Kent State University

Question

Asked 20th Mar, 2023

# Is the multiple correlation coefficient (R) undefined in the case of negative determination coefficients (Artificial Neural Networks)?

I noticed that in some very bad models of neural networks, the value of R² (coefficient of determination) can be negative. That is, the model is so bad that the mean of the data is better than the model.

In linear regression models, the multiple correlation coefficient (R) can be calculated using the root of R². However, this is not possible for a model of neural networks that presents a negative R². In that case, is R mathematically undefined?

I tried calculating the correlation y and y_pred (Pearson), but it is mathematically undefined (division by zero). I am attaching the values.

Obs.: The question is about artificial neural networks.

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## All Answers (5)

Kent State University

R**2 cannot be negative. Adjusted R**2 can sometimes be because of the adjusted number of degrees of freedom in the denominator. See most good linear statistical models text books. Best wishes David Booth

University of West Florida

I don't know, but my first thought here is "why even dwell on it here?"

In regression analysis we often use the coefficient of determination (R**2). It is a useful metric for fit but not for prediction when you need to measure how stable the regression model is. Here, a suggested change to R**2 is to replace SSE with PRESS, and we get R**2(pred)= 1- (PRESS/SSTotal). It can happen that the model is so bad that PRESS>SS Total. R**2(pred)<0 then. I would say, here it is best to either state that R**2(pred) is undefined or set it to zero. The prediction oriented R**2 looks really bad.

Kent State University

Raid PRESS invented 1970s by David Allen of U Kentucky is not a good idea either. See things like AIC and BIC and how to handle prediction models in general. Some references can be found in the attached. IN ANY CASE R**2 cannot be less than 0 but it is well known adjusted R**2 can. Best wishes David Booth

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