Best-Fit Error Measures

The following best-fit error measure algorithms, or error norms, are available. You can specify normalized or unnormalized error for any of the algorithms except R², which is always normalized by definition. Normalized measures use the mean data, rather than the point-wise data, which can help you avoid situations where the error measure calculations divide by zero or by a small number.

The following conventions are used in the discussion below: ${\hat{Y}}_{i}$ is a test data point, $Y_{i}$ is a computed data point, $n$ is the number of data points in a given set of test data, and $\bar{\hat{Y}}$ and $\bar{Y}$ are respectively the mean values of ${\hat{Y}}_{i}$ and $Y_{i}$ , defined as:

\bar{\hat{Y}} \equiv \frac{1}{n} \sum_{i = 1}^{n} {\hat{Y}}_{i}

\bar{Y} \equiv \frac{1}{n} \sum_{i = 1}^{n} Y_{i}

Coefficient of Determination ( $R^{2}$ )

The Coefficient of Determination is defined as:

R^{2} \equiv 1 - \frac{\sum_{i = 1}^{n} {({\hat{Y}}_{i} - Y_{i})}^{2}}{\sum_{i = 1}^{n} {({\hat{Y}}_{i} - \bar{\hat{Y}})}^{2}}

Mean Square Error (MSE)

The Mean Square Error is the sum of the square differences between the test data and computed data points and defined as

M S E \equiv \frac{1}{n} \sum_{i = 1}^{n} {({\hat{Y}}_{i} - Y_{i})}^{2}

The Normalized Mean Square Error (NMSE) is defined as:

N M S E \equiv \frac{M S E}{\bar{\hat{Y}} \bar{Y}}

Mean Absolute Error (MAE)

The Mean Absolute Error measures the average absolute difference between the test data and the computed data points and defined as:

M A E \equiv \frac{1}{n} \sum_{i = 1}^{n} | {\hat{Y}}_{i} - Y_{i} |

The Normalized Mean Absolute Error (NMAE) is:

N M A E \equiv \frac{M A E}{\bar{\hat{Y}}}

Root Mean Square Error (RMSE)

The Root Mean Square Error is the square root of the Mean Square Error and is defined as:

R M S E \equiv \sqrt{M S E}

The Normalized Root Mean Square Error (NRMSE) is:

N R M S E \equiv \frac{R M S E}{\bar{\hat{Y}}}

Relative Square Error (RSE)

The Relative Square Error measure is defined as:

R S E \equiv \sum_{i = 1}^{n} {(1 - \frac{T_{comp} (i)}{T_{test} (i)})}^{2}

This error measure is useful when you are trying to match a curve that has large variations in magnitude, and a close relative match to the low-magnitude part of the response is important.

All of the other error measures above use a mean value of the dataset to normalize the measure. The RSE error measure is the only error measure that uses a point-wise test value to normalize the measure. Mean value normalization is generally a good option and helps avoid numerical problems. Point-wise normalization is problematic when the test data point or computed data point is near zero, causing the point-wise error measure to be extremely large and overwhelm the entire error measure calculation. This typically happens when the response is small early in the test, or for cyclic data as the response crosses from positive to negative (or vice versa). However, the RSE error measure is the best choice when the dataset covers a large range in the response. This is often the case for DMA test data (storage and loss modulus versus frequency).

Best-Fit Error Measures

Coefficient of Determination ( R 2 )

Mean Square Error (MSE)

Mean Absolute Error (MAE)

Root Mean Square Error (RMSE)

Relative Square Error (RSE)

Coefficient of Determination ( $R^{2}$ )