Let \({ n}\) measurements of a process be provided sequentially, where the process follows a sigmoid shape, but the data have lost sigmoidicity due to measuring errors. If we smooth the data by making least the sum of squares of errors subject to one sign change in the second divided differences, then we obtain a sigmoid approximation. It is known that the optimal fit of this calculation is composed of two separate sections, one best convex and one best concave. We propose a method that starts at the beginning of the data and proceeds systematically to construct the two sections of the fit for the current data, step by step as n is increased. Although the minimization calculation at each step may have many local minima, it can be solved in about \({\mathcal {O}}(n^2)\) operations, because of properties of the join between the convex and the concave section. We apply this method to data of daily Covid-19 cases and deaths of Greece, the United States of America and the United Kingdom. These data provide substantial differences in the final approximations. Thus, we evaluate the performance of the method in terms of its capabilities as both constructing a sigmoid-type approximant to the data and a trend detector. Our results clarify the optimization calculation both in a systematic manner and to a good extent. At the same time, they reveal some features of the method to be considered in scenaria that may involve predictions, and as a tool to support policy-making. The results also expose some limitations of the method that may be useful to future research on convex-concave data fitting.

让\({ n}\)过程的测量按顺序提供，其中过程遵循 sigmoid 形状，但数据由于测量错误而失去了 sigmoid 性。如果我们通过使误差平方和至少受到二分差分中一个符号变化的影响来平滑数据，那么我们获得了一个 sigmoid 近似。众所周知，该计算的最佳拟合由两个单独的部分组成，一个最佳凸面和一个最佳凹面。我们提出了一种方法，该方法从数据的开头开始，并随着n的增加逐步构建适合当前数据的两个部分。虽然每一步的最小化计算可能有很多局部最小值，但大约可以在\({\mathcal {O}}(n^2)\)操作，因为凸截面和凹截面之间的连接特性。我们将此方法应用于希腊、美利坚合众国和英国的每日 Covid-19 病例和死亡数据。这些数据提供了最终近似值的实质性差异。因此，我们根据其构建数据的 sigmoid 型近似值和趋势检测器的能力来评估该方法的性能。我们的结果以系统的方式和在很大程度上阐明了优化计算。同时，它们揭示了在可能涉及预测的情景中需要考虑的方法的一些特征，并作为支持决策的工具。结果还暴露了该方法的一些局限性，这些局限性可能对未来凸凹数据拟合的研究有用。