.. _advanced_topics:

Section 3: Advanced Topics (Weeks 7-12)
========================================

**Deep dives into Bayesian inference, fractional viscoelasticity, multi-technique analysis,
constitutive modeling, transient networks, vitrimers, and specialized frameworks**

.. admonition:: Prerequisites
   :class: note

   Sections 1-2 (Fundamentals and Model Usage). Timeline: Weeks 7-16 (30-40 hours).

Learning Objectives
-------------------

By completing this section, you will be able to:

1. Perform Bayesian inference to quantify parameter uncertainty
2. Understand fractional viscoelasticity concepts and applications
3. Fit multiple experimental techniques simultaneously
4. Apply time-temperature superposition (TTS) for mastercurves
5. Interpret credible intervals and posterior distributions
6. Classify soft glassy materials using the SGR framework
7. Extract yield stresses and cage moduli from LAOS using SPP
8. Work with ODE-based constitutive models (Giesekus, IKH, Saramito)
9. Analyze thixotropic and yield stress materials (DMT, Fluidity, HL, STZ, EPM)
10. Apply transient polymer network models (TNT, VLB)
11. Model vitrimer and nanocomposite rheology (HVM, HVNM)
12. Understand dense suspension dynamics via mode-coupling theory (ITT-MCT)
13. Apply all 7 data transforms to experimental data

----

Core Advanced Topics
--------------------

.. toctree::
   :maxdepth: 2

   bayesian_inference
   fractional_viscoelasticity_reference
   multi_technique_fitting
   time_temperature_superposition

Specialized Frameworks
----------------------

.. toctree::
   :maxdepth: 2

   sgr_analysis
   spp_analysis

Constitutive Models & Networks
------------------------------

.. toctree::
   :maxdepth: 2

   constitutive_ode_models
   thixotropy_yielding
   dense_suspensions_glasses
   polymer_network_models
   vitrimer_models

Data Transforms
---------------

.. toctree::
   :maxdepth: 2

   transforms_complete

----

Key Skills Summary
------------------

**Bayesian Workflow** (NLSQ -> NUTS):

.. code-block:: python

   from rheojax.models import Maxwell

   # 1. NLSQ point estimate (fast)
   model = Maxwell()
   model.fit(t, G_t, test_mode='relaxation')

   # 2. Bayesian inference (uncertainty quantification)
   result = model.fit_bayesian(t, G_t, num_samples=2000, num_warmup=1000)

   # 3. Credible intervals
   intervals = model.get_credible_intervals(result.posterior_samples, credibility=0.95)

**Multi-Technique Fitting**:

.. code-block:: python

   from rheojax.pipeline import ModelComparisonPipeline

   pipeline = ModelComparisonPipeline()
   pipeline.fit_multi_technique(
       techniques=['relaxation', 'oscillation'],
       data_dict={'relaxation': (t, G_t), 'oscillation': (omega, G_star)}
   )

**SGR Phase Classification**:

.. code-block:: python

   from rheojax.models import SGRConventional

   model = SGRConventional()
   model.fit(omega, G_star, test_mode='oscillation')

   x = model.parameters.get_value('x')
   print(f"x = {x:.2f}: {'Glass' if x < 1 else 'Fluid'}")

**SPP Yield Stress Extraction**:

.. code-block:: python

   from rheojax.transforms import SPPDecomposer

   decomposer = SPPDecomposer(omega=1.0, gamma_0=1.0, n_harmonics=39)
   result = decomposer.transform(laos_data)
   spp_results = decomposer.get_results()

   print(f"Cage modulus: {spp_results['G_cage']:.1f} Pa")
   print(f"Static yield stress: {spp_results['sigma_sy']:.1f} Pa")

Next Steps
----------

After completing advanced topics, proceed to
**Section 4: Practical Guides** (:doc:`../04_practical_guides/index`) for
production workflows, data I/O, visualization, and batch processing.