====================
Vulnerability Models
====================

Introduction
------------

Vulnerability functions quantify the economic loss ratio—the repair cost of damaged structural 
components normalised by their replacement cost—over a range of intensity measure (IM) levels. 
They are generated by combining damage state probabilities from fragility functions with 
damage-to-loss ratios through the total probability theorem.


Expected Loss Ratio
-------------------

The vulnerability function expresses the expected loss ratio as a function of an intensity 
measure level (IM), obtained by convolving fragility functions with damage-to-loss ratios 
associated with each damage state.

Let :math:`P(DS = ds_i | IM)` denote the probability of the structure being in damage state 
:math:`ds_i` at a given intensity measure level IM, and let :math:`\mu_{LR,i}` be the mean 
loss ratio associated with that damage state. The expected loss ratio is:

.. math::

   E[LR | IM] = \sum_{i=1}^{N_{DS}} P(DS = ds_i | IM) \cdot \mu_{LR,i}

where:

* :math:`N_{DS}` is the total number of discrete damage states
* :math:`P(DS = ds_i | IM)` is derived from the fragility functions
* :math:`\mu_{LR,i}` is the mean loss ratio associated with damage state :math:`ds_i`


Damage-to-Loss Ratios
---------------------

The damage-to-loss ratios for structural components are:

.. list-table:: Damage-to-Loss Ratios
   :header-rows: 1
   :widths: 30 30 40

   * - Damage State
     - Mean Loss Ratio
     - Description
   * - Slight (DS1)
     - 0.05
     - Minor repairs, cosmetic damage
   * - Moderate (DS2)
     - 0.15
     - Moderate repairs needed
   * - Extensive (DS3)
     - 0.60
     - Major structural repairs
   * - Complete (DS4)
     - 1.00
     - Full replacement required


Damage State Probabilities
--------------------------

Fragility functions are expressed in terms of probabilities of exceedance. The probability of 
being in a specific damage state is computed as:

.. math::

   P(DS = ds_i | IM) = \begin{cases}
   P(DS \ge ds_i | IM) - P(DS \ge ds_{i+1} | IM), & i < N_{DS} \\
   P(DS \ge ds_{N_{DS}} | IM), & i = N_{DS}
   \end{cases}


Uncertainty in Loss Estimation
------------------------------

Method 1: Silva (2019) Semi-Empirical
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

When uncertainty in the damage-to-loss relationship is not explicitly modelled, the dispersion 
of the loss ratio can be estimated using the formulation proposed by Silva (2019):

.. math::

   \sigma_{LR|IM} = 0.5 \sqrt{\overline{LR}_{|IM} \left(-0.7 \cdot 2 \overline{LR}_{|IM} - \sqrt{6.8 \overline{LR}_{|IM} + 0.5}\right)}

where :math:`\overline{LR}_{|IM}` is the mean loss ratio conditional on IM.

The coefficient of variation is:

.. math::

   COV(LR | IM) = \frac{\sigma_{LR|IM}}{E[LR | IM]}


Method 2: Explicit Statistical Propagation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Alternatively, uncertainty can be quantified by explicitly propagating uncertainty through the 
convolution. Using the law of total variance:

.. math::

   Var(LR | IM) = \sum_{i=1}^{N_{DS}} P(DS = ds_i | IM) \left[\sigma_{LR,i}^2 + \left(\mu_{LR,i} - E[LR | IM]\right)^2\right]

This formulation captures both:

* Uncertainty **within damage states** due to variability in damage-to-loss ratios
* Uncertainty due to **damage-state mixing** as intensity varies


Beta Distribution Interpretation
--------------------------------

The loss ratio conditional on IM can be modelled as a **Beta distribution** due to its bounded 
support on [0, 1]:

.. math::

   LR | IM \sim Beta(\alpha(IM), \beta(IM))

The Beta parameters are computed from the mean and variance:

.. math::

   \kappa(IM) = \frac{\mu(IM)[1-\mu(IM)]}{\sigma^2(IM)} - 1

.. math::

   \alpha(IM) = \mu(IM) \cdot \kappa(IM), \quad \beta(IM) = [1-\mu(IM)] \cdot \kappa(IM)


Interactive Viewer
------------------

Use the interactive viewer below to explore vulnerability functions for different building classes.
All decision variables are displayed in stacked plots.

.. raw:: html

   <div class="viewer-container" id="vulnerability-viewer">
       <div class="viewer-map" id="vulnerability-region-map"></div>
       <div class="viewer-region-display" id="vulnerability-region-display">Click map to select region</div>

       <div class="viewer-controls">
           <div class="control-row">
               <div class="control-group">
                   <label for="vulnerability-model-select">Model:</label>
                   <select id="vulnerability-model-select" class="viewer-select model-select" onchange="switchModel(this.value).then(() => { vulnerabilitySelections = []; updateVulnerabilitySelections(); initVulnerabilityViewer(); })">
                       <option value="GEM">GEM</option>
                   </select>
               </div>
           </div>
       </div>

       <div class="viewer-controls">
           <div class="control-row">
               <div class="control-group">
                   <label for="vulnerability-building-select">Building Class:</label>
                   <select id="vulnerability-building-select" class="viewer-select">
                       <option value="">-- Select Building Class --</option>
                   </select>
               </div>
               <div class="control-group">
                   <label for="vulnerability-imt-select">IMT:</label>
                   <select id="vulnerability-imt-select" class="viewer-select">
                       <option value="">-- Select IMT --</option>
                       <option value="PGA">PGA</option>
                       <option value="SA(0.3)">SA(0.3)</option>
                       <option value="SA(0.6)">SA(0.6)</option>
                       <option value="SA(1.0)">SA(1.0)</option>
                   </select>
               </div>
               <div class="control-group">
                   <label for="vulnerability-occupancy-select">Occupancy:</label>
                   <select id="vulnerability-occupancy-select" class="viewer-select">
                       <option value="">-- Select Occupancy --</option>
                       <option value="RES">Residential (RES)</option>
                       <option value="COM">Commercial (COM)</option>
                       <option value="IND">Industrial (IND)</option>
                   </select>
               </div>
           </div>
           <div class="control-group">
               <button id="vulnerability-add-btn" class="viewer-btn btn-primary">Add to Plot</button>
               <button id="vulnerability-clear-btn" class="viewer-btn btn-danger">Clear All</button>
           </div>
       </div>
       <div class="viewer-selections" id="vulnerability-selections"></div>
       <div class="viewer-controls slider-controls">
           <div class="control-row">
               <div class="control-group">
                   <label for="vulnerability-xmax-slider">X-Axis Max (g):</label>
                   <input type="range" id="vulnerability-xmax-slider" min="1" max="10" value="5" step="0.5">
                   <span id="vulnerability-xmax-value">5.0</span>
               </div>
               <div class="control-group">
                   <label for="vulnerability-ymax-slider">Y-Axis Max:</label>
                   <input type="range" id="vulnerability-ymax-slider" min="0.1" max="1" value="1" step="0.1">
                   <span id="vulnerability-ymax-value">1.0</span>
               </div>
           </div>
       </div>
       <div id="vulnerability-plots-container">
           <!-- Plots for each decision variable will be dynamically created here -->
       </div>
   </div>

   <script>
       document.addEventListener('DOMContentLoaded', function() {
           if (typeof initVulnerabilityViewer === 'function') {
               initVulnerabilityViewer();
           }
       });
   </script>


Average Annual Loss Ratio
-------------------------

In performance-based earthquake engineering, the **Average Annual Loss Ratio (AALR)** quantifies 
the expected loss ratio due to earthquake-induced damage over one year:

.. math::

   AALR = \int_0^{\infty} LR(IM) \cdot \frac{d\lambda(IM)}{dIM} \, dIM

where:

* :math:`LR(IM)` is the loss ratio from the vulnerability function
* :math:`\lambda(IM)` is the annual frequency of exceedance from the hazard curve

In practice, AALR is approximated using discrete intensity levels:

.. math::

   AALR \approx \sum_{i=1}^{N} LR(IM_i) \cdot [\lambda(IM_i) - \lambda(IM_{i+1})]


Regional Variations
-------------------

Vulnerability functions account for regional variations in:

* Construction practices and material quality
* Seismic design code adoption and enforcement
* Building maintenance and age distributions
* Economic factors affecting repair costs


References
----------

* Silva V. (2019). Uncertainty and Correlation in Seismic Vulnerability Functions of Building 
  Classes. Earthquake Spectra.
  
* Martins L. et al. (2016). Development and assessment of damage-to-loss models for moment-frame 
  reinforced concrete buildings. Earthquake Engineering & Structural Dynamics.
  
* Di Pasquale G. et al. (2005). New developments in seismic risk assessment in Italy. Bulletin 
  of Earthquake Engineering.