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Volatility analysis quantifies asset price variation, critical for risk management, option valuation, and portfolio optimization. For Apple Inc., volatility is driven by investor sentiment, economic fundamentals, and external shocks. This study focuses on classical ARCH/GARCH and Bayesian SV models to analyze Apple’s daily volatility dynamics.

This is the heat map of crime locations in Auckland that we have drawn from June 2024 to June 2025. The higher the heat (the darker the red), the more active the criminal activities.

This scatter plot visualizes the relationship between flipper length (in mm) and body mass (in grams) for three penguin species — Adelie, Chinstrap, and Gentoo — using the Palmer Penguins dataset. Each point represents an individual penguin, with different colors and shapes indicating species. The plot reveals that Gentoo penguins generally have longer flippers and higher body mass, Adelie penguins tend to have shorter flippers and lower mass, while Chinstrap penguins fall in between. This visualization highlights species-specific morphological differences.

Abstract Interval-censored survival data, where events are known only within observation intervals, require methods beyond standard right-censoring approaches. We assess the Expectation–Maximization (EM) nonparametric maximum likelihood estimator (Turnbull) against Kaplan–Meier (KM, pseudo right-censored) and parametric accelerated failure time (AFT) models. Simulations under Weibull and log-normal mechanisms show that EM achieves low integrated squared error (ISE), accurately recovering the true survival shape without distributional assumptions. Weibull AFT models, while assumption-dependent, yielded smoother curves and lower integrated Brier scores (IBS) when covariates were included. A real-world ovarian cancer example demonstrated EM’s robustness and Bayesian AFT’s ability to propagate uncertainty and perform posterior predictive checks. We propose a workflow: use EM to reveal baseline shape, then apply (Bayesian) AFT for covariate-adjusted prediction and uncertainty quantification.