gradient concept
| Definition | The vector of a function's partial derivatives, the direction of fastest increase, whose dot product with a small step is the first-order change in the function. The book calls the gradient of a consumer its sensitivity. Primer L, equations L.19 and L.20. |
|---|---|
| Example | f(x) = x1 squared + 3 x1 x2 has gradient (8, 3) at (1, 2), and a step of 0.01 along the first coordinate changes f by 0.0801. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id gradient, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation L.19. |
| Assumptions and scope | none |
| Prior art | none recorded |
| Evidence | none |
| Reviewed | semantic review 2026-09-09; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation L.19.
\[\nabla f(x)=\Big(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_d}\Big),\qquad f(x+\delta)\approx f(x)+\nabla f(x)\cdot\delta.\]
Book equation L.20.
\[\nabla(w\cdot x)=w,\qquad \nabla\|x\|^{2}=2x,\qquad \nabla\big(x^{\top}Ax\big)=2Ax,\qquad \nabla g(w\cdot x)=g'(w\cdot x)\,w.\]
Conditions
none
Ledger
none
First stated
Primer L section L.9 of Data Mining as Observation, added in draft 0.3 (2026-09-09) for the ECE 514 readers whose first courses are far behind. The idea is standard and TSK Appendix A covers it at length.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
none
Used in
Data Mining as Observation primer L, chapters 0, 4, 6, 7, 8, 11, 12, 14.
Related
sensitivity; partial derivative; finite difference; Jacobian; read operator.
See also
none
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.