## 1 Introduction  
Feature selection is known as one of core concepts in the field of machine learning based fault diagnosis [1; 2; 3]. There are several feature selection approaches available such as information gain, mutual information, and the chi-square test [4; 5; 6]. Among them, the chi-square, also written as \(\chi^{2}\), test statistic can be also used for abnormal signal detection [7; 8]. The chi-square test statistic is developed from the difference between two variables such as the original state and its filtered estimate, and then compared with a precomputed threshold to detect abnormal signals. Meanwhile, in contrast to the recursive infinite memory structure (IMS) filter like the traditionally used Kalman filter [9; 10; 11; 12], the finite memory structure (FMS) filter has been known to have inherent good properties such as bounded-input, bounded-output (BIBO) stability and and more robustness against temporary uncertainties due to its processing manner of finite measurements on the most recent window [13; 14; 15; 16]. Thus, the FMS filter has been applied successfully for various engineering problems [17; 18; 19; 20; 21; 22; 23]. As shown in [13; 14; 15; 16], the FMS filter is known to have better noise suppression as the measurement window length grows. In other words, the noise suppression of the FMS filter is closely related to the measurement window length. However, even if the FMS filter can show greater noise suppression as the window length increases, the tracking speed of the state estimate for the actual state variable worsens in proportion to the window length, which can degrade the estimation performance of the FMS filter. This implies the FMS filter requires a compromise between the noise suppression and the tracking speed of the state estimate. According to this observation, the estimation error of the FMS filter with a short measurement window length is smaller than that of the FMS filter with a long measurement window length, while uncertainty exists. In addition, the convergence of the estimation error for the FMS filter with a short window length is much faster than that of the FMS filter with a long window length when temporary uncertainty is disappearing. This means that the FMS filter with a short window length is superior in terms of the tracking ability. Thus, if the FMS filter with a short window length is applied to temporarily uncertain systems, it can outperform the FMS filter with a long window length, although the FMS filter with a short window length is designed without considering the robustness. This paper proposes an FMS filtering with two kinds of measurement windows using the chi-square test statistic to cover the nominal system as well as the temporarily uncertain system. The simple matrix form for the FMS filter is developed from the conditional density of the current state given finite past measurements. Then, one of the two FMS filters, the primary FMS filter and the secondary FMS filter, with different measurement windows is operated selectively to obtain the valid estimate according to presence or absence of uncertainty. The primary FMS filter is selected for the nominal system and the secondary FMS filter is selected for the temporarily uncertain system, respectively. A declaration rule is defined to indicate the presence or absence of uncertainty, operate the suitable one from two filters, and then obtain the valid filtered estimate. A test variable for the declaration rule is developed using a chi-square test statistic from the estimation error and compared with a precomputed threshold. Finally, computer simulations are performed for a two kinds of dynamic systems such as a F404 gas turbine aircraft engine and electric motor to verify the proposed selective FMS filtering with two kinds of measurement windows and compare with existing FMS filtering and IMS filtering. Through computer simulation works, it is confirmed that the proposed selective FMS filtering works well for the nominal system as well as the temporarily uncertain system.