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</head>
<body>
<div class="visit-counter" title="Page visits">
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<span id="visit-count">Loading...</span>
</div>
<div class="hero" role="banner">
<h1>📊 Mathematics for Data Science & Machine Learning</h1>
<p>A comprehensive, beautifully rendered reference for formulas and concepts.</p>
<div class="hero-stats" aria-hidden="true">
<div class="stat-box"><span class="stat-number">17</span><span class="stat-label">Topics</span></div>
<div class="stat-box"><span class="stat-number">200+</span><span class="stat-label">Formulas</span></div>
<div class="stat-box"><span class="stat-number">∞</span><span class="stat-label">Applications</span></div>
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<button class="nav-btn" onclick="scrollToSection('linear-algebra')">Linear Algebra</button>
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<!-- 1. LINEAR ALGEBRA -->
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<div class="section-number">1</div>
<h2 id="la-title" class="section-title">Linear Algebra</h2>
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<div class="section-content">
<div class="subsection">
<div class="subsection-title">🔢 Vectors & Matrices</div>
<div class="formula-card">
<div class="formula-name">Dot Product</div>
<div class="formula">$$\mathbf{a}\cdot\mathbf{b}=\sum_{i=1}^n a_i b_i$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Matrix Multiplication</div>
<div class="formula">$$(AB)_{ij}=\sum_{k=1}^n A_{ik}B_{kj}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Transpose</div>
<div class="formula">$$(AB)^T=B^T A^T$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Identity Matrix</div>
<div class="formula">$$AI = IA = A$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Inverse</div>
<div class="formula">$$AA^{-1}=A^{-1}A=I$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">📏 Norms</div>
<table class="formula-table">
<tr><th>Norm</th><th>Formula</th></tr>
<tr><td>L1 (Manhattan)</td><td>$$\|x\|_1=\sum_i |x_i|$$</td></tr>
<tr><td>L2 (Euclidean)</td><td>$$\|x\|_2=\sqrt{\sum_i x_i^2}$$</td></tr>
<tr><td>Frobenius</td><td>$$\|A\|_F=\sqrt{\sum_i\sum_j a_{ij}^2}$$</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">🎯 Eigenvalues & Eigenvectors</div>
<div class="code-block">
Eigen equation: \(Av=\lambda v\)<br/>
Characteristic: \(\det(A-\lambda I)=0\)<br/>
Trace: \(\mathrm{tr}(A)=\sum_i \lambda_i = \sum_i a_{ii}\)<br/>
Determinant: \(\det(A)=\prod_i \lambda_i\)
</div>
</div>
</div>
</section>
<!-- 2. CALCULUS -->
<section id="calculus" class="section" aria-labelledby="calc-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">2</div>
<h2 id="calc-title" class="section-title">Calculus</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">📐 Basic Derivatives</div>
<table class="formula-table">
<tr><th>Rule</th><th>Formula</th></tr>
<tr><td>Power Rule</td><td>$$\frac{d}{dx}(x^n) = nx^{n-1}$$</td></tr>
<tr><td>Chain Rule</td><td>$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$</td></tr>
<tr><td>Product Rule</td><td>$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$</td></tr>
<tr><td>Quotient Rule</td><td>$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$$</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">∇ Partial Derivatives</div>
<table class="formula-table">
<tr><th>Concept</th><th>Formula</th></tr>
<tr><td>Gradient</td><td>$$\nabla f=\left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right)$$</td></tr>
<tr><td>Hessian</td><td>$$H_{ij}=\frac{\partial^2 f}{\partial x_i\partial x_j}$$</td></tr>
<tr><td>Jacobian</td><td>$$J_{ij}=\frac{\partial f_i}{\partial x_j}$$</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">🧮 Activation Derivatives</div>
<table class="formula-table">
<tr><th>Function</th><th>Derivative</th></tr>
<tr><td>Sigmoid</td><td>$$\sigma'(x)=\sigma(x)(1-\sigma(x))$$</td></tr>
<tr><td>Tanh</td><td>$$\tanh'(x)=1-\tanh^2(x)$$</td></tr>
<tr><td>ReLU</td><td>$$\text{ReLU}'(x)=\begin{cases}1&x>0\\0&\text{otherwise}\end{cases}$$</td></tr>
<tr><td>Exponential</td><td>$$\frac{d}{dx}(e^x) = e^x$$</td></tr>
<tr><td>Logarithm</td><td>$$\frac{d}{dx}(\ln x) = \frac{1}{x}$$</td></tr>
</table>
</div>
</div>
</section>
<!-- 3. PROBABILITY -->
<section id="probability" class="section" aria-labelledby="prob-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">3</div>
<h2 id="prob-title" class="section-title">Probability</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">🎲 Basic Probability Rules</div>
<div class="formula-card">
<div class="formula-name">Probability</div>
<div class="formula">$$P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Complement Rule</div>
<div class="formula">$$P(A^c)=1-P(A)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Addition Rule</div>
<div class="formula">$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Multiplication Rule</div>
<div class="formula">$$P(A\cap B)=P(A|B)P(B)=P(B|A)P(A)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Conditional Probability</div>
<div class="formula">$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🔮 Bayes' Theorem</div>
<div class="formula-card">
<div class="formula-name">Bayes' Rule</div>
<div class="formula">$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Extended Form</div>
<div class="formula">$$P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^c)P(A^c)}$$</div>
</div>
<div class="info-box"><strong>💡 Key Application:</strong> Fundamental in ML for classification, spam detection, and probabilistic reasoning</div>
</div>
<div class="subsection">
<div class="subsection-title">📊 Expected Value & Variance</div>
<table class="formula-table">
<tr><th>Concept</th><th>Formula</th></tr>
<tr><td>Expected Value</td><td>$$E[X]=\sum x_i P(x_i)\text{ or }\int x f(x)dx$$</td></tr>
<tr><td>Variance</td><td>$$\text{Var}(X)=E[(X-\mu)^2]=E[X^2]-(E[X])^2$$</td></tr>
<tr><td>Standard Deviation</td><td>$$\sigma = \sqrt{\text{Var}(X)}$$</td></tr>
<tr><td>Covariance</td><td>$$\text{Cov}(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]$$</td></tr>
<tr><td>Correlation</td><td>$$\rho(X,Y)=\frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$$</td></tr>
</table>
</div>
</div>
</section>
<!-- 4. STATISTICS -->
<section id="statistics" class="section" aria-labelledby="stat-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">4</div>
<h2 id="stat-title" class="section-title">Statistics</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">📈 Descriptive Statistics</div>
<div class="formula-card">
<div class="formula-name">Mean</div>
<div class="formula">$$\mu=\frac{1}{n}\sum_{i=1}^n x_i$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Sample Variance</div>
<div class="formula">$$s^2=\frac{1}{n-1}\sum_{i=1}^n (x_i-\bar{x})^2$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Population Variance</div>
<div class="formula">$$\sigma^2=\frac{1}{n}\sum_{i=1}^n (x_i-\mu)^2$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Standard Error</div>
<div class="formula">$$SE = \frac{\sigma}{\sqrt{n}}$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🎯 Hypothesis Testing</div>
<div class="formula-card">
<div class="formula-name">Z-score</div>
<div class="formula">$$z=\frac{x-\mu}{\sigma}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">T-statistic</div>
<div class="formula">$$t=\frac{\bar{x}-\mu}{s/\sqrt{n}}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Confidence Interval</div>
<div class="formula">$$CI=\bar{x}\pm z_{\alpha/2}\cdot SE$$</div>
</div>
<div class="info-box">
<strong>Key Terms:</strong><br/>
• <strong>Type I Error (α):</strong> Rejecting true null hypothesis<br/>
• <strong>Type II Error (β):</strong> Failing to reject false null hypothesis<br/>
• <strong>Power:</strong> 1 - β
</div>
</div>
</div>
</section>
<!-- 5. LINEAR & LOGISTIC REGRESSION -->
<section id="regression" class="section" aria-labelledby="regress-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">5</div>
<h2 id="regress-title" class="section-title">Linear & Logistic Regression</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">📉 Linear Regression</div>
<div class="formula-card">
<div class="formula-name">Simple Model</div>
<div class="formula">$$y=\beta_0+\beta_1 x+\varepsilon$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Matrix Form</div>
<div class="formula">$$\mathbf{y}=\mathbf{X}\beta+\varepsilon$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Normal Equation</div>
<div class="formula">$$\beta=(X^TX)^{-1}X^T y$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Predicted Values</div>
<div class="formula">$$\hat{y} = X\beta$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">📊 Loss Functions</div>
<table class="formula-table">
<tr><th>Metric</th><th>Formula</th></tr>
<tr><td>Mean Squared Error</td><td>$$MSE=\frac{1}{n}\sum_{i=1}^n (y_i-\hat{y}_i)^2$$</td></tr>
<tr><td>Root MSE</td><td>$$RMSE=\sqrt{MSE}$$</td></tr>
<tr><td>Mean Absolute Error</td><td>$$MAE=\frac{1}{n}\sum_{i=1}^n |y_i-\hat{y}_i|$$</td></tr>
<tr><td>R² Score</td><td>$$R^2=1-\frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y})^2}$$</td></tr>
<tr><td>Adjusted R²</td><td>$$R^2_{adj}=1-\frac{(1-R^2)(n-1)}{n-p-1}$$</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">🔄 Logistic Regression</div>
<div class="formula-card">
<div class="formula-name">Sigmoid Function</div>
<div class="formula">$$\sigma(z)=\frac{1}{1+e^{-z}}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Logit</div>
<div class="formula">$$z = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Probability</div>
<div class="formula">$$P(y=1|x) = \sigma(w^T x + b)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Odds & Log-Odds</div>
<div class="formula">$$\text{Odds} = \frac{P(y=1)}{P(y=0)} = e^z \quad ; \quad \log(\text{Odds}) = z$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Log Loss (Binary Cross-Entropy)</div>
<div class="formula">$$L=-\frac{1}{n}\sum_{i=1}^n[y_i\log\hat{y}_i+(1-y_i)\log(1-\hat{y}_i)]$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🛡️ Regularization</div>
<table class="formula-table">
<tr><th>Type</th><th>Cost Function</th></tr>
<tr><td>Ridge (L2)</td><td>$$J(\beta)=\sum(y_i-\hat{y}_i)^2+\lambda\sum\beta_j^2$$</td></tr>
<tr><td>Lasso (L1)</td><td>$$J(\beta)=\sum(y_i-\hat{y}_i)^2+\lambda\sum|\beta_j|$$</td></tr>
<tr><td>Elastic Net</td><td>$$J(\beta)=\sum(y_i-\hat{y}_i)^2+\lambda_1\sum|\beta_j|+\lambda_2\sum\beta_j^2$$</td></tr>
</table>
</div>
</div>
</section>
<!-- 6. NEURAL NETWORKS -->
<section id="neural-networks" class="section" aria-labelledby="nn-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">6</div>
<h2 id="nn-title" class="section-title">Neural Networks</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">⚡ Activation Functions</div>
<table class="formula-table">
<tr><th>Function</th><th>Formula</th></tr>
<tr><td>Sigmoid</td><td>$$\sigma(x)=\frac{1}{1+e^{-x}}$$</td></tr>
<tr><td>Tanh</td><td>$$\tanh(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$</td></tr>
<tr><td>ReLU</td><td>$$\text{ReLU}(x)=\max(0,x)$$</td></tr>
<tr><td>Leaky ReLU</td><td>$$f(x)=\begin{cases}x&x>0\\\alpha x&\text{otherwise}\end{cases}$$</td></tr>
<tr><td>Softmax</td><td>$$\text{softmax}(x_i)=\frac{e^{x_i}}{\sum_j e^{x_j}}$$</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">🔄 Forward Propagation</div>
<div class="formula-card">
<div class="formula-name">Linear Combination</div>
<div class="formula">$$z^{(l)}=W^{(l)}a^{(l-1)}+b^{(l)}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Activation</div>
<div class="formula">$$a^{(l)}=g(z^{(l)})$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">⬅️ Backpropagation</div>
<div class="formula-card">
<div class="formula-name">Output Layer Error</div>
<div class="formula">$$\delta^{(L)}=(a^{(L)}-y)\odot g'(z^{(L)})$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Hidden Layer Error</div>
<div class="formula">$$\delta^{(l)}=[(W^{(l+1)})^T\delta^{(l+1)}]\odot g'(z^{(l)})$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Weight Gradient</div>
<div class="formula">$$\frac{\partial L}{\partial W^{(l)}}=\delta^{(l)}(a^{(l-1)})^T$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Bias Gradient</div>
<div class="formula">$$\frac{\partial L}{\partial b^{(l)}}=\delta^{(l)}$$</div>
</div>
</div>
</div>
</section>
<!-- 7. OPTIMIZATION ALGORITHMS -->
<section id="optimization" class="section" aria-labelledby="opt-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">7</div>
<h2 id="opt-title" class="section-title">Optimization Algorithms</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">⬇️ Gradient Descent Variants</div>
<div class="formula-card">
<div class="formula-name">Batch Gradient Descent</div>
<div class="formula">$$\theta:=\theta-\alpha\nabla J(\theta)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Stochastic Gradient Descent</div>
<div class="formula">$$\theta:=\theta-\alpha\nabla J(\theta;x^{(i)},y^{(i)})$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Mini-batch Gradient Descent</div>
<div class="formula">$$\theta:=\theta-\alpha\frac{1}{m}\sum_{i=1}^m\nabla J(\theta;x^{(i)},y^{(i)})$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🚀 Advanced Optimizers</div>
<div class="formula-card">
<div class="formula-name">Momentum</div>
<div class="formula">$$v:=\beta v+(1-\beta)\nabla J(\theta)\\
\theta:=\theta-\alpha v$$</div>
</div>
<div class="formula-card">
<div class="formula-name">RMSprop</div>
<div class="formula">$$s:=\beta s+(1-\beta)(\nabla J)^2\\
\theta:=\theta-\alpha\frac{\nabla J}{\sqrt{s}+\epsilon}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Adam (Adaptive Moment Estimation)</div>
<div class="formula">$$m:=\beta_1 m+(1-\beta_1)\nabla J\\
v:=\beta_2 v+(1-\beta_2)(\nabla J)^2\\
\theta:=\theta-\alpha\frac{\hat m}{\sqrt{\hat v}+\epsilon}$$</div>
</div>
<div class="info-box"><strong>Pro tip:</strong> Adam is widely used in modern deep learning.</div>
</div>
</div>
</section>
<!-- 8. EVALUATION METRICS -->
<section id="metrics" class="section" aria-labelledby="met-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">8</div>
<h2 id="met-title" class="section-title">Evaluation Metrics</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">✅ Classification Metrics</div>
<table class="formula-table">
<tr><th>Metric</th><th>Formula</th></tr>
<tr><td>Accuracy</td><td>$$\text{Accuracy}=\frac{TP+TN}{TP+TN+FP+FN}$$</td></tr>
<tr><td>Precision</td><td>$$\text{Precision}=\frac{TP}{TP+FP}$$</td></tr>
<tr><td>Recall (Sensitivity)</td><td>$$\text{Recall}=\frac{TP}{TP+FN}$$</td></tr>
<tr><td>Specificity</td><td>$$\text{Specificity}=\frac{TN}{TN+FP}$$</td></tr>
<tr><td>F1-Score</td><td>$$F_1=\frac{2\cdot \text{Precision}\cdot \text{Recall}}{\text{Precision}+\text{Recall}}$$</td></tr>
<tr><td>F-beta Score</td><td>$$F_\beta=\frac{(1+\beta^2)\cdot \text{Precision}\cdot \text{Recall}}{\beta^2\cdot\text{Precision}+\text{Recall}}$$</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">📉 Confusion Matrix</div>
<div class="info-box">
<strong>Legend:</strong><br/>
• <strong>TP</strong> = True Positive<br/>
• <strong>TN</strong> = True Negative<br/>
• <strong>FP</strong> = False Positive (Type I Error)<br/>
• <strong>FN</strong> = False Negative (Type II Error)
</div>
</div>
<div class="subsection">
<div class="subsection-title">📈 ROC & AUC</div>
<div class="formula-card">
<div class="formula-name">True Positive Rate (TPR)</div>
<div class="formula">$$TPR=\frac{TP}{TP+FN}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">False Positive Rate (FPR)</div>
<div class="formula">$$FPR=\frac{FP}{FP+TN}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">AUC (Area Under Curve)</div>
<div class="formula">$$AUC = \int_0^1 TPR(FPR^{-1}(x))dx$$</div>
</div>
<div class="info-box"><strong>Note:</strong> ROC curves plot TPR vs FPR. AUC measures the area under the ROC curve.</div>
</div>
</div>
</section>
<!-- 9. CLUSTERING -->
<section id="clustering" class="section" aria-labelledby="clust-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">9</div>
<h2 id="clust-title" class="section-title">Clustering</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">🎯 K-Means Algorithm</div>
<div class="formula-card">
<div class="formula-name">Objective Function</div>
<div class="formula">$$J=\min \sum_{j=1}^k\sum_{x\in C_j}\|x-\mu_j\|^2$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Centroid Update</div>
<div class="formula">$$\mu_j=\frac{1}{|C_j|}\sum_{x\in C_j}x$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">📏 Distance Metrics</div>
<table class="formula-table">
<tr><th>Metric</th><th>Formula</th></tr>
<tr><td>Euclidean</td><td>$$d(x,y)=\sqrt{\sum_i (x_i-y_i)^2}$$</td></tr>
<tr><td>Manhattan</td><td>$$d(x,y)=\sum_i|x_i-y_i|$$</td></tr>
<tr><td>Cosine Similarity</td><td>$$\cos(\theta)=\frac{x\cdot y}{\|x\|\|y\|}$$</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">📊 Silhouette Score</div>
<div class="formula-card">
<div class="formula-name">Silhouette Coefficient</div>
<div class="formula">$$s(i)=\frac{b(i)-a(i)}{\max\{a(i),b(i)\}}$$</div>
</div>
<div class="info-box"><strong>Where:</strong> \(a(i)\) = mean distance to same cluster, \(b(i)\) = mean distance to nearest cluster</div>
</div>
</div>
</section>
<!-- 10. DEEP LEARNING -->
<section id="deep-learning" class="section" aria-labelledby="dl-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">10</div>
<h2 id="dl-title" class="section-title">Deep Learning</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">🔄 Batch Normalization</div>
<div class="formula-card">
<div class="formula-name">Normalize</div>
<div class="formula">$$\hat x=\frac{x-\mu_B}{\sqrt{\sigma_B^2+\epsilon}}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Scale & Shift</div>
<div class="formula">$$y=\gamma\hat x+\beta$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🧠 Convolutional Neural Networks (CNN)</div>
<div class="formula-card">
<div class="formula-name">Convolution Operation</div>
<div class="formula">$$(f * g)(t) = \sum_x f(x)g(t-x)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Output Size</div>
<div class="formula">$$O=\left\lfloor\frac{W-K+2P}{S}\right\rfloor+1$$</div>
</div>
<div class="info-box"><strong>Parameters:</strong> W = input size, K = kernel size, P = padding, S = stride</div>
</div>
<div class="subsection">
<div class="subsection-title">🔁 Recurrent Neural Networks (RNN)</div>
<div class="formula-card">
<div class="formula-name">Hidden State</div>
<div class="formula">$$h_t = \tanh(W_{xh}x_t + W_{hh}h_{t-1} + b_h)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Output</div>
<div class="formula">$$y_t = W_{hy}h_t + b_y$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🧬 Long Short-Term Memory (LSTM)</div>
<div class="formula-card">
<div class="formula-name">Forget Gate</div>
<div class="formula">$$f_t=\sigma(W_f[h_{t-1},x_t]+b_f)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Input Gate</div>
<div class="formula">$$i_t=\sigma(W_i[h_{t-1},x_t]+b_i)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Output Gate</div>
<div class="formula">$$o_t=\sigma(W_o[h_{t-1},x_t]+b_o)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Cell State</div>
<div class="formula">$$C_t=f_t\odot C_{t-1}+i_t\odot\tilde{C}_t$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Hidden State</div>
<div class="formula">$$h_t=o_t\odot\tanh(C_t)$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">💧 Dropout</div>
<div class="info-box">
<strong>Training:</strong> Output = mask ⊙ activation / (1 - p)<br/>
<strong>Testing:</strong> Use all neurons (no dropout)
</div>
</div>
</div>
</section>
<!-- 11. DIMENSIONALITY REDUCTION -->
<section id="dimensionality-reduction" class="section" aria-labelledby="dimred-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">11</div>
<h2 id="dimred-title" class="section-title">Dimensionality Reduction</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">📊 Principal Component Analysis (PCA)</div>
<div class="formula-card">
<div class="formula-name">Covariance Matrix</div>
<div class="formula">$$\Sigma=\frac{1}{n}X^T X$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Principal Components</div>
<div class="formula">$$\text{Eigenvectors of } \Sigma$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Explained Variance Ratio</div>
<div class="formula">$$\text{EVR}=\frac{\lambda_i}{\sum_j \lambda_j}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Projection</div>
<div class="formula">$$Z=XW \quad \text{(where W = eigenvectors)}$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🔢 Singular Value Decomposition (SVD)</div>
<div class="formula-card">
<div class="formula-name">Decomposition</div>
<div class="formula">$$X = U\Sigma V^T$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Reduced Form</div>
<div class="formula">$$X \approx U_k\Sigma_k V_k^T$$</div>
</div>
</div>
</div>
</section>
<!-- 12. INFORMATION THEORY -->
<section id="information-theory" class="section" aria-labelledby="info-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">12</div>
<h2 id="info-title" class="section-title">Information Theory</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">📡 Entropy and Information</div>
<table class="formula-table">
<tr><th>Measure</th><th>Formula</th></tr>
<tr><td>Entropy</td><td>$$H(X)=-\sum P(x_i)\log_2 P(x_i)$$</td></tr>
<tr><td>Cross-Entropy</td><td>$$H(p,q)=-\sum p(x)\log q(x)$$</td></tr>
<tr><td>KL Divergence</td><td>$$D_{KL}(P\|Q)=\sum P(x)\log\frac{P(x)}{Q(x)}$$</td></tr>
<tr><td>Mutual Information</td><td>$$I(X;Y)=H(X)-H(X|Y)=H(Y)-H(Y|X)$$</td></tr>
<tr><td>Conditional Entropy</td><td>$$H(Y|X)=-\sum_x\sum_y P(x,y)\log P(y|x)$$</td></tr>
</table>
</div>
</div>
</section>
<!-- 13. SUPPORT VECTOR MACHINES -->
<section id="svm" class="section" aria-labelledby="svm-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">13</div>
<h2 id="svm-title" class="section-title">Support Vector Machines</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">🎯 Linear SVM</div>
<div class="formula-card">
<div class="formula-name">Decision Function</div>
<div class="formula">$$f(x)=w^Tx+b$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Margin</div>
<div class="formula">$$\text{margin}=\frac{2}{\|w\|}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Optimization</div>
<div class="formula">$$\min \frac{1}{2}\|w\|^2 \quad \text{s.t. } y_i(w^Tx_i+b)\geq 1$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🛡️ Soft Margin SVM</div>
<div class="formula-card">
<div class="formula-name">Objective</div>
<div class="formula">$$\min \frac{1}{2}\|w\|^2 + C\sum_i\xi_i$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Constraint</div>
<div class="formula">$$y_i(w^Tx_i+b)\geq 1-\xi_i, \quad \xi_i\geq 0$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🔧 Kernel Functions</div>
<table class="formula-table">
<tr><th>Kernel</th><th>Formula</th></tr>
<tr><td>Linear</td><td>$$K(x,x')=x^Tx'$$</td></tr>
<tr><td>Polynomial</td><td>$$K(x,x')=(x^Tx'+c)^d$$</td></tr>
<tr><td>RBF (Gaussian)</td><td>$$K(x,x')=\exp(-\gamma\|x-x'\|^2)$$</td></tr>
<tr><td>Sigmoid</td><td>$$K(x,x')=\tanh(\alpha x^Tx'+c)$$</td></tr>
</table>
</div>
</div>
</section>
<!-- 14. DECISION TREES & ENSEMBLES -->
<section id="decision-trees" class="section" aria-labelledby="dt-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">14</div>
<h2 id="dt-title" class="section-title">Decision Trees & Ensembles</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">🌳 Impurity Measures</div>
<table class="formula-table">
<tr><th>Measure</th><th>Formula</th></tr>
<tr><td>Gini Impurity</td><td>$$\text{Gini}=1-\sum_i p_i^2$$</td></tr>
<tr><td>Entropy</td><td>$$H=-\sum_i p_i\log_2(p_i)$$</td></tr>
<tr><td>Classification Error</td><td>$$E=1-\max(p_i)$$</td></tr>
</table>
<div class="formula-card">
<div class="formula-name">Information Gain</div>
<div class="formula">$$IG(D,A)=H(D)-\sum_v\frac{|D_v|}{|D|}H(D_v)$$</div>
</div>
</div>
<div class="subsection">
<div class="subsection-title">🌲 Ensemble Methods</div>
<div class="formula-card">
<div class="formula-name">Bagging (Random Forest)</div>
<div class="formula">$$\hat{y}=\frac{1}{B}\sum_{b=1}^B f_b(x)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">AdaBoost - Sample Weight</div>
<div class="formula">$$w_i^{(t+1)}=w_i^{(t)}\cdot\exp[\alpha_t\cdot\mathbb{1}(y_i\neq h_t(x_i))]$$</div>
</div>
<div class="formula-card">
<div class="formula-name">AdaBoost - Model Weight</div>
<div class="formula">$$\alpha_t=\frac{1}{2}\ln\frac{1-\varepsilon_t}{\varepsilon_t}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Gradient Boosting - Update</div>
<div class="formula">$$F_m(x)=F_{m-1}(x)+\gamma_m h_m(x)$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Gradient Boosting - Residual</div>
<div class="formula">$$r_{im}=-\left[\frac{\partial L(y_i,F(x_i))}{\partial F(x_i)}\right]_{F=F_{m-1}}$$</div>
</div>
</div>
</div>
</section>
<!-- 15. BIAS-VARIANCE TRADEOFF -->
<section id="bias-variance" class="section" aria-labelledby="bv-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">15</div>
<h2 id="bv-title" class="section-title">Bias-Variance Tradeoff</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">⚖️ Error Decomposition</div>
<div class="formula-card">
<div class="formula-name">Total Error</div>
<div class="formula">$$E[(y-\hat{y})^2]=\text{Bias}^2+\text{Variance}+\text{Irreducible Error}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Bias</div>
<div class="formula">$$\text{Bias}=E[\hat{y}]-y$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Variance</div>
<div class="formula">$$\text{Variance}=E[(\hat{y}-E[\hat{y}])^2]$$</div>
</div>
<div class="info-box">
<strong>💡 Key Insights:</strong><br/>
• <strong>High Bias</strong> → Underfitting (model too simple)<br/>
• <strong>High Variance</strong> → Overfitting (model too complex)<br/>
• <strong>Goal:</strong> Find the optimal balance between bias and variance
</div>
</div>
</div>
</section>
<!-- 16. QUICK REFERENCE: LOSS FUNCTIONS -->
<section id="loss-functions" class="section" aria-labelledby="loss-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">16</div>
<h2 id="loss-title" class="section-title">Quick Reference: Loss Functions</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">📉 Regression Losses</div>
<table class="formula-table">
<tr><th>Loss</th><th>Formula</th><th>Use Case</th></tr>
<tr><td>MSE</td><td>$$L=\frac{1}{n}\sum_{i=1}^n(y_i-\hat{y}_i)^2$$</td><td>Standard regression</td></tr>
<tr><td>MAE</td><td>$$L=\frac{1}{n}\sum_{i=1}^n|y_i-\hat{y}_i|$$</td><td>Robust to outliers</td></tr>
<tr><td>Huber</td><td>$$L=\begin{cases}\frac{1}{2}(y-\hat{y})^2&|y-\hat{y}|\leq\delta\\\delta|y-\hat{y}|-\frac{1}{2}\delta^2&\text{otherwise}\end{cases}$$</td><td>Combines MSE & MAE</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">🎯 Classification Losses</div>
<table class="formula-table">
<tr><th>Loss</th><th>Formula</th><th>Use Case</th></tr>
<tr><td>Binary Cross-Entropy</td><td>$$L=-[y\log(\hat{y})+(1-y)\log(1-\hat{y})]$$</td><td>Binary classification</td></tr>
<tr><td>Categorical Cross-Entropy</td><td>$$L=-\sum_i y_i\log(\hat{y}_i)$$</td><td>Multi-class classification</td></tr>
<tr><td>Hinge Loss</td><td>$$L=\max(0,1-y\cdot\hat{y})$$</td><td>SVM classification</td></tr>
</table>
</div>
</div>
</section>
<!-- 17. FEATURE ENGINEERING -->
<section id="feature-engineering" class="section" aria-labelledby="fe-title">
<div class="section-header" onclick="toggleSection(this)">
<div class="section-number">17</div>
<h2 id="fe-title" class="section-title">Feature Engineering</h2>
<div class="section-toggle">▼</div>
</div>
<div class="section-content">
<div class="subsection">
<div class="subsection-title">📊 Normalization Techniques</div>
<table class="formula-table">
<tr><th>Method</th><th>Formula</th><th>Range</th></tr>
<tr><td>Min-Max Scaling</td><td>$$x'=\frac{x-\min}{\max-\min}$$</td><td>[0, 1]</td></tr>
<tr><td>Z-Score Normalization</td><td>$$x'=\frac{x-\mu}{\sigma}$$</td><td>~ [-3, 3]</td></tr>
<tr><td>Max Abs Scaling</td><td>$$x'=\frac{x}{|\max|}$$</td><td>[-1, 1]</td></tr>
</table>
</div>
<div class="subsection">
<div class="subsection-title">🔢 Polynomial Features</div>
<div class="formula-card">
<div class="formula-name">Degree 2</div>
<div class="formula">$$[x_1, x_2] \rightarrow [1, x_1, x_2, x_1^2, x_1 x_2, x_2^2]$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Degree 3</div>
<div class="formula">$$[x_1, x_2] \rightarrow [1, x_1, x_2, x_1^2, x_1 x_2, x_2^2, x_1^3, x_1^2 x_2, x_1 x_2^2, x_2^3]$$</div>
</div>
<div class="info-box">
<strong>💡 Tip:</strong> Polynomial features help capture non-linear relationships in data, but be careful of overfitting with high degrees.
</div>
</div>
<div class="subsection">
<div class="subsection-title">🏷️ Encoding Categorical Variables</div>
<div class="formula-card">
<div class="formula-name">One-Hot Encoding</div>
<div class="formula">$$\text{Category } c \rightarrow [0, 0, \ldots, 1, \ldots, 0] \text{ (1 at position } c\text{)}$$</div>
</div>
<div class="formula-card">
<div class="formula-name">Label Encoding</div>
<div class="formula">$$\text{Categories } \{A, B, C\} \rightarrow \{0, 1, 2\}$$</div>
</div>
</div>
</div>
</section>
<!-- CONTRIBUTOR SECTION -->
<section id="contributor" class="section" style="text-align:center;">
<div class="section-header" onclick="toggleSection(this)">