# Rms Error Calculation Matlab

Support Answers MathWorks Search MathWorks.com MathWorks Answers Support MATLAB Answers™ MATLAB Central Community Home MATLAB Answers File Exchange Cody Blogs Newsreader Link Exchange ThingSpeak Anniversary Home Ask Answer Browse More Contributors Recent Activity Flagged Content how to calculate mean square error in matlab Flagged as Spam Help MATLAB Central Community Home MATLAB Answers File Exchange Cody Blogs

## Normalized Root Mean Square Error Matlab

Newsreader Link Exchange ThingSpeak Anniversary Home Ask Answer Browse More Contributors Recent Activity Flagged Content Flagged as Spam Help Trial software root mean square error formula Joe (view profile) 1 question 0 answers 0 accepted answers Reputation: 0 Vote0 RMSE - Root mean square Error Asked by Joe Joe (view profile) 1 question 0 answers 0 accepted answers Reputation: 0 root mean square error calculation matlab code on 27 Mar 2011 Latest activity Commented on by Lina Eyouni Lina Eyouni (view profile) 35 questions 0 answers 0 accepted answers Reputation: 0 on 25 Jul 2016 Accepted Answer by John D'Errico John D'Errico (view profile) 4 questions 1,893 answers 687 accepted answers Reputation: 4,342 3,990 views (last 30 days) 3,990 views (last 30 days) [EDIT: 20110610 00:17 CDT - reformat - WDR]So i was looking online how to

## Immse Matlab

check the RMSE of a line. found many option, but I am stumble about something,there is the formula to create the RMSE: http://en.wikipedia.org/wiki/Root_mean_square_deviationDates - a VectorScores - a Vectoris this formula is the same as RMSE=sqrt(sum(Dates-Scores).^2)./Datesor did I messed up with something? 0 Comments Show all comments Tags rmseroot mean square error Products No products are associated with this question. Related Content 3 Answers John D'Errico (view profile) 4 questions 1,893 answers 687 accepted answers Reputation: 4,342 Vote5 Link Direct link to this answer: https://www.mathworks.com/matlabcentral/answers/4064#answer_12671 Answer by John D'Errico John D'Errico (view profile) 4 questions 1,893 answers 687 accepted answers Reputation: 4,342 on 10 Jun 2011 Accepted answer

Yes, it is different. The Root Mean Squared Error is exactly what it says.(y - yhat) % Errors (y - yhat).^2 % Squared Error mean((y - yhat).^2) % Mean Squared Error RMSE = sqrt(mean((y - yhat).^2)); % Root Mean Squared Error What you have written is different, in that you have divided by dates, effectively normalizing the result. Also, there is no mean, only a sum. The difference is that a mean divides by the number of elements. It is an average.sqrt(sum(Dates-Scores).^2)./Dates Thus, you have written what could be described as a "normalized sum of the squared errors",

Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle navigation Trial Software Product Updates Documentation Home Image Processing Toolbox Examples Functions and Other Reference Release Notes PDF Documentation Image matlab code for mean square error of two images Analysis Image Quality Image Processing Toolbox Code Generation Image Processing Toolbox Functions immse On rms error excel this page Syntax Description Examples Calculate Mean-Squared Error in Noisy Image Input Arguments X Y Output Arguments err More About

## Rms Matlab

Code Generation MATLAB Function Block See Also This is machine translation Translated by Mouse over text to see original. Click the button below to return to the English verison of the page. Back to https://www.mathworks.com/matlabcentral/answers/4064-rmse-root-mean-square-error English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch English Estonian Finnish French German Greek Haitian Creole Hindi Hmong Daw Hungarian Indonesian Italian Japanese Korean Latvian Lithuanian Malay Maltese Norwegian Polish Portuguese Romanian Russian Slovak Slovenian Spanish Swedish Thai Turkish Ukrainian Vietnamese Welsh MathWorks Machine Translation The automated translation of this page is provided by a general purpose third party translator https://www.mathworks.com/help/images/ref/immse.html tool. MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. Translate immse Mean-squared error collapse all in page Syntaxerr = immse(X,Y) exampleDescriptionexample`err`

` = immse(X,Y)`

calculates the mean-squared error (MSE) between the arrays X and Y. X and Y can be arrays of any dimension, but must be of the same size and class.Code Generation support: Yes.MATLAB Function Block support: Yes.Examplescollapse allCalculate Mean-Squared Error in Noisy ImageOpen Script Read image and display it.ref = imread('pout.tif'); imshow(ref) Create another image by adding noise to a copy of the reference image.A = imnoise(ref,'salt & pepper', 0.02); imshow(A) Calculate mean-squared error between the two images.err = immse(A, ref); fprintf('\n The mean-squared error is %0.4f\n', err); The mean-squared error is 353.7631 Input Argumentscollapse allX -- Input arraynumeric array Input array, specified as a nonsparse, numeric array. Example: err = immse(I,I2); Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32Y -- Input arraynonsparse, numeric array Input arrays, specified as a nonsparse, numeric array. Example: err = immse(I,I2); Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32Output Argumentscollapse allerr -- Mean-squared e

to calculate the root mean square error. Assume you have one set of numbers that represent the Actual values you want to predict. Actual = [1 2 3 4]; Then assume http://kawahara.ca/root-mean-square-error-tutorial-matlab/ you have another set of numbers that Predicted the actual values. Predicted = http://stackoverflow.com/questions/8316916/matlab-fastest-way-to-do-a-root-mean-squared-error-between-a-vector-and-array-o [1 3 1 4]; How do you evaluate how close Predicted values are to the Actual values? Well you could use the root mean square error (RMSE) to give a sense of the Predicted values error. Here's some MATLAB code that does exactly that. % rmse tutorial. % The actual mean square values that we want to predict. Actual = [1 2 3 4]; % The values we actually predicted. Predicted = [1 3 1 4]; % One way is to use the Root Mean Square function and pass in the "error" part. rmse = rms(Predicted-Actual) % That's it! You're done. % But for those of you who are the curious type, % here's mean square error how to calculate the root-mean-square-error by hand. % First calculate the "error". err = Actual - Predicted; % Then "square" the "error". squareError = err.^2; % Then take the "mean" of the "square-error". meanSquareError = mean(squareError); % Then take the "root" of the "mean-square-error" to get % the root-mean-square-error! rootMeanSquareError = sqrt(meanSquareError) % That's it! You have calculated the RMSE by hand. % So, this is true. rootMeanSquareError == rmse Share this:Click to share on Twitter (Opens in new window)Click to share on Facebook (Opens in new window)Click to share on Google+ (Opens in new window) Related Author JeremyPosted on July 10, 2013February 4, 2014Categories MATLAB 2 thoughts on “Root Mean Square Error tutorial - MATLAB” Pingback: Mean absolute error tutorial MATLAB | kawahara.ca Roman says: June 18, 2014 at 2:02 pm Thanks! Comments are closed. Post navigation Previous Previous post: X3D - how to rotate an objectNext Next post: Talk on spinal cord segmentation My Tweets Recent Posts how to fix a blank login screen on a ubuntu laptop with nvidia drivers October 26, 2016 multi-resolution-tract CNN with hybrid pretrained and skin-lesion trai

here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Stack Overflow Questions Jobs Documentation Tags Users Badges Ask Question x Dismiss Join the Stack Overflow Community Stack Overflow is a community of 6.3 million programmers, just like you, helping each other. Join them; it only takes a minute: Sign up MATLAB: fastest way to do a root-mean-squared error between a vector and array of vectors up vote 2 down vote favorite I have a question regarding the fastest way to compute the RMSE between a single vector and an array of vectors. Specifically, I have a vector A representing an point and would like to find the index in a list B of points that A is closest to. Right now I am using: tempmat = bsxfun(@minus,A,B); tempmat1 = sqrt(sum(tempmat.^2,2); index = find(tempmat1 == min(tempmat1)); this takes about 0.058 seconds to calculate the index. Is there a faster way in MATLAB of doing this? I performing this calculations literally millions of times. Many thanks for reading, Joe search matlab vector find share|improve this question asked Nov 29 '11 at 19:58 JoeMcG 5727 A is an [x,y,z] vector and B is an array with each row being an [x,y,z] vector. Somehow that didn't make it past the formatting. –JoeMcG Nov 29 '11 at 21:48 add a comment| 2 Answers 2 active oldest votes up vote 4 down vote accepted tempmat = bsxfun(@minus,A,B); tmpmat1 = sum(tempmat.^2,2); [m,index] = min(tempmat1); m = sqrt(m); %# optional, only if you need the actual numerical value This avoids calculating sqrt on the whole array, since the minumum of the squared differences will have the same index. It also uses the second output of min to avoid the second pass of find. share|improve this answer answered Nov 29 '11 at 20:04 mtrw 15.1k33550 cool, don't know why I didn't see that before! This provides about a 15% increase in speed. Is

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Regression Mean Square Error p deviation MSD of an estimator of a procedure for estimating an unobserved quantity measures the average of the squares of the errors mean squared error example or deviations that is the difference between the estimator and what is p Mean Square Error Matlab p estimated MSE is a risk function corresponding to the expected value of the squared error loss or quadratic mean square error excel loss The difference occurs because of randomness or because the estimator doesn't account for information that could produce a more accurate estimate The MSE is a measure of mse

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Rms Error In Excel p ProductsHomearound the homeproductivityHow to Get the RMS in ExcelHow to Get the RMS in ExcelBy Ron PriceExcel does not include a predefined function to calculate an RMS so manually entered functions must be used individually or in tandem to calculate this value The Root Mean Square calculates the effective rate or measurement of a varying set of values It is the square root of the average how to calculate root mean square error of the squared values in a data set RMS is primarily used in physics and p Root Mean Square Error Using Excel

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Rmse Residual Mean Square Error p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample standard deviation of root mean square error formula the differences between predicted values and observed values These individual differences are called root mean square error in r residuals when the calculations are performed over the data sample that was used for estimation and are called p Root Mean Square Error Interpretation p prediction errors when computed out-of-sample The RMSD serves to aggregate

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rms error versus standard deviation

Rms Error Versus Standard Deviation p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample standard deviation of the differences between predicted values and observed values These individual differences are mean square error formula called residuals when the calculations are performed over the data sample that was used for root mean square error interpretation estimation and are called prediction errors when computed out-of-sample The RMSD serves to aggregate the magnitudes of the errors in predictions for root

rms error excel

Rms Error Excel p ProductsHomearound the homeproductivityHow to Get the RMS in ExcelHow to Get the RMS in ExcelBy Ron PriceExcel does not include a predefined function to calculate an RMS so manually entered functions must be used individually or in tandem to calculate this value The Root Mean Square calculates the effective rate or measurement of a varying set of values It is the square root of how to calculate root mean square error the average of the squared values in a data set RMS is primarily used in p Root Mean Square Error Using Excel Sheet Example p

rms difference error

Rms Difference Error p spread of the y values around that average To do this we use the root-mean-square error r m s error To construct the r m s error you first need to determine the residuals Residuals are the difference between the actual values and the root mean square error formula predicted values I denoted them by where is the observed value for the ith root mean square error matlab observation and is the predicted value They can be positive or negative as the predicted value under or over estimates the actual value root mean square error interpretation

rms error formula

Rms Error Formula p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample standard deviation of the root mean square error interpretation differences between predicted values and observed values These individual differences are called p Root Mean Square Error Excel p residuals when the calculations are performed over the data sample that was used for estimation and are called prediction root mean square error matlab errors when computed out-of-sample The RMSD serves to aggregate the magnitudes of

rmse error wiki

Rmse Error Wiki p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample standard deviation of the differences between root mean square error formula predicted values and observed values These individual differences are called residuals when the p Root Mean Square Error Interpretation p calculations are performed over the data sample that was used for estimation and are called prediction errors when computed out-of-sample root mean square error example The RMSD serves to aggregate the magnitudes of

rms error calculation excel

Rms Error Calculation Excel p Google Het beschrijft hoe wij gegevens gebruiken en welke opties je hebt Je moet dit vandaag nog doen Navigatie overslaan NLUploadenInloggenZoeken Laden Kies je taal Sluiten Meer informatie how to calculate root mean square error View this message in English Je gebruikt YouTube in het Nederlands Je kunt p Root Mean Square Error Using Excel Sheet Example p deze voorkeur hieronder wijzigen Learn more You're viewing YouTube in Dutch You can change this preference below Sluiten Ja nieuwe how to calculate rmse in r versie behouden Ongedaan maken Sluiten Deze video is niet beschikbaar WeergavewachtrijWachtrijWeergavewachtrijWachtrij

rms standard deviation error

Rms Standard Deviation Error p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample standard deviation of the differences between predicted values and observed root mean square error interpretation values These individual differences are called residuals when the calculations are performed over the root mean square error excel data sample that was used for estimation and are called prediction errors when computed out-of-sample The RMSD serves to aggregate the root mean square error matlab magnitudes of the

root mean square error and variance

Root Mean Square Error And Variance p deviation MSD of an estimator of a procedure for estimating an unobserved quantity measures the average of the squares of the errors or deviations that is the difference between the estimator and what is estimated MSE is a risk function p Root Mean Square Error Formula p corresponding to the expected value of the squared error loss or quadratic loss The difference root mean square error example occurs because of randomness or because the estimator doesn't account for information that could produce a more accurate estimate The MSE is a root mean square

root mean square error excel calculator

Root Mean Square Error Excel Calculator p ProductsHomearound the homeproductivityHow to Get the RMS in ExcelHow to Get the RMS in ExcelBy Ron PriceExcel does not include a predefined function to calculate an RMS so manually entered functions must be used individually or in tandem to calculate this value The Root Mean Square calculates the effective rate or measurement of a varying set of how to calculate root mean square error values It is the square root of the average of the squared values root mean square error using excel sheet example in a data set RMS is primarily used

root mean square error meaning

Root Mean Square Error Meaning p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample standard deviation of the differences between predicted values and root mean square error in r observed values These individual differences are called residuals when the calculations are performed root mean square error excel over the data sample that was used for estimation and are called prediction errors when computed out-of-sample The RMSD serves p Root Mean Square Error Matlab p to aggregate

root average squared error

Root Average Squared Error p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample root mean square error interpretation standard deviation of the differences between predicted values and observed values These root mean square error excel individual differences are called residuals when the calculations are performed over the data sample that was used p Root Mean Square Error Matlab p for estimation and are called prediction errors when computed out-of-sample The RMSD serves to aggregate the magnitudes

root mean square error in excel

Root Mean Square Error In Excel p View this message in English YouTube how to calculate root mean square error root mean square error using excel sheet example Learn more You're viewing YouTube in Greek You p How To Calculate Rmse In R p can change this preference below p Rmsd In Excel p count total U V Calculating RMSE in Excel John Saunders rmse calculator playlist p p Maps Linking Movement with Direction Lines Maps Cartography October Rasterization and Vectorization The How-To' Guide GIS Analysis September How to Get Harmonized Environmental p Excel Sumsq p Demographic Data with TerraPop

rms error variance

Rms Error Variance p deviation MSD of an estimator of a procedure for estimating an unobserved quantity measures the average of the squares of the errors or deviations that is the difference between the estimator and what is estimated MSE is p Root Mean Square Error Formula p a risk function corresponding to the expected value of the squared error loss or quadratic root mean square error interpretation loss The difference occurs because of randomness or because the estimator doesn't account for information that could produce a more accurate p Root Mean Square Error Example p estimate The MSE is

root mean square error equation

Root Mean Square Error Equation p spread of the y values around that average To do this we use the root-mean-square error r m s error To construct the r m s error you first need to determine the residuals Residuals are the difference between the actual values and the p Root Mean Square Error Excel p predicted values I denoted them by where is the observed value for the ith root mean square error interpretation observation and is the predicted value They can be positive or negative as the predicted value under or over estimates the actual value root

root mean square error in statistics

Root Mean Square Error In Statistics p spread of the y values around that average To do this we use the root-mean-square error r m s error To construct the r m s error you first need to determine the residuals Residuals are the difference between the actual values and the root mean square error formula predicted values I denoted them by where is the observed value for the ith p Root Mean Square Error Interpretation p observation and is the predicted value They can be positive or negative as the predicted value under or over estimates the actual value

root mean square error excel example

Root Mean Square Error Excel Example p Google Het beschrijft hoe wij gegevens gebruiken en welke opties je hebt Je moet dit vandaag nog doen Navigatie overslaan NLUploadenInloggenZoeken Laden Kies je taal how to calculate root mean square error Sluiten Meer informatie View this message in English Je gebruikt YouTube in het root mean square error using excel sheet example Nederlands Je kunt deze voorkeur hieronder wijzigen Learn more You're viewing YouTube in Dutch You can change this preference rmsd in excel below Sluiten Ja nieuwe versie behouden Ongedaan maken Sluiten Deze video is niet beschikbaar WeergavewachtrijWachtrijWeergavewachtrijWachtrij Alles verwijderenOntkoppelen Laden

root mean square error excel 2007

Root Mean Square Error Excel p View this message in English YouTube p How To Calculate Rmse In Excel p Learn more You're how to calculate root mean square error viewing YouTube in Greek You can change this preference below p Calculate Mean Square Error Excel p root mean square error using excel sheet example count total U V Calculating RMSE in Excel John Saunders rmsd in excel playlist Calculating the root mean squared error using Excel p p one file Read More Free Trial Home Products Tips Demos Support Documentation Blog FAQ Library Service Level Agreement Thank you Beta

root mean square error formula excel

Root Mean Square Error Formula Excel p ProductsHomearound the homeproductivityHow to Get the RMS in ExcelHow to Get the RMS in ExcelBy Ron PriceExcel does not include a predefined function to calculate an RMS so manually entered functions must be used individually or in tandem to calculate this value The Root Mean Square calculates the effective rate or measurement of a varying set of values It how to calculate root mean square error is the square root of the average of the squared values in a data p Root Mean Square Error Using Excel Sheet Example p set RMS is

root mean square error compared to standard deviation

Root Mean Square Error Compared To Standard Deviation p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample standard deviation of the differences between predicted values p Mean Square Error Formula p and observed values These individual differences are called residuals when the calculations are root mean square error example performed over the data sample that was used for estimation and are called prediction errors when computed out-of-sample The RMSD root mean square error interpretation serves to

root mean square error gaussian

Root Mean Square Error Gaussian p deviation MSD of an estimator of a procedure for estimating an unobserved quantity measures the average of the squares of the errors or deviations that is the difference between the estimator and what is estimated MSE is a risk function root mean square error formula corresponding to the expected value of the squared error loss or quadratic loss The difference p Root Mean Square Error Interpretation p occurs because of randomness or because the estimator doesn't account for information that could produce a more accurate estimate The MSE is a root mean square error

root mean square error matlab

Root Mean Square Error Matlab p Support Answers MathWorks Search MathWorks com MathWorks Answers Support MATLAB Answers trade MATLAB Central Community Home MATLAB Answers File Exchange how to calculate mean square error in matlab Cody Blogs Newsreader Link Exchange ThingSpeak Anniversary Home Ask Answer p Normalized Root Mean Square Error Matlab p Browse More Contributors Recent Activity Flagged Content Flagged as Spam Help MATLAB Central Community Home MATLAB root mean square error calculation matlab code Answers File Exchange Cody Blogs Newsreader Link Exchange ThingSpeak Anniversary Home Ask Answer Browse More Contributors Recent Activity Flagged Content Flagged as Spam Help Trial

root mean square error and standard deviation

Root Mean Square Error And Standard Deviation p deviation MSD of an estimator of a procedure for estimating an unobserved quantity measures the average of the squares of the errors or deviations that is the difference between the estimator p Root Mean Square Error Formula p and what is estimated MSE is a risk function corresponding to the expected root mean square error interpretation value of the squared error loss or quadratic loss The difference occurs because of randomness or because the estimator doesn't root mean square error excel account for information that could produce a more accurate estimate The

root mean square error formula in excel

Root Mean Square Error Formula In Excel p RMSE in Excel John Saunders SubscribeSubscribedUnsubscribe Loading Loading Working Add to Want to watch this again later Sign in to add this video to a playlist Sign in Share More Report Need to report the video Sign how to calculate root mean square error in to report inappropriate content Sign in Transcript Statistics views Like root mean square error using excel sheet example this video Sign in to make your opinion count Sign in Don't like this video Sign in to calculate mean square error excel make your opinion count Sign in

root mean square error calculation in matlab

Root Mean Square Error Calculation In Matlab p Support Answers MathWorks Search MathWorks com MathWorks Answers Support MATLAB Answers trade MATLAB Central Community Home MATLAB Answers File how to calculate mean square error in matlab Exchange Cody Blogs Newsreader Link Exchange ThingSpeak Anniversary Home Ask normalized root mean square error matlab Answer Browse More Contributors Recent Activity Flagged Content Flagged as Spam Help MATLAB Central Community root mean square error formula Home MATLAB Answers File Exchange Cody Blogs Newsreader Link Exchange ThingSpeak Anniversary Home Ask Answer Browse More Contributors Recent Activity Flagged Content Flagged as Spam Help root mean square

root mean square error linear regression

Root Mean Square Error Linear Regression p Consulting Quick Question Consultations Hourly Statistical Consulting Results Section Review Statistical Project Services Free Webinars Webinar Recordings Contact Customer Login Statistically Speaking Login Workshop Center Login All Logins Assessing the Fit of Regression Models by Karen A well-fitting regression model results in predicted values close to root mean square error interpretation the observed data values The mean model which uses the mean for every predicted value p Rmse Vs R p generally would be used if there were no informative predictor variables The fit of a proposed regression model should therefore be better

root mean square error function

Root Mean Square Error Function p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the root mean square error formula values actually observed The RMSD represents the sample standard deviation of p Root Mean Square Error Interpretation p the differences between predicted values and observed values These individual differences are called residuals when root mean square error excel the calculations are performed over the data sample that was used for estimation and are called prediction errors when computed out-of-sample The RMSD serves to aggregate p

root mean square error xls

Root Mean Square Error Xls p ProductsHomearound the homeproductivityHow to Get the RMS in ExcelHow to Get the RMS in ExcelBy Ron PriceExcel does not include a predefined function to calculate an RMS so manually entered functions must be used individually or in tandem to calculate this value The Root Mean Square calculates the p How To Calculate Root Mean Square Error In Excel p effective rate or measurement of a varying set of values It root mean square error using excel sheet example is the square root of the average of the squared values in a data set RMS

root mean square error of estimation

Root Mean Square Error Of Estimation p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values actually observed The RMSD represents the sample standard root mean square error interpretation deviation of the differences between predicted values and observed values These individual p Root Mean Square Error Excel p differences are called residuals when the calculations are performed over the data sample that was used for estimation p Root Mean Square Error Matlab p and are called prediction errors when computed out-of-sample The RMSD serves

root mean square error algorithm

Root Mean Square Error Algorithm p RMSE is a frequently used measure of the differences between values sample and population values predicted by a model or an estimator and the values root mean square error example actually observed The RMSD represents the sample standard deviation of p Root Mean Square Error Interpretation p the differences between predicted values and observed values These individual differences are called residuals when rmse formula excel the calculations are performed over the data sample that was used for estimation and are called prediction errors when computed out-of-sample The RMSD serves to aggregate root mean square