Women's basal body temperature (BBT) shows a periodic pattern that associates with menstrual cycle. Although this fact suggests a possibility that daily BBT time series can be useful for estimating the underlying phase state as well as for predicting the length of current menstrual cycle, little attention has been paid to model BBT time series. In this study, we propose a state‐space model that involves the menstrual phase as a latent state variable to explain the daily fluctuation of BBT and the menstruation cycle length. Conditional distributions of the phase are obtained by using sequential Bayesian filtering techniques. A predictive distribution of the next menstruation day can be derived based on this conditional distribution and the model, leading to a novel statistical framework that provides a sequentially updated prediction for upcoming menstruation day. We applied this framework to a real data set of women's BBT and menstruation days and compared prediction accuracy of the proposed method with that of previous methods, showing that the proposed method generally provides a better prediction. Because BBT can be obtained with relatively small cost and effort, the proposed method can be useful for women's health management. Potential extensions of this framework as the basis of modeling and predicting events that are associated with the menstrual cycles are discussed. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Nonlinear phenomena are universal in ecology. However, their inference and prediction are generally difficult because of autocorrelation and outliers. A traditional least squares method for parameter estimation is capable of improving short-term prediction by estimating autocorrelation, whereas it has weakness to outliers and consequently worse long-term prediction. In contrast, a traditional robust regression approach, such as the least absolute deviations method, alleviates the influence of outliers and has potentially better long-term prediction, whereas it makes accurately estimating autocorrelation difficult and possibly leads to worse short-term prediction. We propose a new robust regression approach that estimates autocorrelation accurately and reduces the influence of outliers. We then compare the new method with the conventional least squares and least absolute deviations methods by using simulated data and real ecological data. Simulations and analysis of real data demonstrate that the new method generally has better long-term and short-term prediction ability for nonlinear estimation problems using spawner–recruitment data. The new method provides nearly unbiased autocorrelation even for highly contaminated simulated data with extreme outliers, whereas other methods fail to estimate autocorrelation accurately.

Nonlinear phenomena are universal in ecology. However, their inference and prediction are generally difficult because of autocorrelation and outliers. A traditional least squares method for parameter estimation is capable of improving short-term prediction by estimating autocorrelation, whereas it has weakness to outliers and consequently worse long-term prediction. In contrast, a traditional robust regression approach, such as the least absolute deviations method, alleviates the influence of outliers and has potentially better long-term prediction, whereas it makes accurately estimating autocorrelation difficult and possibly leads to worse short-term prediction. We propose a new robust regression approach that estimates autocorrelation accurately and reduces the influence of outliers. We then compare the new method with traditional least squares and least absolute deviations methods using simulated data and real ecological data. Simulations and analysis of real data demonstrate that the new method generally has better long-term and short-term prediction ability for nonlinear estimation problems using spawner--recruitment data. The new method provides nearly unbiased autocorrelation even for highly contaminated simulated data with extreme outliers, whereas the other methods fail to estimate autocorrelation accurately.