24 modelling

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24 modelling

  1. 1. Hadley Wickham Stat405Intro to modelling Tuesday, 16 November 2010
  2. 2. 1. What is a linear model? 2. Removing trends 3. Transformations 4. Categorical data 5. Visualising models Tuesday, 16 November 2010
  3. 3. What is a linear model? Tuesday, 16 November 2010
  4. 4. Tuesday, 16 November 2010
  5. 5. observed value Tuesday, 16 November 2010
  6. 6. observed value Tuesday, 16 November 2010
  7. 7. predicted value observed value Tuesday, 16 November 2010
  8. 8. predicted value observed value Tuesday, 16 November 2010
  9. 9. predicted value observed value residual Tuesday, 16 November 2010
  10. 10. y ~ x # yhat = b1x + b0 # Want to find b's that minimise distance # between y and yhat z ~ x + y # zhat = b2x + b1y + b0 # Want to find b's that minimise distance # between z and zhat z ~ x * y # zhat = b3(x⋅y) + b2x + b1y + b0 Tuesday, 16 November 2010
  11. 11. X is measured without error. Relationship is linear. Errors are independent. Errors have normal distribution. Errors have constant variance. Assumptions Tuesday, 16 November 2010
  12. 12. Removing trends Tuesday, 16 November 2010
  13. 13. library(ggplot2) diamonds$x[diamonds$x == 0] <- NA diamonds$y[diamonds$y == 0] <- NA diamonds$y[diamonds$y > 30] <- NA diamonds$z[diamonds$z == 0] <- NA diamonds$z[diamonds$z > 30] <- NA diamonds <- subset(diamonds, carat < 2) qplot(x, y, data = diamonds) qplot(x, z, data = diamonds) Tuesday, 16 November 2010
  14. 14. Tuesday, 16 November 2010
  15. 15. Tuesday, 16 November 2010
  16. 16. mody <- lm(y ~ x, data = diamonds, na = na.exclude) coef(mody) # yhat = 0.05 + 0.99⋅x # Plot x vs yhat qplot(x, predict(mody), data = diamonds) # Plot x vs (y - yhat) = residual qplot(x, resid(mody), data = diamonds) # Standardised residual: qplot(x, rstandard(mody), data = diamonds) Tuesday, 16 November 2010
  17. 17. qplot(x, resid(mody), data=dclean) Tuesday, 16 November 2010
  18. 18. qplot(x, y - x, data=dclean) Tuesday, 16 November 2010
  19. 19. Your turn Do the same thing for z and x. What threshold might you use to remove outlying values? Are the errors from predicting z and y from x related? Tuesday, 16 November 2010
  20. 20. modz <- lm(z ~ x, data = diamonds, na = na.exclude) coef(modz) # zhat = 0.03 + 0.61x qplot(x, rstandard(modz), data = diamonds) last_plot() + ylim(-10, 10) qplot(rstandard(mody), rstandard(modz)) Tuesday, 16 November 2010
  21. 21. Transformations Tuesday, 16 November 2010
  22. 22. Can we use a linear model to remove this trend? Tuesday, 16 November 2010
  23. 23. Can we use a linear model to remove this trend? Tuesday, 16 November 2010
  24. 24. Can we use a linear model to remove this trend? Linear models are linear in their parameters which can be any transformation of the data Tuesday, 16 November 2010
  25. 25. Your turn Use a linear model to remove the effect of carat on price. Confirm that this worked by plotting model residuals vs. color. How can you interpret the model coefficients and residuals? Tuesday, 16 November 2010
  26. 26. modprice <- lm(log(price) ~ log(carat), data = diamonds, na = na.exclude) diamonds$relprice <- exp(resid(modprice)) qplot(carat, relprice, data = diamonds) diamonds <- subset(diamonds, carat < 2) qplot(carat, relprice, data = diamonds) qplot(carat, relprice, data = diamonds) + facet_wrap(~ color) qplot(relprice, ..density.., data = diamonds, colour = color, geom = "freqpoly", binwidth = 0.2) qplot(relprice, ..density.., data = diamonds, colour = cut, geom = "freqpoly", binwidth = 0.2) Tuesday, 16 November 2010
  27. 27. log(Y) = a * log(X) + b Y = c . dX An additive model becomes a multiplicative model. Intercept becomes starting point, slope becomes geometric growth. Multiplicative model Tuesday, 16 November 2010
  28. 28. Residuals resid(mod) = log(Y) - log(Yhat) exp(resid(mod)) = Y / (Yhat) Tuesday, 16 November 2010
  29. 29. # Useful trick - close to 0, exp(x) ~ x + 1 x <- seq(-0.2, 0.2, length = 100) qplot(x, exp(x)) + geom_abline(intercept = 1) qplot(x, x / exp(x)) + scale_y_continuous("Percent error", formatter = percent) # Not so useful here because the x is also # transformed coef(modprice) Tuesday, 16 November 2010
  30. 30. Categorical data Tuesday, 16 November 2010
  31. 31. Compare the results of the following two functions. What can you say about the model? ddply(diamonds, "color", summarise, mean = mean(price)) coef(lm(price ~ color, data = diamonds)) Your turn Tuesday, 16 November 2010
  32. 32. Categorical data Converted into a numeric matrix, with one column for each level. Contains 1 if that observation has that level, 0 otherwise. However, if we just do that naively, we end up with too many columns (because we have one extra column for the intercept) So everything is relative to the first level. Tuesday, 16 November 2010
  33. 33. Visualising models Tuesday, 16 November 2010
  34. 34. # What do you think this model does? lm(log(price) ~ log(carat) + color, data = diamonds) # What about this one? lm(log(price) ~ log(carat) * color, data = diamonds) # Or this one? lm(log(price) ~ cut * color, data = diamonds) # How can we interpret the results? Tuesday, 16 November 2010
  35. 35. mod1 <- lm(log(price) ~ log(carat) + cut, data = diamonds) mod2 <- lm(log(price) ~ log(carat) * cut, data = diamonds) # One way is to explore predictions from the model # over an evenly spaced grid. expand.grid makes # this easy grid <- expand.grid( carat = seq(0.2, 2, length = 20), cut = levels(diamonds$cut), KEEP.OUT.ATTRS = FALSE) str(grid) grid grid$p1 <- exp(predict(mod1, grid)) grid$p2 <- exp(predict(mod2, grid)) Tuesday, 16 November 2010
  36. 36. Plot the predictions from the two sets of models. How are they different? Your turn Tuesday, 16 November 2010
  37. 37. qplot(carat, p1, data = grid, colour = cut, geom = "line") qplot(carat, p2, data = grid, colour = cut, geom = "line") qplot(log(carat), log(p1), data = grid, colour = cut, geom = "line") qplot(log(carat), log(p2), data = grid, colour = cut, geom = "line") qplot(carat, p1 / p2, data = grid, colour = cut, geom = "line") Tuesday, 16 November 2010
  38. 38. # Another approach is the effects package # install.packages("effects") library(effects) effect("cut", mod1) cut <- as.data.frame(effect("cut", mod1)) qplot(fit, reorder(cut, fit), data = cut) qplot(fit, reorder(cut, fit), data = cut) + geom_errorbarh(aes(xmin = lower, xmax = upper), height = 0.1) qplot(exp(fit), reorder(cut, fit), data = cut) + geom_errorbarh(aes(xmin = exp(lower), xmax = exp(upper)), height = 0.1) Tuesday, 16 November 2010

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