Statistical Analysis

Statistical models, particularly those considered core subjects like regression and time series analysis, are a practical aspect of statistics. This section introduces the principles and applications of these models, with examples using basic built-in datasets in R for easy copy-pasting and learning.

Regression Analysis

Multiple Regression Analysis

• What is Regression Analysis?
• Simple Regression Analysis
  • Interpreting Simple Regression Results in R
• Multiple Regression Analysis
  • Interpreting Multiple Regression Results in R
• Definition of Regression Coefficients and Derivation of Estimators
  • 🔒(25/02/22) 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 in Model Diagnostics
  • Checking Homoscedasticity of Residuals in Model Diagnostics
  • Checking Independence of Residuals in Model Diagnostics
  • Checking Normality of Residuals in Model Diagnostics
• Estimates 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
  • Performing Principal Component Regression in R
• Solving not defined because of singularities in R Regression Analysis

Derived Models

• Logistic Regression Analysis
  • Interpreting Logistic Regression Results in R
• Nonlinear Regression Analysis: Variable Transformation in Regression
• Sparse Regression
• Ridge Regression
• LASSO Regression
• Compressed Sensing
• Sparse Identification of Nonlinear Dynamics (SINDy)
  • STLSQ

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 $MA$
  • Autocorrelation Function (ACF)
• Autoregressive Process $AR$
  • Partial Autocorrelation Function (PACF)
• ARMA Model $ARMA$
  • Reversibility of ARMA Models
  • Extended Autocorrelation Function (EACF)
  • Selecting ARMA Models Using EACF in R
• ARIMA Model $ARIMA$
  • Analyzing Time Series with ARIMA Models in R
  • Interpreting ARIMA Model Analysis Results in R
  • Forecasting with ARIMA Models in R
  • Residual Analysis for ARIMA Models
  • Drift in ARIMA Models
• Seasonal ARIMA Models

External Factors

• Time Series Regression Analysis
  • Cross-Correlation Function (CCF)
  • Prewhitening prewhiten()
  • Spurious Correlations in Time Series Regression Analysis
• Intervention Analysis
  • Step and Pulse Functions
  • Additive Outliers
  • Innovative Outliers
• Dynamic Regression Model $ARIMAX$

Heteroskedasticity

• GARCH Model $GARCH$
  • Heteroskedasticity and Volatility Clustering in Time Series Analysis
  • ARCH Effect
  • Analyzing Time Series with GARCH Models in R

Spatial Statistics Analysis

• What is Spatial Data Analysis?
• Opening shp Files with QGIS
• PROJ in Geostatistics

Matheron School

• Spatial Processes
• Stationarity of Spatial Processes
• Variogram $2 \gamma \left( \mathbf{h} \right)$
• Isotropy in 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 Model)

• Spatial Autoregressive Model (SAR)
• Spatial Error Model (SEM)
• Spatial Durbin Model (SDM)

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