NMSE Normalized Mean Square Error

The Normalized Mean Square Error (NMSE) is a metric used to evaluate the accuracy of a prediction or estimation model in the field of statistics and signal processing. It is derived from the Mean Square Error (MSE), which measures the average squared difference between the predicted values and the true values of a dataset. The NMSE further normalizes the MSE to provide a relative measure of error that can be compared across different datasets or models.

To understand NMSE, let's first discuss the Mean Square Error (MSE). Suppose we have a dataset with N samples, denoted by {(x₁, y₁), (x₂, y₂), ..., (xN, yN)}, where xi represents the input or independent variable, and yi represents the corresponding true or target output. Additionally, let ȳi denote the predicted output by a model for the input xi.

The MSE is calculated by taking the average of the squared differences between the predicted and true values:

MSE = (1/N) * Σ(yi - ȳi)²

The MSE provides a measure of the overall deviation between the predicted and true values, with larger values indicating a higher level of error. However, the MSE itself does not provide a normalized measure that can be compared across different datasets or models. This is where the NMSE comes into play.

The NMSE normalizes the MSE by dividing it by the variance of the true values. The variance measures the spread or variability of the true values around their mean. By dividing the MSE by the variance, the NMSE scales the error metric to a relative value that can be interpreted as a percentage.

NMSE = (MSE / Var(y)) * 100

Where Var(y) represents the variance of the true values y.

The NMSE is often expressed as a percentage, making it more intuitive to understand. A lower NMSE value indicates better accuracy and a smaller prediction error, while a higher NMSE value indicates poorer accuracy and a larger prediction error.

The normalization factor, Var(y), in the denominator of the NMSE equation accounts for the scale of the true values. By dividing the MSE by the variance, the NMSE becomes independent of the absolute magnitude of the data. This normalization is especially useful when comparing the performance of different models or datasets with varying scales.

The NMSE can be used in various fields, including signal processing, image reconstruction, regression analysis, and machine learning. It provides a quantitative measure of the accuracy of a prediction model, enabling researchers and practitioners to assess and compare different models' performance.

It's important to note that the NMSE is just one of many evaluation metrics used in practice. Depending on the specific problem and requirements, other metrics such as Mean Absolute Error (MAE), Root Mean Square Error (RMSE), or coefficient of determination (R²) may also be used.

In summary, the Normalized Mean Square Error (NMSE) is a metric that provides a normalized measure of prediction error. By dividing the Mean Square Error (MSE) by the variance of the true values, the NMSE enables the comparison of prediction accuracy across different datasets or models. A lower NMSE indicates better accuracy, while a higher NMSE indicates poorer accuracy. It is a widely used metric in statistics and signal processing to assess the performance of prediction or estimation models.