Definition: A Kalman filter is a mathematical algorithm used to estimate the state of a dynamic system based on noisy measurements. It is a type of recursive filter that updates its estimates of the state and its uncertainty over time as new measurements are obtained. The Kalman filter is based on a probabilistic model of the system, where the state of the system is represented as a random variable and the measurements are also subject to noise. The filter estimates the state of the system by combining the predicted state based on the system dynamics with the measurement information, using a weighted average that gives more weight to measurements that are more reliable.The mathematical representation of a Kalman filter for fNIRS studies is similar to that of a general Kalman filter, but with specific parameters and state variables that are tailored to the characteristics of the fNIRS signal. The state vector of an fNIRS Kalman filter typically includes the changes in concentration of oxygenated hemoglobin (HbO) and deoxygenated hemoglobin (HbR) in each brain region being measured. The state vector at time k is denoted as
![x(k) = [HbO(k), HbR(k)]^T](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-ed898ab43e69892ed7218a535cbc328a_l3.png "Rendered by QuickLaTeX.com")
The measurement vector typically includes the measured optical density changes at each wavelength and each source-detector pair, denoted as
![z(k) = [Delta OD1(k), Delta OD2(k), ..., Delta ODn(k)]^T](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-6f880ba4d4dff826c7e3ea7e10cd7d11_l3.png "Rendered by QuickLaTeX.com")
where
represents the change in optical density at the i-th source-detector pair at time k.
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References: https://doi.org/10.1134/S2075108711020076
https://doi.org/10.1109/IMOC.2009.5427541
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