Chapter 6. Classification Algorithm

1. Overview

⑴ Definition

① y ~ X (where | { y } | ＜ ∞ )

② Belongs to the supervised algorithm category

⑵ Calibrated Classification Model

2. Type 1. Linear Classifier

⑴ Overview

① Utilizes non-linear regression analysis

② Drawback 1. Linear classifiers can learn only one template per class

③ Drawback 2. Linear classifiers can implement only linear decision boundaries among decision boundaries

④ (Reference) Hyperplane: Represents multi-dimensional surfaces beyond 3D for multiple regression

⑵ 1-1. SVM (Support Vector Machine)

① Overview

○ Definition: Technique for classifying points into two categories by setting the optimal hyperplane (maximum margin hyperplane; maximum margin hyperplane) that maximizes the margin in high-dimensional or infinite-dimensional space

Figure 1. SVM and Hyperplane

○ Multiple support vectors are possible: Setting multiple hyperplanes can classify points into multiple sets

○ Linearly inseparable classification problems can be solved by mapping from the original space to a higher-dimensional space

○ Advantages: Frequently used method with well-established research. Strong resistance to overfitting compared to other models. High accuracy.

○ Disadvantages: Requires providing a test set for supervised learning. Slower than other models.

② Formula

Figure 2. SVM Formula

○ Significance of the final formula: It is easy to apply an optimization algorithm and can be extended to classification problems that are not linearly separable by applying a kernel.

○ By introducing the slack variable ξ_i = 1 - y_i (wx_i + b), some data points that violate the constraints are allowed.

③ Kernels used in SVM

○ Linear Kernel: Basic type of kernel, 1-dimensional and faster than other functions

○ Polynomial Kernel: A generalized formula for linear kernels, less efficient in terms of effectiveness and accuracy

○ Gaussian Kernel: Generally used kernel, used when there is no prior knowledge of data

○ Gaussian RBF Kernel: The most commonly used kernel, typically used for nonlinear data. In the following equation, the vector ℓ represents a landmark (e.g., the center of a data point).

○ Sigmoid Kernel: Preferred as a kernel in artificial neural networks

④ Implementation in R

## data
print(head(iris))

## training
library(e1071)
train <- sample(1:150, 100)
sv <- svm(Species ~., data= iris, subset = train, type = "C-classification")

## testing
pred = predict(sv, iris[-train, ])
print(head(pred))

## review 
tt <- table(iris$Species[-train], pred)
print(tt) # confusion matrix

⑶ 1-2. nu-SVM

⑷ 1-3. Probability Regression Model: LPM, probit, logit, etc.

① R Code

## R code

# Train the logistic regression model
model <- glm(species ~ ., data = x_train, family = binomial())

# Predict probabilities
prob_estimates <- predict(model, x_test, type = "response")

3. Type 2. K-Nearest Neighbors Algorithm (K-NN)

⑴ Overview

① A method to label a given test data point using K nearest training data points

② Example: YouTube recommendation system

③ Doesn’t work well beyond 5 dimensions

⑵ Step 1. Data Loading and Architecture Design

import numpy as np
import tensorflow as tf

class NearestNeighbor(object):
    ...


(Xtr, Ytr), (Xte, Yte) = tf.keras.datasets.mnist.load_data()

print(Xtr.shape) # (60000, 28, 28)
print(Ytr.shape) # (60000,)
print(Xte.shape) # (10000, 28, 28)
print(Yte.shape) # (10000,)

Xtr_rows = Xtr.reshape(Xtr.shape[0], 28*28)
Xte_rows = Xte.reshape(Xte.shape[0], 28*28)

print(Xtr_rows.shape) # (60000, 784)
print(Xte_rows.shape) # (10000, 784)

nn = NearestNeighbor()
nn.train(Xtr_rows, Ytr)

Yte_predict = nn.predict(Xte_rows, 'L2')
print ('accuracy: %f' + str(np.mean(Yte_predict == Yte)) )

# L1 : accuracy = 47.29%
# L2 : accuracy = 27.24%
# dot : accuracy = 9.9%

⑶ Step 2. NN Class Design

class NearestNeighbor(object):
    def __init__(self):
        pass

    def train(self, X, Y):
        self.Xtr = X
        self.Ytr = Y

    def predict(self, X, dist_metric='L2'):
        num_test = X.shape[0]
        Ypred = np.zeros(num_test, dtype = self.Ytr.dtype)
        
        for i in range(num_test):
            if dist_metric=='L1': ## L1 distance
                distances = np.sum(np.abs(self.Xtr - X[i, :]), axis = 1) 
            elif dist_metric=='L2': ## L2 distance
                distances = np.sum(np.square(self.Xtr - X[i, :]), axis = 1) 
            elif dist_metric=='dot': ## dot distance
                distances = np.dot(self.Xtr, X[i, :])
            
            min_index = np.argmin(distances)
            Ypred[i] = self.Ytr[min_index]
            
            if i%100 == 0:
                print(i)
        return Ypred

① The training phase is merely a data storage phase

② Factor 1. Designing the distance function

○ Distance Function(distance function, metric): Defines distance

○ 1-1. L1 Distance

○ 1-2. L2 Distance: Euclidean distance using the Pythagorean theorem (standard)

○ 1-3. p-norm

○ 1-4. Dot Product

○ 1-5. KL Distance (Kullback-Leibler Divergence)(Kullback-Leibler divergence)

○ 1-6. Other distance functions

○ Results from L1 and L2 distances are similar: L1 38.6%, L2 35.4%

③ Factor 2. 1-NN vs. k-NN

○ The edges of each region become smoother as k increases

○ If k is too small, it is not robust and may respond to outliers or noise when labeling

○ Ambiguous regions appear as k increases

⑷ Step 3. Hyperparameter Tuning: Try various k values and choose the one that yields the best results

Xval_rows = Xtr_rows[:1000, :] # take first 1000 for validation
Yval = Ytr[:1000]
Xtr_rows = Xtr_rows[1000:, :] # keep last 59,000 for train
Ytr = Ytr[1000:]

validation_accuracies = []
for k in [1, 3, 5, 10, 20, 50, 100]:
    nn = NearestNeighbor()
    nn.train(Xtr_rows, Ytr)
    Yval_predict = nn.predict(Xval_rows, k = k)
    acc = np.mean(Yval_predict == Yval)
    print 'accuracy: %f' % (acc,)
    validation_accuracies.append((k, acc))

⑸ Step 4. Improvement

① Drawbacks

○ Using KNN at the pixel level may not result in true similarity, leading to poor performance

○ Test time is very slow

○ Curse of dimensionality: As the dimension increases, the number of data required to maintain the same distance between examples increases geometrically

○ Too sparse examples among examples can lead to sloppy classification results

② Improvements

○ Data Normalization: Normalize data to have a mean of 0 and a standard deviation of 1

○ Dimensionality Reduction: Techniques like PCA, NCA, random projection, etc.

○ Cross-Validation: Computationally expensive, commonly used methods include 3-fold, 5-fold, and 10-fold cross-validation

○ Approximate Nearest Neighbors: Reduces accuracy but improves speed (e.g., FLANN)

③ Despite these improvements, KNN is rarely used in practice

⑹ 2-1. Mutual Nearest Neighbors Algorithm

⑺ Comparison between K-NN and K-Means Clustering

Item	K-NN	K-Means Clustering
Type	Supervised Learning	Unsupervised Learning
Meaning of k	Number of Nearest Neighbors	Number of Clusters
Optimization Techniques	Cross Validation, Confusion Matrix	Elbow Method, Silhouette Method
Application	Classification	Clustering

Table. 1. Comparison of K-NN and K-Means Clustering

4. Type 3. Decision Trees

⑴ Overview

① Models for predicting output values for given input values, including single trees and regression tree models

② Technique of drawing decision trees with decision rules using classification functions for analysis

③ The purity of child nodes increases compared to the purity of parent nodes

④ Calculation results are directly shown in decision trees

⑤ Disadvantage: Predictive errors are significant near the boundary points of separation because continuous variables are treated as discontinuous values

⑵ 3-1. Simple Decision Trees

① Overview

○ Advantage 1. Good representation for given data

○ Advantage 2. Interpretable and fast

○ Advantage 3. Easily handles the following data: categorical, mixed data, missing data, multiple classes

○ Drawback 1. Potential for overfitting, meaning it is vulnerable to outliers.

○ Drawback 2. The tree becomes too deep.

○ Solution: Tree pruning

② Terminology

○ Root Node

○ Internal Node (Node): Forms conditional statements for attributes

○ Branching: Splits from internal nodes based on attribute values

○ Leaf Node (Terminal Node, Decision Node): Outputs (class assignment)

③ Learning Decision Trees

○ Condition 1. Not too big or too small

○ Condition 2. Occam’s Razor Theory: Simplicity is the best

○ Condition 3. Complexity depends on depth

○ Learning the smallest and simplest decision tree is an NP-complete problem (Hyafil & Rivest (1976))

○ Decision trees can be learned greedily using heuristic methods

### BuildTree (D): Greedy training of a decision tree ###
# Input: A data set D = (X, Y).
# Output: A decision tree.

if LeafCondition(D) then
    fn = FindBestPrediction(D)
else
    jn, tn = FindBestSplid(D)
    DL = {(x(i), y(i)) :  x(i)_(jn) < tn} and
    DR = {(x(i), y(i)) :  x(i)_(jn) ≥ tn}
    leftChild = BuildTree(DL)
    rightChild = BuildTree(DR)
end if

○ Step 1. Score all splits

○ Step 2. Greedily find the condition with the maximum information gain

○ Step 3. Divide the dataset with that condition and recursively perform Step 1 and Step 2 in each subtree

○ Step 4. Stopping criteria

④ R Code

## R code

library(rpart)

# Train the decision tree model
model <- rpart(species ~ ., data = x_train, method = "class")

# Predict probabilities
prob_estimates <- predict(model, x_test)

⑤ Application 1: Pruning

⑥ Application 2: Information Gain Ratio

⑶ 3-2. Bagging (Bootstrap Aggregation)

① Definition: Determines classification results for different datasets obtained by sampling with replacement by voting

Figure 3. Bagging Illustration

② Bootstrap: A series of processes that proceed automatically once started, not limited to decision trees

③ Advantage 1. Reduces the variance of unstable processes like trees, improving predictability

④ Advantage 2. Smoothes the boundaries of decision trees

⑷ 3-3. Random Forest

① Definition: Utilizes m features randomly selected from each sub-dataset for bagging

○ Example: When learning, randomly select only some features such as city, population, average income, number of newborns, and number of newlyweds among various features

② Feature 1. De-correlating: Increases individuality of each sub-dataset by using only some features

③ Feature 2. Unused datasets in each sub-dataset can be used to validate out-of-sample errors

④ Performance: Random forest ＞ bagging ＞ decision tree

○ Accuracy: Comparable or higher than Adaboost

○ Robustness: Relatively robust against outliers and noise

○ Speed: Faster than bagging and boosting

○ Simplicity and easy parallelization

○ Currently the best-performing algorithm in classification algorithm research

⑤ R Code

## R code

library(randomForest)

# Train the random forest model
model <- randomForest(species ~ ., data = x_train)

# Predict probabilities
prob_estimates <- predict(model, x_test, type = "prob")

⑸ 3-4. C4.5 and C5.0

① Developed by Australian researcher J. Ross Quinlan

② The initial version was developed as ID 3 (Iterative Dichotomizer 3) in 1986

③ Uses training data when using pruning

④ Uses entropy index as a measure of impurity with discrete dependent variables

⑹ 3-5. CHAID

① Uses the chi-squared statistic and employs multiway splits in the decision tree algorithm

⑺ 3-6. CART (Classification and Regression Tree)

① Performs classification by repeatedly bifurcating each independent variable to form a binary tree

⑻ 3-7. AdaBoost

⑼ 3-8. QUEST

5. Type 4. Base Classifier

⑴ Naïve Bayes classifier

① Overview

○ An algorithm that assumes conditional independence between variables in the data.

○ An algorithm that combines prior information about classes with information extracted from data, using Bayes’ theorem to classify which class a given data point belongs to.

○ It can be used as a solution for problems in text classification, such as determining which category (e.g., spam, economics, sports, etc.) a document belongs to.

② Formalization

○ Premise: There are K classes C₁, C₂, ···, C_K.

○ Objective: Given a new sample x = (x₁, ···, x_N) composed of multiple features, determine which class it belongs to.

○ Formalization: Similar to MLE (Maximum Likelihood Estimation).

③ Limitations

○ Violation of conditional independence: If there are significant correlations between features (e.g., age and history of heart disease), related variables may be multiplied multiple times, leading to an overestimation of their effects.

○ Inability to exclude the influence of irrelevant features.

○ Solution: By using methods such as PCR or mRMR to discard unnecessary variables and retain only the essential ones, these limitations can be mitigated.

⑵ Bayesian Network

6. Type 5. Domain Adaptation Classification

⑴ Overview

① Definition : Obtaining universal knowledge from one domain and applying it to another domain.

② One domain : Referred to as the source domain, tangential domain, etc. Large dataset size.

③ Another domain : Referred to as the target domain, specific domain, etc. Small dataset size.

④ Differences from generalization

○ Domain adaptation classification : Differences in format, data structure, etc., between source and target.

○ Generalization : Format, data structure, etc., are the same between source and target.

⑵ Types

① UDA (Unsupervised Domain Adaptation) : When there are no labels in the target domain.

Figure 4. Diagram of UDA

○ In other words, the structure of applying features obtained from the source domain directly to the target domain.

② SSL (Supervised Domain Adaptation) : When there are labels in the target domain.

③ SSDA (Semi-Supervised Domain Adaptation) : When the target domain has labels for only some cases.

Figure 5. Diagram of SSDA

○ Process 1: Augmentation

○ Process 2: Random logit interpolation

○ Process 3: Distribution alignment

○ Process 4: Relative confidence threshold

○ Process 5: Loss function

○ Reason for being state-of-the-art (SOTA): Because of the trade-off, as can be confirmed in the ‘motivation’ section in the link.

⑶ Objective Functions

① Aims to reduce the discrepancy between domains.

② Types of discrepancies

○ MMD (Maximum Mean Discrepancy)

○ DANN (Domain Adversarial Neural Network)

○ MCD (Maximum Classifier Discrepancy)

⑷ Applications

① Divergence Domain Adaptation

② Adversarial Domain Adaptation

○ Generative Adversarial Network (GAN)

○ Definition : Two AI systems learning while competing with each other (game learning).

○ Generative Model : Utilizes knowledge learned based on various features of existing data to generate new data.

○ Discriminative Model : Evaluates how similar the generated new data is to the existing data.

○ Automatically creates results that are close to reality by exchanging ideas between the generative and discriminative models.

③ Reconstruction Domain Adaptation

7. Type 6. Deep Learning-Based Classification

⑴ Deep Learning (Part 1, Part 2, Part 3)

⑵ 6-1. 1D CNN

8. Other Algorithms

⑴ Faiss

① A similarity-based search and dense vector clustering algorithm developed by Meta in 2023.

② Fast search algorithm: capable of obtaining the nearest neighbor and the k-th nearest neighbor.

③ Allows for the search of multiple vectors at once (batch processing).

④ There is a trade-off between accuracy and speed.

⑤ Instead of minimizing Euclidean distance, it calculates by maximizing the maximum inner product.

⑥ Returns all elements contained within a given radius.

Input : 2021.12.12 11:37

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Chapter 6. Classification Algorithm

1. Overview

2. Type 1. Linear Classifier

3. Type 2. K-Nearest Neighbors Algorithm (K-NN)

4. Type 3. Decision Trees

5. Type 4. Base Classifier

6. Type 5. Domain Adaptation Classification

7. Type 6. Deep Learning-Based Classification

8. Other Algorithms

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