Sanna Passino and Heard’s contribution to the Discussion of ‘Statistical exploration of the Manifold Hypothesis’ by Whiteley et al.
本文讨论了潜在度量空间模型中潜在变量独立假设在时间序列数据中的局限性，提出了一种时间依赖的潜在轨迹模型，并通过环面上的分数布朗运动示例展示了动态主成分分析的有效性。
We congratulate the authors on their excellent contribution and the thoughtful perspectives they brought to this topic. In this contribution, we focus on the assumption of independence between the latent variables Z1,…,Zn in the Latent Metric Space (LMS) model in Section 2. Data often arrive sequentially, leading to intrinsic row-wise dependencies between data matrices observed at T>1 time points, Yt=(Yi,j,t)1≤i≤n,1≤j≤p∈Rn×p,t=1,…,T. This scenario is commonly encountered with longitudinal data, or dynamic networks. A temporal LMS model. We could define latent trajectories Zi=(Zi,1,…,Zi,T)∈ZT, sampled independently for each individual i=1,…,n. Observed data matrices Yt∈Rn×p would then be generated as imposing the same assumptions as Section 2. Temporal dependence within each Zi induces dependence between corresponding rows of the matrices Yt, while the latent geometry is preserved through time by the shared metric space Z. Joint dimension reduction. Dimension reduction of the matrices Y1,…,YT can proceed using the unfolded data matrix Y~=[Y1⊤∣…∣YT⊤]⊤∈RnT×p. Let s≤min{p,nT}, and let the columns of the matrix V~Y~∈Rp×s be the orthonormal eigenvectors associated with the s largest eigenvalues of Y~⊤Y~∈Rp×p. In the terminology of the paper, the dimension-s dynamic PCA embedding is: These quantities can be interpreted as stable time-indexed principal component scores summarizing the evolution of each latent trajectory in a shared low-dimensional space. Illustrative example: latent trajectories on a torus. Consider the setting of Section 3.5, in which the latent space Z corresponds to a torus embedded in R3, parameterized by two radii ρ1>ρ2>0. A point z∈Z can be expressed via two angles (θ1,θ2) and a map h:R2→R3: Each individual follows a latent trajectory on the torus, described by Zi,t=h(θ1,i,t,θ2,i,t). Here we consider paths obtained via fractional Brownian motion (fBm) processes with Hurst parameter Hi∈(0,1). For each angle index k=1,2 and individual i=1,…,n, the fBm model with normally distributed initial angles assumes for σ0>0, where Ki is the T×T fBm covariance matrix with entries Following the data-generating structure from Section 3.5 for Xj,j=1,…,p, and Et, we sample observations from model (1) with T=30,p=50,σ=0.01, for n=1,000 individuals evolving as an fBm process on the torus (ρ1=0.75,ρ2=0.15) with Hi=0.9. The latent trajectories are then estimated by the principal component scores (2). Results for the six highest time-indexed principal component scores, for six randomly selected trajectories, are given in Figure 1b, c, along with their true counterparts on the torus in Figure 1a. Latent fBm trajectories on a torus and resulting principal component scores. (a) Six sampled Zi paths. (b) Dimensions 1–3 of ζ~i,t and (c) Dimensions 4–6 of ζ~i,t. As in Section 3.5, whilst the global shapes of the true latent positions and the estimates differ, the inter-point distances seem relatively well-preserved. In the temporal case, we also observe realistic within-individual distances, with relative path lengths approximately preserved. Discussion. In conclusion, we invite the authors to further comment on where the requirement for Z1,…,Zn to be independent is exploited. Clarifying this point could shed further light on the proposed methodology and provide additional intuition on how the results might extend to settings where the latent variables are dependent. Such an extension could, in our opinion, considerably enhance the applicability of the model in temporal contexts, as we attempted to demonstrate in this brief example. Not applicable to this article.