Short notes and write-ups I publish as I explore ideas — mostly on Bayesian methods, time-series modeling, and traffic behavior.
This post is part of a series that supports and promotes our recent document. It collects three parts into one page so readers can follow the full narrative from basic HMMs to a factorial...
Autoregressive (AR) processes are a class of time series models used to describe a variable that is correlated with its past values. AR models are widely applied in various fields such as economics, engineering,...
Probabilistic graphical models (PGMs) provide a compact, visual language for reasoning about joint distributions over many random variables. In directed acyclic graphs (DAGs), three elementary three-node structures — tail-to-tail, head-to-tail, and head-to-head — serve...
In autonomous driving, modeling and understanding the interactions between the ego vehicle and its surrounding vehicles is crucial for safe and efficient navigation. One important challenge is dealing with varying traffic densities and dynamic...
Time-series forecasting is a critical application of Gaussian Processes (GPs), as they offer a flexible and probabilistic framework for predicting future values in sequential data. GPs not only provide point predictions but also quantify...
In this post, we’ll explore four important concepts in time series modeling and stochastic processes: Autoregressive processes, Cochrane-Orcutt correction, Ornstein-Uhlenbeck (OU) processes, and Gaussian processes (GPs). After explaining each concept, we will also examine...
Autocorrelation is a key property of time series data, describing the dependency of a variable on its past values. Both the Fourier Transform (FT) and Gaussian Processes (GP) can model autocorrelation, but they operate...
Ordinary Least Squares (OLS) is one of the most widely used methods for linear regression. It provides unbiased estimates of the model parameters under the assumption that the error terms are independent and identically...
Consider the linear regression model: \(\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon},\) where: \(\mathbf{y}\) is an \(n \times 1\) vector of observations. \(\mathbf{X}\) is an \(n \times p\) design matrix (full column rank). \(\boldsymbol{\beta}\) is a...
In many driving behavior studies, we model how a following vehicle responds to the movement of a lead vehicle. For example, the Intelligent Driver Model (IDM) uses a set of parameters \(\boldsymbol{\theta} = (v_0,...
Hierarchical models are powerful tools in statistical modeling and machine learning, enabling us to represent data with complex dependency structures. These models are particularly useful in contexts where data is naturally grouped or exhibits...
The log-sum-exp trick is a critical technique in numerical computations involving logarithms and exponentials. It is widely used in machine learning, especially in algorithms like the forward-backward procedure in Hidden Markov Models ( HMMs)....
Problem: Solve $\frac{\partial\left|\boldsymbol{A}\circ (\boldsymbol{Y}-\boldsymbol{W}^\top\boldsymbol{X})\right|_ {F}^{2}}{\partial\boldsymbol{W}}$ and $\frac{\partial\left|\boldsymbol{A}\circ ( \boldsymbol{Y}-\boldsymbol{W}^\top\boldsymbol{X})\right|_{F}^{2}}{\partial\boldsymbol{X}}$, where $\circ$ denotes the Hadamard product, and all variables are matrices.