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SI&DA - Recap of Previous Courses - Mind Map

Jul 31, 20244 min read

Random Variables

SI&DA - Definition of ‘CDF (Cumulative Distribution Function)‘
SI&DA - Definition of ‘PDF (Probability Density Function)‘
Uniform PDF
Gaussian PDF
SI&DA - Properties of the CDF and PDF
SI&DA - Multivariate Distributions
SI&DA - Definition of ‘Joint CDF’
SI&DA - Definition of ‘Joint PDF’
SI&DA - Definition of ‘Marginal PDF’
SI&DA - Mean & Variance (and Standard Deviation)
SI&DA - Definition of ‘Confidence Interval’
SI&DA - Definition of ‘Covariance Matrix’
SI&DA - Definition of ‘Independent Random Variables’
SI&DA - Definition of ‘Uncorrelated Random Variables’
SI&DA - Theorem ‘Independent RVs are also Uncorrelated’
SI&DA - Gaussian Random Variables
SI&DA - Theorem ‘Independent or Uncorrelated Gaussian RVs’
SI&DA - Property of Multivariate Gaussian RVs
SI&DA - Theorem ‘Affine RV’
SI&DA - Definition of ‘Cross-Covariance’
SI&DA - Central Limit Theorem
SI&DA - Functions of RVs
SI&DA - Multivariate Functions of RVs
SI&DA - Definition of ‘Conditional Distribution’
SI&DA - Definition of ‘Conditional Mean’
SI&DA - Definition of ‘Conditional Variance and Covariance’
Important Formulas for Gaussian Conditional Distribution

NOTE: The new mean depends on the observation , while the new covariance does not it depends only on the covariance and cross-covariance of and .

Estimation Theory

SI&DA - Difference from Parametric and Bayesian Estimation

SI&DA - Parametric Estimation

SI&DA - Definition of ‘Estimator’
SI&DA - Definition of ‘Unbiased Estimator’
SI&DA - Definition of ‘Consistent Estimator’
SI&DA - Theorem ‘Sequence of Unbiased Estimators’
SI&DA - Theorem ‘Law of Large Numbers’
SI&DA - Definition of ‘MSE (Mean Square Error)‘
SI&DA - Definition of ‘MSE Estimator Comparison’
SI&DA - Definition of ‘Uniformly Minimum Variance Unbiased Estimator (UMVUE)‘
SI&DA - Definition of ‘BLUE (Best Linear Unbiased Estimator)‘
SI&DA - Theorem ‘The Cramer-Rao Bound’
SI&DA - Definition of ‘Efficient Estimator’
SI&DA - Property the Fisher Information Matrix for i.i.d. RVs
SI&DA - Properties of the Sample Mean
SI&DA - Definition of ‘Maximum Likelihood (ML) Estimators’
SI&DA - Definition of ‘Logarithmic Likelihood Function’
SI&DA - Definition of ‘Gauss-Markov Estimator’
SI&DA - Least Square Estimator
SI&DA - Summary of GM and LS estimator Properties

Bayesian Estimation

SI&DA - Definition of ‘Bayes Risk Function’
SI&DA - Definition of ‘Minimum Square Error Estimator’
SI&DA - Definition of ‘Linear Minimum Square Error Estimator’
SI&DA - Theorem ‘Schur Complement’
SI&DA - Discrete solution to the Linear Minimum Square Error Estimate

SI&DA - Definition of ‘Stochastic Process’

SI&DA - Difference between Random Variable and Realization of a Stochastic Process
SI&DA - CDF and PDF of a Stochastic Process
SI&DA - First and Second Order Statistics
SI&DA - Mean and Covariance of an SP
SI&DA - Definition of ‘Strong Stationarity’
SI&DA - Definition of ‘Weak Stationarity’
SI&DA - Theorem ‘Covariance Function of a Stationary SP’
SI&DA - Definition of ‘Joint Stationarity’
SI&DA - Properties of the Covariance Matrix of SPs
SI&DA - Definition of ‘Normalized Covariance’
Type of Stochastic Processes
SI&DA - Definition of ‘Purely Deterministic SP’
SI&DA - Definition of ‘White SP’
SI&DA - Definition of ‘Wiener Process (or Brownian Motion)‘
SI&DA - Definition of ‘Exponentially Correlated SPs’

SI&DA - Linear Stochastic Systems

SI&DA - Time Shift Operator
SI&DA - Characteristic Polynomial
SI&DA - Free and Forced Response of the System
SI&DA - Transformation of an SP
SI&DA - Definition of ‘Spectrum’
SI&DA - Definition of ‘Spectral Density’
SI&DA - Definition of ‘Cross-Spectrum’
SI&DA - Properties of the Spectrum, Spectral Density and Cross-Spectrum
SI&DA - A General Result for Linear Stochastic Systems
SI&DA - Definition of ‘Moving Average (MA) SP’
SI&DA - Definition of ‘Auto-regressive (AR) SP’
SI&DA - Definition of ‘Auto-regressive Moving Average (ARMA) SP’

SI&DA - Time Series Prediction

SI&DA - Spectral Factorization
SI&DA - k-Step Ahead Predictor
SI&DA - k-Step Ahead Prediction Error

Graph View

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