Delta-rational chord: Difference between revisions

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This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''1:''r''2:...:''r''''n'' or the same chord linearly shifted to have root α. Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms.

~~==== Naive <math>L^p</math> metrics on delta signatures ====~~

To impose a metric space structure on delta signatures of length <math>k</math> we can first normalize their respective cumulative forms and then compare the signatures by using the <math>L^p</math> metric.

Given normalized delta signatures <math>1 = \delta_0, \delta_1, ..., \delta_n</math> and <math>1 = \epsilon_0, \epsilon_1,..., \epsilon_n,</math> we rewrite them as <math>\mathbf{D} = (D_1, ..., D_n), \ 1 < D_1 < D_2 < \cdots < D_n</math> and <math>\mathbf{E} = (E_1, ..., E_n), \ 1 < E_1 < E_2 < \cdots < E_n</math> where <math>D_k = \sum_{i=0}^k \delta_i</math> and <math>E_k = \sum_{i=0}^k \epsilon_i.</math> Then we take the ''p''-norm of the differences where <math>1 \le p \le \infty</math>:

~~<math>\displaystyle{\left\| \mathbf{D} - \mathbf{E} \right\|_p = \Bigg( \sum_{i=1}^n (D_i - E_i)^{p} \Bigg)^{1/p}.}</math>~~

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==== Partially DR ====

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 This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> or the same chord linearly shifted to have root α. Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms.
-==== Naive <math>L^p</math> metrics on delta signatures ====
-To impose a metric space structure on delta signatures of length <math>k</math> we can first normalize their respective cumulative forms and then compare the signatures by using the <math>L^p</math> metric.
-Given normalized delta signatures <math>1 = \delta_0, \delta_1, ..., \delta_n</math> and <math>1 = \epsilon_0, \epsilon_1,..., \epsilon_n,</math> we rewrite them as <math>\mathbf{D} = (D_1, ..., D_n), \ 1 < D_1 < D_2 < \cdots < D_n</math> and <math>\mathbf{E} = (E_1, ..., E_n), \ 1 < E_1 < E_2 < \cdots < E_n</math> where <math>D_k = \sum_{i=0}^k \delta_i</math> and <math>E_k = \sum_{i=0}^k \epsilon_i.</math> Then we take the ''p''-norm of the differences where <math>1 \le p \le \infty</math>:
-<math>\displaystyle{\left\| \mathbf{D} - \mathbf{E} \right\|_p = \Bigg( \sum_{i=1}^n (D_i - E_i)^{p} \Bigg)^{1/p}.}</math>
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 ==== Partially DR ====