## 4 Discussion and Conclusions  
The Kalman filter is an adequate recursive filter that evaluates the internal capacity of a linear dynamic system (LDS) using a series of error estimates. Possible applications of the KF are in economics, biology, and the environmental sciences, and it is an important topic in control systems engineering and control theory. Together with the LQR, the KF solves the linear quadratic Gaussian (LQG) control problem. The linear quadratic Gaussian controller, linear quadratic regulator, and Kalman filter are solutions to some of the most important problems in control theory. In many applications, the few \"detectable\" measured parameters are smaller than the internal state. In this situation, the Kalman filter could estimate the entire internal state by combining a series of measurements. The Kalman filter could be adapted to a wide variety of economic applications. It is mainly an ordinal least-square (OLS) procedure that gives the minimum mean square estimators (MMSE) under the Gaussian assumption. Even when the normality assumption  
Figure 3: Orientation estimate errors for 100 Monte Carlo trials. Figure 2: Measurement errors and position estimates of 100 Monte Carlo trials. is waived, the Kalman filter minimizes any symmetric loss function. The KF is not only used directly in economic problems but it could also be used as a background to other estimation techniques, such as the estimation of Markov switching models or the quasi-maximum likelihood estimation procedure. In the environmental sciences, the Kalman filter has wide applications in the areas of the pollution of aquatic systems, measuring atmospheric pollution, environmental policy management, and forecasting hurricanes [33; 34]. The KF could also be used to estimate the temporal variations in associations between the radial growth of black oak and physiologically based climate indices [35; 36]. In biology, biological samples and bacteriological infections could be detected using a multi-process Kalman filter to provide an online estimate of the inflection point of changes in the growth of an organism [37]. A wide collection of Kalman filters have been developed in the past few decades from the original formulation of the Kalman filter. The information filter, Schmidt's extended filter, or the Kalman-Bucy filter are a wide variety of filters that were developed by [PERSON], [PERSON], and many others. Probably the most used type of Kalman filter is the phase-locked loop (PLL), which is now omnipresent in television sets, radios, satellite communications receivers, and almost any other communications equipment. In this paper, we presented the KF model, the general case of the KF, and the Gaussian case. The semigroup of the Hamiltonian matrices and contraction property were used in the KF algorithm to estimate the Gaussian errors. The main result of this study was the use of a recursive algorithm of the Kalman filter to estimate the Gaussian errors. Several possible variations of our approach exist. Model herring inflow as a state variable and model herring inflow as pure noise with a non-zero mean or model noise as pure white (non-geometric) noise assume perfect observations of the control variables (catch and inflow). These variations were all tested and led to only small changes in the recursive algorithm fit and small variations in the parameter estimates. The main limitation of this study came from the fact that there were not many applications of the algorithms described in the previous sections. Future work, as well as possible applications of the methodology, could be applied in mathematics, physics, and engineering research areas. Moreover, a future research topic is how to treat more general noise terms, for example, terms including unknown parameters. Conceptualization, M.B. ; Data curation, M.B. ; Formal analysis, M.B. ; Funding acquisition, C.B. ; Methodology, M.B. ; Project administration, C.B. and M.B. ; Resources, C.; Software, M.B. ; Supervision, C.B. ; Validation, C.B. and M.B. ; Writing - original draft, M.B.