How is numparam calculated for smoothing splines?

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Jo
Jo 2014-3-13
回答： Archoran Encor 2021-8-23
I am using the function: [fitobject,gof,output]=fit(x,y,'smoothingspline','smoothingparam',s); where 's' is the chosen level of smoothing.
I understand that output.numparam gives "Number of unknown parameters (coefficients) to fit", but I am not sure how MATLAB calculates this number for the (cubic) smoothing spline. It changes for each level of 's' - from 2 with maximum smoothing (makes sense since it is a linear function), to >40, and with the exception of maximum smoothing it usually includes decimals, which I'm finding hard to interpret. I would be really grateful if anyone could tell me what this number indicates and/or how it is calculated. Thank you.

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Archoran Encor
Archoran Encor 2021-8-23
It's the same question I had to answer for a research project of mine today. It is basically the trace of the smoothing matrix - being a value between 2 and your number of input points. That's some hell of a formula using the smoothing value p and the sum of all spline leverages. But I think Matlab is only being stuck with their own definitions there...
But basically you can refer to this value as maximum degrees of freedom of your interpolation.

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