On high-dimensional wavelet eigenanalysis

Abstract

In this paper, we characterize the asymptotic and large scale behavior of the eigenvalues of wavelet random matrices in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. We show that the $r$ largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge in probability to scale-invariant functions in the high-dimensional limit. By contrast, the remaining $p-r$ eigenvalues remain bounded in probability. Under additional assumptions, we show that the $r$ largest log-eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference.

Figures (3)

• Figure 1: The convergence of the rescaled wavelet log-eigenvalues in the three-way limit $\frac{p\space2^j}{n} \rightarrow c \in [0,\infty)$. In this simulation exercise, $X$ is an ofBm and $Z$ is a vector of Gaussian white noise processes (see Section \ref{['s:examples']} for a discussion of these models). $X$ and $Z$ are generated independently. For each $n$, the coordinates matrix ${\mathbf P} {\mathbf P}_H$ is randomly drawn based on i.i.d. standard Gaussian entries, and then normalized to have unit-norm columns. For notational simplicity, we reexpressed the scaling factor as $a(n) = 2^j$, where $j = j_n \rightarrow \infty$. For $r=6$ and $h_q \in \{0.1,0.3, 0.5,0.6,0.8,0.9\}$, $q = 1,\hdots,r$, the plots display the asymptotic behavior of $\frac{1}{2}[(\log \Lambda_{\ell}(2^j))/j-1]$, where $\Lambda_{\ell}(2^j):=\lambda_{\ell}(\mathbf{W}(2^j))$, $\ell = 1,\hdots,p$. The six dashed lines correspond to the values $\{0.1,0.3, 0.5,0.6,0.8,0.9\}$. In all plots, $p$, $n$ and $j$ increase while their ratio remains fixed at $p \space 2^j/n =: p/n_j= c = 1/2$ (left column) and $c=1/4$ (right column). In light of Theorem \ref{['t:lim_n_a_times_lambda/a^(2h+1)']}, for $j = j_n \rightarrow \infty$, $(1/j)\log \Lambda_{p-r+q}(2^j) \stackrel{{\Bbb P}}\rightarrow 2h_q+1$, $q=1,\hdots,6$, and $(1/j)\log \Lambda_{p-r}(2^j) \stackrel{{\Bbb P}}\rightarrow 0$ in the three-way limit. As expected, the $r=6$ largest $\frac{1}{2}[(\log \Lambda_{\ell}(2^j))/j-1]$ approach $h_1,\hdots,h_6$ in the plots as $n$, $p$ and $j$ grow. By contrast, the remaining $\frac{1}{2}[(\log \Lambda_{\ell}(2^j))/j-1]$ tend toward zero as $j$ increases in all instances. For a fixed $c$ (column), going from the top to the bottom row, the magnitudes of $p$ and $2^j$ are larger and smaller, respectively, by a factor of 2 for each value of $n$. As a result, we observe near-convergence at smaller octaves $j$ (n.b.: axes have been shifted to align the curves). Similarly, going from the left to the right column (i.e., as $c$ decreases), the near-convergence also occurs at smaller$j$, which is reflective of a less extreme high-dimensional regime $c$.
• Figure 2: The fluctuations of wavelet log-eigenvalues in the three-way limit $\frac{p\space2^j}{n} \rightarrow c=1/2$. In the same simulation framework as for Figure \ref{['fig:logeig']}, Theorem \ref{['t:asympt_normality_lambdap-r+q']} predicts that the joint distribution of $\{\log \Lambda_{p-r+q}(2^j))\}_{q=1,\hdots,6}$ is asymptotically Gaussian in the three-way limit, after centering and rescaling. In fact, this convergence can be seen in the so-called Gamma plots displayed above, which are expected to look close to a straight line under joint Gaussianity. The plots show the empirical quantiles of the squared Mahalanobis distance statistic vs. the theoretical quantiles of a $\chi^2_6$ distribution based on 5000 realizations (e.g., Johnson and Wichern johnson:wichern:2002). The (effective) sample size $n_j=n/2^j$ increases from left-to-right and $p/n_j$ is set equal to $1/2.$ The plots also display Kolmogorov-Smirnov distance statistics $d_{KS}$, which tend to shrink for larger values of $n_j$.
• Figure 3: Schematic representation of the behavior of wavelet eigenvectors vis-à-vis the coordinate vectors in the three-way limit \ref{['e:three-fold_lim']}. The plot depicts the case $r = 3$ and $r_1 = r_2 = r_3 = 1$ (i.e., $h_1 < h_2 < h_3$). It visually represents the asymptotic behavior of the angles among the high-dimensional vectors. In the limit, we observe that $\textnormal{span}_{\ell=1,2,3}\{{\mathbf u}_{p-r+\ell}(n)\}\sim \textnormal{span}_{\ell=1,2,3}\{{\mathbf p}_{\ell}(n)\}$, $|\langle {\mathbf u}_{p-1}(n),{\mathbf p}_r(n)\rangle| \sim 0$ and $\max\{|\langle {\mathbf u}_{p-2}(n),{\mathbf p}_r(n)\rangle|, |\langle {\mathbf u}_{p-2}(n),{\mathbf p}_{r-1}(n)\rangle| \}\sim 0$. In particular, $\langle {\mathbf u}_p(n),{\mathbf p}_r(n)\rangle^2 \sim 1$ and, approximately, ${\mathbf u}_{p-1}(n) \in \textnormal{span}\{{\mathbf p}_{r-1}(n),{\mathbf p}_r(n)\}$. In the general case, the former three relations are given by \ref{['e:subseq_condition_1_top_explanation']}, \ref{['e:subseq_condition_2_top_explanation']} and \ref{['e:subseq_condition_2_top_explanation-2']}, respectively.