中心和比例预测器

Question

我有一个包含5个变量的数据框：批/晶圆/序列号/电压/放大。在该数据帧中，有1020个子集由Serial_number分组。每个子集都有一定数量的测量数据点（放大电压）。

我用

拟合数据

summary(fit2.lme <- lmer(log(log(Amplification)) ~ poly(Voltage, 3) + (poly(Voltage, 1) | Serial_number),
+  data = APD))

产生：

Linear mixed model fit by REML ['lmerMod']
Formula: log(log(Amplification)) ~ poly(Voltage, 3) + (poly(Voltage, 1) |      Serial_number)
   Data: APD

REML criterion at convergence: 35286.1

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-20.7724  -0.2438  -0.1297   0.2434  13.2663 

Random effects:
 Groups        Name             Variance  Std.Dev. Corr 
 Serial_number (Intercept)      1.439e-02  0.1199       
               poly(Voltage, 1) 2.042e+03 45.1908  -0.76
 Residual                       8.701e-02  0.2950       
Number of obs: 76219, groups:  Serial_number, 1020

Fixed effects:
                    Estimate Std. Error t value
(Intercept)        5.944e-02  3.914e-03    15.2
poly(Voltage, 3)1  5.862e+02  1.449e+00   404.5
poly(Voltage, 3)2 -1.714e+02  3.086e-01  -555.4
poly(Voltage, 3)3  4.627e+01  3.067e-01   150.8

Correlation of Fixed Effects:
            (Intr) p(V,3)1 p(V,3)2
ply(Vlt,3)1 -0.713                
ply(Vlt,3)2  0.015 -0.004         
ply(Vlt,3)3  0.004  0.012   0.018

当我在随机效果中添加更高的多项式时，我收到警告：

>  summary(fit3.lme <- lmer(log(log(Amplification)) ~ poly(Voltage, 3) + (poly(Voltage, 2) | Serial_number),
+   data = APD))
Linear mixed model fit by REML ['lmerMod']
Formula: log(log(Amplification)) ~ poly(Voltage, 3) + (poly(Voltage, 2) |      Serial_number)
   Data: APD

REML criterion at convergence: 16285.9

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-20.5042  -0.2393  -0.0697   0.3165  13.9634 

Random effects:
 Groups        Name              Variance  Std.Dev. Corr       
 Serial_number (Intercept)       1.584e-02  0.1259             
               poly(Voltage, 2)1 1.777e+03 42.1536  -0.67      
               poly(Voltage, 2)2 1.579e+03 39.7365   0.87 -0.95
 Residual                        6.679e-02  0.2584             
Number of obs: 76219, groups:  Serial_number, 1020

Fixed effects:
                    Estimate Std. Error t value
(Intercept)        5.858e-02  4.062e-03    14.4
poly(Voltage, 3)1  5.938e+02  1.351e+00   439.5
poly(Voltage, 3)2 -1.744e+02  1.276e+00  -136.7
poly(Voltage, 3)3  5.719e+01  2.842e-01   201.2

Correlation of Fixed Effects:
            (Intr) p(V,3)1 p(V,3)2
ply(Vlt,3)1 -0.641                
ply(Vlt,3)2  0.825 -0.906         
ply(Vlt,3)3 -0.001  0.030  -0.004 
convergence code: 1
Model failed to converge with max|grad| = 2.22294 (tol = 0.002, component 1)
Model is nearly unidentifiable: large eigenvalue ratio
 - Rescale variables?

Warning messages:
1: In optwrap(optimizer, devfun, getStart(start, rho$lower, rho$pp),  :
  convergence code 1 from bobyqa: bobyqa -- maximum number of function evaluations exceeded
2: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  Model failed to converge with max|grad| = 2.22294 (tol = 0.002, component 1)
3: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  Model is nearly unidentifiable: large eigenvalue ratio
 - Rescale variables?

数据如下（如果需要，可以提供完整的数据）。它包括674个变量的77479个可观察量：

'data.frame':   77479 obs. of  6 variables:
 $ Serial_number: num  9.12e+08 9.12e+08 9.12e+08 9.12e+08 9.12e+08 ...
 $ Lot          : int  9 9 9 9 9 9 9 9 9 9 ...
 $ Wafer        : int  912 912 912 912 912 912 912 912 912 912 ...
 $ Amplification: num  1 1 1.01 1.01 1.01 ...
 $ Voltage      : num  25 30 34.9 44.9 49.9 ...

，数据本身如下：

Serial_number    Lot    Wafer    Amplification    Voltage
1    912009913    9    912    1.00252    24.9681
2    912009913    9    912    1.00452    29.9591
3    912009913    9    912    1.00537    34.9494
(...)
73    912009913    9    912    918.112    375.9850
74    912009913    9    912    1083.74    377.9990
75    912009897    9    912    1.00324    19.9895
76    912009897    9    912    1.00449    29.9777
(...)

警告意味着什么？根据anova，fit3.lme模型更好地描述了数据：

> anova(fit3.lme, fit2.lme)
refitting model(s) with ML (instead of REML)
Data: APD
Models:
fit2.lme: log(log(Amplification)) ~ poly(Voltage, 3) + (poly(Voltage, 1) | 
fit2.lme:     Serial_number)
fit3.lme: log(log(Amplification)) ~ poly(Voltage, 3) + (poly(Voltage, 2) | 
fit3.lme:     Serial_number)
         Df   AIC   BIC   logLik deviance Chisq Chi Df Pr(>Chisq)    
fit2.lme  8 35294 35368 -17638.9    35278                            
fit3.lme 11 16264 16366  -8121.1    16242 19036      3  < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Warning message:
In optwrap(optimizer, devfun, x@theta, lower = x@lower, calc.derivs = TRUE,  :
  convergence code 1 from bobyqa: bobyqa -- maximum number of function evaluations exceeded

因此我想使用该模型，但我坚持警告。

更新

中心和比例预测器

ss.CS<- transform(APD, Voltage=scale(Voltage)) 
> fit31.lme<- update(fit3.lme, data=ss.CS) 
Error in poly(dots[[i]], degree, raw = raw, simple = raw) : 
'degree' must be less than number of unique points

另一个变量（不知道哪个有意义）

> ss.CS<- transform(APD, Amplitude=scale(Amplitude))
Error in scale(Amplitude) : object 'Amplitude' not found
> ss.CS<- transform(APD, Amplification=scale(Amplification))
> fit31.lme<- update(fit3.lme, data=ss.CS)
Warning messages:
1: In log(Amplification) : NaNs produced
2: In log(log(Amplification)) : NaNs produced
3: In log(Amplification) : NaNs produced
4: In log(log(Amplification)) : NaNs produced
5: In log(Amplification) : NaNs produced
6: In log(log(Amplification)) : NaNs produced
7: Some predictor variables are on very different scales: consider rescaling

检查奇点

> diag.vals<- getME(fit3.lme, "theta")[getME(fit3.lme, "lower")==0]
> any(diag.vals<- 1e-6)
[1] TRUE
Warning message:
In any(diag.vals <- 1e-06) : coercing argument of type 'double' to logical

使用Richardson外推计算梯度和Hessian

> devfun<- update(fit3.lme, devFunOnly=TRUE)
>  if(isLMM(fit3.lme)){
+     pars<- getME(fit3.lme, "theta")
+     } else {
+             pars<- getME(fit3.lme, c("theta", "fixef"))
+            }
> if (require("numDeriv")) {
+    cat("hess:\n"); print(hess <- hessian(devfun, unlist(pars)))
+    cat("grad:\n"); print(grad <- grad(devfun, unlist(pars)))
+    cat("scaled gradient:\n")
+    print(scgrad <- solve(chol(hess), grad))
+  }
Loading required package: numDeriv
hess:
             [,1]          [,2]          [,3]         [,4]         [,5]         [,6]
[1,] 39137.840764 -56.189442277 -1.348127e+02  3.789427141 25.456612941 -3.806942811
[2,]   -56.189442   0.508077776  6.283795e-01 -0.068882737 -0.056159369  0.003228274
[3,]  -134.812704   0.628379462  1.061584e+00 -0.079620905 -0.152816413  0.007457255
[4,]     3.789427  -0.068882737 -7.962090e-02  0.516054976  0.534346634  0.001513168
[5,]    25.456613  -0.056159369 -1.528164e-01  0.534346634  0.901191745 -0.002344407
[6,]    -3.806943   0.003228274  7.457255e-03  0.001513168 -0.002344407  0.179283416
grad:
[1] -22.9114985   2.2229416  -0.2959238   0.6790044  -0.2343368  -0.4020556
scaled gradient:
[1] -0.1123624  4.4764140 -0.8777938  1.3980054 -0.4223921 -0.9508207
> fit3.lme@optinfo$derivs
$gradient
[1] -22.9118920   2.2229424  -0.2959264   0.6790037  -0.2343360  -0.4020605

$Hessian
             [,1]         [,2]          [,3]        [,4]        [,5]        [,6]
[1,] 39137.915527 -56.20745850 -134.87176514  3.74780273 25.47540283 -3.79016113
[2,]   -56.207458   0.44262695    0.61462402 -0.04736328 -0.06585693  0.02130127
[3,]  -134.871765   0.61462402    1.04296875 -0.10467529 -0.23223877  0.05438232
[4,]     3.747803  -0.04736328   -0.10467529  0.52026367  0.50909424 -0.02130127
[5,]    25.475403  -0.06585693   -0.23223877  0.50909424  0.68481445 -0.02044678
[6,]    -3.790161   0.02130127    0.05438232 -0.02130127 -0.02044678  0.07617188

4。从原始值（或略微扰动的值）重新启动拟合：

> fit3.lme.restart <- update(fit3.lme, start=pars)
> summary(fit3.lme.restart)
Linear mixed model fit by REML ['lmerMod']
Formula: log(log(Amplification)) ~ poly(Voltage, 3) + (poly(Voltage, 2) |      Serial_number)
   Data: APD

REML criterion at convergence: 16250.3

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-20.4868  -0.2404  -0.0697   0.3166  13.9464 

Random effects:
 Groups        Name              Variance  Std.Dev. Corr       
 Serial_number (Intercept)       1.823e-02  0.1350             
               poly(Voltage, 2)1 2.124e+03 46.0903  -0.77      
               poly(Voltage, 2)2 1.937e+03 44.0164   0.90 -0.96
 Residual                        6.668e-02  0.2582             
Number of obs: 76219, groups:  Serial_number, 1020

Fixed effects:
                    Estimate Std. Error t value
(Intercept)          0.05823    0.00434    13.4
poly(Voltage, 3)1  593.83396    1.47201   403.4
poly(Voltage, 3)2 -174.61257    1.40711  -124.1
poly(Voltage, 3)3   57.15901    0.28427   201.1

Correlation of Fixed Effects:
            (Intr) p(V,3)1 p(V,3)2
ply(Vlt,3)1 -0.735                
ply(Vlt,3)2  0.868 -0.927         
ply(Vlt,3)3 -0.001  0.028  -0.003

5。尝试所有可用的优化器

> source(system.file("utils", "allFit.R", package="lme4"))
Loading required package: optimx
Loading required package: dfoptim
> fit3.lme.all <- allFit(fit3.lme)
bobyqa : [OK]
Nelder_Mead : [OK]
nlminbw : [OK]
nmkbw : [OK]
optimx.L-BFGS-B : [OK]
nloptwrap.NLOPT_LN_NELDERMEAD : [OK]
nloptwrap.NLOPT_LN_BOBYQA : [OK]
Warning messages:
1: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  unable to evaluate scaled gradient
2: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  Model failed to converge: degenerate  Hessian with 1 negative eigenvalues
3: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  unable to evaluate scaled gradient
4: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  Model failed to converge: degenerate  Hessian with 1 negative eigenvalues
5: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  unable to evaluate scaled gradient
6: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  Model failed to converge: degenerate  Hessian with 1 negative eigenvalues
7: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  unable to evaluate scaled gradient
8: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  Model failed to converge: degenerate  Hessian with 1 negative eigenvalues

 ss <- summary(fit3.lme.all)

> ss$ fixef               ## extract fixed effects
                              (Intercept) poly(Voltage, 3)1 poly(Voltage, 3)2 poly(Voltage, 3)3
bobyqa                         0.05822789          593.8340         -174.6126          57.15901
Nelder_Mead                    0.05822787          593.8340         -174.6126          57.15902
nlminbw                        0.05822787          593.8340         -174.6126          57.15902
nmkbw                          0.05841966          593.7804         -174.4999          57.17107
optimx.L-BFGS-B                0.05822845          593.8336         -174.6116          57.16183
nloptwrap.NLOPT_LN_NELDERMEAD  0.05823870          593.8330         -174.6076          57.16039
nloptwrap.NLOPT_LN_BOBYQA      0.05823870          593.8330         -174.6076          57.16039

> ss$ llik                ## log-likelihoods
                       bobyqa                   Nelder_Mead                       nlminbw                         nmkbw               optimx.L-BFGS-B 
                    -8125.125                     -8125.125                     -8125.125                     -8129.827                     -8125.204 
nloptwrap.NLOPT_LN_NELDERMEAD     nloptwrap.NLOPT_LN_BOBYQA 
                    -8125.137    

> ss$ sdcor               ## SDs and correlations
                              Serial_number.(Intercept) Serial_number.poly(Voltage, 2)1.(Intercept) Serial_number.poly(Voltage, 2)2.(Intercept)
bobyqa                                        0.1350049                                    46.09013                                    44.01631
Nelder_Mead                                   0.1350064                                    46.09104                                    44.01705
nlminbw                                       0.1350065                                    46.09106                                    44.01707
nmkbw                                         0.1347214                                    46.11336                                    43.81219
optimx.L-BFGS-B                               0.1356576                                    46.32849                                    44.27171
nloptwrap.NLOPT_LN_NELDERMEAD                 0.1347638                                    45.97995                                    43.91054
nloptwrap.NLOPT_LN_BOBYQA                     0.1347638                                    45.97995                                    43.91054
                              Serial_number.poly(Voltage, 2)1 Serial_number.poly(Voltage, 2)2.poly(Voltage, 2)1 Serial_number.poly(Voltage, 2)2
bobyqa                                             -0.7665898                                         0.9042387                      -0.9608662
Nelder_Mead                                        -0.7665981                                         0.9042424                      -0.9608680
nlminbw                                            -0.7665980                                         0.9042425                      -0.9608680
nmkbw                                              -0.7658163                                         0.9076551                      -0.9649999
optimx.L-BFGS-B                                    -0.7713801                                         0.9067725                      -0.9617129
nloptwrap.NLOPT_LN_NELDERMEAD                      -0.7645748                                         0.9034336                      -0.9606020
nloptwrap.NLOPT_LN_BOBYQA                          -0.7645748                                         0.9034336                      -0.9606020
                                  sigma
bobyqa                        0.2582156
Nelder_Mead                   0.2582156
nlminbw                       0.2582156
nmkbw                         0.2584714
optimx.L-BFGS-B               0.2582244
nloptwrap.NLOPT_LN_NELDERMEAD 0.2582207
nloptwrap.NLOPT_LN_BOBYQA     0.2582207

> ss$ theta               ## Cholesky factors
                              Serial_number.(Intercept) Serial_number.poly(Voltage, 2)1.(Intercept) Serial_number.poly(Voltage, 2)2.(Intercept)
bobyqa                                        0.5228377                                   -136.8323                                    154.1396
Nelder_Mead                                   0.5228438                                   -136.8364                                    154.1428
nlminbw                                       0.5228439                                   -136.8365                                    154.1429
nmkbw                                         0.5212237                                   -136.6278                                    153.8521
optimx.L-BFGS-B                               0.5253478                                   -138.3947                                    155.4631
nloptwrap.NLOPT_LN_NELDERMEAD                 0.5218936                                   -136.1436                                    153.6293
nloptwrap.NLOPT_LN_BOBYQA                     0.5218936                                   -136.1436                                    153.6293
                              Serial_number.poly(Voltage, 2)1 Serial_number.poly(Voltage, 2)2.poly(Voltage, 2)1 Serial_number.poly(Voltage, 2)2
bobyqa                                               114.6181                                         -71.06063                    1.578418e+01
Nelder_Mead                                          114.6186                                         -71.06067                    1.578354e+01
nlminbw                                              114.6187                                         -71.06067                    1.578351e+01
nmkbw                                                114.7270                                         -71.14411                    3.440466e-42
optimx.L-BFGS-B                                      114.1731                                         -70.65227                    1.527854e+01
nloptwrap.NLOPT_LN_NELDERMEAD                        114.7688                                         -71.19817                    1.568481e+01
nloptwrap.NLOPT_LN_BOBYQA                            114.7688                                         -71.19817                    1.568481e+01

> ss$ which.OK            ## which fits worked
                       bobyqa                   Nelder_Mead                       nlminbw                         nmkbw               optimx.L-BFGS-B 
                         TRUE                          TRUE                          TRUE                          TRUE                          TRUE 
nloptwrap.NLOPT_LN_NELDERMEAD     nloptwrap.NLOPT_LN_BOBYQA 
                         TRUE                          TRUE

由于用户的注意，我添加了以下内容：

> bam(log(log(Amplification)) ~ s(Voltage) + s(Serial_number, bs="re") + s(Voltage, Serial_number, bs="re"), data=APD, discrete = TRUE)

Family: gaussian 
Link function: identity 

Formula:
log(log(Amplification)) ~ s(Voltage) + s(Serial_number, bs = "re") + 
    s(Voltage, Serial_number, bs = "re")

Estimated degrees of freedom:
  9 993 987  total = 1990.18 

fREML score: -226.8182   


> summary(bam(log(log(Amplification)) ~ s(Voltage) + s(Serial_number, bs="re") + s(Voltage, Serial_number, bs="re"), data=APD, discrete = TRUE))

Family: gaussian 
Link function: identity 

Formula:
log(log(Amplification)) ~ s(Voltage) + s(Serial_number, bs = "re") + 
    s(Voltage, Serial_number, bs = "re")

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.11500    0.01896   6.066 1.31e-09 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                             edf Ref.df     F p-value    
s(Voltage)                 8.998      9 89229  <2e-16 ***
s(Serial_number)         993.441   1019 55241  <2e-16 ***
s(Voltage,Serial_number) 986.741   1019 36278  <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.989   Deviance explained =   99%
fREML = -226.82  Scale est. = 0.051396  n = 76219

在https://uploadfiles.io/n7h9z上，您可以下载r脚本和数据。