High Dimensional Model Representation
======================================

Description
----------------

High-dimensional model representation (HDMR) is a method of representing complex systems in
high-dimensional spaces. It is a technique to decompose a high-dimensional function into a sum of
lower-dimensional functions. These lower-dimensional functions are called "HDMR components".

.. math::

	f\left( \mathbf{x}\right) =\sum_{i=1}^{p}f_{i}\left( x_{i}\right)
	+\sum_{1\leq i<j\leq p}f_{i,j}\left( x_{i},x_{j}\right)
	+...+f_{1,2,...,p}\left( x_{1},x_{2},...,x_{p}\right) 
	
	f_{0}=\int_{\mathbb{R}^{p}}f\left( \mathbf{x}\right) d\mathbf{x}
	
	f_{i}\left( x_{i}\right) =\int_{\mathbb{R}^{p}}f\left( \mathbf{x}\right) d%
	\mathbf{x}_{\sim i}-f_{0}
	
	f_{i,j}\left( x_{i},x_{j}\right) =\int_{\mathbb{R}^{p}}f\left( \mathbf{x}%
	\right) d\mathbf{x}_{\sim\left\{ i,j\right\} }-f_{0}-f_{i}\left(
	x_{i}\right) -f_{j}\left( x_{j}\right)
	
It is particularly useful in cases where the number of variables or parameters is large, 
making it difficult to understand the underlying relationships between the variables.

HDMR is commonly used in various fields such as engineering, physics, and chemistry, to model
complex systems and predict their behavior. Here are an example/application to fix the ideas:

	- *A Likelihood function*: Conditional on a dataset, the likelihood function 
	  is a function of the parameters of the model. But the exact relationship between
	  the likelihood and the parameters is unknown. The reason we may be interested in such
	  a problem is that the likelihood might be very costly to evaluate. HDMR could be used
	  in this case to decompose the likelihood into lower-dimensional components. The resulting
	  metamodel could be far less costly to evaluate.

The HDMR components are chosen such that they capture the most important features of the system,
while still being computationally tractable. HDMR can also be used to estimate the sensitivity
of a system to changes in its inputs, which can be useful for optimization and design.

Variances

.. math::

	D=\int_{\mathbb{R}^{p}}f^{2}\left( \mathbf{x}\right) d\mathbf{x}-f_{0}^{2}
	
	D_{i}=\int_{\mathbb{R}^{p}}f_{i}^{2}\left( x_{i}\right) dx_{i}
	
	D_{i,j}=\int_{\mathbb{R}^{p}}\int_{\mathbb{R}^{p}}f_{i,j}^{2}\left(
	x_{i},x_{j}\right) dx_{i}dx_{j}
	  
Sensitivities

.. math::

	S_{i}=\frac{D_{i}}{D}
	
	S_{i,j}=\frac{D_{i,j}}{D}


HDMR is a powerful tool for understanding and predicting the behavior of complex systems,
but it also requires large computational resources and can be difficult to implement in practice.
	  
Quick-start examples
----------------------

A known Polynomial function
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
::

	% Define the polynomial function
	f = @(x) x.^3 + 2*x.^2 - 5*x + 3;
	
	% Define the lower-dimensional components of the polynomial
	f1 = @(x) x.^3;
	f2 = @(x) 2*x.^2;
	f3 = @(x) -5*x + 3;
	
	% Plot the polynomial and its lower-dimensional components
	x = linspace(-2, 2, 100);
	y = f(x);
	y1 = f1(x);
	y2 = f2(x);
	y3 = f3(x);
	plot(x, y, x, y1, '--', x, y2, ':', x, y3, '-.','linewidth',2);
	legend('f(x)', 'f1(x)', 'f2(x)', 'f3(x)');

Details
-------------

.. toctree::
   :maxdepth: 2
   :caption: Contents:

   high_dimensional_model_representation