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In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference position over time.
The particle's Mean squared displacement from its original position is: =, where is the dimension of the particle's Brownian motion. For example, the diffusion of a molecule across a cell membrane 8 nm thick is 1-D diffusion because of the spherical symmetry; However, the diffusion of a molecule from the membrane to the center of a eukaryotic ...
Physical scientists often use the term root mean square as a synonym for standard deviation when it can be assumed the input signal has zero mean, that is, referring to the square root of the mean squared deviation of a signal from a given baseline or fit. [8] [9] This is useful for electrical engineers in calculating the "AC only" RMS of a signal.
The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled).
Simulated squared displacements of free Brownian particles (semi-transparent wiggly lines) as a function of time, for three selected choices of initial squared velocity which are 0, 3k B T/m, and 6k B T/m respectively, with 3k B T/m being the equipartition value in thermal equilibrium. The colored solid curves denote the mean squared ...
The RMSD of predicted values ^ for times t of a regression's dependent variable, with variables observed over T times, is computed for T different predictions as the square root of the mean of the squares of the deviations:
Mean squared displacement for different types of anomalous diffusion Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), r 2 ( τ ) {\displaystyle \langle r^{2}(\tau )\rangle } , and time.
In mathematics and its applications, the mean square is normally defined as the arithmetic mean of the squares of a set of numbers or of a random variable. [ 1 ] It may also be defined as the arithmetic mean of the squares of the deviations between a set of numbers and a reference value (e.g., may be a mean or an assumed mean of the data), [ 2 ...