Estimation, imputation and prediction for the functional linear model with scalar response with responses missing at random

Predicting rainfall and irrigation requirements of corn in Ecuador

This work is a case study whose objective is prediction of irrigation needs of corn crops in different regions of Ecuador; being this a fundamental basic food for the country's economy, as in the remaining countries of the Andean area. The proposed methodology seeks to help improving the quality of corn crop. Specifically, we propose the application of regression models, within the framework of Functional Data Analysis (FDA), to predict the amount of rainfall (scalar response variable) in the places with the highest production of corn in Ecuador, as a function of functional covariates such as temperature and wind speed. From the estimation of the amount of rainfall, effective precipitation is calculated. This is the fraction of water used by the crops, from which the value of real evapotranspiration or ETc is obtained and, more importantly, the irrigation requirements at each stage of the corn crop, for its adequate physiological development. Application of regression models based on functional basis, Functional Principal Components (FPC) or Functional Partial Least Squares (FPLS) for scalar response variable, allows us to use the information of variables such as wind speed and temperature (of functional nature) in a better way than using multivariate models, for predicting the amount of rainfall, obtaining, as a result, very explicative models, defined by a high goodness of fit (R 2 = 0.97 , with 6 significant parameters and an error of 0.14) and practical utility. The model has been also applied to North Peru regions, obtaining rainfall prediction errors between 9% and 22%. Thus, the geographical limitations of the model could be the Andean regions with similar climate. In addition, this study proposes the application of FDA exploratory analysis and FDA outlier detection techniques as a common and useful practice in the specific domain of rainfall prediction studies, prior to applying the regression models.

Kernel machine learning methods to handle missing responses with complex predictors. Application in modelling five-year glucose changes using distributional representations

BACKGROUND AND OBJECTIVES

Missing data is a ubiquitous problem in longitudinal studies due to the number of patients lost to follow-up. Kernel methods have enriched the machine learning field by successfully managing non-vectorial predictors, such as graphs, strings, and probability distributions, and have emerged as a promising tool for the analysis of complex data stemming from modern healthcare. This paper proposes a new set of kernel methods to handle missing data in the response variables. These methods will be applied to predict long-term changes in glycated haemoglobin (A1c), the primary biomarker used to diagnose and monitor the progression of diabetes mellitus, making emphasis on exploring the predictive potential of continuous glucose monitoring (CGM).

METHODS

We propose a new framework of non-linear kernel methods for testing statistical independence, selecting relevant predictors, and quantifying the uncertainty of the resultant predictive models. As a novelty in the clinical analysis, we used a distributional representation of CGM as a predictor and compared its performance with that of traditional diabetes biomarkers.

RESULTS

The results show that, after the incorporation of CGM information, predictive ability increases from R²=0.61 to R²=0.71. In addition, uncertainty analysis is useful for characterising some subpopulations where predictivity is worsened, and a more personalised clinical follow-up is advisable according to expected patient uncertainty in glucose values.

CONCLUSIONS

The proposed methods have proven to deal effectively with missing data. They also have the potential to improve the results of predictive tasks by including new complex objects as explanatory variables and modelling arbitrary dependence relations. The application of these methods to a longitudinal study of diabetes showed that the inclusion of a distributional representation of CGM data provides greater sensitivity in predicting five-year A1c changes than classical diabetes biomarkers and traditional CGM metrics.

Elucidating age and sex-dependent association between frontal EEG asymmetry and depression: An application of multiple imputation in functional regression

Frontal power asymmetry (FA), a measure of brain function derived from electroencephalography, is a potential biomarker for major depressive disorder (MDD). Though FA is functional in nature, it is typically reduced to a scalar value prior to analysis, possibly obscuring its relationship with MDD and leading to a number of studies that have provided contradictory results. To overcome this issue, we sought to fit a functional regression model to characterize the association between FA and MDD status, adjusting for age, sex, cognitive ability, and handedness using data from a large clinical study that included both MDD and healthy control (HC) subjects. Since nearly 40% of the observations are missing data on either FA or cognitive ability, we propose an extension of multiple imputation (MI) by chained equations that allows for the imputation of both scalar and functional data. We also propose an extension of Rubin's Rules for conducting valid inference in this setting. The proposed methods are evaluated in a simulation and applied to our FA data. For our FA data, a pooled analysis from the imputed data sets yielded similar results to those of the complete case analysis. We found that, among young females, HCs tended to have higher FA over the θ, α, and β frequency bands, but that the difference between HC and MDD subjects diminishes and ultimately reverses with age. For males, HCs tended to have higher FA in the β frequency band, regardless of age. Young male HCs had higher FA in the θ and α bands, but this difference diminishes with increasing age in the α band and ultimately reverses with increasing age in the θ band.