wpmu dev
Table of Contents
Table 1. Probabilistic approaches comparison
| Name | Overview | Pros & Cons |
| Bayesian Methods | These methods incorporate prior knowledge along with observed data to update beliefs and quantify uncertainty. Example: Bayesian Structural Time Series (BSTS) Data requirements: – Time series data: Regularly spaced observations over time. – Covariates: Optional external variables that can be included as regressors. – Prior information: Optional but beneficial for informing the model. Example: PyMC Data requirements: – Custom data: Flexible to work with any dataset as long as it can be modeled probabilistically. – Prior information: Required for Bayesian inference. – Time series/Cross-sectional data: Depends on the specific model being implemented. | Advantages: Flexible, incorporates prior information, and provides a full distribution of possible outcomes. Disadvantages: Computationally intensive, and requires expertise in Bayesian statistics. |
| Ensemble Methods | Combines multiple models to improve predictive performance and quantify uncertainty. Example: Bootstrap Aggregating (Bagging) Data requirements: – Large dataset: Sufficiently large to benefit from resampling. – Independent observations: The underlying assumption is that observations are independent. Example: Quantile Regression Forests Data requirements: – Predictors and response variable: Both continuous and categorical predictors can be used. | Advantages: Robust, and often improves accuracy and reliability of forecasts. Disadvantages: Can be complex to implement and interpret, and is computationally expensive. |
| State Space Models | Models that describe the evolution of a system’s state over time, incorporating both observations and hidden states. Example: Kalman Filter Data requirements: – Linear time series data: Suitable for linear Gaussian models. – Observations and control inputs: Required to define state transitions and measurements. – Noise parameters: Assumptions about process and measurement noise. Example: Particle Filter Data requirements: – Non-linear time series data: Suitable for non-linear and non-Gaussian models. – Observations and control inputs: To define state transitions and measurements. | Advantages: Well-suited for handling time series with underlying state changes, and can model various types of noise and dynamics. Disadvantages: Can be complex to implement, especially for non-linear systems. |
| Generalized Additive Models (GAMs) | Flexible models that allow for non-linear relationships between the predictors and the response variable. Example: Prophet Data requirements: – Daily observations: Time series data with daily frequency, though it can handle missing days. – Seasonality indicators: Optional indicators for yearly, weekly, and daily seasonality. – Holiday data: Optional but can improve accuracy for business-related forecasting. | Advantages: Interpretable, handles missing data and outliers well, and is suitable for time series with seasonality and trends. Disadvantages: Less flexible than fully custom Bayesian models, and makes strong assumptions about the data structure. |
| Quantile Regression | Predicts specific quantiles (percentiles) of the response variable distribution, providing a full picture of the potential outcomes. Example: Quantile Regression Data requirements: – Predictors and response variable: Both continuous and categorical predictors can be used. – Sufficient observations: Enough data to estimate different quantiles robustly. | Advantages: Simple to implement, and provides clear quantiles for uncertainty estimates. Disadvantages: Assumes a fixed quantile structure, and may not capture complex dependencies as well as other methods. |
| Deep Learning Approaches | Use neural networks to model complex patterns in data and generate probabilistic forecasts. Example: Bayesian Neural Networks DeepAR Data requirements: – Large dataset: Typically requires a substantial amount of data to train effectively. | Advantages: Can capture complex patterns and interactions in large datasets, and is scalable. Disadvantages: Requires large amounts of data, and can be a “black box” with less interpretability. |
| Gaussian Processes | Models the distribution over possible functions that fit the data. Example: Gaussian Process Regression Data requirements: – Moderate dataset size: Computationally expensive, so best suited for moderate-sized datasets. – Kernel choice: Requires an appropriate choice of kernel function to define the covariance structure. | Advantages: Flexible, provides a natural way to quantify uncertainty, and is good for small datasets. Disadvantages: Computationally expensive for large datasets, and can be complex to implement. |
Table 2. Probabilistic forecasting tools
| Tool name | Overview | Pros & Cons |
| PyMC | Probabilistic programming library in Python focused on Bayesian statistical modeling and inference. Suitable use case: Complex hierarchical models, custom probabilistic models, and detailed uncertainty quantification. | Advantages: – Flexibility: Allows for building complex custom models. – Bayesian inference: Comprehensive tools for MCMC and variational inference. – Visualization: Strong support for model diagnostics and posterior visualization. – Integration: Works well with NumPy, SciPy, and pandas. Disadvantages: – Learning curve: Requires understanding of Bayesian statistics. – Performance: MCMC can be slow for large datasets or complex models. – Verbose: More code required for simple models compared to specialized libraries. |
| Prophet | Forecasting tool developed by Facebook designed for time series data with strong seasonal effects and holiday impacts. Suitable use case: Business forecasting, time series data with strong seasonality and holiday effects, quick and interpretable results. | Advantages: – Ease of use: User-friendly with minimal parameter tuning. – Automatic handling: Deals with missing data, outliers, and holidays. – Interpretability: Clear and interpretable model components (trend, seasonality, holidays). Disadvantages: – Limited flexibility: Fixed model structure not suitable for complex custom models. – Assumptions: Makes strong assumptions about seasonality and trends. |
| Pyro | Probabilistic programming library built on PyTorch, designed for deep probabilistic modeling. Suitable use case: Deep probabilistic models, large-scale data, and integration with neural networks. | Advantages: – Flexibility: Supports deep probabilistic models and variational inference. – Integration with PyTorch: Leverages PyTorch’s capabilities for deep learning. – Scalability: Handles large datasets and complex models efficiently. Disadvantages: – Complexity: Requires knowledge of both probabilistic modeling and PyTorch. – Learning curve: Steeper learning curve due to its flexibility and power. |
| Orbit | Open-source package developed by Uber for Bayesian time series forecasting. Suitable use case: Time series forecasting, and business applications needing scalable and specialized tools. | Advantages: – Specialization: Designed specifically for time series forecasting. – Ease of use: High-level interface for common forecasting tasks. – Scalability: Optimized for performance with large datasets. Disadvantages: – Flexibility: Less flexible for custom probabilistic models compared to general-purpose libraries. – Community support: Smaller user community compared to more established libraries. |
| NumPyro | Lightweight probabilistic programming library that leverages JAX for accelerated computations. Suitable use case: High-performance probabilistic modeling, large datasets, and leveraging JAX’s capabilities. | Advantages: – Performance: Very fast due to JAX’s just-in-time compilation and automatic differentiation. – Flexibility: Supports a range of probabilistic models. – Scalability: Efficient handling of large datasets. Disadvantages: – Ecosystem: Less mature ecosystem compared to PyMC or Stan. – Learning curve: Requires familiarity with JAX and probabilistic modeling. |
| Stan | State-of-the-art platform for statistical modeling and high-performance statistical computation. Suitable use case: High-performance Bayesian inference, complex and precise statistical models, and applications requiring rigorous statistical accuracy. | Advantages: – Performance: Highly optimized for Bayesian inference with efficient sampling algorithms. – Flexibility: Supports a wide range of models. Disadvantages: – Complexity: Requires understanding of Stan’s modeling language and Bayesian statistics. – Learning curve: Steeper learning curve compared to more user-friendly libraries. |
