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Cc analysis: Difference between revisions
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It further turns out that the resulting arrangement of vectors (that minimize the function <math>\Phi(\mathbf{x})</math> given above) has the properties | It further turns out that the resulting arrangement of vectors (that minimize the function <math>\Phi(\mathbf{x})</math> given above) has the properties | ||
# 0 <= length <= 1 for each vector; short vectors have a low signal-to-noise ratio, long vectors a good signal-to-noise ratio, and vectors of length 1 represent ideal (prototypical) data sets. In fact, the length of each vector is CC* (as defined in [http://dx.doi.org/10.1126/science.1218231 Karplus & Diederichs (2012)]), and its | # 0 <= length <= 1 for each vector; short vectors have a low signal-to-noise ratio, long vectors a good signal-to-noise ratio, and vectors of length 1 represent ideal (prototypical) data sets. In fact, the length of each vector is CC* (as defined in [http://dx.doi.org/10.1126/science.1218231 Karplus & Diederichs (2012)]), and its relation to the signal-to-noise ratio is given in eqn. 4. | ||
# vectors point in the same direction (i.e. they lie on a radial line) if their data sets only differ by random noise | # vectors point in the same direction (i.e. they lie on a radial line) if their data sets only differ by random noise | ||
# vectors representing systematically different (heterogeneous) data sets enclose an angle whose cosine is the CC of their respective ideal data sets (eqn. 5). | # vectors representing systematically different (heterogeneous) data sets enclose an angle whose cosine is the CC of their respective ideal data sets (eqn. 5). | ||
# if all CCs are known, the solution is unique in terms of lengths of vectors, and angles between them. However, a rotated (around the origin) or inverted (through the origin) arrangement of the vectors leaves the functional unchanged, because these transformations do not change lengths and angles. | # if all CCs are known, the solution is unique in terms of lengths of vectors, and angles between them. However, a rotated (around the origin) or inverted (through the origin) arrangement of the vectors leaves the functional unchanged, because these transformations do not change lengths and angles. | ||
# as long as the problem is | # as long as the problem is well-determined, the vectors can be calculated. Unknown CCs between data sets (e.g. in case of crystallographic data sets that don't have common reflections) can be estimated from the dot product of their vectors. Well-determination means: each data set has to be related (directly or indirectly i.e through others) to any other by at least as many CCs as the desired number of dimensions is. A necessary condition for this is that each data set has at least as many relations to others (input lines to cc_analysis involving this data set) as the number of dimensions is. It is of course better if more relations are specified! | ||
== The program == | == The program == | ||