Distinctive aspects of non-parametric fitting

Transcription

1 5. Introduction to nonparametric curve fitting: Loess, kernel regression, reproducing kernel methods, neural networks Distinctive aspects of non-parametric fitting Objectives: investigate patterns free of strictures imposed by parametric models Can produce surprising results Regression coefficients appear but (typically) do not have an obvious interpretation Often have very good predictive performance in cross-validation Tuning methods similar to those for parametric methods 1

2 Example: thin-plate splines Risk of heart attack after 19 years as a function of cholesterol level and blood pressure. Left: logistic regression model. Right: thin plate spline fit. Wahba (007) log ) ( j i j i N j j i j i j i i i x x x x x x x x x x f x LOESS REGRESSION: Non-parametric exploration of inbreeding depression for yield and somatic cell count in Jersey cattle

3 AN OVERVIEW OF LOWESS REGRESSION 1) DATA POINTS x i, y i ; i 1,,...,n ) SPANNING PARAMETER f; 0 f 1 k fn; k LARGEST INTEGER fn 3) FOR EACH x 0 FIND k POINTS x i CLOSEST TO x 0 Nx 0 NEIGHBORHOOD OF k POINTS 4) COMPUTE Δx 0 max x i Nx 0 x o x i 5) TO EACH x i, y i ; x i Nx 0 ASSIGN WEIGHT w i x 0 1 x o x i Δx ) FIT BY WEIGHTED LEAST-SQUARES k w i x 0y i 0 1 x i x i i1 RETURN yx x i x i 7) REPEAT FOR EACH OF THE x 0 3

4 ROBUST LOWESS STANDARD LOWESS NOT ROBUST BASED ON LEAST-SQUARES WEIGHTS BI-SQUARE LOWESS RE-WEIGHT POINTS ACCORDING TO RESIDUAL IF RESIDUAL LARGE, WEIGHT IS DECREASED 1) FIT DATA USING STANDARD LOESS ) CALCULATE LOESS RESIDUALS y i y i 3) COMPUTE q 1 median y i y i r i 4) CALCULATE BI-SQUARE ROBUST WEIGHTS 1 y i y i 6 q 1 5) REPEAT LOESS WITH WEIGHTS r i w ix 0 6) REPEAT -5 UNTIL LOESS CURVE CONVERGES 4

5 Example Birth rate in US population (U. S. Department of Health, Education and Welfare) n=96 births per 1000 US population during Top > Ordinary Least Squares with 1 st, nd & 3 rd degree polynomial Bottom > LOWESS fit with f =., f=.4 & f=.6 5

6 GALTON S BEND (Wachsmuth et al. 003, Am. Stat.) A possibility is that Galton ignored concealed heterogeneity 6

7 Does the bend disappear by disaggregation of the sample? Analysis of data from Pearson and Lee (1903) BEND STILL THERE! Wachsmuth et al. (003) write: 7

8 INBREEDING DEPRESSION Examine relationships of yield (milk, protein, fat) and somatic cell score (SCS) with inbreeding coefficient (F) using field data from US Jerseys Use REML, BLUP and local regression method (LOESS) for this purpose LEVEL OF INBREEDING IN HOLSTEINS, USA 8

9 Relationship between mean value of a quantitative trait and inbreeding coefficient (F) expected to be linear under dominance Not so ifepistatic interactions between dominance effects exist (Crow & Kimura, 1970) ONE-LOCUS MODEL GENOTYPE X A 1 A 1 A 1 A A A FREQUENCY p 1 1 F p 1 F p 1 p 1 F p 1 F p F PHENOTYPE A D A EX Ap p 1 p 1 p D p 1 p DF F 1 F 1 %Heterozygosity ADDITIVE MODEL WITH F (or H) AS COVARIATECONTRADICTORY 9

10 TWO (UNLINKED) LOCI: NO EPISTASIS Joint frequencies are product of marginal frequencies GENOTYPE A 1 A 1 A 1 A A A FREQUENCY p 1 1 F p 1 F p 1 p 1 F p 1 F p F B 1 B 1 r 1 1 F r 1 F A B D A B A B B 1 B r 1 r 1 F A DB D A DB A DB B B r 1 F r F A B D A B A B EX Ap p 1 Br r 1 p 1 p D A r 1 r DB p 1 p D A r 1 r D BF F TWO (UNLINKED) LOCI: EPISTASIS GENOTYPE A 1 A 1 A 1 A A A FREQUENCY p 1 1 F p 1 F p 1 p 1 F p 1 F p F B 1 B 1 r 1 1 F r 1 F A B I D A B L A B I B 1 B r 1 r 1 F A D B K D A D B J A D B K B B r 1 F r F A B I D A B L A B I ALLELES AT A and B LOCI SAME SUBSCRIPTADD I (ADDITIVE X ADDITIVE) HOMOZYGOUS AT A HETEROZYGOUS AT BSUBSTRACT AND ADD K HOMOZYGOUS AT B HETEROZYGOUS AT ASUBSTRACT AND ADD L (ADDITIVE X DOMINANCE) HETEROZYGOUS AT A AND BADD J (DOMINANCE X DOMINANCE) I,J,K,L: parameters (4 d. freedom) 10

11 Mean value under dominance x dominance epistasis EX Ap p 1 Br r 1 p 1 p D A r 1 r D B Ip 1 p r 1 r Lp 1 p r 1 r Kr 1 r p 1 p 4Jp 1 p r 1 r p 1 p D A r 1 r D B Lp 1 p r 1 r Kr 1 r p 1 p 4Jp 1 p r 1 r F 4Jp 1 p r 1 r F F F Dominance, additive x dominance, and dominance x dominance intervene in linear regression Epistasis without dominance does not enter into mean-f relationship Dominance x dominance intervenes in second-order regression DATA First lactation records (herds) on 59,778 (1,14) Jersey cows 6 generations of known pedigree First calving between 1995 and

12 Distribution of F F calculated from all known pedigree information F ranged between 0 and 34% Median F = 6.5% Histogram of F values F(%) 1

13 Procedures Fit linear models without F as covariate Compute EBLUP residuals from these models Fit nonparametric regression to EBLUP residuals in order to obtain nonparametric lines describing relationship between performance and inbreeding level Linear Models Model yijk HYSi AGEj 1 ( Dijk D) ak eijk y ijk = somatic cell score (SCS), milk, protein, or fat yield; HYS i = fixed effect of herd-year-season (i = 1,,.,176 for DS; for DS4 or 6406 for DS6, with seasons classes January April, May August, September December); AGE j = fixed effect of age at calving class; j = 1,,.,6 (< 617, , , , , or >1016 days of age); = fixed regression coefficient of performance on days in milk; 1 D ijk = days in milk for animal k in herd-year-season i and age of calving class j; D = 63; a k e ijk = random additive genetic effect of animal k, and = random residual. 13

14 Linear Model Assumptions Genetic and residual effects assumed mutually independent, with e ~ N( 0, I e ) and a ~ N( 0, Aσ a ) where A is the additive relationship matrix (1 + F k in the k th diagonal position, F k is the inbreeding coefficient of animal k) Nonparametric regression Fit LOESS regression to BLUP residuals with F as covariate Vary spanning parameter & degree of local polynomial Plot fitted values of residuals against F 14

15 ~ ij LOESS (Fitting done by locally weighted least squares) is LOESS fit using only residuals in the neighborhood of F i, i=1,, n (i=1,,,n animals; j=1,,4 traits) Size of neighborhood determined by f q = number of points in neighborhood n = total number of points q n Robust LOESS Weights assigned to : ˆijk I) II) w w [ t 1] [ t ] [ t ] ijk ijk ijk t=1,,3,4 [1] Fk Fi 3 3 w ijk [1 ( ) ] l 1,,... q max( Fl Fi ) ~ ijk ˆ [ t] ijk ijk [1 ( ) ] 6 med ~ med median of all ( ˆ ) ijk ijk 15

16 Cows withat least6 generations of known pedigree f 1 residual σ a nd degree local polynomial F(%) Robust original (black) with bootstrap (light blue) LOESS curves of yields for US Jerseys with at least 6 generations of known pedigree, based on medians of EBLUP residuals (y-axis = eˆ / ˆ ) ijk a f=0.9 f=0.5 f=0.9 nd degree local polynomial 16

17 Conclusions LOESS analysis suggested local relationships. Effects of inbreeding seem nil, until for F values up to ~7% Effects of inbreeding not accounted well by additive models Results may be confounded by effects of selection that are unaccounted for Kernel Regression 17

18 y i gx i e i ; i 1,,...,n where: y i is the measurement taken on individual i x i is a p 1 vector of observed SNP genotypes g. is some unknown function relating genotypes to phenotypes. Set gx i Ey i x i conditional expectation function e i 0, is a random residual 18

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