Análisis de componentes principales y Clúster funcional general por Estación

We again consider the simultaneously diagonalizable setup described in Subsection 4.6.1, where X = L2_{[0,1] with Dirichlet boundary conditions. As in the previous}

subsection we defineA₀ to be the negative Laplacian with Dirichlet boundary cond- tions in [0,1]. We consider the case where K = (I + ₁₀1_π2A0)

−1_{, C}

0 = A−₀1 and

C₁ =A−

4 5

0 . In the language of Subsection 4.6.1, we have that Assumptions 4.6.1 are

satisfied with α = 1, β = 4/5 and ` = 1, hence, since 2α+ 4`−2β = 22/5 > 1, Assumption 4.6.2 is also satisfied. We assume that we have data of the form

y =Ku¯+ ¯σ−12C 1 2 1ξ, (4.7.1) where ¯ u(x) = 0.75·_1[0.1,0.25](x) + 0.25·_1[0.35,0.38]+ sin4(2πx)·_1[0.5,1](x), x∈[0,1],

is the true underlying signal, ¯σ = 256 and ξ is a Gaussian white noise. Noticing β−2`

hence have that the infinite dimensional assumptions on the underlying model are satisfied and our intuition presented in Subsection 4.3.1 applies.

Instead of working in the frequency domain and truncating the Karhunen- Loeve expansion, we discretize the domain [0,1] using a uniform grid ofN points and use finite differences to discretizeA0 hence also K,C0 and C1, [7, 8]. In particular,

we replaceA₀ by the N×N matrix

A0=N2           2 −1 0 . . . 0 −1 2 −1 . .. ... 0 . .. . .. . .. 0 .. . . .. −1 2 −1 0 . . . 0 −1 2           ,

and the operators K,C₀ and C₁ by the corresponding N ×N matrices calculated through the appropriate functions of the matrixA0. In definingK, we also replace

the identity operator by the N ×N identity matrix. We define the inner product and norm inRN u, v RN = 1 N N X j=1 ujvj, and u RN = 1 N N X j=1 u2_j 1₂ .

Since we consider u to be discretized on the grid, we have uj = u(_Nj), and hence have a discrete approximation of X with norm and inner product which are the discrete analogues of the L2-norm and inner product. We do not prove that this discretization scheme satisfies Assumptions 4.2.1 and 4.2.4, however we expect this to be the case. In fact, instead of discretizing y in (4.7.1) by discretizing ¯u on the grid and replacing the operatorsK and C1 by the corresponding matrices and ξ by

a white noise in RN, we do this only for N = 8192 and produce the data at the lower discretization levels N = 32,128,512 andN = 2048 by subsampling. That is we treat the data at levelN = 8192 as our infinite dimensional data and discretize it by subsampling. This is not exactly what we assume in (4.2.6), however it is very closely related.

presented in Section 4.1 and Section 4.4, with hyper-parameters α₀ = α₁ = r0 =

1,β0 = β1 = q02 = 104 chosen to give uninformative hyper-priors, that is, hyper-

priors with variance which is much larger than their mean. We use 104 _{iterations of}

the two Gibbs samplers and choseσ(0)= 1 in both cases,δ(0)= 10 andτ(0)= 1/√10. In the calculation of the sample mean and variance of the unknown, we again use a constant burn-in time of 1000 iterations.

In Figure 4.6 we have in the left panel the true solution (dashed black) and discretized noisy data (blue continuous), and in the middle and right pan- els the true solution (dashed black), the sample mean (red continuous) and 87.5% credibility bounds (shaded area) using the standard hierarchical algorithm and the reparametrized algorithm respectively, for dimensionN = 8192. The sample means and credibility bounds at other discretization levels are similar, hence omitted.

Figure 4.6: Left panel: true solution (dashed black) and blurred noisy data (blue continuous). Middle and right panels: true solution (dashed black), sample mean (red continuous) and 87.5% credibility bounds (shaded area) for standard hierarchi- cal (middle) and reparametrized algorithm (right). Dimension isN = 8192.

In Figure 4.7 we have the plots of theσ-chains on the left and the δ-chains on the right, in the standard algorithm, for increasing dimension as we move from top to bottom. As predicted by Theorems 4.2.2 and 4.2.5, the plots show that while in small dimensions both chains appear to have a healthy mixing, asN increases the

σ-chain moves to the true value ¯σ= 256 and fluctuates independently around it with fluctuations which decrease as N increases, while the δ-chain becomes slower and exhibits diffusive behaviour. In Figure 4.8 we have the plots of theσ-chains on the left and theτ2-chains on the right, in the reparametrized algorithm for increasing dimension top to bottom. As expected, theσ-chain exhibits the same behaviour as in the standard algorithm but theτ2-chain appears to be robust with respect to the increase in dimension.

Our observations in Figures 4.7 and 4.8 are also supported by the autocor- relation plots presented in Figure 4.9. We have four panels with the plots of the autocorrelation functions for time lag 1−20 of the four chains at the different dis- cretization levels N. On the left column we have the autocorrelation functions of theσ-chains using the standard algorithm (top) and the reparametrized algorithm

Figure 4.7: Standard algorithm: σ-chains (left column) and δ-chains (right column) for dimensionsN = 32,128,512,2048 and 8192 top to bottom.

Figure 4.8: reparametrized algorithm: σ-chains (left column) and τ2-chains (right column) for dimensionsN = 32,128,512,2048 and 8192 top to bottom.

(bottom) which are practically the same; in both cases the rate of decay of corre- lations seems to increase as N increases, and indeed for N ≥512 even consecutive samples are practically independent. On the right column we have the autocorrela- tion function of the δ-chain in the case of the standard algorithm (top) and of the

τ2-chain in the case of the reparametrized algorithm (bottom). The rate of decay of correlations in theδ-chain appears to decrease as the dimension increases, and in particular for large N the correlations seem to decay very slowly. On the contrary, the rate of decay of correlations in theτ2-chain does not seem to be affected by the increase in dimension.

Figure 4.9: Autocorrelation functions for dimensions 32 (black), 128 (blue), 512 (red), 2048 (green) and 8192 (violet). Top left is forσ-chain in standard algorithm, top right forδ-chain in standard algorithm, bottom left forσ-chain in reparametrized algorithm and bottom right forτ2-chain in reparametrized algorithm.

The fact that in low dimensions the rate of decay of correlations is slower in the τ2-chain than in the δ-chain, is due to the small noise effect explained in Section 4.4. To highlight this effect, we run the reparametrized algorithm again in the case of a much smaller noise, namely ¯σ= 2562, and plot theσ and τ2-chains in Figure 4.10. As expected, the σ-chain exhibits the same behaviour as before, but theτ2-chain mixes very poorly. We once more highlight that new work is required

to produce effective hierarchical algorithms in this small noise limit.

Figure 4.10: reparametrized algorithm for small noise, ¯σ = 2562: σ-chain (left column) andτ2-chain (right column) for dimension N = 512.

In conclusion, our numerical simulations again support the results on the standard hierarchical algorithm presented in Section 4.2 and our intuition on the reparametrized algorithm discussed in Section 4.4. Once more, they suggest that it should be possible to improve Theorem 4.2.5 on the behaviour of the σ-chain to a result formulated almost surely with respect to the data.

4.7.3 Linear Bayesian inverse problem with coarse data using finite