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Linear Models

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  1. Linear Models Tony Dodd

  2. Overview • Linear models. • Parameter estimation. • Linear in the parameters. • Classification. • The nonlinear bits. An Overview of State-of-the-Art Data Modelling

  3. Linear models • Linear model has general form where is the th component of input . • Assume and therefore is the bias. • Can represent lines and planes. • Should ALWAYS try a linear model first! An Overview of State-of-the-Art Data Modelling

  4. Parameter estimation • Least squares estimation. • Choose parameters that minimise • Unique minimum… • Optimum when noise is Gaussian. An Overview of State-of-the-Art Data Modelling

  5. Least squares cost function An Overview of State-of-the-Art Data Modelling

  6. Least squares parameters • Define the design matrix • Then the optimal parameters given by An Overview of State-of-the-Art Data Modelling

  7. Example An Overview of State-of-the-Art Data Modelling

  8. Example An Overview of State-of-the-Art Data Modelling

  9. Example An Overview of State-of-the-Art Data Modelling

  10. How can we generalise this? • Consider instead • Where is a nonlinear function of the inputs. • Nonlinear transform of the inputs and then form a linear model (more tomorrow). An Overview of State-of-the-Art Data Modelling

  11. Linear in the parameters • A nonlinear model that is often called linear. • Can apply simple estimation to the parameters. • But… it is nonlinear in the basis functions. An Overview of State-of-the-Art Data Modelling

  12. Parameter estimation • Define the design matrix • Then the optimal parameters given by An Overview of State-of-the-Art Data Modelling

  13. Example An Overview of State-of-the-Art Data Modelling

  14. Example An Overview of State-of-the-Art Data Modelling

  15. Example An Overview of State-of-the-Art Data Modelling

  16. Example – how does it work? An Overview of State-of-the-Art Data Modelling

  17. Example – how does it work? An Overview of State-of-the-Art Data Modelling

  18. Example – how does it work? To get the function estimate Add all these together An Overview of State-of-the-Art Data Modelling

  19. Example – when it all goes wrong More on this later. An Overview of State-of-the-Art Data Modelling

  20. Linear classification How do we apply linear models to classification – output is now categorical? • Discriminant analysis. • Probit analysis. • Log-linear regression. • Logistic regression. An Overview of State-of-the-Art Data Modelling

  21. Logistic regression • A regression model for Bernoulli-distributed targets. • Form the linear model where An Overview of State-of-the-Art Data Modelling

  22. Can we generalise it? • Instead of use a linear in the parameters model An Overview of State-of-the-Art Data Modelling

  23. Parameter estimation • Maximum likelihood. • Maximise the probability of getting the observed results given the parameters. • Although unique minimum need to use iterative techniques (no closed form solution). An Overview of State-of-the-Art Data Modelling

  24. Example An Overview of State-of-the-Art Data Modelling

  25. Example An Overview of State-of-the-Art Data Modelling

  26. Example An Overview of State-of-the-Art Data Modelling

  27. Example An Overview of State-of-the-Art Data Modelling

  28. Example – class probabilities An Overview of State-of-the-Art Data Modelling

  29. But… An Overview of State-of-the-Art Data Modelling

  30. Basis function optimisation Need to estimate: • Type of basis functions. • Number of basis functions. • Positions of basis functions. These are nonlinear problems – difficult! An Overview of State-of-the-Art Data Modelling

  31. Types of basis functions • Usually choose a favourite! • Examples include: Polynomials: Gaussians: … An Overview of State-of-the-Art Data Modelling

  32. Number of basis functions • How many basis functions? • Slowly increase number until overfit data. • Exploratory vs optimal. • More on this in the next talk. An Overview of State-of-the-Art Data Modelling

  33. Positions of basis functions • This is really difficult! • One easy possibility is to put one basis function on each data point. • Uniform grid (but curse of dimensionality). • Advantage of global basis functions e.g. polynomials – don’t need to optimise positions. An Overview of State-of-the-Art Data Modelling

  34. Concluding remarks • Always try a linear model first. • Can make nonlinear in the input but linear in the parameters. • But becomes nonlinear optimisation. • Is least squares/maximum likelihood the best way? An Overview of State-of-the-Art Data Modelling

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