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13 Pages

### feb23

Course: STA 450, Fall 2009
School: Toledo
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Word Count: 1179

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of Degrees freedom for splines degrees of freedom for cubic splines, 1 covariate K knots, K + 1 intervals, 4(K + 1) parameters 3K restrictions leaves K + 4 parameters, omit constant term, leaves K + 3 degrees of freedom for natural splines linear on (-, 1 ] and [K , ): 4 pars cubic in interior intervals 4(K - 1) 3K restrictions, leaves K parameters, but 2 knots added at x(1) , x(n) : K + 2 omit constant...

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of Degrees freedom for splines degrees of freedom for cubic splines, 1 covariate K knots, K + 1 intervals, 4(K + 1) parameters 3K restrictions leaves K + 4 parameters, omit constant term, leaves K + 3 degrees of freedom for natural splines linear on (-, 1 ] and [K , ): 4 pars cubic in interior intervals 4(K - 1) 3K restrictions, leaves K parameters, but 2 knots added at x(1) , x(n) : K + 2 omit constant term get K + 1 what does omit constant term mean? Example S0 (x) = a0 + b0 x + c0 x 2 + d0 x 3 , 0x 1 2 S1 (x) = a1 + b1 (x - 1) + c1 (x - 1) + d1 (x - 1)3 , e.g. a0 = b0 = c0 = d0 = 1; fixes a1 = 4, b1 = 6, c1 = 8 or, force a0 =0, add it in later STA 450/4000 S: February 23 2005: , 1x 2 1 Degrees of freedom for splines degrees of freedom for smoothing splines min {yi - f (xi )}2 + f {f (t)}2 dt solution is natural cubic splines with knots at unique xi f (x) = N Nj (x)j , say j=1 min {yi - j Nj (xi )j }2 + jk j k jk min (y - N)T (y - N) + T N jk = Nj (t)Nk (t)dt ^ just like ridge regression: = (N T N + N )-1 N T y ^ = N = n(N T N + N )-1 N T y = S y, say ^ f degrees of freedom defined to be traceS by analogy Same formula works for regression splines (actually easier) cf (5.15) STA 450/4000 S: February 23 2005: , 2 Multidimensional splines (5.7) Suppose we have X1 , X2 , and E(y | X ) = f (X1 , X2 ) one solution is to combine separate spline bases for X1 and X2 e.g. additively: f (X1 , X2 ) = f1 (X1 ) + f2 (X2 ) (this is what was done for heart data) doesn't permit interactions alternative is to use all possible cross products: called tensor products f (X1 , X2 ) = M1 M2 jk h1j (X1 )h2k (X2 ) j=1 k=1 analogous to forming quadratic functions in regression 2 2 using, e.g., x1 , x1 x2 , x2 STA 450/4000 S: February 23 2005: , 3 Multidimensional splines (5.7) alternative to derive smoothing splines in two dimensions: min N {yi - f (x i )}2 + J(|f |) i=1 f J(|f |) = 2 2 (1 f + 2 f + 212 f )2 dxdy as in univariate case, solution exists in a spline basis similar to natural splines (5.39): f (x) = 0 + T x + N j hj (x) j=1 hj (x) = (||x - x j ||), (z) = z 2 log z called radial basis functions: take this form because of symmetry of penalty note uses N knots; reduced in implementation by regularization STA 450/4000 S: February 23 2005: , 4 kernel methods for regression Kernel methods for regression: univariate model: E(Y | x) = f (x) ("smooth") data: yi = f (xi ) + i running mean smoother: ^(x0 ) = ave(yi | xi Nk (x0 )) f Nk (x0 ) set of k "nearest neighbours": k smallest values of |xi - x0 | smoother estimate using kernel weighted average ^(x0 ) = f [Figure 6.1] kernel K (x0 , x) = D |x - x0 | or D |x - x0 | h (x0 ) N i=1 K (x0 , yi )yi N i=1 K (x0 , xi ) STA 450/4000 S: February 23 2005: , 5 kernel methods for regression determines the width of the neighbourhood, hence smoothness increasing gives smoother function (higher bias, lower variance) metric window width (h (x0 ) = ) - constant bias, variance 1/local density nearest neighbour window width (h (x0 ) depends on x0 ) constant variance, bias 1/local density Choice of kernel: 3 2 Epanichakov 4 (1 - t ), |t| 1 D(t) = 0 (1 - |t|3 )3 , |t| 1 tri - cube 0 1 Gaussian = (t) = exp(-t 2 /2) 2 = STA 450/4000 S: February 23 2005: , 6 kernel methods for regression R or Splus: ksmooth(x,y,kernel=c("box","normal"),bandwidth=0.5,range loess(formula) more later > > > > > > > > > > > > eps<-rnorm(100,0,1/3) x<-runif(100) sin4x <- function(x){sin(4*x)} y<-sin4(x)+eps plot(sin4,0,1,type="l",ylim=c(-1.0,1.5),xlim=c(0,1)) points(x,y) lines(ksmooth(x,y,"box",bandwidth=.2),col="blue") lines(ksmooth(x,y,"normal",bandwidth=.2),col="green") plot(sin4,0,1,type="l",ylim=c(-1.0,1.5),xlim=c(0,1)) lines(ksmooth(x,y,"normal",bandwidth=.2),col="green") lines(ksmooth(x,y,"normal",bandwidth=0.6),col="red") 7 (Figure lines(ksmooth(x,y,"normal",bandwidth=0.4),col="blue") 6.1) STA 450/4000 S: February 23 2005: , kernel methods for regression Local linear regression replace weighted average of xi 's with weighted linear (or polynomial) regression: better endpoint behaviour min (x0 ),(x0 ) K (x0 , xi ){yi - (x0 ) - (x0 )xi }2 ^(x0 ) = (1, x0 )(X T W (x0 )X )-1 X T W (x0 )y f X = W (x0 ) = diag K (x0 , xi ) STA 450/4000 S: February 23 2005: , 8 1 x1 1 x2 . . =B . . . . 1 xn kernel methods for regression Notes Recall weighted least squares: min wi (yi - 0 - 1 xi )2 or min(y - X )T W (y - X ) ^ = (X T WX )-1 X T Wy can combine the least squares weights with the kernel weights; see Figure 6.4 and pp. 169, 170. can also do local quadratic regression (and higher) but increases bias at endpoints for extrapolation book recommends local linear fits; for good fits in middle local quadratic In R there are several smoothers: ksmooth and loess are built in The first uses kernel smoothing, the second uses local linear regression (robustified) STA 450/4000 S: February 23 2005: , 9 kernel methods for regression scatter.smooth fits a loess curve to a scatter plot loess takes a family argument : family = gaussian gives weighted least squares using K as weights and family=symmetric gives a robust version using Tukey's biweight supsmu implements "Friedman's super smoother": a running lines smoother with elaborate adaptive choice of bandwidth Library KernSmooth has locpoly for local polynomial fits, and by setting degree = 0 gives a kernel smooth STA 450/4000 S: February 23 2005: , 10 kernel methods for regression > lo1 <- loess(y~x, degree=1, span=0.75) > attributes(lo1) \$names [1] "n" "fitted" "residuals" "enp" [7] "two.delta" "trace.hat" "divisor" "pars" [13] "terms" "xnames" "x" "y" "s" "kd" "weights" "one. "call \$class [1] "loess" > plot(sin4,0,1,type="l",ylim=c(-1.0,1.5),xlim=c(0,1)) > points(x,lo1\$fitted,pch=".",col="red") > plot(x,lo1\$fitted,pch=".",col="red",ylim=c(...

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STA 442F / STA 2101 F: Applied Statistics I This course teaches methods of applied statistics, with the applications studied motivating the sets of methods taught. The undergraduate calendar description is: Advanced topics in statistics and data anal
Toledo - STA - 442
Toledo - STA - 442
&gt; library(boot) &gt; data(calcium) &gt; plot(calcium\$time,calcium\$cal)&gt; calcium.nls= nls(cal ~ b0*(1-exp(-time/b1), start=c(b0=5,b1=5),data=calcium) &gt; summary(calcium.nls) Formula: cal ~ b0 * (1 - exp(-time/b1) Parameters: Estimate Std. Error t value Pr(
Toledo - STA - 442
Toledo - STA - 442
150qq qqx100xqqtimex xqxx xx qx x xxq qq50qxqqxqq01.00.5 logwbc0.00.5
Toledo - STA - 442
sort(time/exu.glm\$fitted.values) 0.0 0.0qq q qq q q0.51.01.52.02.53.03.50.5q q q q1.0q q1.5qqexpq2.0 2.5 3.0 3.5q q
Toledo - STA - 442
Toledo - STA - 442
Toledo - STA - 442
R version 2.5.1 (2007-06-27)Copyright (C) 2007 The R Foundation for Statistical ComputingISBN 3-900051-07-0R is free software and comes with ABSOLUTELY NO WARRANTY.You are welcome to redistribute it under certain conditions.Type 'license()' or
Toledo - STA - 442
R version 2.5.0 (2007-04-23) Copyright (C) 2007 The R Foundation for Statistical Computing ISBN 3-900051-07-0 R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or '
Toledo - BIO - 349
CSB349 - Eukaryotic Gene ExpressionProf. Vince Tropepe Department of Cell &amp; Systems Biology 20091CSB349 Course Manual 2009 There are signicant changes from last year, so please ensure that you have access to a copy of the 2009 manual. Detaile
Toledo - BIO - 349
CSB349 Lecture 2 Genome structure: Duplication and repetitive DNA1Acquiring new genes and changing the genomic landscape It is generally observed that genomes have become more complex over time and one indication of this complexity is the variat
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CSB349 Lecture 4 Genome structure: Nucleosome structure and remodeling1Packaging genomic DNA2Fundamental unit of chromatin is the nucleosome Eukaryotic DNA is packed into a nucleoprotein complex called chromatin The fundamental repeating un
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CSB349 Lecture 8RNA processing: Splicing1RNA processing2The primary transcript Unlike the vast majority of bacterial, phage and other viral genes, most of the coding sequences (exons) in eukaryotic genes are interrupted by non-coding sequ
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CSB349 Lecture 9 RNA processing: Cleavage and polyadenylation1Transcription termination At the end of transcription the mRNA is cleaved at the poly(A) site by an endonuclease and a poly(A) tail is added to the exposed 3 end The release of RNAP
Toledo - BIO - 349
Time/DayTuesdayWednesdayT0301A - ESC2087 - DonnaThursdayT0601A ESC3088 Jacqueline T0601B ESC4076 - Graham10 12 pmT0301B - SS1087 - GrahamT0101A - RW122 - IrinaT0401A - RW122 - Donna T0401B - ESC4076 - SadiaT0701A - ESC4076 Jacque
Toledo - BIO - 349
A Guide To Evaluating Your Own WritingIn assigning grades, instructors concentrate on the four areas of topic, ideas, organization, and expression. You can use this checklist to help you preview how an instructor might read your workor you can ask a
Toledo - STA - 3103
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Toledo - STA - 3103
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: edgeworth.dvi %CreationDate: Fri Feb 2 13:48:18 1996 %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips edgeworth %DVIPSPa
Toledo - STA - 3103
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: tail1.dvi %CreationDate: Wed Mar 13 11:29:56 1996 %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips tail1 %DVIPSParameter
Toledo - STA - 3103
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: references.dvi %CreationDate: Thu Apr 11 14:11:07 1996 %Pages: 6 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips references %DVIP
Toledo - STA - 410
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Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.497 Copyright 1986, 1992 Radical Eye Software %Title: lec5.dvi %CreationDate: Fri Oct 20 17:14:45 1995 %Pages: 3 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec5 %DVIPSSource: Te
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.497 Copyright 1986, 1992 Radical Eye Software %Title: lec6.dvi %CreationDate: Mon Oct 23 21:28:07 1995 %Pages: 3 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec6 %DVIPSSource: Te
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.497 Copyright 1986, 1992 Radical Eye Software %Title: lec7.dvi %CreationDate: Tue Oct 31 13:58:27 1995 %Pages: 3 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec7.dvi %DVIPSSource
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.497 Copyright 1986, 1992 Radical Eye Software %Title: lec8.dvi %CreationDate: Mon Nov 6 21:31:46 1995 %Pages: 4 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec8.dvi %DVIPSSource:
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.497 Copyright 1986, 1992 Radical Eye Software %Title: lec9.dvi %CreationDate: Tue Nov 14 13:07:11 1995 %Pages: 4 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec9.dvi %DVIPSSource
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.497 Copyright 1986, 1992 Radical Eye Software %Title: lec11.dvi %CreationDate: Tue Nov 28 13:41:53 1995 %Pages: 5 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec11.dvi %DVIPSSour
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.497 Copyright 1986, 1992 Radical Eye Software %Title: lec12.dvi %CreationDate: Tue Dec 5 13:00:32 1995 %Pages: 4 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: odvips lec12.dvi %DVIPSSour
Toledo - SCI - 199
Got a new papersize ator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: lec13.dvi %CreationDate: Tue Jan 9 16:40:44 1996 %Pages: 2 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec13.dvi %DVIPSParam
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: lec14.dvi %CreationDate: Mon Jan 22 16:41:50 1996 %Pages: 2 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec14 %DVIPSParameter
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: lec16.dvi %CreationDate: Mon Feb 12 15:18:08 1996 %Pages: 2 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec16 %DVIPSParameter
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: lec21.dvi %CreationDate: Tue Mar 26 16:50:43 1996 %Pages: 3 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec21 %DVIPSParameter
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: lec22.dvi %CreationDate: Thu May 23 17:15:37 1996 %Pages: 3 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips lec22 %DVIPSParameter
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: prop1.dvi %CreationDate: Mon Jan 22 16:17:12 1996 %Pages: 2 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips prop1 %DVIPSParameter
Toledo - SCI - 199
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: prop3.dvi %CreationDate: Mon Jan 22 16:17:19 1996 %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips prop3 %DVIPSParameter
Toledo - STA - 3000
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Toledo - STA - 3000
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Toledo - STA - 3000
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Toledo - STA - 3000
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Toledo - STA - 3000
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Toledo - STA - 3000
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Toledo - STA - 3000
%!PS-Adobe-2.0 %Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %Title: hw8.dvi %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %DocumentPaperSizes: Letter %EndComments %DVIPSCommandLine: dvips hw8 %DVIPSParameters: dpi=600, co
Toledo - STA - 200498
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Toledo - STA - 200498
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Toledo - STA - 200498
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Toledo - STA - 200498
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Toledo - STA - 200498
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Toledo - STA - 200498
%!PS-Adobe-2.0 %Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %Title: ems.dvi %Pages: 3 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %DocumentPaperSizes: Letter %EndComments %DVIPSCommandLine: dvips ems %DVIPSParameters: dpi=600, co
Toledo - PHY - 138
Toledo - PHY - 138
Toledo - PHY - 138
Toledo - PHY - 138
Toledo - PHY - 138
Toledo - PHY - 138
Quiz 1A dentist uses a mirror to look at the back of a second molar (A). Next, she wishes to look at the back of a lateral incisor (B), which is 90 away. By what angle should she rotate her mirror? A. 90 B. 45 C. 180MirrorABClass Vote:Corre
Toledo - PHY - 138
Toledo - PHY - 138