What kind of RMSE should I choose to show the capability

2 min readJul 28, 2021

What kind of RMSE should I choose to show the capability of function approximation with Neural Networks ?

This Data set shown below is one of My Data set for function approximation with Neural Network in MATLAB. The left column is the Target vector and the right column is the model output vector.

-0.29039678  -0.212428117
  0.149798734  0.268486256
  0.851351352  0.563358782
  -0.694364578  -0.652148947
  -0.326624496  -0.468058466
  -0.019838988  0.209298441
  -0.765669926  -0.828415796
  -0.133410006  0.003000269
  -0.122484186  -0.073538329
  -0.471535366  -0.376903371
  0.181426106  0.365548334
  0.186889016  0.37947765
  -0.260782058  -0.024155069
  -0.215353652  -0.371639751
  -0.062967222  0.092607096
  -0.461759632  -0.56834877
  0.074468086  0.154155435
  0.624496838  0.529777368
  -0.197814836  -0.096543508
  -0.247268546  -0.136442585
  0.106382978  0.28668492
  -0.791259344  -0.718599416
  -0.532489936  -0.572352826
  -0.006325474  -0.010056861
  0.48706153  0.485766679
  -0.631397354  -0.619362709
  -0.815698678  -0.823629139
  0.272857964  0.354840024
  0.080506038  0.220713202

and I want to show the capability of my model by calculating the RMSE value between the Target & Output vectors in MATLAB .

my question is that what kind of RMSE should I use ?

The RMSE obtained by fitlm function

lm=fitlm(x,y,'linear');

or RMSE obtained by the formula shown below?

RMSE = sqrt(sum((data(:) - estimate (:)).^2) / numel(data));

and if I use the RMSE obtained by fitlm function, Is the RMSE value true ??

because the vector created by

lm.predict(x)

and my Output vector are different.

ANSWER

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1. The default performance function of the regression NNs NEWFIT (calls the generic NEWFF; both are obsolete but still available) and FITNET (current: calls the generic FEEDFORWARDNET) is mean-square-error, MSE, which is scale dependent.

2. However, it is better to use the SCALE INDEPENDENT NORMALIZED MSE, NMSE. MSE is normalized by the MSE of the simplest NN model: the one whose output is just a constant, INDEPENDENT OF THE INPUT! In order to minimize that MSE, the constant must be the average target variance.

In recent threads I have used the notation vart1. In earlier threads I have used the notation MSE00:

NMSE = MSE/MSE00 = MSE/vart1

where

MSE00 = vart1 = mean(var(target',1))

3. This is not a frivolous choice: NMSE is the fraction of the average target variance that is NOT modelled by the net. Conversely, the “Coefficient of Variation” also known as “R-squared” defined by

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