## 5.3 Survey design and optimization

Although the topic of survey design is still in its infancy, the basic idea is to carry out an optimization of
survey parameters (such as for example choice of targets, depth of field, number of spectroscopic fibers, etc.)
in order to identify the configuration that is more likely to return a high FoM for the scientific question
being considered. Example of this approach applied to dark-energy parameters can be found
in [96, 707, 708, 103], while [590] discussed a more general methodology. In [96] a method is defined to
optimize future surveys, in the framework of Bayesian statistics and without necessarily assuming a
dark-energy model. In [103], [707] and [708] this method is used to produce forecasts for future weak lensing
and galaxy redshift surveys.
The optimization process is carried out subject to constraints, such as for example design parameter
ranges and/or cost constraints. This is generally a numerically complex and computationally expensive
procedure. It typically requires to explore the design parameters space (e.g., via MCMC), generating at each
point a set of pseudo-data that are analysed as real data would, in order to compute their FoM. Then the
search algorithm moves on to maximize the FoM.

In order to carry out the optimization procedure, it might be useful to adopt a principal component
analysis (PCA) to determine a suitable parametrization of [468, 838]. The redshift range of the
survey can be split into bins, with the equation of state taking on a value in the -th bin:

where the basis functions are top-hats of value 1 inside the bin, and 0 elsewhere. If is the Fisher
matrix for the parameters , one can diagonalize it by writing , where is a
diagonal matrix, and the rows of are the eigenvectors or the so-called principal components.
These define a new basis (in which the new coefficients are uncorrelated) so the equation of state can
be written as
The diagonal elements of are the eigenvalues and define the variance of the new parameters,
.
One can now reconstruct by keeping only a certain number of the most accurately determined
modes, i.e., the ones with largest eigenvalues. The optimal number of modes to retain can be estimated by
minimizing the risk, defined as the sum of the bias squared (how much the reconstructed equation of
state departs from the true one by neglecting the more noisy modes) plus the variance of the
estimate [468].