R - Levenberg Marquardt非线性最小二乘拟合Heligman Pollard模型参数

时间：2013-07-28 10:41:02

标签： r nonlinear-functions nonlinear-optimization model-fitting levenberg-marquardt

我试图重现Kostakis的纸张解决方案。在本文中，使用de Heligman-Pollard模型将删节死亡率表扩展为完整的生命表。该模型有8个参数必须安装。作者使用了改进的Gauss-Newton算法;该算法（E04FDF）是NAG计算机程序库的一部分。 Levenberg Marquardt不应该产生相同的参数集吗？我的代码或LM算法的应用有什么问题？

library(minpack.lm)


## Heligman-Pollard is used to expand an abridged table.
## nonlinear least squares algorithm is used to fit the parameters on nqx observed over 5 year   intervals (5qx)
AGE <- c(0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70)
MORTALITY <- c(0.010384069, 0.001469140, 0.001309318, 0.003814265, 0.005378395, 0.005985625,     0.006741766, 0.009325056, 0.014149626, 0.021601755, 0.034271934, 0.053836246, 0.085287751, 0.136549522, 0.215953304)

## The start parameters for de Heligman-Pollard Formula (Converged set a=0.0005893,b=0.0043836,c=0.0828424,d=0.000706,e=9.927863,f=22.197312,g=0.00004948,h=1.10003)
## I modified a random parameter "a" in order to have a start values. The converged set is listed above. 
parStart <- list(a=0.0008893,b=0.0043836,c=0.0828424,d=0.000706,e=9.927863,f=22.197312,g=0.00004948,h=1.10003)

## The Heligman-Pollard Formula (HP8) = qx/px = ...8 parameter equation
HP8 <-function(parS,x)
ifelse(x==0, parS$a^((x+parS$b)^parS$c) + parS$g*parS$h^x, 
             parS$a^((x+parS$b)^parS$c) + parS$d*exp(-parS$e*(log(x/parS$f))^2) +
                 parS$g*parS$h^x)

## Define qx = HP8/(1+HP8)
qxPred <- function(parS,x) HP8(parS,x)/(1+HP8(parS,x))

## Calculate nqx predicted by HP8 model (nqxPred(parStart,x))
nqxPred <- function(parS,x)
(1 -(1-qxPred(parS,x)) * (1-qxPred(parS,x+1)) *
    (1-qxPred(parS,x+2)) * (1-qxPred(parS,x+3)) *
    (1-qxPred(parS,x+4))) 

##Define Residual Function, the relative squared distance is minimized  
ResidFun <- function(parS, Observed,x) (nqxPred(parS,x)/Observed-1)^2

## Applying the nls.lm algo. 
nls.out <- nls.lm(par=parStart, fn = ResidFun, Observed = MORTALITY, x = AGE,
                  control = nls.lm.control(nprint=1,
                                           ftol = .Machine$double.eps,
                                           ptol = .Machine$double.eps,
                                           maxfev=10000, maxiter = 500))

summary(nls.out)


## The author used a modified Gauss-Newton algorithm, this alogorithm (E04FDF) is part of the NAG library of computer programs
## Should not Levenberg Marquardt yield the same set of parameters

2 个答案:

答案 0 :(得分：12)

这里的底线是@Roland是绝对正确的，这是一个非常不适合的问题，你不应该期望获得可靠的答案。我已经

了
  • 以一些小方式清理代码（这只是审美）
  • 将ResidFun更改为返回残差，而不是残差平方。 （前者是正确的，但这并没有太大区别。）
  • 探索了几个不同优化器的结果。它实际上看起来你得到的答案更好比你上面列出的“融合参数”，我假设它是原始研究中的参数（你能提供参考吗？）

加载包：

library(minpack.lm)

数据，作为数据框：

d <- data.frame(
   AGE = seq(0,70,by=5),
   MORTALITY=c(0.010384069, 0.001469140, 0.001309318, 0.003814265,
               0.005378395, 0.005985625, 0.006741766, 0.009325056,
               0.014149626, 0.021601755, 0.034271934, 0.053836246,
               0.085287751, 0.136549522, 0.215953304))

首先查看数据：

library(ggplot2)
(g1 <- ggplot(d,aes(AGE,MORTALITY))+geom_point())
g1+geom_smooth()  ## with loess fit

参数选择：

据推测，这些是原始论文的参数...... 

parConv <- c(a=0.0005893,b=0.0043836,c=0.0828424,
             d=0.000706,e=9.927863,f=22.197312,g=0.00004948,h=1.10003)

扰动参数：

parStart <- parConv
parStart["a"] <- parStart["a"]+3e-4

公式：

HP8 <-function(parS,x)
    with(as.list(parS),
         ifelse(x==0, a^((x+b)^c) + g*h^x, 
                a^((x+b)^c) + d*exp(-e*(log(x/f))^2) + g*h^x))
## Define qx = HP8/(1+HP8)
qxPred <- function(parS,x) {
    h <- HP8(parS,x)
    h/(1+h)
}
## Calculate nqx predicted by HP8 model (nqxPred(parStart,x))
nqxPred <- function(parS,x)
    (1 -(1-qxPred(parS,x)) * (1-qxPred(parS,x+1)) *
     (1-qxPred(parS,x+2)) * (1-qxPred(parS,x+3)) *
     (1-qxPred(parS,x+4))) 
##Define Residual Function, the relative squared distance is minimized  
ResidFun <- function(parS, Observed,x) (nqxPred(parS,x)/Observed-1)

n.b。这与OP的版本略有不同; nls.lm想要残差，而不是残差平方。

与其他优化器一起使用的平方和函数：

ssqfun <- function(parS, Observed, x) {
   sum(ResidFun(parS, Observed, x)^2)
}

应用nls.lm。 （不确定为什么ftol和ptol被降低了 从sqrt(.Machine$double.eps)到.Machine$double.eps - 前者通常是实际对精度的限制...... 

nls.out <- nls.lm(par=parStart, fn = ResidFun,
                  Observed = d$MORTALITY, x = d$AGE,
                  control = nls.lm.control(nprint=0,
                                           ftol = .Machine$double.eps,
                                           ptol = .Machine$double.eps,
                                           maxfev=10000, maxiter = 1000))

parNLS <- coef(nls.out)

pred0 <- nqxPred(as.list(parConv),d$AGE)
pred1 <- nqxPred(as.list(parNLS),d$AGE)

dPred <- with(d,rbind(data.frame(AGE,MORTALITY=pred0,w="conv"),
               data.frame(AGE,MORTALITY=pred1,w="nls")))

g1 + geom_line(data=dPred,aes(colour=w))

线条难以区分，但参数有些大 差异：

round(cbind(parNLS,parConv),5)
##     parNLS  parConv
## a  1.00000  0.00059
## b 50.46708  0.00438
## c  3.56799  0.08284
## d  0.00072  0.00071
## e  6.05200  9.92786
## f 21.82347 22.19731
## g  0.00005  0.00005
## h  1.10026  1.10003

d，f，g，h接近，但a，b，c的数量级不同，e的差异为50％。

查看原始方程式，这里发生的是a^((x+b)^c)被设置为常量，因为a接近1：a约为1，{{1} }和b基本上无关。

让我们检查相关性（我们需要一个广义逆，因为 矩阵是如此强烈相关）：

c

这实际上并没有那么有用 - 它确实只是确认了很多 变量是强相关的... 

obj <- nls.out
vcov  <- with(obj,deviance/(length(fvec) - length(par)) * 
              MASS::ginv(hessian))

cmat <- round(cov2cor(vcov),1)
dimnames(cmat) <- list(letters[1:8],letters[1:8])

##      a    b    c    d    e    f    g    h
## a  1.0  0.0  0.0  0.0  0.0  0.0 -0.1  0.0
## b  0.0  1.0 -1.0  1.0 -1.0 -1.0 -0.4 -1.0
## c  0.0 -1.0  1.0 -1.0  1.0  1.0  0.4  1.0
## d  0.0  1.0 -1.0  1.0 -1.0 -1.0 -0.4 -1.0
## e  0.0 -1.0  1.0 -1.0  1.0  1.0  0.4  1.0
## f  0.0 -1.0  1.0 -1.0  1.0  1.0  0.4  1.0
## g -0.1 -0.4  0.4 -0.4  0.4  0.4  1.0  0.4
## h  0.0 -1.0  1.0 -1.0  1.0  1.0  0.4  1.0

这警告了不良的缩放，也发现了各种各样的不同 答案：只有library(optimx) mvec <- c('Nelder-Mead','BFGS','CG','L-BFGS-B', 'nlm','nlminb','spg','ucminf') opt1 <- optimx(par=parStart, fn = ssqfun, Observed = d$MORTALITY, x = d$AGE, itnmax=5000, method=mvec,control=list(kkt=TRUE)) ## control=list(all.methods=TRUE,kkt=TRUE)) ## Boom! ## fvalues method fns grs itns conv KKT1 KKT2 xtimes ## 2 8.988466e+307 BFGS NA NULL NULL 9999 NA NA 0 ## 3 8.988466e+307 CG NA NULL NULL 9999 NA NA 0 ## 4 8.988466e+307 L-BFGS-B NA NULL NULL 9999 NA NA 0 ## 5 8.988466e+307 nlm NA NA NA 9999 NA NA 0 ## 7 0.3400858 spg 1 NA 1 3 NA NA 0.064 ## 8 0.3400858 ucminf 1 1 NULL 0 NA NA 0.032 ## 1 0.06099295 Nelder-Mead 501 NA NULL 1 NA NA 0.252 ## 6 0.009275733 nlminb 200 1204 145 1 NA NA 0.708 声称已收敛，但ucminf获得了答案 更好的答案 - nlminb参数似乎被忽略了...... 

itnmax

完成，但收到错误的收敛警告...... 

opt2 <- nlminb(start=parStart, objective = ssqfun,
         Observed = d$MORTALITY, x = d$AGE,                   
               control= list(eval.max=5000,iter.max=5000))

parNLM <- opt2$par

看起来round(cbind(parNLS,parConv,parNLM),5) ## parNLS parConv parNLM ## a 1.00000 0.00059 1.00000 ## b 50.46708 0.00438 55.37270 ## c 3.56799 0.08284 3.89162 ## d 0.00072 0.00071 0.00072 ## e 6.05200 9.92786 6.04416 ## f 21.82347 22.19731 21.82292 ## g 0.00005 0.00005 0.00005 ## h 1.10026 1.10003 1.10026 sapply(list(parNLS,parConv,parNLM), ssqfun,Observed=d$MORTALITY,x=d$AGE) ## [1] 0.006346250 0.049972367 0.006315034 和nlminb得到了类似的答案，并且实际上比最初声明的参数做了更好（相当多）：

minpack.lm

enter image description here 

pred2 <- nqxPred(as.list(parNLM),d$AGE)

dPred <- with(d,rbind(dPred,
               data.frame(AGE,MORTALITY=pred2,w="nlminb")))

g1 + geom_line(data=dPred,aes(colour=w))
ggsave("cmpplot.png")

enter image description here

其他可以尝试的事情是：

  • 适当的缩放 - 虽然对此的快速测试似乎没有那么多帮助
  • 提供分析梯度
  • 使用AD Model Builder
  • 使用ggplot(data=dPred,aes(x=AGE,y=MORTALITY-d$MORTALITY,colour=w))+ geom_line()+geom_point(aes(shape=w),alpha=0.3) ggsave("residplot.png") 中的slice函数来探索旧参数和新参数是否代表不同的最小值，或者旧参数是否仅仅是错误的收敛...
  • 从bbmle获取KKT（Karsh-Kuhn-Tucker）标准计算器或用于类似检查的相关包

PS：最大的偏差（到目前为止）是最老的年龄组，可能也有小样本。从统计学的角度来看，可能值得做一个由各个点的精度加权的拟合...

答案 1 :(得分：0)

@BenBolker，使用整个数据集（基础qx）值拟合参数。仍然无法重现参数

library(minpack.lm)

library(ggplot2)

library(optimx)

getwd()

d <- data.frame(AGE = seq(0,74), MORTALITY=c(869,58,40,37,36,35,32,28,29,23,24,22,24,28,
                                           33,52,57,77,93,103,103,109,105,114,108,112,119,
                                           125,117,127,125,134,134,131,152,179,173,182,199,
                                           203,232,245,296,315,335,356,405,438,445,535,594,
                                           623,693,749,816,915,994,1128,1172,1294,1473,
                                           1544,1721,1967,2129,2331,2559,2901,3203,3470,
                                           3782,4348,4714,5245,5646))


d$MORTALITY <- d$MORTALITY/100000

ggplot(d,aes(AGE,MORTALITY))+geom_point()  

##Not allowed to post Images

g1 <- ggplot(d,aes(AGE,MORTALITY))+geom_point()

g1+geom_smooth()## with loess fit

报告的参数：

parConv <- c(a=0.0005893,b=0.0043836,c=0.0828424,d=0.000706,e=9.927863,f=22.197312,
             g=0.00004948,h=1.10003)

parStart <- parConv

parStart["a"] <- parStart["a"]+3e-4


## Define qx = HP8/(1+HP8)

HP8 <-function(parS,x)
with(as.list(parS),
ifelse(x==0, a^((x+b)^c) + g*h^x, a^((x+b)^c) + d*exp(-e*(log(x/f))^2) + g*h^x))



qxPred <- function(parS,x) {
  h <- HP8(parS,x)
  h/(1+h)
}



##Define Residual Function, the relative squared distance is minimized,
ResidFun <- function(parS, Observed,x) (qxPred(parS,x)/Observed-1)

ssqfun <- function(parS, Observed, x) {
  sum(ResidFun(parS, Observed, x)^2)
}

nls.out <- nls.lm(par=parStart, fn = ResidFun, Observed = d$MORTALITY, x = d$AGE, 
                  control = nls.lm.control(nprint=1, ftol = sqrt(.Machine$double.eps), 
                  ptol = sqrt(.Machine$double.eps), maxfev=1000, maxiter=1000))


parNLS <- coef(nls.out)

pred0 <- qxPred(as.list(parConv),d$AGE)
pred1 <- qxPred(as.list(parNLS),d$AGE)


#Binds Row wise the dataframes from pred0 and pred1
dPred <- with(d,rbind(data.frame(AGE,MORTALITY=pred0,w="conv"),
      data.frame(AGE,MORTALITY=pred1,w="nls")))


g1 + geom_line(data=dPred,aes(colour=w))

round(cbind(parNLS,parConv),7)

mvec <- c('Nelder-Mead','BFGS','CG','L-BFGS-B','nlm','nlminb','spg','ucminf')
opt1 <- optimx(par=parStart, fn = ssqfun,
    Observed = d$MORTALITY, x = d$AGE,
    itnmax=5000,
    method=mvec, control=list(all.methods=TRUE,kkt=TRUE,)
## control=list(all.methods=TRUE,kkt=TRUE)) ## Boom

get.result(opt1, attribute= c("fvalues","method", "grs", "itns",
           "conv", "KKT1", "KKT2", "xtimes"))

##       method       fvalues  grs itns conv KKT1 KKT2 xtimes
##5         nlm 8.988466e+307   NA   NA 9999   NA   NA      0
##4    L-BFGS-B 8.988466e+307 NULL NULL 9999   NA   NA      0
##2          CG 8.988466e+307 NULL NULL 9999   NA   NA   0.02
##1        BFGS 8.988466e+307 NULL NULL 9999   NA   NA      0
##3 Nelder-Mead     0.5673864   NA NULL    0   NA   NA   0.42
##6      nlminb     0.4127198  546   62    0   NA   NA   0.17


opt2 <- nlminb(start=parStart, objective = ssqfun,
    Observed = d$MORTALITY, x = d$AGE,
    control= list(eval.max=5000,iter.max=5000))

parNLM <- opt2$par

检查参数：

round(cbind(parNLS,parConv,parNLM),5)

##    parNLS  parConv   parNLM
##a  0.00058  0.00059  0.00058
##b  0.00369  0.00438  0.00369
##c  0.08065  0.08284  0.08065
##d  0.00070  0.00071  0.00070
##e  9.30948  9.92786  9.30970
##f 22.30769 22.19731 22.30769
##g  0.00005  0.00005  0.00005
##h  1.10084  1.10003  1.10084

SSE评论：

sapply(list(parNLS,parConv,parNLM),
  ssqfun,Observed=d$MORTALITY,x=d$AGE)  

 ##[1] 0.4127198 0.4169513 0.4127198

无法上传图片，但代码在此处。仍然看来，当使用完整的死亡率数据（未删节或子集）时，文章中找到的参数不是最合适的

##pred2 <- qxPred(as.list(parNLM),d$AGE)

##dPred <- with(d,rbind(dPred,
    data.frame(AGE,MORTALITY=pred2,w="nlminb")))

##g1 + geom_line(data=dPred,aes(colour=w))
ggplot(data=dPred,aes(x=AGE,y=MORTALITY-d$MORTALITY,colour=w))
        + geom_line()+geom_point(aes(shape=w),alpha=0.3)
相关问题
最新问题