# Describing the goodness of fit of a model

1. Jun 23, 2010

### McKendrigo

Describing the "goodness of fit" of a model

Hi there,

I would like to ask advice on an appropriate way to define how well a measurement 'matches up' to the predicted response. In other words, I have a set of data for bandwidth measurement for an LED (amplitude vs. frequency). I also have a predicted response, from a simple equation:

$$M(f) = \sqrt{3}/2* \pi *\tau *f$$

Where M(f) is the amplitude at a given frequency, and Tau is the LED time constant.

I'd like to know a good way to quantify how well the curve of measured values matches the curve of predicted values, so that I can quantify the 'goodness' of the model depending on different Tau values and so on.

Any guidance would be appreciated!

2. Jun 23, 2010

### EnumaElish

Re: Describing the "goodness of fit" of a model

Can you post your model? What are the variables, and what are the parameters?

[Edit: the "usual" goodness-of-fit measure is the R-squared statistic, the ratio of variance explained by the model to the total variance of the "left-hand side" variable.]

Last edited: Jun 23, 2010
3. Jun 23, 2010

### statdad

Re: Describing the "goodness of fit" of a model

The classical $$R^2$$ is not useful in situations where there is no intercept term. However, I don't get the sense that you did a regression to get this equation? If this is not a regression problem, you might look at the maximum absolute error between your predictions and actual values.
What other information can you give about this problem?

4. Jul 1, 2010

### McKendrigo

Re: Describing the "goodness of fit" of a model

Hi guys,

Thanks for your replies, sorry for taking so long to get back to you!

I don't think I was totally clear with my question - I have an equation which describes the frequency response of a light-emitting material (shown above). A value for Tau has been found for this material, so using the equation I can predict the frequency response of the material.

I have separately measured the actual frequency response of the material. I am not fitting the equation to the data, in fact, I am merely plotting the measured and predicted responses to see how well they agree. In other words, I'd like a way of quantifying how well the 'guess curve' and the 'measured curve' agree. At the moment, the error between the predicted and measured -3dB points taken from the curves is the best way I can think of quantifying the agreement.

Specifying the maximum absolute error sounds like a sensible approach.

5. Jul 1, 2010

### Ygggdrasil

Re: Describing the "goodness of fit" of a model

Perhaps you can use the root mean square deviation (RMSD) between your model and the data. That is, for every data point, take the difference between that data point and its expected value from the model and square that difference. Then average these squares of differences across the data set and take the square root of the average.

Alternatively, for a dimensionless quantity, you could divide the differences by the observed measurement prior to squaring.

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