Root-mean-square

INTRODUCTION. The root mean square (RMS/rms) value of a set of data, aka quadratic mean, is a statistical measure - a type of average - defined as the square root of the mean square value. This in turn is defined as the arithmetic mean of the squares of the individual values. It is a particular case (i.e. with exponent 2) of the `generalised` mean (see diagram). The RMS value can be calculated for a sequence of discrete values or for a continuous function. The value for a continuous function (e.g. a signal) can be approximated by taking the RMS of a series of equally spaced samples. One justification for using RMS values as an indicator of average is that we are often interested only in the magnitudes of data values, not their signs. Whereas use of the arithmetic mean would result in a cancelling-out effect between negatives and positives - which is unwanted in many contexts - the squaring involved in calculating RMS values produces only positive outcomes, thus avoiding this problem. Why do we not use the mean of the absolute values of a data set - a parameter which is more intuitive and easier to calculate than RMS - as an indicator of magnitude-average? This question is at the centre of an ongoing debate about the traditional and almost ubiquitous use - and teaching - of standard deviation (SD) as opposed to mean (absolute) deviation (MD). The principal argument deployed by proponents of SD is that it is much easier to incorporate SD into higher-level calculations than it would be for MD. This is countered by the comment that we are now in an era where the power of computing negates what would otherwise be problematical any advanced mathematical analysis using dispersion data. An advantage of using RMS values, as far as science is concerned, is that these values may well correspond to actual meaningful and measurable physical parameters. For example, an RMS alternating current is equivalent to a measurable direct current of the same value. ROOT MEAN SQUARE EXAMPLES In the formulae (see diagram), the 𝒚 quantities being rms-averaged can refer to many different physical situations. (1a) y = displacement (net distance travelled) in a “random walk” of N steps each of mean length L. RMS displacement = L√N. (1b) y = end-to-end distance of a freely jointed linear polymer chain averaged over all conformations and consisting of N segments each of length L. RMS overall length = L√N. (2) y = speed of a gas molecule. RMS speed of all molecules in an ideal gas = √{3RT/M}: R=molar gas constant, T=absolute temperature, M=molar mass. (3) y = acceleration G measured by accelerometer. RMS acceleration (GRMS) = √A : A = area under acceleration spectral density (ASD) curve. GRMS expresses the overall energy of a particular random vibration event and may be a statistical value used in mechanical engineering. (4) y = instantaneous voltage or current of cyclically varying signal. RMS value turns out to be its “effective” value, i.e. value of the direct voltage or current that would produce the equivalent power dissipation in a resistive load. For a sinusoidally varying AC, the RMS voltage or current = yp/root(2) (subscri pt ‘p’ denotes ‘peak’). RMS voltage or current when there is a DC offset y0 and AC amplitude 𝑦p to the sinusoid = √{y0^2+yp^2/2}. (5) y = stress or strain or sound pressure. (6) y = the pairwise separation of corresponding atoms along the backbones of two similar superimposed protein molecules. RMSD denotes RMS Deviation. (7) y = the pairwise differences between two corresponding data items in two sets of equal size and status, e.g. two time series. ROOT MEAN SQUARE DEVIATION EXAMPLES Here, the 𝒚 quantity represents the DEVIATION of a member of a set of values from some single reference quantity, eg the mean. Key: RMSD = RMS Deviation RMSE = RMS Error RMSF = RMS Fluctuation. (8) y = deviation of a selected particle’s position from some reference position as a function of time, e.g. backbone atoms during the molecular dynamics simulations. RMSF = measure of average fluctuation. (9) 𝒚 = deviation of a measurement from a theoretically predicted value ÿ (this type of deviation is called a residual). The RMSE (aka RMSD) is a measure this ‘error’. RMSE = √<|𝑦-y ̂|2>. RMSE amplifies large deviations (errors), giving them much higher weight than does the mean absolute error. This means RMSE is very useful when large errors particularly need to be underlined. (10) y = deviation of a value from population mean. RMS value is the population standard deviation √<|𝑦-µ|2> (11) y = deviation of displacement or momentum from mean for harmonic oscillator. RMSD = 𝑦m/𕔆 : 𝑦m = maximum displacement or momentum (cf RMS voltage or current). Applies to both classical and quantum mechanical harmonic oscillators (though with different interpretations). (12) y = deviation of the internal energy of a canonical ensemble from the mean. RMSF = √{kT²Cv}: k=Boltzmann constant, T=absolute temperature, Cv=specific heat capacity at constant volume. Other applications of RMSD include geographic information systems, hydrogeology, computational neuroscience, protein nuclear magnetic resonance spectroscopy, economics and experimental psychology.