Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/astro193/VLK_slides_apr01.pdf Дата изменения: Wed Apr 1 23:53:38 2015 Дата индексирования: Sun Apr 10 11:57:39 2016 Кодировка:

Astro 193 : 2015 Apr 1
·

Follow-up
· · · ·

Spearman's and Kendall's Class on Apr 10 Gibbs sampler acceptance rates: third time's the charm Adaptive MCMC

· ·

Bias-Variance Tradeoff / Kernel Density Estimation Signal Processing: Fourier Transforms


Adaptive MCMC
·

Run a chain for a short number of iterations in the kth block with modified step size (Hyungsuk Tak) compute acceptance rate if >0.4/npar, inflate step size by f+=e+min{0.07,1/k} if <0.2/npar, deflate step size by f­=e­min{0.07,1/k} f± 1 as k throw out iterations from before f± converges

· · · · ·


Bias-variance tradeoff
· ·

Given sample x = {x1, x2, ..., xn} ~ f(x), obtain estimate f(x) Least disruptive solution: histogram · f(x) = (1/h) {Ii: |xi-x|h}
·

f(x) = mean({Yi: |xi-x|h}) with regression Y = f(x) +

· ·

Affected by noise fluctuations. Also, how to choose h? Minimize Risk, mean squared error R(f(x),f(x)) = E[(f(x)-f(x))І] = E[fІ] - 2E[ff] + E[fІ] = fІ - 2fE[f] + E[fІ] + E[f]І - E[f]І = E[f]І - 2fE[f] + fІ + E[fІ] -E[f]І = E[f]І - 2fE[f] + fІ + E[fІ] -E[f]І = (E[f]-f)І + var[f] = bias2 + variance

·

When h is small, bias is small and variance is large; when h is large, bias is large and variance is small. Optimize h to minimize Risk. As h0, R ~ o(h4 + 1/nh), h
·
optimal

·

~ o(1/n1/5)
­1/5

in Gaussian case, hoptimal = 1.06 f n


Kernel Density Estimation
·

A "kernel" K(x) is a non-negative real-valued function such that xR dx K(x) = 1 ; E[x] = x
R

dx x K(x) = 0 ; E[x2] = xR dx x K(x) <

·

The Kernel density estimator for a bandwidth h, fh(x) = [1/nh]
i=1..n

K([x-xi]/h)

·

Types of Kernels
· ·

Boxcar, Gaussian, Triangle, Tophat RMF
i=1..n li(x)Yi

·

For linear smoothers, fh(x) = li = K([x-xi]/h] /
i=1..n

K([x-xi]/h), centered on xi, weighting all of the sample

Can write fh = LY in matrix notation Effective degrees of freedom, = trace(L) =
·
i=1..n

Lii

Choosing the bandwidth via cross-validation (Jackknife/leave-one-out) 2 CV|h = (1/n) i=1..n (Yi - fh(-i)(xi)) Generalized CV, GCV(h) (1/n) [1-/n]
-2 i=1..n

(Yi-fh(xi))

2