# Bruce

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 May12 awarded Student May11 comment How did the author of the following paper compute the curvature matrix?@Willie Wong. Thanks for the additional details. I now get it. I really appreciate your help. May10 comment How did the author of the following paper compute the curvature matrix?@Willie Wong. I was hoping you could expand on what you said a few comments ago. Pretend I'm even MORE clueless than I appear ;-) May10 comment How did the author of the following paper compute the curvature matrix?@Willie Wong. What I understand is that the output is a sum of weighted basis functions, that the basis function at each node is the product of the individual basis functions in each dimension, and that I need the weights (derived through a training process) in order to compute the curvature (penalty term). What I am not grasping is how the individual k vectors are computed in order to compute the K matrix, in order to use the weights to compute the penalty term. You don't have the weights that I'm using, but you were still able to compute the k vectors. How EXACTLY did you do this? Thanks May9 comment How did the author of the following paper compute the curvature matrix?@Willie Wong. Aren't the values of the f() just the basis functions evaluated at the knots? That is, they take on a value of 1 or 0 at each node location? If so, then what I've been doing is $\Delta^2 f(0,0) = f(0,0) + f(0,0) = 1 + 1 = 2$ and $\Delta^2 f(1,0) = f(1,0) + (f(0.0) + f(2,0) - 2f(1,0) = 1 + 0 + 0 -2(1) = 1 - 2 = -1$. Obviously, I have a fundamental thinking error. May9 awarded Commentator May9 comment How did the author of the following paper compute the curvature matrix?@Willie I haven't been able to reproduce the values you arrived at for the $k_0^T$ and $k_1^T$ equations. I appear to be getting some signs wrong. For example, I'm getting [2, -1,...] as the first two values of $k_0^T$ (with the 'boundary condition' you laid out). Would it be possible for you to show an example of how you arrived at those values? I hope this isn't too much of an imposition. May5 awarded Scholar May5 accepted How did the author of the following paper compute the curvature matrix? May5 comment How did the author of the following paper compute the curvature matrix?@Joriki Here, p = the number of multivariate basis functions in the jth submodel, w = the weight of the basis function, $\mu$ = the membership x within that basis function May5 comment How did the author of the following paper compute the curvature matrix?@joriki The paragraph that the formula is in goes on to say "due to the lattice structure of neurofuzzy models, the cross product terms should be sufficiently regularized by only constraining the curvature parallel to the inputs. Therefore the output's curvature can be approximated by . This approximation is now a linear function of the input. May5 comment How did the author of the following paper compute the curvature matrix?@joriki Here is the formula ${d^2 y (x,w) \over dx^2} = \sum_{i=1}^{n} \sum_{b=1}^{p} [ \prod_{j=1,j\ne i} \mu_{A_{k_{j}} ^{b_{j}} } (x_{j}) ] w_{b} {\delta^2 \mu_{A_{k_{i}} ^{b_{i}} } (x_{i}) \over \delta x_{i}^2}$ The y on the left hand side should have a hat over it, but I didn't know how to do that... May5 comment How did the author of the following paper compute the curvature matrix?@joriki I've been confused about how the second derivative was computed because it doesn't exist at the knots of these linear basis functions. I came across what I believe is the methodology used in the book "Adaptive Modelling, Estimation, and Fusion from Data: A Neurofuzzy Approach". I was going to reproduce the formula given, but don't know how to embed LaTex into these comments. I suppose if I could embed an image, I could just scan it in as well... May5 comment How did the author of the following paper compute the curvature matrix?@Willie Wong. I tried to send a note to the author, but this paper is something like 15 years old, and he has moved on. I haven't been able to track him down. May4 comment How did the author of the following paper compute the curvature matrix?Hi Willie. Thanks for responding. What I don't understand is how the author arrived at the values that he did for the curvature matrix. For instance, how did the author arrive at a value of 6 for the (0,0) position in the matrix K on page 7? I don't understand how to compute the curvature at the centre of each basis function. May4 asked How did the author of the following paper compute the curvature matrix?