Las relaciones interculturales como relaciones de pertenencia compartida

In nonparametric methods, priors are assigned directly on the space of functions, and function parameterisation has in principle no role. In practice however, the functions are discretised so that they can be represented by a finite-dimensional parameter vector. After discretisation, the input-estimation problem can be treated as a parameter-estimation problem.

The function parameterisation should ideally be flexible enough so that it can accurately represent any function that has a significant probability under the posterior. This can be easily accomplished by making a fine-grained parameterisation with a large number of parameters. On the other hand, such a high-dimensional parameter estimation problem can result in long computation times.

Inbasis function models, the input function is represented as a linear combin- ation of a set ofNB fixed basis functionsBj(t):

u(t) = NB−1

X

j=0

ajBj(t). (2.18)

Here, the basis function coefficientsaj are the parameters to be estimated.

In this model, there is a linear relationship between the parameters and the input function. It is also possible to allow nonlinear relationships. As an example, each basis function could be parameterised though a nonlinear mapping.

The number of basis functions can be fixed, or can be inferred from the data, as is done in the genetic algorithm presented by Madden et al. (1996). In Bayesian inference, this results in atransdimensional problem, where the number of model parameters is one of the parameters to be estimated. Inference algorithms that can handle these problems include Reversible Jump MCMC (Green 1995; Green 2003) and product-space MCMC (Carlin and Chib 1995).

A common choice of basis functions is piecewise constant functions. The time-interval [ti, tf] is divided into several subintervals, and the input function is

assumed to be constant over each interval. In this case, thejth basis function has the value1in thejth time interval and0elsewhere, and thejth basis function coefficient is the function value over that interval. This choice of basis functions makes the problem sparse in the sense that, at any time point, the dynamics of the system are governed by a single basis function coefficient, something that can be exploited by optimal-control methods (Andersson 2013). The disadvantage of these functions is that to make realistic-looking functions, small discretisation steps are needed. This results in a high-dimensional problem which can cause computational difficulties. Alternatively, piecewise linear functions can be used. They carry similar advantages and disadvantages as piecewise constant functions.

Basis splines (B-splines) (Boor 1986; Schumaker 2007) can be used to obtain functions that are differentiable to any desired degree. Each basis function is a polynomial of degreek−1, where k is called the order of the spline. To represent a function over the time interval[ti, tf], a set of time pointst0 =ti < t1 < t2. . . tn−1, calledknots, is defined. Thejth B-spline basis function of orderkis defined recursively by: Bj,1(t) =    1 if tj ≤t≤tj+1 0 otherwise (2.19) Bj,k(t) = t−tj tj+k−1−tj Bj,k−1(t) + tj+k−t tj+k−tj+1 Bj+1,k−1(t). (2.20)

The resulting function is continuously differentiablek−1 times. It can be seen that piecewise constant and piecewise linear basis functions are special cases of B-splines, of order1 and2. B-splines of higher order can represent realistic-looking functions with a relatively small number of knots. They can therefore be preferable to simpler basis functions if high-dimensional problems are to be avoided. However, when the number of knots is small, it is difficult to assess whether this parameterisation can represent any function that has a non-negligible probability under the posterior.

One way to select a set of basis functions is to consider the following: given that only a finite set of basis functions is used, it is desirable to select these such that functions that are probable under the prior can be represented with small discretisation error, at the expense of a higher discretisation error for improbable functions. TheKarhunen-Lo`eve basis functions are derived from these considerations. These functions have the additional advantage that the prior over the function itself can be translated to a prior over the basis function coefficients. This way, the function- estimation problem can be easily converted to a parameter-estimation problem. The Karhunen-Lo`eve basis functionsφj(t), corresponding to a zero-mean Gaussian process

prior with covariance functionK(s, t), are defined as the solutions to the eigenvalue problem (Levy 2008; Wang 2008):

Z tf

ti

K(s, t)φj(s) ds=λjφj(t). (2.21)

The basis function coefficients are independent zero-mean Gaussian random variables with varianceλj. The basis functions have to be derived separately for each

prior. For Gaussian processes with non-zero means, additional basis functions can be added. A more detailed explanation is given in Section 3.1.