# NRMSE

AcronymDefinition
NRMSENormalized Root Mean Square Error
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Results Train Test Overall MSE 79671.423 253835.159 114918.846 RMSE 282.261 503.821 338.997 NRMSE 0.045 40.908 18.520 MAE 196.618 352.406 228.146 MAPE 47.503 136.486 65.512 MSE: mean square error; RMSE: root mean square error; NRMSE: normalized root mean square error; MAE: mean absolute error; MAPE: mean absolute percentage error
Its NRMSE is less than traditional and novel localization algorithms under various node densities.
The GCM performances for time series are evaluated by the NRMSE [11, 12] and defined as
NRMSE = [square root of [[summation].sub.j] [[absolute value of [u.sub.j] - [u.sup.exact.sub.j]].sup.2]/ [[summation].sub.j] [[absolute value of [u.sup.exact.sub.j]].sup.2]].
In the results of the 5-cross validation, means and standard deviation of [R.sup.2] and nRMSE for each muscle ranged from 0.0199 [+ or -] 0.0062 (ECR) to 0.6333 [+ or -] 0.0033 (FDP) and 0.1303 [+ or -] 0.0053 (APL) to 0.1825 [+ or -] 0.0098 (FCU) as shown in Table 1.
NRMSE = [square root of ([[summation].sup.n.sub.i=1][([pred.sub.i]-[obs.sub.i]).sup.2])/n/[X.sub.max]-[X.sub.min] x 100, (1)
Because the NRMSE was very small and the step size seemed quite big (for each parameter we have less than 10 different values), we decided to further decrease the step size from 0.5 to 0.1 (and 0.01 respectively), and try to predict the entire new space shown in Table V.
For example, correlation coefficient in training data is [R.sup.2] = 0.94 and in testing data is [R.sup.2] = 0.91 and improvement of NRMSE comparing to scenario 2 (NRMSE = 0.3023).
Method NRMSE Linear model 1.504 Cascade correlation NN 0.170 Fuzzy system  0.103 (*) RBFNN(a)  0.097 (*) Six order polynomial 0.085 Back-propagation NN 0.060 Genetic algorithm and RBFNN(a)  0.050 (*) Neural gas  0.050 This work case 1 0.038 (*) long-term: x(t+85) (a) Radial basis function neural network Table 3 Summary of data used in this study.
Nomenclature [Q.sub.L]: Liquid flow rate [Q.sub.O]: Oil flow rate DCA: Decline curve analysis [Q.sub.W]: Water flow rate WC: Water cut BHP: Bottomhole pressure GOR: Gas-to-oil ratio NRMSE: Normalized root mean square error PRBS: Pseudorandom binary sequence MISO: Multiple input-single output.
The NRMSE curves under the conditions where SNR = 8 dB to -4 dB are shown in Figure 4.
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