.. toctree::
.. _DexWeight:

Weights
-------

*Weights* are commonly used in :ref:`Decision Analysis <MethDecisionAnalysis>`
to model the importance of :ref:`attributes <DexAttribute>`. Weights
are numbers, usually normalized to the sum or maximum of 1 or 100, which
define the relative contribution of the corresponding attribute to the final
:ref:`evaluation <DexEvaluation>`. In Decision Analysis, :ref:`aggregation
functions <DexAggregation>` are commonly defined using some form of
the weighted sum, for example:

f(X\ :sub:`1`, X\ :sub:`2`,..., X\ :sub:`n`) = w\ :sub:`0`\ +
w\ :sub:`1`\ ×X\ :sub:`1` + w\ :sub:`2`\ ×X\ :sub:`2` + ... +
w\ :sub:`n`\ ×X\ :sub:`n`

Here, w\ :sub:`i` denote weights and X\ :sub:`i` denote attributes.

In :ref:`qualitative DEX models <DexModel>`, there is
natively no room for weights: attributes are symbolic and aggregation
functions are defined by decision rules. However, to bridge the gap
between qualitative and quantitative models, it is possible to introduce
weights - in a somewhat approximate and imprecise way - also in qualitative models.

Principle
~~~~~~~~~

.. image:: images/CarWei.png
   :alt: Utility function and hyperplane

The figure above illustrates the basic approach. It shows the
CAR aggregation function as defined in the :ref:`Car Evaluation <ExampleCar>` model,
represented by points (dots) in a three-dimensional space.
Each point represents one defined decision rule. To find out the weights,
we place a (hyper)plane (shown in red) into this space so that it matches
the points as closely as possible (using the least squares measure).
Once done, weights can be determined directly from the slopes of the hyperplane:
the higher the slope in the direction of an attribute, the higher the corresponding
relative weight. In the above figure, the weights of PRICE and
TECH.CHAR. are both 50. These are *local normalized* weights (see the definition below).

In DEX, weights are used for two purposes:

-  as an approximate representation of aggregation functions, used primarily
   for verification and overview, and
-  for defining aggregation functions or their parts (see :ref:`using weights <DexUseWeights>`).
   
.. _DexWeightTypes:

Weight Types
~~~~~~~~~~~~

Actually, DEX uses four types of weights, as illustrated with the
following weights from the :ref:`Car Evaluation <ExampleCar>` model:

.. image:: images/CarWei4.png
   :alt: Car: Weights

The difference between *local* and *global* weights is due to the :ref:`tree of
attributes <DexTree>`:

- *Local* weights always refer to a single aggregate attribute and a single corresponding aggregation function,
  so that the sum of weights of the attribute's immediate descendants (function arguments) is 100%.
   
- *Global* weights, on the other hand, take into account the structure
  of the tree and relative importance of its sub-trees. A global weight of
  an attribute is calculated as a product of the local weight and the global weight
  of the parent attribute. A global weight of the root attribute is 100%.
  For example: the global normalized weight of BUY.PRICE is 50% (its local normalized
  weight) × 50% (global normalized weight of PRICE), which gives 25%.

Weights can also be *normalized* or not. This is because some
:ref:`scales <DexScale>` can have more values than others.
Geometrically, larger scales appear longer, they have lower slopes and,
consequently, smaller weights. *Normalization* refers to the procedure
in which all scales are adjusted to the same length (unit interval)
before determining the weights. Usually, this is the better method of
weight assessment and comparison of attributes.