Machine learning: Hypothesis testing. Anders Hildeman

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2 Location of trees 0 Observed trees Figur: Observed points pattern of the tree specie Beilschmiedia pendula.

3 Location of trees discretized and counted 0 Gridded observations Figur: Two-dimensional histogram of the observed point pattern.

4 Model Cox mixture model Y D i {X, Θ} Po(λ i ) K log λ(s i ) {X }, Θ = π k (s i )X k(s i ) k=1 X k Θ N(Bβ, Σ(σ k, r k, ν k )) Bayesian model with a non-homogeneous Poisson distribution and latent Gaussian random eld priors. B are covariates and β are the linear regression coecients for those covariates.

5 MCMC sampling of marginal posteriors Sample from a Markov chain that has the posterior as its stationary distribution. Approximate the marginal posteriors from the sample histograms.

6 Posterior mean Poisson intensity Intensity Figur: Estimated Poisson intensity.

7 Posterior mean Gaussian eld X Figur: Estimated Gaussian eld.

8 Posterior mean classication eld Z Figur: Estimated classication eld.

9 Posterior linear xed eects Figur: Histogram of linear xed eects.

10 Multiple hypothesis testing Which covariates have an eect on the distribution of trees? Draw inference from the sampled data. p-values acquired from sample quantiles. Let us rst look on simple Poisson regression. λ(s) = e 16 i=1 B i (s)β i

11 Bonferroni Accept if p i α N. Holm Accept for all (i) up until the rst (i) such that p (i) > up until the rst (i) with a rejection. α N+1 i Benjamini-Hochberg Accept for all (i) up until the last (i) for which p (i) q i N.

12 p-values B-H Holm Bonf Figur: Ordered p-values in comparison with hypothesis thresholds in log-scale.

13 Covariate Bonf. Holm B-H Intercept x x x Elevation x x x Slope x x x Al B x x x Ca Cu x x x Fe x x K Mg Mn x N Nmin x x x P x x x Zn x x x ph x x x

14 Let us look on Poisson regression for the intensity eld with two mixture classes. λ(s) = e π 1(s)γ+π 2 (s) 16 i=1 B i (s)β i

15 p-values B-H Holm Bonf Figur: Ordered p-values in comparison with hypothesis thresholds in log-scale.

16 Covariate Bonf. Holm B-H Intercept x x x Elevation x x x Slope Al B x x x Ca Cu Fe K Mg Mn N Nmin P Zn x x ph

17 Let us look on a log-gaussian Cox with linear regression. λ(s) = e X (s)+ 16 i=1 B i (s)β i

18 p-values B-H Holm Bonf Figur: Ordered p-values in comparison with hypothesis thresholds in log-scale.

19 Covariate Bonf. Holm B-H Intercept x x x Elevation Slope x x x Al B Ca Cu Fe K Mg Mn N Nmin P Zn ph

20 Let us look on a log-gaussian Cox mixture model with linear regression. λ(s) = e π 1(s)γ+π 2 (s)(x (s)+ 16 i=1 B i (s)β i)

21 p-values B-H Holm Bonf Figur: Ordered p-values in comparison with hypothesis thresholds in log-scale.

22 Covariate Bonf. Holm B-H Intercept x x x Elevation Slope Al B Ca Cu Fe K Mg Mn N Nmin P Zn ph

23 There are probably some collinearity present among the covariates. Hence, some signicant eects might not be visible. Use principal component analysis. Assuming that we only have a second order dependency structure we will now look at 16 independent covariates instead.

24 Considering the Holm method. Poisson regression: Signicant covariates: Poisson regression mixture model: Signicant covariates: 4 9. log-gaussian Cox with linear regression: Signicant covariates: 2 3. Full model: Signicant covariates: 1 1.

25 Question Can we decide which covariates that are signicant for the PCA analysis?