非线性建模起始值

Question

I have a huge time series datasets involving many loops. But, the model couldn't be converged. My model is given as follows..

Model <- nls(Y~b+m*X^D+Z, start = list(b=0.5,m=0.5, D=1),data=logar)

I just want a clear cut determination of starting value of nonlinear modelling!

Data(logar)

X         Y                  Z
135      -0.171292376      85  
91      0.273954718        54   
171     -0.288513438       107
88       -0.17363066       54
59     -1.770852012        50
1        0                 37
1       0                  32
1       0.301029996        36
2       -0.301029996       39
1       1.041392685        30
11      -0.087150176       42
9        0.577236408       20
34       -0.355387658      28
15        0.329058719      17
32        -0.182930683     24
21        0.196294645      21
33        0.114954516      91
43       -0.042403849      111
39       -0.290034611       88
20       -0.522878746       76
6        -0.301029995       108
3         0.477121254       78
9          0                63
9          0.492915522      51
28       -0.243038048       88
16        -0.028028724      17
15      -0.875061263        29
2       -0.301029996        44
1        0                  52
1        1.531478917        65

Answer 1

Here is a rough guide based on what I would do.

1. Choosing starting values for parameters m and b .

   You can narrow down starting parameters by plugging in values X, Y, Z from a few observations into your model; for example, take the measurement with Y=0 , X=1 and Z=37 , leading to 0 = b + m + 37 , or b = -37 - m .

2. Choosing starting value for parameter D .

   As suggested by @jogo, plotting (Y - Z) as a function of X will give you a good idea about a sensible starting parameter for D . In this case, a linear dependence D = 1 seems to be a good initial guess.

    plot((Y - Z) ~ X, data = logar) 

   

If we now modify the starting parameters accordingly, nlm reaches convergence

fit <- nls(Y ~ b + m * X ^ D + Z, start = list(b = -37 - 0.5, m = 0.5, D = 1), data = logar)
fit
#Nonlinear regression model
#  model: Y ~ b + m * X^D + Z
#   data: df
#       b        m        D
#-46.1134  -0.2118   1.0704
# residual sum-of-squares: 19846
#
#Number of iterations to convergence: 7
#Achieved convergence tolerance: 3.761e-06

Answer 2

We can write the model as:

Y - Z ~ b + m * X^D

With algorithm="plinear" only the nonlinear parameters, in this case only D , need starting values and the starting value may not be particularly critical. (For example, at least in this case if we use a starting value of -1 instead of 1 then a few more iterations are needed but we get the same answer.) Note that with the plinear algorithm the right hand side should be specified as a matrix whose columns are implicitly multiplied by the linear parameters.

o <- order(logar$X)
fit.nls <- nls(Y - Z ~ cbind(1, X^D), logar[o, ], start = list(D = 1),
  algorithm = "plinear")

which converges to the following in 6 iterations:

> fit.nls
Nonlinear regression model
  model: Y - Z ~ cbind(1, X^D)
   data: logar[o, ]
       D    .lin1    .lin2 
  1.0703 -46.1135  -0.2118 
 residual sum-of-squares: 19846

Number of iterations to convergence: 6 
Achieved convergence tolerance: 8.814e-06

Given that D is so close to 1 you may wish to simplify the model to just:

fit.lm <- lm(Y - Z ~ X, logar)

Visually these model fits seem indistinguishable:

plot(Y-Z ~ X, logar)
lines(fitted(fit.nls) ~ X, logar[o, ], col = "red", lty = 2)
abline(fit.lm, col = "blue", lty = 3)

Note

The input in reproducible form is:

logar <- structure(list(X = c(135L, 91L, 171L, 88L, 59L, 1L, 1L, 1L, 2L, 
1L, 11L, 9L, 34L, 15L, 32L, 21L, 33L, 43L, 39L, 20L, 6L, 3L, 
9L, 9L, 28L, 16L, 15L, 2L, 1L, 1L), Y = c(-0.171292376, 0.273954718, 
-0.288513438, -0.17363066, -1.770852012, 0, 0, 0.301029996, -0.301029996, 
1.041392685, -0.087150176, 0.577236408, -0.355387658, 0.329058719, 
-0.182930683, 0.196294645, 0.114954516, -0.042403849, -0.290034611, 
-0.522878746, -0.301029995, 0.477121254, 0, 0.492915522, -0.243038048, 
-0.028028724, -0.875061263, -0.301029996, 0, 1.531478917), Z = c(85L, 
54L, 107L, 54L, 50L, 37L, 32L, 36L, 39L, 30L, 42L, 20L, 28L, 
17L, 24L, 21L, 91L, 111L, 88L, 76L, 108L, 78L, 63L, 51L, 88L, 
17L, 29L, 44L, 52L, 65L)), class = "data.frame", row.names = c(NA, 
-30L))