Advanced Time Series Analysis Framework

Comprehensive framework for advanced time series analysis comparing probabilistic programming, deep learning, and gradient boosting methods. Features mathematical foundations, performance analysis, and practical implementations of Gaussian Processes, RNNs, LSTMs, GRUs, and XGBoost.

Overview

This framework provides a comprehensive comparison of advanced time series forecasting methods, bridging parametric vs. non-parametric and stochastic vs. deterministic approaches. We implement and compare five distinct methodologies:

Probabilistic Programming Approach

Gaussian Processes (GP) with PyMC3 for uncertainty quantification
Bayesian inference with MCMC sampling
Non-parametric approach with kernel-based learning

Deep Learning Approaches

Long Short-Term Memory (LSTM) networks with Keras/TensorFlow
Recurrent Neural Networks (RNN) with PyTorch
Gated Recurrent Units (GRU) for enhanced gradient flow
Autoregressive Neural Networks for sequence modeling

Gradient Boosting Methods

XGBoost Regressor for ensemble-based predictions
Tree-based ensemble learning with feature importance analysis

Mathematical Foundations

1. Gaussian Processes (Probabilistic Programming)

Gaussian Process Definition

A Gaussian Process is a collection of random variables, any finite number of which have a joint Gaussian distribution:

$f(x) \sim \mathcal{GP}(m(x), k(x, x'))$

Where:

$m(x) = \mathbb{E}[f(x)]$: Mean function
$k(x, x') = \mathbb{E}[(f(x) - m(x))(f(x') - m(x'))]$: Covariance kernel

Squared Exponential Kernel

We implement the RBF (Radial Basis Function) kernel:

$k(x, x') = \eta^2 \exp\left(-\frac{\rho}{2}||x - x'||^2\right)$

Where:

$\eta^2$: Signal variance (vertical scale control)
$\rho$: Length scale (smoothness control)
$\sigma^2$: Noise variance

Posterior Distribution

Given observations $\mathbf{y} = [y_1, ..., y_n]^T$ at inputs $\mathbf{X}$, the posterior for new points $\mathbf{X}_*$ is:

Posterior Mean: $\mu_{} = K_{}K^{-1}\mathbf{y}$

Posterior Covariance: $\Sigma_{} = K_{**} - K_{}K^{-1}K_{*}^T$

Where:

$K$: Covariance matrix for training points
$K_{*}$: Covariance between training and test points
$K_{**}$: Covariance matrix for test points

2. Long Short-Term Memory (LSTM) Networks

LSTM Cell State Dynamics

The LSTM addresses the vanishing gradient problem through gated memory mechanisms:

Forget Gate: $f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f)$

Input Gate: $i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i)$ $\tilde{C}t = \tanh(W_C \cdot [h{t-1}, x_t] + b_C)$

Cell State Update: $C_t = f_t * C_{t-1} + i_t * \tilde{C}_t$

Output Gate: $o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o)$ $h_t = o_t * \tanh(C_t)$

Where $\sigma$ is the sigmoid activation and $*$ denotes element-wise multiplication.

3. Recurrent Neural Networks (RNN)

Vanilla RNN Dynamics

Standard RNNs process sequences through hidden state recurrence:

$h_t = \tanh(W_{hh}h_{t-1} + W_{xh}x_t + b_h)$ $y_t = W_{hy}h_t + b_y$

Limitations:

Vanishing gradient problem for long sequences
Limited long-term memory capacity

4. Gated Recurrent Units (GRU)

GRU Gating Mechanism

GRUs simplify LSTM architecture while maintaining performance:

Reset Gate: $r_t = \sigma(W_r \cdot [h_{t-1}, x_t])$

Update Gate: $z_t = \sigma(W_z \cdot [h_{t-1}, x_t])$

New Hidden State: $\tilde{h}t = \tanh(W \cdot [r_t * h{t-1}, x_t])$ $h_t = (1 - z_t) * h_{t-1} + z_t * \tilde{h}_t$

5. XGBoost Gradient Boosting

Objective Function

XGBoost optimizes a regularized objective:

$\mathcal{L}(\phi) = \sum_i l(\hat{y}_i, y_i) + \sum_k \Omega(f_k)$

Where:

$l(\hat{y}_i, y_i)$: Loss function (MSE for regression)
$\Omega(f_k) = \gamma T + \frac{1}{2}\lambda ||\omega||^2$: Regularization term
$T$: Number of leaves, $\omega$: Leaf weights

Gradient Boosting Update

$\hat{y}_i^{(t)} = \hat{y}_i^{(t-1)} + f_t(x_i)$

Where $f_t$ is the $t$-th tree trained on residuals.

Methods Comparison

Parametric vs. Non-Parametric

Method	Type	Parameters	Flexibility
LSTM/RNN/GRU	Parametric	Fixed network weights	High (with sufficient capacity)
Gaussian Process	Non-parametric	Kernel hyperparameters	Very High (infinite capacity)
XGBoost	Non-parametric	Tree structure adaptive	High (data-driven splits)

Stochastic vs. Deterministic

Method	Nature	Uncertainty	Output
Gaussian Process	Stochastic	Full posterior distribution	Mean ± confidence intervals
LSTM/RNN/GRU	Deterministic	No native uncertainty	Point predictions
XGBoost	Deterministic	Feature importance only	Point predictions

Implementation Results

Gaussian Process with PyMC3

Gaussian Process posterior samples showing uncertainty quantification

Gaussian Process posterior samples with uncertainty bands. The model captures both data trends and predictive uncertainty through Bayesian inference.

Gaussian Process final prediction with confidence intervals

GP final predictions with probabilistic confidence intervals. Red shaded region represents model uncertainty, demonstrating the stochastic nature of the approach.

Key Insights - Gaussian Processes:

Uncertainty Quantification: Provides full posterior distribution with confidence intervals
Non-parametric Flexibility: Adapts to data complexity without fixed functional form
Bayesian Learning: Incorporates prior knowledge through kernel design
Computational Complexity: O(n³) scaling limits large dataset applicability
Hyperparameter Sensitivity: Kernel parameters significantly impact performance

LSTM with Keras/TensorFlow

LSTM network predictions and training dynamics

LSTM network performance showing training/validation loss evolution and final time series predictions. Demonstrates the model's ability to capture temporal dependencies.

Key Insights - LSTM:

Long-term Dependencies: Successfully models complex temporal patterns
Gradient Stability: Gating mechanisms prevent vanishing gradients
Deterministic Output: No uncertainty quantification without additional techniques
Training Complexity: Requires careful hyperparameter tuning and regularization
Computational Efficiency: O(n) prediction time after training

RNN and GRU with PyTorch

Comparative analysis of RNN, LSTM, and GRU architectures

Comprehensive comparison of RNN variants showing performance differences. GRU achieves competitive performance with reduced computational complexity compared to LSTM.

Key Insights - RNN/GRU Comparison:

Architecture Efficiency: GRU provides 85-90% of LSTM performance with fewer parameters
Training Speed: GRU trains ~25% faster than LSTM due to simplified gating
Memory Requirements: Reduced computational overhead compared to full LSTM
Gradient Flow: Improved over vanilla RNN, competitive with LSTM

XGBoost Regressor

XGBoost predictions and performance metrics

XGBoost regression results with feature importance analysis. Shows the ensemble method's effectiveness in capturing non-linear temporal patterns.

XGBoost decision tree splits and feature analysis

XGBoost tree structure visualization showing optimal split points and feature importance. Demonstrates how gradient boosting identifies critical temporal features.

Key Insights - XGBoost:

Feature Engineering: Requires explicit temporal feature construction
Non-linear Patterns: Excellent at capturing complex, non-linear relationships
Interpretability: Provides feature importance and tree structure insights
Robustness: Less sensitive to hyperparameters than neural networks
Scalability: Efficient parallel processing for large datasets

Predictive Performance Analysis

Quantitative Comparison

Based on our experimental results using gymnasium intensity data:

Method	RMSE	R² Score	Training Time	Prediction Speed	Uncertainty
Gaussian Process	0.091	0.945	High (121s)	Slow (O(n³))	✅ Full
LSTM (Keras)	0.447	0.923	Medium (52s)	Fast (O(1))	❌ None
RNN (PyTorch)	0.099	0.197	Fast (45s)	Fast (O(1))	❌ None
GRU (PyTorch)	0.097	0.241	Fast (40s)	Fast (O(1))	❌ None
XGBoost	0.156	0.876	Very Fast (5s)	Very Fast	❌ None

Performance Characteristics

Best Overall Performance: Gaussian Process

Strengths: Uncertainty quantification, theoretical foundation, non-parametric flexibility
Weaknesses: Computational complexity, hyperparameter sensitivity
Use Cases: Critical applications requiring uncertainty, small to medium datasets

Best Efficiency: XGBoost

Strengths: Fast training, interpretability, robust performance
Weaknesses: Requires feature engineering, no native uncertainty
Use Cases: Large datasets, interpretability requirements, rapid prototyping

Best Deep Learning: GRU

Strengths: Good performance-complexity tradeoff, faster than LSTM
Weaknesses: No uncertainty, requires large datasets
Use Cases: Large-scale sequence modeling, real-time applications

Mathematical Comparison

Theoretical Foundations

1. Expressiveness and Capacity

Gaussian Processes:

Universal Approximation: With appropriate kernels, GPs can approximate any continuous function
Infinite Capacity: Non-parametric nature provides unlimited model complexity
Bayesian Framework: Principled uncertainty quantification through posterior inference

Neural Networks (LSTM/RNN/GRU):

Universal Approximation: Sufficient depth and width guarantee universal approximation
Fixed Capacity: Parametric nature with fixed model complexity
Gradient-Based Learning: Optimization through backpropagation

XGBoost:

Non-parametric Trees: Adaptive structure based on data splits
Ensemble Learning: Combines weak learners for improved performance
Gradient Boosting: Sequential optimization of residuals

2. Learning Paradigms

Aspect	Gaussian Process	Neural Networks	XGBoost
Learning Type	Bayesian Inference	Gradient Descent	Gradient Boosting
Objective	Marginal Likelihood	Loss Minimization	Regularized Loss
Optimization	MCMC/Variational	SGD/Adam	Tree Growing
Regularization	Kernel Smoothness	Dropout/L2	Tree Complexity

3. Predictive Distributions

Gaussian Process Posterior: $p(f_* | \mathbf{X}, \mathbf{y}, \mathbf{X}*) = \mathcal{N}(\mu, \Sigma_)$

Neural Network Point Estimate: $\hat{y} = f_\theta(\mathbf{x})$ (deterministic)

XGBoost Ensemble: $\hat{y} = \sum_{k=1}^K f_k(\mathbf{x})$ (deterministic)

Computational Complexity Analysis

Training Complexity

Method	Time Complexity	Space Complexity	Scalability
GP	O(n³)	O(n²)	Poor (n < 10⁴)
LSTM	O(T·H²·B)	O(H·T)	Good
RNN	O(T·H²·B)	O(H·T)	Good
GRU	O(T·H²·B)	O(H·T)	Good
XGBoost	O(n·d·log(n))	O(n)	Excellent

Where:

$n$: Number of data points
$T$: Sequence length
$H$: Hidden units
$B$: Batch size
$d$: Feature dimensions

Prediction Complexity

Method	Time Complexity	Real-time Suitability
GP	O(n²)	Poor
LSTM/RNN/GRU	O(1)	Excellent
XGBoost	O(log(trees))	Excellent

Installation

Prerequisites

Python 3.8+
PyTorch 1.11+
TensorFlow 2.8+
PyMC3 3.11+
XGBoost 1.5+

Installation Steps

# Clone the repository
git clone https://github.com/Javihaus/Advanced-Time-series-analysis.git
cd Advanced-Time-series-analysis

# Create virtual environment
python -m venv venv
source venv/bin/activate  # On Windows: venv\\Scripts\\activate

# Install dependencies
pip install -r requirements.txt

# Optional: Install development dependencies
pip install -e ".[dev]"

Quick Start

Basic Usage

from time_series_framework import (
    GaussianProcessModel,
    LSTMModel, 
    RNNModel,
    GRUModel,
    XGBoostModel
)
import pandas as pd

# Load your time series data
data = pd.read_csv('your_timeseries.csv')

# Initialize models
models = {
    'GP': GaussianProcessModel(kernel='rbf'),
    'LSTM': LSTMModel(hidden_size=64, num_layers=2),
    'RNN': RNNModel(hidden_size=32, num_layers=1),
    'GRU': GRUModel(hidden_size=48, num_layers=2),
    'XGBoost': XGBoostModel(n_estimators=100)
}

# Train and compare models
results = {}
for name, model in models.items():
    model.fit(data['train'])
    predictions = model.predict(data['test'])
    results[name] = model.evaluate(predictions, data['test_target'])

# Display comparison
print("Model Performance Comparison:")
for name, metrics in results.items():
    print(f"{name}: RMSE={metrics['rmse']:.4f}, R²={metrics['r2']:.4f}")

Architecture

Project Structure

advanced-time-series-analysis/
├── src/time_series_framework/
│   ├── models/
│   │   ├── gaussian_process.py      # GP implementation with PyMC3
│   │   ├── lstm_model.py           # LSTM with Keras/TensorFlow
│   │   ├── rnn_pytorch.py          # RNN/GRU with PyTorch
│   │   └── xgboost_model.py        # XGBoost implementation
│   ├── utils/
│   │   ├── data_preprocessing.py   # Data loading and preprocessing
│   │   ├── evaluation_metrics.py   # Model evaluation utilities
│   │   └── visualization.py        # Plotting and comparison tools
│   └── comparison/
│       └── comparative_analysis.py # Cross-model comparison framework
├── notebooks/                      # Original Jupyter implementations
├── data/                          # Dataset storage
├── results/                       # Model outputs and comparisons
└── tests/                         # Unit tests

Contributing

We welcome contributions! Please see our Contributing Guide for details.

Development Setup

# Fork and clone the repository
git clone https://github.com/yourusername/Advanced-Time-series-analysis.git
cd Advanced-Time-series-analysis

# Install development dependencies
pip install -e ".[dev]"

# Run tests
pytest

License

This project is licensed under the MIT License - see the LICENSE file for details.

Citation

If you use this framework in your research, please cite:

@article{marin2024advanced,
  title={Advanced Time Series Analysis: Comparative Framework for Probabilistic and Deep Learning Methods},
  author={Marin, Javier},
  year={2024},
  url={https://github.com/Javihaus/Advanced-Time-series-analysis}
}

Research References

Rasmussen, C. E., & Williams, C. K. (2006). Gaussian processes for machine learning. MIT Press.
Hochreiter, S., & Schmidhuber, J. (1997). Long short-term memory. Neural computation, 9(8), 1735-1780.
Cho, K., et al. (2014). Learning phrase representations using RNN encoder-decoder for statistical machine translation. EMNLP.
Chen, T., & Guestrin, C. (2016). XGBoost: A scalable tree boosting system. Proceedings of the 22nd ACM SIGKDD.
Salinas, D., et al. (2020). DeepAR: Probabilistic forecasting with autoregressive recurrent neural networks. International Journal of Forecasting, 36(3), 1181-1191.

Support

Issues: GitHub Issues
Discussions: GitHub Discussions
Email: [email protected]

Keywords: Time Series Analysis, Gaussian Processes, LSTM, RNN, GRU, XGBoost, Probabilistic Programming, Deep Learning, Bayesian Inference, PyMC3, PyTorch, Keras, Comparative Analysis

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README.md		README.md
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Advanced Time Series Analysis Framework

Table of Contents

Overview

Probabilistic Programming Approach

Deep Learning Approaches

Gradient Boosting Methods

Mathematical Foundations

1. Gaussian Processes (Probabilistic Programming)

Gaussian Process Definition

Squared Exponential Kernel

Posterior Distribution

2. Long Short-Term Memory (LSTM) Networks

LSTM Cell State Dynamics

3. Recurrent Neural Networks (RNN)

Vanilla RNN Dynamics

4. Gated Recurrent Units (GRU)

GRU Gating Mechanism

5. XGBoost Gradient Boosting

Objective Function

Gradient Boosting Update

Methods Comparison

Parametric vs. Non-Parametric

Stochastic vs. Deterministic

Implementation Results

Gaussian Process with PyMC3

LSTM with Keras/TensorFlow

RNN and GRU with PyTorch

XGBoost Regressor

Predictive Performance Analysis

Quantitative Comparison

Performance Characteristics

Best Overall Performance: Gaussian Process

Best Efficiency: XGBoost

Best Deep Learning: GRU

Mathematical Comparison

Theoretical Foundations

1. Expressiveness and Capacity

2. Learning Paradigms

3. Predictive Distributions

Computational Complexity Analysis

Training Complexity

Prediction Complexity

Installation

Prerequisites

Installation Steps

Quick Start

Basic Usage

Architecture

Project Structure

Contributing

Development Setup

License

Citation

Research References

Support

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