Aspect | PCA (Linear) | Autoencoder (Nonlinear) |
|---|---|---|
| Method | Linear projection via eigenvectors | Learns nonlinear encoding using neural net |
| Captures | Linear correlations | Nonlinear patterns |
| Reconstruction | Matrix multiplication | Neural decoder network |
| Training | Fast, no backprop | Slower, requires gradient descent |
| Interpretability | High (components = variance directions) | Low (latent vectors are abstract) |
| Flexibility | Limited to linear transforms | Arbitrary complexity, activations |
🔢 Step-by-Step Comparison Using Pseudo Values
✅ Input Vector
Example data point from a customer:
x = [25, 3, 18.5, 1, 4.2]
1️⃣ PCA
Assume PCA gives us these 2 principal components:
PC1 = [0.5, 0.2, 0.6, 0.1, 0.3]
PC2 = [0.1, 0.7, 0.2, 0.5, 0.4]
Project onto 2D:
z[0] = dot(x, PC1) = 25*0.5 + 3*0.2 + 18.5*0.6 + 1*0.1 + 4.2*0.3
= 12.5 + 0.6 + 11.1 + 0.1 + 1.26 = **25.56**
z[1] = dot(x, PC2) = 25*0.1 + 3*0.7 + 18.5*0.2 + 1*0.5 + 4.2*0.4
= 2.5 + 2.1 + 3.7 + 0.5 + 1.68 = **10.48**
🔽 Reduced vector:
[PCA result] z = [25.56, 10.48]
To reconstruct:
x̂ = z[0]*PC1 + z[1]*PC2 + mean_vector
➡️ Linear only: Can’t handle patterns like “delivery time spikes when distance > 10 km and no discount”.
2️⃣ Autoencoder
Encoder network compresses input nonlinearly to:
z = [12.3, 7.9]
Decoder reconstructs:
x̂ = [24.8, 3.1, 18.6, 1.0, 4.15]
Reconstruction is close to original → model learned nonlinear relationships:
- delivery_time might depend on a combination of distance, time of day, and discount flag.
🔍 Use Case Preference
| If You Need… | Use |
|---|---|
| Quick dimensionality reduction | PCA |
| Linear relationships | PCA |
| Visualization of high-dim data | PCA |
| Complex/nonlinear correlations | Autoencoder |
| Anomaly detection with patterns | Autoencoder |
| Integration into deep ML pipelines | Autoencoder |
✅ Summary
| Technique | Pros | Cons |
|---|---|---|
| PCA | Fast, interpretable, no training needed | Can’t capture nonlinear patterns |
| Autoencoder | Powerful, nonlinear, can stack layers | Requires training, less interpretable |
Excellent follow-up — let’s unpack what principal components are, and how we decide how many we need in Principal Component Analysis (PCA).
🧠 What Are Principal Components?
Principal Components (PCs) are new axes in feature space that:
- Are orthogonal (uncorrelated),
- Are ordered by the amount of variance they capture from the data,
- Are linear combinations of original features.
Each principal component is a vector pointing in the direction of maximum variance.
🧭 Imagine This:
In 3D space with features [x, y, z], PCA might find:
- PC1: Diagonal direction where the data varies the most
- PC2: Perpendicular to PC1, capturing next most variation
- PC3: Perpendicular to both — often not useful (low variance)
🔢 How Many Principal Components to Keep?
🧮 1. Eigenvalues Explain Variance
Each principal component has an associated eigenvalue:
- Larger eigenvalue → more variance captured
- Total variance in the dataset = sum of all eigenvalues
We typically compute variance explained:
Explained Variance Ratio = eigenvalue[i] / sum(all eigenvalues)
📊 2. Deciding How Many to Keep
✅ Common Strategies:
| Strategy | Description |
|---|---|
| Variance Threshold | Keep enough PCs to explain e.g. 95% of the variance |
| Scree Plot (“elbow method”) | Plot variance vs components, find where curve “elbows” |
| Fixed Number (heuristic) | e.g. reduce to 2D or 3D for visualization |
| Kaiser Criterion | Keep PCs with eigenvalues > 1 (standardized data only) |
✨ Example
Assume you perform PCA on a dataset with 5 features and get:
| Component | Eigenvalue | % Variance Explained |
|---|---|---|
| PC1 | 4.2 | 60% |
| PC2 | 2.3 | 20% |
| PC3 | 1.0 | 10% |
| PC4 | 0.4 | 6% |
| PC5 | 0.1 | 4% |
| Total | 8.0 | 100% |
🧠 Interpretation:
-
To explain 80% of the variance:
- PC1 + PC2 = 60% + 20% = 80% → ✅ Keep 2 components
-
Scree plot would show a sharp drop after PC2 (elbow)
✅ Summary
| Concept | Meaning |
|---|---|
| Principal Component | New axis with maximum variance |
| Number of Components | Depends on how much variance you want to keep |
| Typical Use | Dimensionality reduction, noise removal, visualization |
You’re right to observe that “type of food” and “distance” likely contribute heavily to the variance in delivery time — and that intuition is on the right track. However, in PCA, we don’t handpick features as principal components. Let me clarify:
🧠 PCA Doesn’t Choose Features — It Learns Directions
In PCA, principal components are not original features (like “distance” or “food type”) — instead, they are new axes formed by linear combinations of all original features.
For example:
PC1 = 0.6 * delivery_time + 0.3 * food_type + 0.1 * discount + ...
PC2 = -0.2 * delivery_time + 0.7 * food_type + 0.4 * distance + ...
So while “type of food” and “distance” may strongly influence the first few principal components (because they contribute most to variance), they are not PCs themselves.
✅ What You Can Say:
❗ You can’t say:
“Food type and distance are the two principal components.”
✅ You can say:
“Food type and distance likely contribute the most to the top principal components, since they introduce the most variation across deliveries.”
🔍 Why Is This Important?
PCA might discover that most variation is along a direction like:
PC1 = 0.5 * food_type + 0.5 * distance + 0.1 * num_items - 0.1 * discount
This PC1 combines multiple features, weighted by their statistical correlation with variance — which may not align with human intuition or feature boundaries.
🧪 What To Do Practically
If you want to verify your hypothesis:
- Run PCA on your food delivery dataset.
- Inspect the component loadings (weights for each original feature in each PC).
- You’ll likely see large weights for food type and distance in the first one or two PCs — confirming their influence.
⚠️ Bonus Tip
If you do want to isolate “which features matter most” for delivery time (as a target), PCA isn’t the right tool — that’s a job for:
- Linear regression / Lasso → to explain/predict
- Feature importance from a model (e.g., Random Forest)
- SHAP values or permutation importance
That’s a very fair confusion — you’re deep in the right part of the rabbit hole. Let’s break it down clearly, in two parts:
🔢 Part 1: Where Do the PCA Weights Come From?
The weights (aka loadings) in PCA are computed through eigen decomposition of the covariance matrix of the data.
Step-by-step:
-
Start with data matrix
X(rows = samples, columns = features). -
Standardize each feature to mean=0 and variance=1.
-
Compute covariance matrix:
C = (Xᵀ · X) / (n - 1) -
Perform eigen decomposition:
C · v = λ · vv= eigenvector (this becomes a principal component direction)λ= eigenvalue (tells how much variance that direction captures)
-
The eigenvectors (columns of matrix
V) give the weights you’re asking about:PC₁ = v₁ = [w₁, w₂, ..., wₙ] ← weights on original features
📌 Example: Say
PC1 = [0.6, 0.3, 0.7]means:PC1 = 0.6 * feature1 + 0.3 * feature2 + 0.7 * feature3
🔢 Part 2: How Do We Decide How Many PCs to Keep?
Now that we’ve got the components, we need to pick how many to keep (i.e., reduce to).
Each eigenvalue λᵢ tells how much variance its component vᵢ captures. You can:
✅ Use Variance Explained
-
Sort eigenvalues in descending order:
λ₁ = 3.2, λ₂ = 1.8, λ₃ = 0.5, ... -
Compute explained variance ratio:
Total variance = λ₁ + λ₂ + λ₃ + ... PC1 explains = λ₁ / total PC2 explains = λ₂ / total -
Cumulative variance:
PC1 = 60% PC1 + PC2 = 90% PC1 + PC2 + PC3 = 98% -
Choose number of PCs based on threshold:
- Keep enough to explain ≥95% variance (very common)
- Or stop at “elbow” in scree plot
✍️ Visual Summary
Imagine original 5D data → PCA reduces:
| PC | Eigenvalue | Variance Explained | Cumulative |
|---|---|---|---|
| PC1 | 4.2 | 60% | 60% |
| PC2 | 2.3 | 20% | 80% |
| PC3 | 1.0 | 10% | 90% |
| PC4 | 0.4 | 6% | 96% |
| PC5 | 0.1 | 4% | 100% |
✅ So keep PC1 + PC2 + PC3 to capture 90% of the variance.
✅ Final Intuition
- PCA finds directions (PCs) where data spreads most.
- Each direction is a mix of original features (weighted by eigenvectors).
- Keep enough directions (PCs) to explain most variance.
Great — let’s walk through a concrete mini PCA example using fake but simple numbers. You’ll see exactly:
- How to compute the covariance matrix
- How PCA gets the principal components (weights)
- How to determine how many to keep
🍕 Example: Mini Food Delivery Dataset
We’ll use 3 features for simplicity:
| Observation | Food Prep Time (min) | Distance (km) | Discount Used |
|---|---|---|---|
| A | 10 | 5 | 0 |
| B | 12 | 7 | 1 |
| C | 14 | 6 | 0 |
✅ Step 1: Standardize the Data
Compute mean and std deviation for each column:
- Prep: mean = 12, std = 2
- Dist: mean = 6, std = 1
- Discount: mean = 1/3 ≈ 0.33, std ≈ 0.58
Standardized X:
| Obs | Prep | Dist | Discount |
|---|---|---|---|
| A | (10–12)/2 = -1.0 | (5–6)/1 = -1.0 | (0–0.33)/0.58 ≈ -0.57 |
| B | 0.0 | 1.0 | 1.15 |
| C | 1.0 | 0.0 | -0.57 |
So standardized matrix X:
X =
[
[-1.0, -1.0, -0.57],
[ 0.0, 1.0, 1.15],
[ 1.0, 0.0, -0.57]
]
✅ Step 2: Compute Covariance Matrix
Cov(X) = (Xᵀ · X) / (n - 1)
Manually:
Cov = [
[1.0, 0.5, 0.0],
[0.5, 1.0, 0.5],
[0.0, 0.5, 1.0]
]
Rows and columns correspond to: [Prep, Dist, Discount]
✅ Step 3: Compute Eigenvalues and Eigenvectors
You’d normally use a library here, but let’s just assume the results:
| PC | Eigenvalue (λ) | Eigenvector (weights) | Variance % |
|---|---|---|---|
| PC1 | 1.9 | [0.58, 0.58, 0.58] | 63% |
| PC2 | 0.9 | [-0.71, 0.0, 0.71] | 30% |
| PC3 | 0.2 | [0.41, -0.82, 0.41] | 7% |
🔎 Notice: PCs are weighted combinations of all 3 original features.
✅ Step 4: Decide How Many PCs to Keep
Let’s look at cumulative variance:
- PC1 = 63%
- PC1 + PC2 = 93% ✅
- PC1 + PC2 + PC3 = 100%
So: keeping just 2 components retains 93% of all the variation. That’s enough in most practical scenarios.
🧠 Intuition
- PC1 mixes all features equally — it’s a “general magnitude” component.
- PC2 contrasts discount vs prep time — shows tension between those.
- PC3 is minor — very little added info.
✅ Final Summary
| Step | What It Does |
|---|---|
| Standardization | Removes scale bias |
| Covariance matrix | Finds correlated directions |
| Eigen decomposition | Gives variance + new axes |
| Eigenvectors | Define the weights (your question!) |
| Eigenvalues | Tell how many components to keep |