Read the equation
Expanding the two outer products shows that the operator difference equals a changed output factor times the later input factor plus the earlier output factor times a changed input factor. This rearrangement makes small factor differences explicit and can avoid cancellation in a norm subtraction. It is algebraically exact for the shared scale s.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| U_j | Later effective update |
| U_i | Earlier effective update |
| a | Input-side factors |
| b | Output-side factors |
| s | Shared scale |
How it works
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Assumptions & limits
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Original context
Read the surrounding source passage
4.3. Numerical stability and principal components The primary calculation uses small Gram matrices formed from the factors. Subtracting nearly equal entries can lose precision for small late movements. I crosscheck every available effective corner using the exact identity Uj−Ui=s[(bj−bi)ajT+bi(aj−ai)T]. (4.7) Inner products of these sums reduce to vector dot products and avoid subtracting large, nearly equal matrix norms. The comparison covers 462 available combinations of center and lag, comprising 156, 154, and 152 centers at the three lags. It checks algebraically equivalent calculations, not a second empirical model. For principal component analysis, or PCA, I stack and center the 158 regularly sampled output vectors. Eigenanalysis of the centered Gram matrix gives component scores and explained variance fractions. Component signs are arbitrary; the implementation fixes a deterministic convention. Because the components use the whole sampled trajectory, their coordinates are retrospective rather than available to an online detector before future checkpoints exist.
Exact original notation
{U}_{j}-{U}_{i}=s[({b}_{j}-{b}_{i}){a}_{j}^{T}+{b}_{i}{({a}_{j}-{a}_{i})}^{T}].Related builds
Sources & provenance
{U}_{j}-{U}_{i}=s[({b}_{j}-{b}_{i}){a}_{j}^{T}+{b}_{i}{({a}_{j}-{a}_{i})}^{T}].
{U}_{j}-{U}_{i}=s[({b}_{j}-{b}_{i}){a}_{j}^{T}+{b}_{i}{({a}_{j}-{a}_{i})}^{T}].
{U}_{j}-{U}_{i}=s[({b}_{j}-{b}_{i}){a}_{j}^{T}+{b}_{i}{({a}_{j}-{a}_{i})}^{T}].
{U}_{j}-{U}_{i}=s[({b}_{j}-{b}_{i}){a}_{j}^{T}+{b}_{i}{({a}_{j}-{a}_{i})}^{T}].