Math 2.0

Statistics Definition

Response surface methodology

Builds on Design of experimentsFactorial experimentLeast squaresHessian matrix

Definition

Let x=(x1,…,xk)x = (x_1, \dots, x_k) be the settings of kk controllable factors in a region R⊆ℝkR \subseteq \R^k. Assume that the observed response is y=η(x)+εy = \eta(x) + \varepsilon, where η ⁣:R→ℝ\eta \colon R \to \R is an unknown smooth function and ε\varepsilon is a random error with mean 00. Response surface methodology is the sequential strategy for locating an optimum of η\eta that repeats these steps:

  1. Run an experiment, planned by the methods of design of experiments, in a small neighbourhood of the current point.
  2. While curvature is negligible, fit by least squares a first-order model y^=b0+∑ibixi\hat y = b_0 + \sum_i b_i x_i. Then move the current point along the path of steepest ascent, the direction (b1,…,bk)(b_1, \dots, b_k).
  3. When the first-order model shows lack of fit, fit a second-order model y^=b0+∑ibixi+∑i≤jbijxixj\hat y = b_0 + \sum_i b_i x_i + \sum_{i \le j} b_{ij} x_i x_j, and locate and classify its stationary point.

Remarks

Example: a fitted second-order model can be written y^=b0+xTb+xTBx\hat y = b_0 + x^T b + x^T B x with a symmetric k×kk \times k matrix BB. When BB is invertible, its stationary point is xs=−12B−1bx_s = -\frac12 B^{-1} b, since the gradient b+2Bxb + 2 B x vanishes there. The Hessian matrix of y^\hat y is 2B2B, so if all eigenvalues of BB are negative, xsx_s is the maximum of the fitted surface. This eigenvalue check is called canonical analysis.

Non-example: a single factorial experiment analysed once, with no move to new settings, is not response surface methodology, because step 2 is missing.

Typical designs are two-level factorial designs with centre points for the first-order stage, and central composite designs or Box–Behnken designs for the second-order stage.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first.

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Design of experiments, concept, no evidence levelDesign of experimentsconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFactorial experiment, concept, no evidence levelFactorial experimentconceptFunction, concept, no evidence levelFunctionconceptMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionNorm, definition, no evidence levelNormdefinitionPartial derivative, concept, no evidence levelPartial derivativeconceptInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionHigher partial derivatives, concept, no evidence levelHigher partial deriva…conceptNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionHessian matrix, concept, no evidence levelHessian matrixconceptA space is the direct sum of a subspace and its orthogonal complement, statement, no evidence levelA space is the direct…statementProjection, definition, no evidence levelProjectiondefinitionOrthogonal projection, definition, no evidence levelOrthogonal projectiondefinitionOrthogonal projection is the closest point, statement, no evidence levelOrthogonal projection…statementLeast squares, concept, no evidence levelLeast squaresconceptResponse surface methodology, definition, no evidence levelResponse surface meth…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Design of experiments concept
  2. Euclidean space object
  3. Factorial experiment concept
  4. Function concept
  5. Matrix definition
  6. Vector space definition
  7. Function of several variables concept
  8. Transpose definition
  9. Inner product definition
  10. Linear map definition
  11. Linear subspace definition
  12. Norm definition
  13. Partial derivative concept
  14. Inner product space definition
  15. Orthogonal vectors definition
  16. Endomorphism definition
  17. Sum of subspaces definition
  18. Higher partial derivatives concept
  19. Norm induced by an inner product definition
  20. Orthogonal complement definition
  21. Direct sum of subspaces definition
  22. Hessian matrix concept
  23. A space is the direct sum of a subspace and its orthogonal complement statement
  24. Projection definition
  25. Orthogonal projection definition
  26. Orthogonal projection is the closest point statement
  27. Least squares concept
  28. Response surface methodology definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Response_surface_methodology
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DefinesResponse surface methodology
UsesDesign of experiments, Factorial experiment, Least squares, Hessian matrix

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