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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Identifiability Diagnostic Framework — Regime-Switching State-Space Models</title>
<meta name="description" content="Enterprise-grade unsupervised diagnostic system for identifying parameter identifiability in 2-regime switching state-space models using SVD-based analysis.">
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</head>
<body class="bg-white text-gray-900">
<!-- Hero Section -->
<div class="hero">
<div class="max-w-6xl mx-auto px-6 sm:px-8">
<div class="flex flex-wrap gap-2 mb-8">
<span class="badge">🔍 Identifiability</span>
<span class="badge">📐 State-Space Models</span>
<a href="https://doi.org/10.5281/zenodo.20517274" class="badge" target="_blank" rel="noopener">🔬 Zenodo</a>
<a href="https://medium.com/@prakulhiremath/how-we-broke-the-circularity-trap-of-latent-regime-validation-6ecae61b54a3" class="badge" target="_blank" rel="noopener">✍️ Medium</a>
<span class="badge">🐍 Python 3.8+</span>
<span class="badge">📦 PyPI Ready</span>
</div>
<div class="max-w-3xl">
<h1 class="text-5xl sm:text-6xl font-display font-bold mb-4 text-slate-900">Identifiability Diagnostic</h1>
<p class="text-xl text-slate-600 mb-2 font-display font-semibold">Framework for Regime-Switching State-Space Models</p>
<p class="text-2xl font-display font-semibold mb-8">
<span class="text-slate-700">Can you learn the rules</span>
<span class="hero-accent"> from the data alone?</span>
</p>
<p class="text-lg text-slate-600 mb-8 max-w-2xl leading-relaxed">
Enterprise-grade unsupervised pre-training diagnostic system for identifying practical identifiability boundaries in 2-regime switching state-space models. Uses Observable Moment Matrix + SVD to determine whether regime parameters are identifiable from observations alone.
</p>
<div class="flex flex-wrap gap-4">
<a href="#algorithm" class="btn">Learn Algorithm</a>
<a href="#installation" class="btn btn-outline">Install →</a>
</div>
<p class="text-sm text-slate-500 mt-6 italic">>85% test coverage. Full type hints. Production-ready.</p>
</div>
</div>
</div>
<!-- Core Problem -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section">
<div class="bg-slate-50 border-l-4 border-blue-600 p-8 rounded-sm">
<p class="text-lg text-slate-800 font-semibold mb-4">"The circularity trap: You need to know the true model to train a model that learns the true model."</p>
<p class="text-slate-700 leading-relaxed">In regime-switching systems, you observe transitions but never directly observe which regime you're in. This creates a fundamental question: <strong>Given your data and interventions, can you actually identify the regime parameters?</strong> Or are you chasing a ghost? Identifiability Diagnostic answers this before you waste compute.</p>
</div>
</div>
<div class="divider"></div>
<!-- Algorithm -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section" id="algorithm">
<h2>Seven-Step Diagnostic Pipeline</h2>
<p>The framework implements a rigorous mathematical pipeline to determine identifiability:</p>
<div class="architecture-diagram">
<pre>[1] Compute First-Order Innovations
dy_t = y_t - y_{t-1}
[2] Lift to Feature Space
v_t = [dy_t, dy_{t-1}]^T
[3] Align Interventional Context
Match intervention levels u to innovation timeline
[4] Construct Moment Matrix
M ∈ ℝ^{|U| × 4} with conditional covariances
M_ij = Cov(v_t | u_t = u_i)
[5] Execute SVD
M = U Σ V^T (Singular Value Decomposition)
[6] Compute Metrics
σ_2 (second singular value)
Effective rank
Condition number
[7] Deploy Gate
if σ_2 > τ (5.12e-17): IDENTIFIABLE ✓
else: NON-IDENTIFIABLE ✗</pre>
</div>
<h3>Mathematical Foundation</h3>
<p>For regime-switching model:</p>
<div class="math-block">
$$y_t = a_{s_t} y_{t-1} + b_{s_t} u_t + c_{s_t} u_t y_{t-1} + \eta_t$$
</div>
<p>where <code>s_t ∈ {0, 1}</code> denotes regime, and <code>u_t</code> is exogenous intervention. The Observable Moment Matrix captures the covariance structure stratified by intervention level.</p>
<div class="info-box">
<p><strong>Key Decision Rule:</strong> The second singular value σ₂ is the identifiability gating metric. If σ₂ exceeds threshold, regime parameters are identifiable from data alone.</p>
</div>
<h3>Why This Works</h3>
<table>
<thead>
<tr>
<th>Approach</th>
<th>Strength</th>
<th>Weakness</th>
</tr>
</thead>
<tbody>
<tr>
<td><strong>Train & Pray</strong></td>
<td>No pre-checks needed</td>
<td>100 hours of compute wasted on unidentifiable problem</td>
</tr>
<tr>
<td><strong>Theoretical Analysis</strong></td>
<td>Mathematically rigorous</td>
<td>Requires hand-solving for each problem (weeks of work)</td>
</tr>
<tr style="background: rgba(16, 185, 129, 0.05);">
<td><strong>Observable Moment Matrix</strong></td>
<td>Automatic, <130ms per check</td>
<td>Limited to SSM regime switching (but that's the target domain)</td>
</tr>
</tbody>
</table>
</div>
<div class="divider"></div>
<!-- Features -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section">
<h2>Key Features</h2>
<div class="grid grid-cols-1 md:grid-cols-2 gap-6 my-8">
<div class="feature-card">
<h4 class="font-bold text-slate-900 mb-2">🎯 SVD-Based Analysis</h4>
<p class="text-sm text-slate-700">Automatically detects parameter identifiability via singular value spectrum of Observable Moment Matrix.</p>
</div>
<div class="feature-card">
<h4 class="font-bold text-slate-900 mb-2">🔄 Regime-Switching Support</h4>
<p class="text-sm text-slate-700">Full support for multi-regime autoregressive processes with exogenous interventions.</p>
</div>
<div class="feature-card">
<h4 class="font-bold text-slate-900 mb-2">📊 Comprehensive Metrics</h4>
<p class="text-sm text-slate-700">Effective rank, condition numbers, singular values, and deployment decisions all computed automatically.</p>
</div>
<div class="feature-card">
<h4 class="font-bold text-slate-900 mb-2">🛡️ Production-Grade</h4>
<p class="text-sm text-slate-700">Full type hints, error handling, logging, and >85% test coverage. Ready for enterprise deployment.</p>
</div>
<div class="feature-card">
<h4 class="font-bold text-slate-900 mb-2">🚀 CI/CD Ready</h4>
<p class="text-sm text-slate-700">GitHub Actions workflows, automated testing across Python 3.8–3.11, multi-platform (Linux, macOS, Windows).</p>
</div>
<div class="feature-card">
<h4 class="font-bold text-slate-900 mb-2">📦 PyPI Package</h4>
<p class="text-sm text-slate-700">Easy installation, full dependency management, semantic versioning, and regular updates.</p>
</div>
</div>
</div>
<div class="divider"></div>
<!-- Performance -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section">
<h2>Performance & Benchmarks</h2>
<p>Benchmark results on Intel i7 @ 3.6GHz:</p>
<table>
<thead>
<tr>
<th>Task</th>
<th>Data Size</th>
<th>Time</th>
</tr>
</thead>
<tbody>
<tr>
<td>Data Generation (T=100k)</td>
<td>100,000 observations</td>
<td>~50 ms</td>
</tr>
<tr>
<td>Pipeline Execution</td>
<td>100,000 observations</td>
<td>~80 ms</td>
</tr>
<tr style="background: rgba(16, 185, 129, 0.05);">
<td><strong>Full Diagnostic Cycle</strong></td>
<td><strong>100,000 observations</strong></td>
<td><strong>~130 ms</strong></td>
</tr>
<tr>
<td>Parametric Sweep (5 ε values)</td>
<td>500,000 observations</td>
<td>~650 ms</td>
</tr>
</tbody>
</table>
<h3>Quality Metrics</h3>
<div class="grid grid-cols-1 sm:grid-cols-2 md:grid-cols-4 gap-4 my-8">
<div class="metric-card">
<p class="metric-label">Test Coverage</p>
<p class="metric-value">>85%</p>
</div>
<div class="metric-card">
<p class="metric-label">Type Hints</p>
<p class="metric-value">100%</p>
</div>
<div class="metric-card">
<p class="metric-label">Python Versions</p>
<p class="metric-value">3.8–3.11</p>
</div>
<div class="metric-card">
<p class="metric-label">Platforms</p>
<p class="metric-value">3/3 ✓</p>
</div>
</div>
</div>
<div class="divider"></div>
<!-- Installation & Usage -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section" id="installation">
<h2>Installation & Quick Start</h2>
<h3>Install from PyPI</h3>
<div class="code-block mb-8"><pre><code class="language-bash">pip install identifiability-diagnostic</code></pre></div>
<h3>Or Install from Source</h3>
<div class="code-block mb-8"><pre><code class="language-bash">git clone https://github.com/prakulhiremath/identifiability-diagnostic.git
cd identifiability-diagnostic
pip install -e .</code></pre></div>
<h3>Development Installation</h3>
<div class="code-block mb-8"><pre><code class="language-bash">pip install -e ".[dev]"</code></pre></div>
<h3>Basic Usage</h3>
<div class="code-block mb-8"><pre><code class="language-python">from identifiability_diagnostic import (
RegimeSwitchingGenerator,
PretrainingDiagnosticPipeline
)
from identifiability_diagnostic.utils.config import load_config
# Load configuration
config = load_config('config/parameters.yaml')
# Generate synthetic data
generator = RegimeSwitchingGenerator(config)
y, u = generator.generate(epsilon=1e-3, symmetric=False)
# Run diagnostic pipeline
pipeline = PretrainingDiagnosticPipeline(config)
metrics = pipeline.run(y, u)
# Interpret results
print(f"σ_2 = {metrics['sigma_2']:.4e}")
print(f"Decision: {metrics['deploy_decision']}")
print(f"Identifiable: {metrics['identifiable']}")</code></pre></div>
<h3>Command-Line Interface</h3>
<div class="code-block"><pre><code class="language-bash"># Run parametric sweep
python main.py --config config/parameters.yaml --log-level INFO
# Run with custom epsilon values
python main.py --epsilon 1e-5 1e-4 1e-3 1e-2 1e-1
# Test mode (smaller dataset)
python main.py --test --verbose
# Symmetric control only
python main.py --symmetric-only</code></pre></div>
</div>
<div class="divider"></div>
<!-- Configuration -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section">
<h2>Configuration</h2>
<p>Default configuration in <code>config/parameters.yaml</code>:</p>
<div class="code-block"><pre><code class="language-yaml">simulation:
T: 100000 # Time series length
seed: 42 # Random seed
sigma_n: 0.5 # Process noise std dev
regime_0:
a: 0.90 # AR coefficient regime 0
b: 1.0 # Intervention coupling
c: 0.0 # Bilinear term
regime_1:
a: 0.90 # AR coefficient regime 1
b: 1.0
c: 0.2 # Asymmetric bilinear!
diagnostic:
u_levels: [-2, -1, 0, 1, 2] # Intervention grid
threshold_tau: 5.12e-17 # Gating threshold</code></pre></div>
</div>
<div class="divider"></div>
<!-- Testing -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section">
<h2>Testing & Code Quality</h2>
<h3>Run All Tests</h3>
<div class="code-block mb-8"><pre><code class="language-bash">pytest tests/ -v --cov=src/identifiability_diagnostic</code></pre></div>
<h3>Test Suite Coverage</h3>
<ul class="space-y-2 text-slate-700">
<li>✓ Data generation tests</li>
<li>✓ Pipeline execution tests</li>
<li>✓ Metrics computation tests</li>
<li>✓ Configuration validation tests</li>
<li>✓ End-to-end workflow tests</li>
<li>✓ Edge case handling</li>
<li>✓ Numerical stability tests</li>
</ul>
<h3>Code Quality</h3>
<ul class="space-y-2 text-slate-700 mt-4">
<li><strong>Type Hints:</strong> Full type annotations throughout</li>
<li><strong>Linting:</strong> flake8, black, isort, pylint</li>
<li><strong>Static Analysis:</strong> mypy type checking</li>
<li><strong>Documentation:</strong> Comprehensive docstrings and comments</li>
</ul>
</div>
<div class="divider"></div>
<!-- Project Structure -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section">
<h2>Project Structure</h2>
<div class="architecture-diagram">
<pre>identifiability-diagnostic/
├── .github/workflows/ # CI/CD configurations
├── src/
│ └── identifiability_diagnostic/
│ ├── __init__.py
│ ├── core/ # Core algorithms
│ │ ├── data_generator.py
│ │ ├── pipeline.py
│ │ └── metrics.py
│ ├── utils/ # Utilities
│ │ ├── config.py
│ │ └── logging.py
│ └── exceptions.py
├── tests/
│ ├── unit/ # Unit tests (20+)
│ └── integration/ # Integration tests (10+)
├── examples/ # Usage examples
├── config/
│ └── parameters.yaml # Default config
├── main.py # CLI entry point
├── setup.py # Package setup
├── pyproject.toml # Modern packaging
├── requirements.txt # Dependencies
└── README.md</pre>
</div>
</div>
<div class="divider"></div>
<!-- Citation -->
<div class="max-w-6xl mx-auto px-6 sm:px-8 section">
<h2>Citation</h2>
<div class="code-block">
<pre><code>@software{identifiability_diagnostic_2024,
title={Identifiability Diagnostic Framework for
Regime-Switching Models},
author={Prakul S. Hiremath},
year={2026},
url={https://github.com/prakulhiremath/
identifiability-diagnostic}
}</code></pre>
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<h4 class="font-semibold text-slate-900 mb-4">Resources</h4>
<ul class="space-y-2 text-sm">
<li><a href="https://github.com/prakulhiremath/identifiability-diagnostic" class="text-blue-600 hover:underline" target="_blank" rel="noopener">GitHub Repository</a></li>
<li><a href="https://pypi.org/project/identifiability-diagnostic/" class="text-blue-600 hover:underline" target="_blank" rel="noopener">PyPI Package</a></li>
<li><a href="https://doi.org/10.5281/zenodo.20517274" class="text-blue-600 hover:underline" target="_blank" rel="noopener">Zenodo DOI</a></li>
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<h4 class="font-semibold text-slate-900 mb-4">Learn</h4>
<ul class="space-y-2 text-sm">
<li><a href="#algorithm" class="text-blue-600 hover:underline">Algorithm Details</a></li>
<li><a href="#installation" class="text-blue-600 hover:underline">Installation Guide</a></li>
<li><a href="https://medium.com/@prakulhiremath/how-we-broke-the-circularity-trap-of-latent-regime-validation-6ecae61b54a3" class="text-blue-600 hover:underline" target="_blank" rel="noopener">Medium Article</a></li>
</ul>
</div>
<div>
<h4 class="font-semibold text-slate-900 mb-4">Connect</h4>
<ul class="space-y-2 text-sm">
<li><a href="https://github.com/prakulhiremath/identifiability-diagnostic/issues" class="text-blue-600 hover:underline" target="_blank" rel="noopener">Report Issues</a></li>
<li><a href="https://github.com/prakulhiremath" class="text-blue-600 hover:underline" target="_blank" rel="noopener">GitHub Profile</a></li>
<li>License: MIT</li>
</ul>
</div>
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<div class="border-t border-slate-200 pt-8">
<p class="text-sm text-slate-600 text-center mb-4">
<strong>Identifiability Diagnostic Framework</strong> — Regime-Switching State-Space Model Analysis
</p>
<p class="text-xs text-slate-500 text-center">
MIT License · Python 3.8+ · SVD-Based Analysis · <a href="https://github.com/prakulhiremath/identifiability-diagnostic" class="text-blue-600 hover:underline" target="_blank" rel="noopener">Open Source</a>
</p>
<p class="text-xs text-slate-500 text-center mt-4">
<em>"Know before you train."</em>
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