Read the equation
Both displacement vectors point outward from the center checkpoint: one toward t−k and the other toward t+k. Their normalized inner product measures the angle between those directions. Straight passage through the center gives −1, a right-angle turn gives 0, and return along the same side can give +1; this orientation is the reverse of an incoming-versus-outgoing velocity cosine.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| C_Z | Centered displacement cosine; c_Z in another version |
| Z | Factor or effective-operator path |
| t | Center checkpoint |
| k | Lag on each side |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
The expression is undefined if either displacement has zero norm.
Original context
Read the surrounding source passage
4.4 Local path bend and amplitude eligibility
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}\left(t;k\right)=\frac{⟨{Z}_{t-k}-{Z}_{t},{Z}_{t+k}-{Z}_{t}⟩}{\Vert {Z}_{t-k}-{Z}_{t}\Vert \Vert {Z}_{t+k}-{Z}_{t}\Vert }.Related builds
Sources & provenance
{c}_{Z}\left(t;k\right)=\frac{⟨{Z}_{t-k}-{Z}_{t},{Z}_{t+k}-{Z}_{t}⟩}{\Vert {Z}_{t-k}-{Z}_{t}\Vert \Vert {Z}_{t+k}-{Z}_{t}\Vert }.
{c}_{Z}\left(t;k\right)=\frac{⟨{Z}_{t-k}-{Z}_{t},{Z}_{t+k}-{Z}_{t}⟩}{\Vert {Z}_{t-k}-{Z}_{t}\Vert \Vert {Z}_{t+k}-{Z}_{t}\Vert }.