User:MathKnight-at-TAU




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 * Help:formula

Useful mathematics

 * Classification Theory
 * Binary classifier
 * Sensitivity and Specificity
 * Receiver operating characteristic (ROC)
 * Basic Probability and Statistics
 * Probability $$\left( \Omega, \mathcal{A}, \mathbb{P} \right) $$
 * Statistics
 * Expected value $$\mathbb{E}(X) = \sum_i x_i \cdot \mathbb{P}(X=x_i) = \int x \, \mathrm{d}\mathbb{P}(X=x) = \int x f(x) \mathrm{d}x$$
 * Variance and Standard deviation $$ \sigma^2 = \operatorname{Var}(X) = \operatorname{cov}(X,X) = \mathbb{E}\big[ (X-\mathbb{E}[X])^2 \big] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$$
 * Covariance $$\sigma_{X,Y}^2 = \operatorname{cov}(X,Y) = \mathbb{E}\big[ (X-\mathbb{E}[X])(Y-\mathbb{E}[Y]) \big] = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]$$
 * Covariance matrix $$\Sigma = \mathbb{E}\left[ (\mathbf{X} - \mathbb{E}[\mathbf{X}])(\mathbf{X} - \mathbb{E}[\mathbf{X}])^{\rm T} \right]$$, if $$\mathbf{X} = (X_i)_{i=1}^n$$ then $$\Sigma_{i,j} = \operatorname{cov}(X_i,X_j)$$
 * Conditional probability $$ \mathbb{P}(A \mid B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} $$
 * Bayes' theorem $$\mathbb{P}(A \mid B) = \frac{\mathbb{P}(B \mid A) \, \mathbb{P}(A)}{\mathbb{P}(B)}$$
 * Independence (probability theory) $$A \perp\!\!\!\perp B \iff \mathbb{P}(A \cap B) = \mathbb{P}(A)\mathbb{P}(B) \iff \mathbb{P}(A) = \mathbb{P}(A \mid B) \iff \mathbb{P}(B) = \mathbb{P}(B \mid A)$$
 * Conditional expectation
 * Normal distribution (Gaussian) $$ X \sim \mathcal{N}(\mu,\sigma^2) $$ means $$ \mathrm{d}\mathbb{P}(x) = f(x \mid \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2} } e^{ -\frac{(x-\mu)^2}{2\sigma^2} } = \frac{1}{\sqrt{2\pi\sigma^2} } \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right) $$
 * Statistics
 * Statistics
 * Statistical hypothesis testing
 * p-value
 * Linear regression
 * Spearman's rank correlation coefficient
 * Likelihood function $$\mathcal{L}(\theta \mid x) = p_\theta (x) = \mathbb{P}_\theta (X=x) $$
 * Maximum likelihood estimation (MLE) $$\hat{\theta} = \operatorname{arg \, \max} \{ \mathcal{L}(\theta \mid x) \ \mid \ \theta \in \Theta \}$$
 * Signal Processing
 * Signal processing
 * Time series
 * Fourier analysis
 * Fourier transform and Laplace transform
 * Discrete-time Fourier transform (DTFT) $$X_{2\pi}(\omega) = \sum_{n=-\infty}^{\infty} x[n] \,e^{-i \omega n} \ \implies \ X_{1/T}(f) = X_{2\pi}(2\pi f T)\ \stackrel{\mathrm{def}}{=} \sum_{n=-\infty}^{\infty} \underbrace{T\cdot x(nT)}_{x[n]}\ e^{-i 2\pi f T n}\; \stackrel{\mathrm{Poisson\;f.}}{=} \; \sum_{k=-\infty}^{\infty} X\left(f - k/T\right)$$
 * Convolution $$ (f \ast g)(\tau) = \int_{-\infty}^{\infty} f(t)g(\tau-t)\mathrm{d}t = \int_{-\infty}^{\infty} f(\tau-t)g(t)\mathrm{d}t$$
 * Cross-correlation  $$ (f \star g)(\tau) = \int_{-\infty}^{\infty} f^*(t)g(t+\tau)\mathrm{d}t = \int_{-\infty}^{\infty} f(t-\tau)g(t)\mathrm{d}t$$
 * Auto-correlation
 * Autocovariance: for $$(X_t)_{t \in \mathbb{N}}$$: $$C_{XX}(t,s) = \operatorname{cov}(X_t,X_s) = \mathbb{E}[X_t X_s] - \mathbb{E}[X_t]\mathbb{E}[X_s]$$
 * Low-pass filter
 * Recursive least squares filter (RLS)
 * Kalman filter (excellent introduction here)
 * Autoregressive–moving-average model (ARMA) = Autoregressive model (AR) and Moving-average model (MA)
 * Savitzky-Golay filter
 * Window function
 * Differential equations
 * Partial differential equation (PDE)
 * Finite element method (FEM)
 * Mathematical programming and Numerical analysis
 * Numerical analysis
 * MATLAB
 * Principal Components Analysis
 * Delaunay triangulation
 * COMSOL
 * Fast Fourier transform (FFT)
 * Gradient descent

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Useful Links

 * Help:formula
 * Gothic metal: List of gothic metal bands
 * Triceratops
 * Dinosaur size
 * User:MathKnight/Sandbox

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