|
Home
Introduction
Example
Problems
Screen Shots
Download
Order
|
|
1. General
For nonlinear regression, all statistical software
packages available today, for example, SPSS,
SAS, Statistical,Origin
Pro, DataFit,
Stata and Systat, need
end-users to provide initial guess/start values, and the successful
of regression computation is depended on those start values haveily.
Unfortunately, for general users in most cases, the guess of
the start values are nightmare. They have to try and try, and
even so, the results are not guaranteed to be optimal ones since
the drawbacks of local optimization algorithms adopted (the
most used algorithms are Levenberg-Marquardt (LM), BFGS, Simplex)
With our revolution algorithms
of Global Levenberg-Marquardt (GLM) and Global BFGS, as well
as many other global optimation algorithms, Auto2Fit is now
no long to need end-users to guess/provide start values, but
use freely random values.
2. NIST Test
The nonlinear data set
form NIST
are widely used for the purpose of performance test. Almost
all software developers in statistical/nonlinear areas claimed that
they have passed the test data sets from NIST,
but they have to use one of two start value sets provided already
by NIST.
Auto2Fit, however, use only free random values as initial guess,
but with almost 100% successful, amazing? No any others may achieve so!!!.
The test results are as followings:
Table.1
Results of Nonlinear Test Data Set from NIST
|
No.
|
Name
of Test Set
|
Levels
|
No.
of Parameters
|
Start
Values
|
Algorithms
|
Rate
of Success (%)
|
|
1
|
Misra1a
|
|
2
|
Default
random values between 0 to 5
|
LM + UGO
|
100
|
|
2
|
Chwirut2
|
3
|
100
|
|
3
|
Chwirut1
|
3
|
100
|
|
4
|
Lanczos3
|
6
|
100
|
|
5
|
Gauss1
|
8
|
> 90
|
|
6
|
Gauss2
|
8
|
> 90
|
|
7
|
DanWood
|
2
|
100
|
|
8
|
Misra1b
|
2
|
100
|
|
9
|
Kirby2
|
|
5
|
100
|
|
10
|
Hahn1
|
7
|
100
|
|
11
|
Nelson
|
3
|
100
|
|
12
|
MGH17
|
5
|
100
|
|
13
|
Lanczos1
|
6
|
100
|
|
14
|
Lanczos2
|
6
|
100
|
|
15
|
Gauss3
|
8
|
> 90
|
|
16
|
Misra1c
|
2
|
100
|
|
17
|
Misra1d
|
2
|
100
|
|
18
|
Roszman1
|
4
|
100
|
|
19
|
ENSO
|
9
|
100
|
|
20
|
MGH09
|
|
4
|
100
|
|
21
|
Thurber
|
7
|
100
|
|
22
|
BoxBod
|
2
|
100
|
|
23
|
Rat42
|
3
|
100
|
|
24
|
MGH10
|
3
|
100
|
|
25
|
Eckerle4
|
3
|
100
|
|
26
|
Rat43
|
4
|
100
|
|
27
|
Bennett5
|
3
|
>90
|
*：SM：Simplex Method；MIO：Maximum
Inheirt OPtimization；
3.
Challenge Test Data
The test data sets from
NIST are
too simple for Auto2Fit. Here we provide below some more difficult
test data. There is only one unique
optimal solution of each data set. The results of RMSE and R-Square for
each data set are given for reference.You may try any other
softwares or tools. If
anyone can get correct answers using tools
other than Auto2Fit, please write to here.
Some of those test data are
very hard, and may never get right answers without using Auto2Fit.
Even for Auto2Fit, it does not ensure every run will be
successful. The suggested algorithms for solving those problems
in Auto2Fit are Global Levenberg-Marquard or Global BFGS. In
some cases, you may try to change the control parameter of "Population
Size" from default 30 to 50 or more
|
Test Data
|
Regression Equations
|
Variables
|
Parameters
|
RMSE
|
R^2
|
|
1
|
|
x, y
|
p1 to p5
|
|
0.99678004
|
|
2
|
y
= (p1+p2*x1+p3*x2+p4*x3+p5*x4)/(1+a1*x1+a2*x2+a3*x3+a4*x4)
|
x1 to x4, y
|
p1 to p5, a1 to a4
|
|
0.9346422
|
|
3
|
|
x, y
|
p1 to p3
|
|
0.969929562
|
|
4
|
y = (a0+a1*x1+a2*x2+a3*x3+a4*x4)/(1+b1*x1+b2*x2+b3*x3+b4*x4)
|
x1 to x4, y
|
a0 to a4, b1 to b4
|
|
0.80514286
|
|
5
|
z = p1+p2*x^p3+p4*y^p5+p6*x^p7*y^p8
|
x, y, z
|
p1 to p8
|
0.2703296
|
0.994632848
|
|
6
|
y = a0+a1*x^k1+a2*x^k2+a3*x^k3
|
x, y
|
a0 to a3, k1 to k3
|
0.0214726136
|
0.999644261
|
|
7
|
z = (p1+p2*x+p3*y+p4*x*y)/(1+p5*x+p6*y+p7*x*y)
|
x, y, z
|
p1 to p7
|
1.00626078
|
0.9715471
|
|
8
|
y=p1/((p2+x1)*(1+p3*x2)*(x3-p4)^2)+p5*x3^p6
|
x, y
|
p1 to p6
|
0.01977698
|
0.995372
|
|
9
|
y=a*exp(b*abs(x+c)^d)
|
x, y
|
a, b, c, d
|
1.1546909
|
0.9704752
|
Test
Data 1
|
No x y
|
No x y
|
No x y
|
No x y
|
No x y
|
No x y
|
|
1 160.73 6266.7
2 159.82 6151.9
3 158.84 6035.1
4 157.86 5920.9
5 156.87 5812.6
6 155.88 5702.2
7 154.89 5594.9
8 153.96 5491.3
9 152.97 5385
10 151.98 5282.2
11 150.99 5181.3
12 150.06 5084.8
13 149.08 4988.8
14 148.09 4892.2
15 147.1 4796.9
16 146.17 4701
17 145.18 4608
18 144.2 4515.2
19 143.2 4429.6
20 142.21 4342.9
21 141.25 4255.6
22 140.2 4167.1
23 139.14 4077.6
24 138.05 3987.9
25 136.96 3898.9
26 135.94 3808.5
27 134.84 3717.7
28 133.74 3628.9
29 132.65 3543
30 131.57 3456.3
|
31 130.55 3372.5
32 129.47 3292.8
33 128.4 3212.7
34 127.33 3133.6
35 126.34 3056.6
36 125.29 2985.5
37 124.26 2912.5
38 123.23 2842.9
39 122.21 2774.1
40 121.21 2708.3
41 120.27 2642.1
42 119.27 2580.2
43 118.29 2518.7
44 117.32 2459.1
45 116.42 2401.1
46 115.48 2344.3
47 114.55 2290.9
48 113.62 2237.5
49 112.7 2189
50 111.85 2138.8
51 110.94 2089.4
52 110.03 2042.4
53 109.13 1998.1
54 108.28 1953.6
55 107.38 1906.6
56 106.48 1867.8
57 105.6 1824.5
58 104.72 1784.3
59 103.91 1745
60 103.05 1704.5
|
61 102.2 1668.7
62 101.35 1629
63 100.51 1590.4
64 99.739 1552.5
65 98.913 1514.7
66 98.103 1476.4
67 97.308 1444.3
68 96.513 1411.4
69 95.78 1378.5
70 95.002 1344.8
71 94.239 1307.8
72 93.482 1276.1
73 92.776 1243.5
74 92.039 1212.9
75 91.314 1178.7
76 90.604 1148.4
77 89.942 1115.9
78 89.244 1084.5
79 88.559 1051.5
80 87.889 1029.7
81 87.226 996.16
82 86.569 965.86
83 85.963 937.72
84 85.323 907.87
85 84.694 877.58
86 84.081 838.17
87 83.473 819.48
88 82.876 797.76
89 82.287 768.54
90 81.811 749.96
|
91 81.178 724.39
92 80.614 697.24
93 80.118 674.67
94 79.574 649.49
95 79.011 629.83
96 78.478 614.6
97 78.012 591.87
98 77.494 573.43
99 76.927 558.94
100 76.512 539.45
101 75.962 526.99
102 75.472 514.14
103 75.014 504.11
104 74.566 484.4
105 74.123 473.23
106 73.608 468.93
107 73.183 453.77
108 72.774 448.58
109 72.369 447.73
110 71.897 431.79
111 71.503 432.45
112 71.116 432.15
113 70.741 420.71
114 70.3 427.26
115 69.935 419.76
116 69.572 407.28
117 69.148 408.04
118 68.796 393.71
119 68.448 403.74
120 68.114 408.8
|
121 67.717 401.26
122 67.374 400.81
123 67.037 401.89
124 66.741 408.68
125 66.416 398.49
126 66.015 414.14
127 65.373 419.78
128 64.769 426.82
129 64.109 418.42
130 63.44 446.32
131 62.772 451.55
132 62.111 473.27
133 61.508 499.69
134 60.908 523.66
135 60.219 551.47
136 59.699 593.53
137 59.119 608.69
138 58.547 658.08
139 57.992 712.27
140 57.483 769.4
141 56.969 826.48
142 56.472 896.05
143 55.989 957.57
144 55.513 1065.1
145 55.088 1114.1
146 54.651 1195
147 54.237 1271.5
148 53.836 1355.6
149 53.318 1483.2
150 52.701 1690
|
151 52.08 2245.9
152 51.431 2470.4
153 50.877 2719.1
154 50.298 2957.5
155 49.74 3155.2
156 49.2 3279.4
157 48.702 3546.4
158 48.182 3741
159 47.681 4021
160 47.213 4015.1
161 46.768 4304.7
162 46.368 4127.9
163 45.956 4530.9
164 45.55 4802.9
165 45.157 5047.4
166 44.799 4804.3
167 44.43 5164.1
168 44.078 4781
169 43.727 5175.5
170 43.384 5708.6
171 43.079 5679.6
172 42.899 5161.8
173 42.719 5399.1
174 42.547 5483
175 42.253 4839.4
176 41.962 5360.7
177 41.691 5622
178 40.602 5772.3
|
|
Test
Data 2
|
Test
Data 3
|
Test
Data 4
|
Test Data 5
|
|
No x1 x2
x3
x4 y
|
No x
y
|
No x1
x2 x3
x4 y
|
No.
x y
z
|
|
1 15100 29000 508.0 180 3.40
2 20500 43350 453.7 141 3.00
3 80000 92610 487.9 132 2.70
4 91500 142775 572.3 182 3.37
5 82500 2123160 455.7 113 6.89
6 20000 227800 481.3 170 5.03
7 17800 140000 541.3 179 3.55
8 3900 15980 538.6 186 2.72
9 17300 223200 460.6 100 4.05
10 25700 229400 393.1 133 3.22
11 49400 424500 373.9 106 2.65
12 40700 561700 482.8 107 1.91
13 77000 563600 482.1 140 3.00
14 92900 557600 415.1 121 1.31
15 63300 528300 536.7 144 2.33
16 51600 488940 385.1 154 3.55
17 60000 480500 412.2 111 3.37
18 70000 530500 567.2 139 2.55
|
1 80.0 6.64
2 140.9 11.54
3 204.7 15.89
4 277.9 20.16
5 356.8 21.56
6 453.0 21.69
7 505.6 22.66
8 674.5 23.15
9 802.32 18.16
10 936.04 16.81
|
1 14 1.38 -34 16 582
2 10 0.52 -29 2 458
3 13 1.70 -32 13 559
4 24 0.80 24 1 322
5 12 1.83 41 11 399
6 6 1.77 -50 7 523
7 18 1.23 27 4 322
8 -10 0.28 -8 6 358
9 0 1.20 66 6 354
10 14 1.75 -60 6 574
11 12 1.78 -70 7 489
12 -18 1.37 -15 0 232
13 16 1.38 0 4 440
14 -4 0.29 -9 -7 421
15 -23 1.12 -12 -14 181
16 5 1.52 0 10 426
17 -16 0.63 34 1 364
18 -1 1.32 22 -7 375
19 -18 1.18 4 -11 224
20 8 1.50 -11 5 514
21 -8 1.43 4 -12 381
22 -11 0.74 10 0 275
23 -19 1.07 -5 0 426
|
1 500 25 1.5
2 500 50 2.25
3 500 100 3.15
4 500 200 4.0
5 500 300 4.2
6 500 400 4.3
7 1000 25 1.45
8 1000 50 2.35
9 1000 100 3.95
10 1000 200 6.95
11 1000 300 8.15
12 1000 400 8.4
13 1500 25 1.45
14 1500 50 2.45
15 1500 100 4.15
16 1500 200 7.45
17 1500 300 10.65
18 1500 400 11.85
19 2000 25 1.45
20 2000 50 2.5
21 2000 100 4.2
22 2000 200 7.75
23 2000 300 11.45
24 2000 400 14.3
|
|
Test
Data 6
|
Test
Data 7
|
Test
Data 8
|
Test
Data 9
|
|
No x y
|
No x
y z
|
No x1
x2 x3
y
|
No x y
|
|
1 1.0 8.2
2 2.0 4.6
3 3.0 4.3
4 4.0 4.6
5 5.0 5.1
6 6.0 5.5
7 7.0 5.7
8 8.0 5.5
9 9.0 5.0
10 10.0 3.8
|
1 4332 26.94 43.70
2 4697 23.64 44.50
3 5062 25.19 47.70
4 5428 28.60 52.30
5 5793 28.74 54.21
6 6158 29.33 55.58
7 6523 32.92 55.70
8 6889 31.87 57.70
9 7254 33.07 58.60
10 7619 29.36 60.24
11 7984 27.96 59.13
12 8350 30.18 57.00
13 8715 37.84 57.30
14 9080 38.86 59.00
15 9445 35.18 60.20
16 9811 28.81 61.80
17 10176 34.57 63.17
18 10541 37.49 66.23
19 10906 29.30 61.80
20 11272 32.47 62.03
21 11637 38.24 65.30
|
1 34.9 0.043378 8 0.996556
2 34.9 0.216888 8 0.985708
3 34.9 0.433776 8 0.973826
4 58.2 0.026027 8 0.999409
5 58.2 0.130133 8 0.99817
6 58.2 0.260265 8 1.000176
7 93.1 0.016267 8 0.995131
8 93.1 0.081333 8 1.009887
9 93.1 0.162666 8 1.008251
10 34.9 0.043378 20 0.835576
11 34.9 0.216888 20 0.777734
12 34.9 0.433776 20 0.715483
13 58.2 0.026027 20 0.854949
14 58.2 0.130133 20 0.822743
15 58.2 0.260265 20 0.784273
16 93.1 0.016267 20 0.85902
17 93.1 0.081333 20 0.841512
18 93.1 0.162666 20 0.81895
19 34.9 0.043378 40 0.387322
20 34.9 0.216888 40 0.338941
21 34.9 0.433776 40 0.293558
22 58.2 0.026027 40 0.342388
23 58.2 0.130133 40 0.311761
24 58.2 0.260265 40 0.280112
25 93.1 0.016267 40 0.308071
26 93.1 0.081333 40 0.287257
27 93.1 0.162666 40 0.264443
|
1 27.25 1
2 27.75 3
3 28.25 6
4 28.75 13
5 29.25 18
6 & | |