We introduce the shape module of the Python package Geomstats to analyze shapes of objects represented as landmarks, curves, and surfaces across fields of natural sciences and engineering. The shape module first implements widely used shape spaces, such as the Kendall shape space, as well as elastic spaces of discrete curves and surfaces. The shape module further implements the abstract mathematical structures of group actions, fiber bundles, quotient spaces, and associated Riemannian metrics which allow users to build their own shape spaces. The Riemannian geometry tools enable users to compare, average, interpolate between shapes inside a given shape space. These essential operations can then be leveraged to perform statistics and machine learning on shape data. We present the object-oriented implementation of the shape module along with illustrative examples and show how it can be used to perform statistics and machine learning on shape spaces.

Crowdsourced speed test measurements, such as those by Ookla® and Measurement Lab (M-Lab), offer a critical view of network access and performance from the user's perspective. However, we argue that taking these measurements at surface value is problematic. It is essential to contextualize these measurements to understand better what the attained upload and download speeds truly measure. To this end, we develop a novel Broadband Subscription Tier (BST) methodology that associates a speed test data point with a residential broadband subscription plan. Our evaluation of this methodology with the FCC's MBA dataset shows over 96% accuracy. We augment approximately 1.5M Ookla and M-Lab speed test measurements from four major U.S. cities with the BST methodology. We show that many low-speed data points are attributable to lower-tier subscriptions and not necessarily poor access. Then, for a subset of the measurement sample (80k data points), we quantify the impact of access link type (WiFi or wired), WiFi spectrum band and RSSI (if applicable), and device memory on speed test performance. Interestingly, we observe that measurement time of day only marginally affects the reported speeds. Finally, we show that the median throughput reported by Ookla speed tests can be up to two times greater than M-Lab measurements for the same subscription tier, city, and ISP due to M-Lab's employment of different measurement methodologies. Based on our results, we put forward a set of recommendations for both speed test vendors and the FCC to con-textualize speed test data points and correctly interpret measured performance.

Principal component analysis (PCA) is a well-established tool in machine learning and data processing. The principal axes in PCA were shown to be equivalent to the maximum marginal likelihood estimator of the factor loading matrix in a latent factor model for the observed data, assuming that the latent factors are independently distributed as standard normal distributions. However, the independence assumption may be unrealistic for many scenarios such as modeling multiple time series, spatial processes, and functional data, where the outcomes are correlated. In this paper, we introduce the generalized probabilistic principal component analysis (GPPCA) to study the latent factor model for multiple correlated outcomes, where each factor is modeled by a Gaussian process. Our method generalizes the previous probabilistic formulation of PCA (PPCA) by providing the closed-form maximum marginal likelihood estimator of the factor loadings and other parameters. Based on the explicit expression of the precision matrix in the marginal likelihood that we derived, the number of the computational operations is linear to the number of output variables. Furthermore, we also provide the closed-form expression of the marginal likelihood when other covariates are included in the mean structure. We highlight the advantage of GPPCA in terms of the practical relevance, estimation accuracy and computational convenience. Numerical studies of simulated and real data confirm the excellent finite-sample performance of the proposed approach.