Formulas for Data Wizards
In my graduate program the calculus we learn is specifically geared towards data science. I’ve encounter several formulas that are relevant to analyzing and modeling data.
Key Equations:
1. Derivative Rules: To calculate derivatives in Python, you can use various libraries, such as SymPy or SciPy.
- Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1).
- Chain Rule: If g(x) = f(h(x)), then g'(x) = f'(h(x)) * h'(x).
- Product Rule: (f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x).
- Quotient Rule: (f(x) / g(x))' = (f'(x) * g(x) - f(x) * g'(x)) / g(x)^2.
2. Integral Techniques:
- Integration by Parts: ∫ u * v dx = u * ∫ v dx - ∫ (u' * ∫ v dx) dx.
- Integration by Substitution: ∫ f(g(x)) * g'(x) dx = ∫ f(u) du.
- Partial Fractions: Decomposing a rational function into simpler fractions.
3. Taylor Series:
- Taylor Series Expansion: f(x) = f(a) + f'(a)(x - a) + (f''(a)/2!)(x - a)^2 + ... + (f^n(a)/n!)(x - a)^n + ...
- Maclaurin Series: Taylor series expansion centered around a = 0.
- First-Order Linear Differential Equation: dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x.
- Separable Differential Equation: dy/dx = g(x) * h(y).
- Initial Value Problem (IVP): A differential equation with an initial condition.
- Partial Derivatives: ∂f/∂x, ∂f/∂y, etc., for functions of multiple variables.
- Gradient: ∇f = (∂f/∂x, ∂f/∂y, ...), representing the vector of partial derivatives.
- Hessian Matrix: A matrix of second partial derivatives of a function.
- Multiple Integrals: Integration over regions in multiple dimensions.
6. Optimization:
- Critical Points: Points where the derivative is zero or undefined.
- Extreme Values: Determining maximum and minimum values of a function.
- Lagrange Multipliers: Method for constrained optimization.
These are a few examples of the equations you may encounter in graduate-level calculus for data science. It's always a good practice to refer to your course materials and textbooks for a comprehensive list of formulas relevant to your specific coursework.
Common Complex Equations:
1. Linear Regression Formula:
y = β₀ + β₁x₁ + β₂x₂ + ... + βₙxₙ + ɛ
Where y is the dependent variable, β₀, β₁, β₂, ..., βₙ are the coefficients, x₁, x₂, ..., xₙ are the independent variables, and ɛ represents the error term.
2. Logistic Regression Formula:
p = 1 / (1 + e^(-z))
Where p is the probability of the binary outcome, e is Euler's number (approximately 2.71828), and z is the linear combination of the predictors.
P(c|x₁, x₂, ..., xₙ) = P(c) * P(x₁|c) * P(x₂|c) * ... * P(xₙ|c) / P(x₁, x₂, ..., xₙ)
Where P(c|x₁, x₂, ..., xₙ) is the posterior probability of class c given the features x₁, x₂, ..., xₙ, P(c) is the prior probability of class c, P(x₁|c), P(x₂|c), ..., P(xₙ|c) are the conditional probabilities of each feature given class c, and P(x₁, x₂, ..., xₙ) is the probability of observing the features x₁, x₂, ..., xₙ.
4. Support Vector Machines (SVM) Formula:
w^T x + b = 0
Where w is the weight vector, x is the input vector, b is the bias term, and the equation represents the decision boundary that separates different classes.
5. K-means Clustering Formula:
J = ∑ ||xᵢ - μⱼ||²
Where J is the objective function representing the sum of squared distances between each data point xᵢ and its nearest cluster centroid μⱼ.
6. Principal Component Analysis (PCA) Formula:
Z = XW
Where Z is the matrix of principal components, X is the centered data matrix, and W is the matrix of eigenvectors.
θₙ₊₁ = θₙ - α * ∇J(θ)
Where θₙ is the current parameter value, α is the learning rate, ∇J(θ) is the gradient of the cost function J with respect to θ, and θₙ₊₁ is the updated parameter value.
8. Bayesian Inference Formula:
P(H|D) = P(D|H) * P(H) / P(D)
Where P(H|D) is the posterior probability of hypothesis H given the observed data D, P(D|H) is the likelihood of observing the data D given the hypothesis H, P(H) is the prior probability of hypothesis H, and P(D) is the probability of observing the data D.
These formulas represent a range of concepts and techniques used in data science, including regression, classification, clustering, dimensionality reduction, optimization, and probabilistic inference.
