Statistical Analysis

This section introduces the principles and applications of practical statistical models, especially the core subjects of regression analysis and time series analysis. For most topics we provide R example code using basic built-in datasets that require no downloads, so feel free to copy and paste to learn.

Regression Analysis

Multiple Regression Analysis

• What is Regression Analysis?
• Simple Regression Analysis
  • How to Read Simple Regression Results in R
• Multiple Regression Analysis
  • How to Read Multiple Regression Results in R
• Definition of Regression Coefficients and Derivation of Estimator Formulas
  • Proof of $\text{SST} = \text{SSR} + \text{SSE}$
• Variable Selection Procedures in Statistical Analysis
• Variable Selection Criteria in Statistical Analysis

Residual Analysis

• Model Diagnostics in Regression Analysis
  • Checking Linearity of Residuals via Model Diagnostics
  • Checking Homoscedasticity of Residuals via Model Diagnostics
  • Checking Independence of Residuals via Model Diagnostics
  • Checking Normality of Residuals via Model Diagnostics
• Estimators of Residual Variance and Standard Errors of Regression Coefficients in Multiple Regression
• Regression Analysis with Qualitative Variables
  • Interaction in Regression Analysis

Multicollinearity

• Multicollinearity
  • Variance Inflation Factor VIF
• Principal Component Analysis in Statistics
  • How to Perform Principal Component Regression in R
• Resolving not defined because of singularities in R Regression

Derived Models

• Logistic Regression Analysis
  • How to Read Logistic Regression Results in R
• Nonlinear Regression Analysis: Variable Transformation in Regression
• Sparse Regression
  • Uniform Uncertainty Principle: Restricted Isometry Condition
• Ridge Regression
• LASSO Regression
• Compressed Sensing
• SINDy
  • STLSQ
• 🔒(26/01/06)WSINDy

Time Series Analysis

• What is Time Series Analysis?
• White Noise
• Stationarity
  • Differencing $\nabla$ in Time Series Analysis
  • Transformations in Time Series Analysis

Internal Factors

• Moving Average Process $\text{MA}$
  • Autocorrelation Function ACF
• Autoregressive Process $\text{AR}$
  • Partial Autocorrelation Function PACF
• ARMA Model $\text{ARMA}$
  • Invertibility of ARMA Models
  • Extended Autocorrelation Function EACF
  • Selecting ARMA Models with EACF in R
• ARIMA Model $\text{ARIMA}$
  • How to Analyze Time Series with ARIMA Models in R
  • How to Read ARIMA Model Results in R
  • How to Forecast with ARIMA Models in R
  • Residual Analysis for ARIMA Models
  • Drift in ARIMA Models
• Seasonal ARIMA Model $\text{SARIMA}$
• Fractional ARIMA Model $\text{FARIMA}$

External Factors

• Time Series Regression Analysis
  • Cross-Correlation Function CCF
  • Prewhitening prewhiten()
  • Spurious Correlation in Time Series Regression
• Intervention Analysis
  • Step Function and Pulse Function
  • Additive Outlier
  • Innovative Outlier
• Dynamic Regression Model $\text{ARIMAX}$

Heteroskedasticity

• GARCH Model $\text{GARCH}$
  • Heteroskedasticity and Volatility Clustering in Time Series Analysis
  • ARCH Effect
  • How to Analyze Time Series with GARCH Models in R

Spatial Statistics Analysis

• What is Spatial Data Analysis?
• How to Open shp Files in QGIS
• PROJ in Geostatistics

Matheron School^{Matheron school}

• Spatial Process
• Stationarity of Spatial Processes
• Variogram $2 \gamma \left( \mathbf{h} \right)$
• Isotropy of Variograms $d = \left| \mathbf{h} \right|$
  • Semivariogram Models $\gamma \left( h \right)$
• Empirical Variogram $\gamma^{\ast}, \hat{\gamma}$
• Ordinary Kriging
• Universal Kriging

Areal Data Analysis

• Proximity Matrix
• Moran’s I
• Geary’s C
• Conditional Autoregressive Model CAR
• Spatial Autoregressive Model SAR
• Spatial Error Model SEM
• Spatial Durbin Model SDM

Main References

• Banerjee. (2015). Hierarchical Modeling and Analysis for Spatial Data(2nd Edition)
• Brunton. (2022). Data-Driven Science and Engineering : Machine Learning, Dynamical Systems, and Control(2nd Edition)
• Cryer. (2008). Time Series Analysis: With Applications in R(2nd Edition)
• Hadi. (2006). Regression Analysis by Example(4th Edition)
• James. (2013). An Introduction to Statistical Learning with Applications in R