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Nonlinear Regression

NATIONAL VETERINARY SCHOOL Toulouse. Nonlinear Regression. Didier Concordet. An example. Questions. What does nonlinear mean ? What is a nonlinear kinetics ? What is a nonlinear statistical model ? For a given model, how to fit the data ? Is this model relevant ?.

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Nonlinear Regression

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  1. NATIONAL VETERINARY SCHOOL Toulouse Nonlinear Regression Didier Concordet

  2. An example

  3. Questions • What does nonlinear mean ? • What is a nonlinear kinetics ? • What is a nonlinear statistical model ? • For a given model, how to fit the data ? • Is this model relevant ?

  4. What does nonlinear mean ? • Definition : An operator (P) is linear if : • for all objects x, y on which it operates • P(x+y) = P (x) + P(y) • for all numbers a and all objects x • P (ax) = a P(x) When an operator is not linear, it is nonlinear

  5. Examples Among the operators below which one are nonlinear ? • P (t) = a  t • P(t) = a • P(t) = a + b t • P(t) = a  t + b  t² • P(a,b) = a  t + b  t² • P(A,a) = A exp (- a t) • P(A) = A exp (- 0.1 t) • P(t) = A exp (- a t)

  6. What is a nonlinear kinetics ? Concentration at time t, C(t,D) For a given dose D The kinetics is linear when the operator : is linear When P(D) is not linear, the kinetics is nonlinear

  7. What is a nonlinear kinetics ? Examples :

  8. What is a nonlinear statistical model ? A statistical model Observation : Dep. variable Parameters Covariates : indep. variables Error : residual function

  9. What is a nonlinear statistical model ? A statistical model is linear when the operator : is linear. When is not linear the model is nonlinear

  10. What is a nonlinear statistical model ? Example : Y = Concentration t = time The model : is linear

  11. Examples Among the statistical models below which one are nonlinear ?

  12. Questions • What does nonlinear mean ? • What is a nonlinear kinetics ? • What is a nonlinear statistical model ? • For a given model, how to fit the data ? • Is this model relevant ?

  13. How to fit the data ? Proceed in three main steps • Write a (statistical) model • Choose a criterion • Minimize the criterion

  14. Write a (statistical) model • Find a function of covariate(s) to describe the mean variation of the dependent variable (mean model). • Find a function of covariate(s) to describe the dispersion of the dependent variable about the mean (variance model).

  15. Example is assumed gaussian with a constant variance homoscedastic model

  16. How to choose the criterion to optimize ? Homoscedasticity : Ordinary Least Squares (OLS) When normality OLS are equivalent to maximum likelihood Heteroscedasticity: Weight Least Squares (WLS) Extended Least Squares (ELS)

  17. Homoscedastic models The Ordinary Least-Squares criterion Define :

  18. Heteroscedastic models : Weight Least-Squares criterion Define :

  19. How to choose the weights ? When the model is heteroscedastic (ie is not constant with i) It is possible to rewrite it as where does not depend on i The weights are chosen as

  20. Example with The model can be rewritten as with The weights are chosen as

  21. Extended (Weight) Least Squares Define :

  22. Balance sheet

  23. The criterion properties It converges It leads to consistent (unbiased) estimates It leads to efficient estimates It has several minima

  24. It converges When the sample size increases, it concentrates about a value of the parameter Example : Consider the homoscedastic model The criterion to use is the Least Squares criterion

  25. It converges Small sample size Large sample size

  26. It leads to consistent estimates The criterion concentrates about the true value

  27. It leads to efficient estimates For a fixed n, the variance of an consistent estimator is always greater than a limit (Cramer-Rao lower bound). For a fixed n, the "precision" of a consistent estimator is bounded An estimator is efficient when its variance equals this lower bound

  28. criterion Geometric interpretation This ellipsoid is a confidence region of the parameter

  29. It leads to efficient estimates For a given large n, it does not exist a criterion giving consistent estimates more "convex" than - 2 ln(likelihood) - 2 ln(likelihood) criterion

  30. It has several minima criterion

  31. Minimize the criterion Suppose that the criterion to optimize has been chosen We are looking for the value of denoted which achieve the minimum of the criterion. We need an algorithm to minimize such a criterion

  32. Example Consider the homoscedastic model We are looking for the value of denoted which achieve the minimumof the criterion

  33. Isocontours

  34. Different families of algorithms • Zero order algorithms : computation of the criterion • First order algorithms : computation of the first derivative of the criterion • Second order algorithms : computation of the second derivative of the criterion

  35. Zero order algorithms • Simplex algorithm • Grid search and Monte-Carlo methods

  36. Simplex algorithm

  37. Monte-carlo algorithm

  38. First order algorithms • Line search algorithm • Conjugate gradient

  39. First order algorithms The derivatives of the criterion cancel at its optima Suppose that there is only one parameter to estimate The criterion (e.g. SS) depends only on How to find the value(s) of where the criterion cancels ?

  40. Line search algorithm Derivative of the criterion 1 0 q 2

  41. Second order algorithms Gauss-Newton (steepest descent method) Marquardt

  42. Second order algorithms The derivatives of the criterion cancel at its optima. When the criterion is (locally) convex there is a path to reach the minimum : the steepest direction.

  43. 0 q Gauss Newton (one dimension) Derivative of the criterion 3 2 1 The criterion is convex

  44. Gauss Newton (one dimension) Derivative of the criterion 0 q 1 2 The criterion is not convex

  45. Gauss Newton

  46. Marquardt Allows to deal with the case where the criterion is not convex When the second derivative <0 (first derivative decreases) it is set to a positive value Derivative of the criterion 0 q 3 2 1

  47. Balance sheet

  48. Questions • What does nonlinear mean ? • What is a nonlinear kinetics ? • What is a nonlinear statistical model ? • For a given model, how to fit the data ? • Is this model relevant ?

  49. Is this model relevant ? • Graphical inspection of the residuals • mean model ( f ) • variance model ( g ) • Inspection of numerical results • variance-correlation matrix of the estimator • Akaike indice

  50. Graphical inspection of the residuals For the model Calculate the weight residuals : and draw vs

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