Delta-rational chord: Difference between revisions
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We can replace the 1 with x, vary x and ask, "By at least how much do the deltas have to be off for any x?" | We can replace the 1 with x, vary x and ask, "By at least how much do the deltas have to be off for any x?" | ||
=== | === Direct least-squares error === | ||
Rewriting a bit, if 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> has delta signature +ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub> (where the chord is written to start on 1, i.e. 1:1+ε<sub>1</sub>:...), let <math>D_i = \sum_{k=1}^i \delta_i</math> (the ''target'' delta signature) and <math>E_i = \sum_{k=1}^i \epsilon_i</math> (the ''approximant'' delta signature). Then the resulting linear least-squares optimization problem is | Rewriting a bit, if 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> has delta signature +ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub> (where the chord is written to start on 1, i.e. 1:1+ε<sub>1</sub>:...), let <math>D_i = \sum_{k=1}^i \delta_i</math> (the ''target'' delta signature) and <math>E_i = \sum_{k=1}^i \epsilon_i</math> (the ''approximant'' delta signature). Then the resulting linear least-squares optimization problem is | ||
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which can be plugged back into the error formula to obtain the error. (We multiply the target DR chord by x in order to compare it to the approximation on the same isodifferential series.) This error measure is invariant under scaling the target delta signature, but not under scaling the approximation, which is why we require the approximating chord to start on 1. | which can be plugged back into the error formula to obtain the error. (We multiply the target DR chord by x in order to compare it to the approximation on the same isodifferential series.) This error measure is invariant under scaling the target delta signature, but not under scaling the approximation, which is why we require the approximating chord to start on 1. | ||
This error measure is called ''' | This error measure is called '''direct least-squares error''' (DLS error). This error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric. | ||
This error measure was found by Inthar and groundfault. | This error measure was found by Inthar and groundfault. | ||