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Because both vectors in the centered bend statistic point away from the center, a straight trajectory places them in opposite directions and gives cosine −1. This is the reference for an unbent path under that convention. It does not mean the successive motion directions are opposed.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| C_Z | Centered path cosine |
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Original context
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The local metric used in the released analysis compares displacements from a center checkpoint to earlier and later checkpoints [3]. For Zt=bt or Zt=Ut, define
cZt;k=⟨Zt−k−Zt,Zt+k−Zt⟩∥Zt−k−Zt∥ ∥Zt+k−Zt∥.
This is not the cosine between Zt and Zt+k. A straight, continuing path gives cZ=−1, since the backward and forward displacements point oppositely. Larger values indicate a sharper local bend. The complete windows are evaluated on five-step checkpoint centers for lags k∈{5,10,15}. Each lag uses every center for which its own two neighbors exist; this need not reproduce identical edge trimming in every source plotting call.
An angle can emphasize a tiny late-training movement. We therefore also apply the source program’s factor-amplitude eligibility rule. Let mA be the larger of the preceding and succeeding input-factor displacement norms, and mB the corresponding output-factor quantity. A center is excluded only when both are below threshold η. The sweep is η∈{0,10−4,5×10−4,10−3,2×10−3,5×10−3}.
[Context excerpt; full source linked.]Exact original notation
{c}_{Z}=-1