Estimating Derivatives for Samples of Sparsely Observed Functions, With Application to Online Auction Dynamics
稀疏观测函数样本的导数估计及在线拍卖动态应用AI 解读 ↓
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稀疏观测函数样本的导数估计及在线拍卖动态应用
- 研究问题
- 在每条函数仅有少量含噪观测时，如何恢复其导数并刻画动态过程？研究以在线拍卖的稀疏出价记录及价格变化为应用。
- 方法与思路
- 作者估计特征函数的导数，并结合随机函数展开和稀疏数据的函数主成分得分恢复个体导数，再研究高斯过程下的一阶微分方程。
- 材料陈述的贡献
- 摘要称建立一致性，并在模拟中优于所比较的样条及核平滑方法；拍卖应用展示个体价格曲线导数估计及数据驱动的动态描述。
- 阅读关注点
- 应核对观测稀疏度与噪声条件、导数估计所需光滑性、主成分截断选择，以及高斯过程微分方程结论的具体适用范围。
- 阅读边界
- OpenAlex索引摘要未注明稿件版本；模拟优势仅针对所列比较设定，拍卖价格曲线的动态描述不能直接解释为竞价行为的因果机制。
原文摘要
It is often of interest to recover derivatives of a sample of random functions from sparse and noise-contaminated measurements, especially when the dynamics of underlying processes is of interest. We propose a novel approach based on estimating derivatives of eigenfunctions and expansions of random functions into their eigenfunctions to obtain a representation for derivatives. In combination with estimates for functional principal component scores for sparse data, this leads to a viable solution of the challenging problem to recover derivatives for sparsely observed functions. We establish consistency results and demonstrate in simulations that the method is superior to alternative approaches (derivative estimation with random effects models based on B-spline bases, kernel smoothing, smoothing splines, or P-splines). Our study is motivated by an analysis of bidding histories for eBay auctions, for which bids are typically very sparse in the middle and somewhat more frequent near the beginning and end of an auction. We demonstrate the estimation of derivatives of price curves for individual auctions from the sparsely observed bidding histories and also derive a model-free first-order differential equation that applies in the case of Gaussian processes. This provides a data-driven dynamic model that we use to elucidate auction dynamics.