## 1 Introduction  
The lithium-ion (Li-ion) battery is an integral part of many of our available systems, such as those used in laptops, cell phones, and the aerospace and power industries, and it has become crucial for the overall performance of such systems. As the demand for batteries is rising over the years, researchers are paying more attention to the health monitoring and prognostics of Li-ion batteries to avoid unexpected failures. Scientists are exploring state-of-the-art technologies to control the operating conditions and predict replacement intervals for batteries, aiming to improve the system reliability and consistency.  
Over the years, several studies have been conducted to develop prediction algorithms for the state-of-health (SOH) of Li-ion batteries by the prognostics research community. Model-based and data-driven approaches are the main fields of study. The data-driven model is a method for mapping the relationship between the available observations and hidden variables using a framework built on obtainable performance measures [1]. Conventional approaches to battery prognostics mostly focus on exploring the SOH and state-of-charge (SOC) by employing battery capacity data [2, 3]. These traditional approaches using capacity degradation data and theoretical assumptions frequently contribute to measurement inaccuracies, which may result in a battery failure [4]. However, these inaccuracies have compelled researchers to develop advanced prediction methods such as simulation, physical cell degradation, and data-driven prediction approaches [5]. Following the substantial prospects of Li-ion batteries, extensive research has been conducted to predict the remaining life cycles of Li-ion batteries using capacity degradation to improve battery performance and reliability. A comparative study was conducted for predicting Li-ion battery residual lifetime based on the Bayesian approach [6]. The study developed a non-parametric model which considered the mean of residual lifetimes. The study concluded that the Bayesian approach more accurately provides prediction results based on the confidence interval. However, researchers have seldom tried using other performance measures, such as temperature, voltage, and current observational data, to predict the capacity, although such measurements are as equally significant as capacity degradation for data-driven model development [7].  
The study presented in this article focused on these issues by developing an empirical model based on Li-ion battery capacity degradation that is affected by the ambient environment and load condition. An effective prognostic algorithm should be able to predict the remaining useful life or capacity using the past life cycle data of battery performance. In this study, we aimed to predict the capacity of Li-ion batteries using a data-driven prognostics algorithm supporting uncertainty representation and management. During the study, the monitored data of the Li-ion battery were extracted as a form of a function due to the Li-ion battery's cyclic orders. While developing the model, the study addressed the effect of current, voltage, and temperature on the capacity degradation by applying the functional principal component analysis (fPCA) to predict the remaining capacity of a battery. Unlike the study of [PERSON] and [PERSON] [8], where such monitoring data, i.e., current, voltage, and temperature, were included in the prediction model as a form of averaged quantities for each cycle, we identified the dominant models of functional variation using the framework of dimension reduction techniques. The battery monitoring observations collected from NASA Ames Prognostics Center of Excellence repository are used to illustrate our method. A notable contribution of the study is the proposal of a data-driven model that includes battery condition indicators, such as voltage, current, and temperature, and employs the dimension reduction statistical technique fPCA.