Edge devices have limited resources, which inevitably leads to situations where stream processing services cannot satisfy their needs. While existing autoscaling mechanisms focus entirely on resource scaling, Edge devices require alternative ways to sustain the Service Level Objectives (SLOs) of competing services. To address these issues, we introduce a Multi-dimensional Autoscaling Platform (MUDAP) that supports fine-grained vertical scaling across both service- and resource-level dimensions. MUDAP supports service-specific scaling tailored to available parameters, e.g., scale data quality or model size for a particular service. To optimize the execution across services, we present a scaling agent based on Regression Analysis of Structural Knowledge (RASK). The RASK agent efficiently explores the solution space and learns a continuous regression model of the processing environment for inferring optimal scaling actions. We compared our approach with two autoscalers, the Kubernetes VPA and a reinforcement learning agent, for scaling up to 9 services on a single Edge device. Our results showed that RASK can infer an accurate regression model in merely 20 iterations (i.e., observe 200s of processing). By increasingly adding elasticity dimensions, RASK sustained the highest request load with 28% less SLO violations, compared to baselines.