R help - Mar 2008 - ask for help on nonlinear fitting

I have a table like the following. I want to fit Cm to Vm like this:
Cm ~
Cl+Q1*b1*38.67*exp(-b1*(Vm-Vp1)*0.03867)/(1+exp(-b1*(Vm-Vp1)*0.03867))^2+Q2*b2*38.67*exp(-b2*(Vm-Vp2)*0.03867)/(1+exp(-b2*(Vm-Vp2)*0.03867))^2

I use nls, with start=list(Q1=2e-3, b1=1, Vp1=-25, Q2=3e-3, b2=1,
Vp2=200). But I always get 'singlular gradient' error like this. But
in SigmaPlot I can get the result. How can I get with R.

Thanks!

The table:

"Vm" "Cm" "Ih"
-147.715 8.15 -0.107
-146.944 8.081 -0.106
-146.173 8.089 -0.106
-145.409 8.114 -0.108
-144.638 8.105 -0.107
-143.873 8.085 -0.105
-143.102 8.151 -0.102
-142.338 8.084 -0.1
-141.567 8.079 -0.101
-140.796 8.077 -0.101
-140.032 8.077 -0.102
-139.261 8.126 -0.101
-138.497 8.153 -0.099
-137.726 8.129 -0.097
-136.962 8.073 -0.096
-136.191 8.088 -0.097
-135.426 8.119 -0.098
-134.655 8.121 -0.098
-133.885 8.062 -0.096
-133.12 8.135 -0.093
-132.349 8.089 -0.091
-131.585 8.065 -0.091
-130.814 8.064 -0.092
-130.05 8.071 -0.094
-129.279 8.139 -0.092
-128.515 8.15 -0.09
-127.744 8.119 -0.087
-126.973 8.074 -0.086
-126.208 8.083 -0.087
-125.438 8.084 -0.088
-124.673 8.102 -0.089
-123.902 8.083 -0.087
-123.138 8.132 -0.084
-122.367 8.108 -0.082
-121.596 8.072 -0.083
-120.832 8.07 -0.083
-120.061 8.083 -0.085
-119.297 8.134 -0.084
-118.532 8.14 -0.081
-117.762 8.119 -0.078
-116.997 8.078 -0.077
-116.226 8.102 -0.079
-115.455 8.112 -0.08
-114.691 8.104 -0.08
-113.92 8.093 -0.078
-113.156 8.126 -0.075
-112.385 8.109 -0.073
-111.621 8.066 -0.073
-110.85 8.054 -0.074
-110.085 8.077 -0.076
-109.315 8.134 -0.075
-108.544 8.133 -0.073
-107.779 8.131 -0.069
-107.008 8.093 -0.068
-106.244 8.113 -0.069
-105.473 8.094 -0.071
-104.709 8.106 -0.071
-103.938 8.1 -0.069
-103.174 8.135 -0.066
-102.403 8.115 -0.064
-101.632 8.061 -0.065
-100.868 8.05 -0.066
-100.097 8.091 -0.067
-99.332 8.122 -0.066
-98.561 8.137 -0.064
-97.797 8.123 -0.061
-97.026 8.098 -0.061
-96.255 8.117 -0.061
-95.491 8.108 -0.063
-94.72 8.107 -0.063
-93.956 8.108 -0.062
-93.185 8.144 -0.059
-92.421 8.115 -0.057
-91.65 8.078 -0.057
-90.885 8.053 -0.058
-90.114 8.093 -0.06
-89.344 8.134 -0.059
-88.579 8.147 -0.057
-87.815 8.122 -0.054
-87.044 8.08 -0.053
-86.28 8.128 -0.054
-85.509 8.094 -0.055
-84.745 8.114 -0.056
-83.974 8.119 -0.054
-83.203 8.135 -0.051
-82.438 8.122 -0.049
-81.667 8.062 -0.049
-80.903 8.072 -0.05
-80.132 8.094 -0.052
-79.368 8.142 -0.051
-78.597 8.131 -0.049
-77.826 8.122 -0.046
-77.062 8.097 -0.045
-76.291 8.138 -0.046
-75.527 8.094 -0.048
-74.756 8.124 -0.048
-73.991 8.116 -0.046
-73.221 8.137 -0.044
-72.456 8.118 -0.042
-71.685 8.082 -0.042
-70.914 8.051 -0.043
-70.15 8.089 -0.044
-69.379 8.143 -0.044
-68.615 8.14 -0.042
-67.844 8.113 -0.039
-67.08 8.099 -0.038
-66.309 8.139 -0.039
-65.544 8.084 -0.041
-64.774 8.119 -0.041
-64.003 8.144 -0.039
-63.238 8.142 -0.037
-62.467 8.142 -0.035
-61.703 8.076 -0.035
-60.932 8.069 -0.036
-60.168 8.093 -0.037
-59.397 8.133 -0.037
-58.626 8.131 -0.035
-57.862 8.107 -0.032
-57.098 8.106 -0.031
-56.327 8.146 -0.032
-55.562 8.091 -0.033
-54.791 8.12 -0.034
-54.027 8.141 -0.032
-53.256 8.138 -0.03
-52.485 8.143 -0.028
-51.721 8.071 -0.028
-50.95 8.058 -0.029
-50.186 8.097 -0.03
-49.415 8.146 -0.03
-48.651 8.128 -0.028
-47.88 8.125 -0.025
-47.115 8.116 -0.024
-46.344 8.15 -0.025
-45.573 8.092 -0.026
-44.809 8.142 -0.027
-44.038 8.159 -0.026
-43.274 8.164 -0.023
-42.503 8.161 -0.021
-41.739 8.078 -0.021
-40.968 8.079 -0.022
-40.204 8.105 -0.024
-39.433 8.157 -0.023
-38.662 8.15 -0.021
-37.897 8.131 -0.018
-37.127 8.122 -0.017
-36.362 8.149 -0.018
-35.591 8.092 -0.02
-34.827 8.138 -0.02
-34.056 8.16 -0.019
-33.285 8.152 -0.017
-32.521 8.164 -0.014
-31.75 8.092 -0.014
-30.986 8.074 -0.015
-30.215 8.094 -0.017
-29.45 8.148 -0.016
-28.68 8.151 -0.014
-27.915 8.121 -0.012
-27.144 8.125 -0.01
-26.38 8.162 -0.011
-25.609 8.094 -0.012
-24.845 8.132 -0.013
-24.074 8.17 -0.011
-23.31 8.158 -0.01
-22.539 8.153 -0.007
-21.774 8.081 -0.007
-21.004 8.09 -0.008
-20.233 8.103 -0.01
-19.468 8.151 -0.009
-18.697 8.124 -0.007
-17.933 8.132 -0.005
-17.162 8.122 -0.003
-16.398 8.149 -0.004
-15.627 8.081 -0.005
-14.863 8.13 -0.006
-14.092 8.157 -0.004
-13.321 8.175 -0.003
-12.557 8.157 0
-11.786 8.078 0
-11.021 8.085 -0.001
-10.25 8.106 -0.002
-9.486 8.145 -0.002
-8.715 8.13 0
-7.944 8.136 0.003
-7.18 8.108 0.005
-6.409 8.132 0.004
-5.645 8.099 0.002
-4.874 8.134 0.002
-4.11 8.158 0.003
-3.339 8.165 0.005
-2.574 8.162 0.008
-1.803 8.082 0.008
-1.033 8.087 0.007
-0.268 8.096 0.005
0.503 8.137 0.006
1.267 8.121 0.008
2.038 8.122 0.01
2.802 8.12 0.012
3.566 8.124 0.011
4.337 8.091 0.01
5.108 8.125 0.009
5.873 8.147 0.011
6.643 8.172 0.013
7.408 8.146 0.015
8.179 8.081 0.016
8.943 8.086 0.015
9.714 8.114 0.013
10.485 8.131 0.014
11.249 8.103 0.016
12.02 8.127 0.018
12.784 8.108 0.02
13.555 8.107 0.019
14.32 8.087 0.018
15.09 8.106 0.017
15.855 8.142 0.019
16.626 8.164 0.021
17.397 8.138 0.024
18.161 8.068 0.024
18.932 8.075 0.023
19.696 8.088 0.022
20.467 8.126 0.022
21.231 8.103 0.025
22.002 8.127 0.027
22.766 8.091 0.029
23.537 8.1 0.028
24.308 8.082 0.027
25.073 8.106 0.026
25.844 8.159 0.028
26.608 8.166 0.03
27.379 8.138 0.033
28.143 8.07 0.033
28.914 8.092 0.033
29.685 8.101 0.031
30.449 8.104 0.031
31.22 8.08 0.033
31.984 8.125 0.036
32.755 8.101 0.038
33.52 8.099 0.038
34.284 8.085 0.037
35.055 8.095 0.036
35.826 8.155 0.037
36.59 8.169 0.039
37.361 8.128 0.042
38.125 8.078 0.043
38.896 8.085 0.042
39.66 8.102 0.041
40.431 8.115 0.041
41.196 8.092 0.043
41.967 8.139 0.046
42.737 8.098 0.048
43.502 8.084 0.048
44.273 8.09 0.047
45.037 8.095 0.046
45.808 8.147 0.048
46.572 8.169 0.05
47.343 8.132 0.053
48.107 8.084 0.054
48.878 8.087 0.053
49.649 8.11 0.052
50.414 8.116 0.052
51.184 8.085 0.054
51.949 8.152 0.057
52.72 8.1 0.059
53.484 8.081 0.059
54.255 8.09 0.058
55.026 8.085 0.057
55.79 8.152 0.059
56.561 8.173 0.061
57.325 8.128 0.064
58.096 8.08 0.065
58.86 8.1 0.064
59.631 8.116 0.063
60.396 8.108 0.063
61.167 8.093 0.065
61.938 8.135 0.068
62.702 8.092 0.07
63.473 8.078 0.07
64.237 8.09 0.07
65.001 8.079 0.068
65.772 8.154 0.07
66.537 8.18 0.072
67.307 8.135 0.075
68.078 8.067 0.076
68.843 8.093 0.076
69.614 8.102 0.075
70.378 8.116 0.074
71.149 8.077 0.077
71.913 8.133 0.08
72.684 8.103 0.082
73.455 8.073 0.082
74.219 8.078 0.082
74.99 8.086 0.081
75.754 8.134 0.082
76.525 8.163 0.084
77.29 8.125 0.088
78.061 8.081 0.089
78.825 8.089 0.088
79.596 8.115 0.087
80.367 8.11 0.087
81.131 8.087 0.089
81.902 8.158 0.093
82.666 8.106 0.095
83.437 8.069 0.095
84.201 8.074 0.094
84.972 8.088 0.093
85.737 8.139 0.094
86.507 8.161 0.097
87.278 8.127 0.1
88.043 8.087 0.101
88.814 8.097 0.1
89.578 8.11 0.1
90.349 8.131 0.1
91.113 8.092 0.102
91.884 8.141 0.105
92.648 8.121 0.108
93.419 8.068 0.108
94.19 8.074 0.107
94.954 8.071 0.106
95.719 8.134 0.108
96.49 8.131 0.11
97.254 8.127 0.113
98.025 8.092 0.115
98.796 8.111 0.114
99.56 8.108 0.113
100.331 8.111 0.113
101.095 8.098 0.115
101.866 8.148 0.118
102.631 8.104 0.121
103.401 8.074 0.121
104.166 8.074 0.121
104.937 8.079 0.12
105.708 8.131 0.121
106.472 8.16 0.123
107.243 8.124 0.127
108.007 8.093 0.128
108.778 8.105 0.127
109.542 8.103 0.127
110.313 8.131 0.127
111.077 8.096 0.129
111.848 8.138 0.132
112.619 8.123 0.134
113.384 8.077 0.134
114.155 8.056 0.134
114.919 8.093 0.133
115.69 8.131 0.134
116.454 8.13 0.137
117.225 8.128 0.141
117.996 8.098 0.142
118.76 8.119 0.141
119.531 8.086 0.141
120.295 8.124 0.14
121.066 8.117 0.143
121.831 8.143 0.146
122.602 8.118 0.148
123.366 8.086 0.149
124.137 8.045 0.149
124.908 8.08 0.148
125.672 8.131 0.149
126.436 8.144 0.151
127.207 8.122 0.155
127.971 8.102 0.156
128.742 8.13 0.155
129.507 8.094 0.155
130.278 8.121 0.154
131.048 8.125 0.157
131.813 8.125 0.16
132.584 8.118 0.162
133.348 8.068 0.163
134.119 8.051 0.162
134.883 8.073 0.162
135.654 8.129 0.163
136.418 8.141 0.165
137.189 8.126 0.169
137.96 8.088 0.17
138.725 8.123 0.17
139.495 8.094 0.169
140.26 8.125 0.169
141.031 8.138 0.171
141.795 8.135 0.174
142.566 8.12 0.177
143.337 8.06 0.177
144.101 8.063 0.177
144.872 8.092 0.176
145.636 8.122 0.177
146.407 8.13 0.18
147.171 8.121 0.183
147.942 8.122 0.185
148.707 8.13 0.184

-- 
Ruqiang Liang, Ph. D.
Department of Surgery-Otolaryngology,
University of Kentucky Chandler Medical Center,
800 Rose Street, Lexington, KY 40536
Tel: (859)-323-1110

Ruqiang Liang <ruqiang.liang <at> gmail.com> writes:
> 
> I have a table like the following. I want to fit Cm to Vm like this:
> Cm
~Cl+Q1*b1*38.67*exp(-b1*(Vm-Vp1)*0.03867)/(1+exp(-b1*(Vm-Vp1)*0.03867))^2+
Q2*b2*38.67*exp(-b2*(Vm-Vp2)*0.03867)/(1+exp(-b2*(Vm-Vp2)*0.03867))^2
> 
> I use nls, with start=list(Q1=2e-3, b1=1, Vp1=-25, Q2=3e-3, b2=1,
> Vp2=200). But I always get 'singular gradient' error like this. But
> in SigmaPlot I can get the result. How can I get with R.
I remember a similar case in gastric emptying fitting where my colleague tried
to show me that SigmaPlot is superior to R/nls. When he looked at the standard
deviations of the coefficients, giving estimated gastric empty times of 40
minutes plus minus 800 minutes, I was reminded of Douglas Bates' philosophy:
better nothing than wrong. (Even if I would love to have estimated p-values back
in lmer fits.)

This said, there are two solutions: first, simplify your expression to see what
you are doing. The first part of the expression can be written as

R1*exp(-c1*(Vm-Vp1))/(1+exp(-c1*(Vm-Vp1))^2

where you are free to compute Q1 and b1 later. You will also note that your fit
cannot have a unique solution because of the symmetry in the second term.

As a next step, try to fit 

R1*exp(-exp(d1)*(Vm-Vp1))/(1+exp(-exp(d1)*(Vm-Vp1))^2

This gives a working solution in most cases, effectively forcing a positive
c1=exp(d1). 

However, even that may fail in degenerate cases. If it is not a single fit, but
from a planned pharmacology related experiment, you should try to fit the whole
experiment with all repeats instead of single curves. This often gives excellent
results even when some fits are disastrous. Check package nlme, the book by
Pinheiro/Bates, and some related examples on http://www.menne-biomed.de/gastempt

Dieter