Gauss2: Generated data
In NISTnls: Nonlinear least squares examples from NIST
Description
Format
Details
Source
Examples
Description
The Gauss2 data frame has 250 rows and 2 columns giving
Format
This data frame contains the following columns:
- y
-
A numeric vector of generated response values.
- x
-
A numeric vector of generated input values.
Details
The data are two slightly-blended Gaussians on a
decaying exponential baseline plus normally
distributed zero-mean noise with variance = 6.25.
Source
Rust, B., NIST (1996)
Examples
1
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Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val
plot(y ~ x, data = Gauss2)
Try(fm1 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
+ b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE,
start = c(b1 = 96, b2 = 0.009, b3 = 103, b4 = 106, b5 = 18,
b6 = 72, b7 = 151, b8 = 18)))
Try(fm1a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
+ b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE,
start = c(b1 = 96, b2 = 0.009, b3 = 103, b4 = 106, b5 = 18,
b6 = 72, b7 = 151, b8 = 18), alg = "port"))
Try(fm2 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
+ b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE,
start = c(b1 = 98, b2 = 0.0105, b3 = 103, b4 = 105, b5 = 20,
b6 = 73, b7 = 150, b8 = 20)))
Try(fm2a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
+ b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE,
start = c(b1 = 98, b2 = 0.0105, b3 = 103, b4 = 105, b5 = 20,
b6 = 73, b7 = 150, b8 = 20), alg = "port"))
Try(fm3 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)),
data = Gauss2, trace = TRUE,
start = c(b2 = 0.009, b4 = 106, b5 = 18, b7 = 151, b8 = 18),
algorithm = "plinear"))
Try(fm4 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)),
data = Gauss2, trace = TRUE,
start = c(b2 = 0.0105, b4 = 105, b5 = 20, b7 = 150, b8 = 20),
algorithm = "plinear"))
Example output
9158.14 : 96.000 0.009 103.000 106.000 18.000 72.000 151.000 18.000
1613.56 : 98.45253280 0.01051603 99.38850229 106.48525660 22.32370980 72.41003489 152.15181890 20.45197551
1248.662 : 98.99126952 0.01097356 101.86556798 107.03941891 23.57426325 72.01257296 153.26301970 19.56177338
1247.528 : 99.01805803 0.01099481 101.88020181 107.03103141 23.57853652 72.04581716 153.27000686 19.52559244
1247.528 : 99.01833206 0.01099495 101.88022788 107.03095328 23.57858242 72.04558537 153.27010170 19.52597962
Nonlinear regression model
model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2)
data: Gauss2
b1 b2 b3 b4 b5 b6 b7 b8
99.01833 0.01099 101.88023 107.03095 23.57858 72.04559 153.27010 19.52598
residual sum-of-squares: 1248
Number of iterations to convergence: 4
Achieved convergence tolerance: 1.941e-06
0: 4579.0698: 96.0000 0.00900000 103.000 106.000 18.0000 72.0000 151.000 18.0000
1: 3885.4263: 96.0460 0.00898644 103.141 106.075 18.2944 72.1118 150.945 18.3009
2: 1328.0042: 96.8155 0.00984281 102.331 106.324 20.4943 72.0499 151.310 20.0025
3: 640.87434: 98.8571 0.0108578 101.193 106.850 23.2101 72.2779 152.977 19.8476
4: 623.76939: 99.0151 0.0109923 101.882 107.031 23.5743 72.0464 153.266 19.5296
5: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5785 72.0456 153.270 19.5260
6: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5786 72.0456 153.270 19.5260
7: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5786 72.0456 153.270 19.5260
Nonlinear regression model
model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2)
data: Gauss2
b1 b2 b3 b4 b5 b6 b7 b8
99.01833 0.01099 101.88023 107.03096 23.57858 72.04559 153.27010 19.52597
residual sum-of-squares: 1248
Algorithm "port", convergence message: both X-convergence and relative convergence (5)
4683.131 : 98.0000 0.0105 103.0000 105.0000 20.0000 73.0000 150.0000 20.0000
1381.816 : 99.06960234 0.01095555 100.60411638 106.47040277 22.97605594 71.93900304 152.51045630 20.55444025
1247.944 : 99.01184313 0.01099244 101.92671701 107.06144403 23.60363847 71.96538483 153.30523084 19.50014812
1247.528 : 99.01836275 0.01099502 101.87990857 107.03121551 23.57917927 72.04543633 153.27053360 19.52563078
1247.528 : 99.01832878 0.01099495 101.88021963 107.03095945 23.57859459 72.04558762 153.27010974 19.52596578
Nonlinear regression model
model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2)
data: Gauss2
b1 b2 b3 b4 b5 b6 b7 b8
99.01833 0.01099 101.88022 107.03096 23.57859 72.04559 153.27011 19.52597
residual sum-of-squares: 1248
Number of iterations to convergence: 4
Achieved convergence tolerance: 3.023e-06
0: 2341.5654: 98.0000 0.0105000 103.000 105.000 20.0000 73.0000 150.000 20.0000
1: 1638.6179: 98.1294 0.0103405 103.082 105.233 20.3072 73.1871 150.192 20.5139
2: 1051.9250: 98.2969 0.0105585 102.705 105.674 21.0771 72.6786 150.907 21.1880
3: 652.33375: 98.5643 0.0108103 101.472 106.680 23.0275 71.7470 152.677 20.0730
4: 623.77872: 99.0146 0.0109922 101.888 107.033 23.5753 72.0218 153.275 19.5309
5: 623.76411: 99.0183 0.0109949 101.880 107.031 23.5786 72.0457 153.270 19.5259
6: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5786 72.0456 153.270 19.5260
7: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5786 72.0456 153.270 19.5260
Nonlinear regression model
model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2)
data: Gauss2
b1 b2 b3 b4 b5 b6 b7 b8
99.01833 0.01099 101.88023 107.03096 23.57858 72.04559 153.27010 19.52597
residual sum-of-squares: 1248
Algorithm "port", convergence message: both X-convergence and relative convergence (5)
8866.014 : 0.00900 106.00000 18.00000 151.00000 18.00000 95.96598 105.75578 74.22943
1530.342 : 0.01049287 106.44248073 22.07937985 152.06773601 20.39981995 97.99785791 101.80779903 71.49880586
1248.21 : 0.01097101 107.00572248 23.51555256 153.24897146 19.57794034 98.96295874 101.87954364 71.99181084
1247.528 : 0.01099465 107.03066676 23.57755988 153.26937113 19.52603325 99.01791651 101.88068859 72.04622122
1247.528 : 0.01099494 107.03094632 23.57856542 153.27008950 19.52599140 99.01833202 101.88023739 72.04558783
1247.528 : 0.01099495 107.03095508 23.57858371 153.27010169 19.52597272 99.01832832 101.88022541 72.04558961
Nonlinear regression model
model: y ~ cbind(exp(-b2 * x), exp(-(x - b4)^2/b5^2), exp(-(x - b7)^2/b8^2))
data: Gauss2
b2 b4 b5 b7 b8 .lin1 .lin2 .lin3
0.01099 107.03096 23.57858 153.27010 19.52597 99.01833 101.88023 72.04559
residual sum-of-squares: 1248
Number of iterations to convergence: 5
Achieved convergence tolerance: 2.795e-07
4284.417 : 0.01050 105.00000 20.00000 150.00000 20.00000 98.93239 104.69502 75.76338
1364.82 : 0.01094259 106.42733492 22.89080469 152.39077575 20.54810931 98.91377303 101.74111247 71.71053368
1247.775 : 0.01099405 107.05255932 23.59552897 153.30814069 19.50747579 99.01575775 101.88267666 72.02700682
1247.528 : 0.01099498 107.03121621 23.57908192 153.27051044 19.52553428 99.01827563 101.87995094 72.04552691
1247.528 : 0.01099495 107.03095858 23.57859303 153.27010901 19.52596820 99.01832968 101.88022108 72.04558683
Nonlinear regression model
model: y ~ cbind(exp(-b2 * x), exp(-(x - b4)^2/b5^2), exp(-(x - b7)^2/b8^2))
data: Gauss2
b2 b4 b5 b7 b8 .lin1 .lin2 .lin3
0.01099 107.03096 23.57859 153.27011 19.52597 99.01833 101.88022 72.04559
residual sum-of-squares: 1248
Number of iterations to convergence: 4
Achieved convergence tolerance: 7.344e-06
NISTnls documentation built on May 2, 2019, 2:37 a.m.
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Description Format Details Source Examples
Description
The Gauss2 data frame has 250 rows and 2 columns giving
Format
This data frame contains the following columns:
- y
-
A numeric vector of generated response values.
- x
-
A numeric vector of generated input values.
Details
The data are two slightly-blended Gaussians on a decaying exponential baseline plus normally distributed zero-mean noise with variance = 6.25.
Source
Rust, B., NIST (1996)
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val
plot(y ~ x, data = Gauss2)
Try(fm1 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
+ b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE,
start = c(b1 = 96, b2 = 0.009, b3 = 103, b4 = 106, b5 = 18,
b6 = 72, b7 = 151, b8 = 18)))
Try(fm1a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
+ b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE,
start = c(b1 = 96, b2 = 0.009, b3 = 103, b4 = 106, b5 = 18,
b6 = 72, b7 = 151, b8 = 18), alg = "port"))
Try(fm2 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
+ b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE,
start = c(b1 = 98, b2 = 0.0105, b3 = 103, b4 = 105, b5 = 20,
b6 = 73, b7 = 150, b8 = 20)))
Try(fm2a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
+ b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE,
start = c(b1 = 98, b2 = 0.0105, b3 = 103, b4 = 105, b5 = 20,
b6 = 73, b7 = 150, b8 = 20), alg = "port"))
Try(fm3 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)),
data = Gauss2, trace = TRUE,
start = c(b2 = 0.009, b4 = 106, b5 = 18, b7 = 151, b8 = 18),
algorithm = "plinear"))
Try(fm4 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)),
data = Gauss2, trace = TRUE,
start = c(b2 = 0.0105, b4 = 105, b5 = 20, b7 = 150, b8 = 20),
algorithm = "plinear"))
|
Example output
9158.14 : 96.000 0.009 103.000 106.000 18.000 72.000 151.000 18.000
1613.56 : 98.45253280 0.01051603 99.38850229 106.48525660 22.32370980 72.41003489 152.15181890 20.45197551
1248.662 : 98.99126952 0.01097356 101.86556798 107.03941891 23.57426325 72.01257296 153.26301970 19.56177338
1247.528 : 99.01805803 0.01099481 101.88020181 107.03103141 23.57853652 72.04581716 153.27000686 19.52559244
1247.528 : 99.01833206 0.01099495 101.88022788 107.03095328 23.57858242 72.04558537 153.27010170 19.52597962
Nonlinear regression model
model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2)
data: Gauss2
b1 b2 b3 b4 b5 b6 b7 b8
99.01833 0.01099 101.88023 107.03095 23.57858 72.04559 153.27010 19.52598
residual sum-of-squares: 1248
Number of iterations to convergence: 4
Achieved convergence tolerance: 1.941e-06
0: 4579.0698: 96.0000 0.00900000 103.000 106.000 18.0000 72.0000 151.000 18.0000
1: 3885.4263: 96.0460 0.00898644 103.141 106.075 18.2944 72.1118 150.945 18.3009
2: 1328.0042: 96.8155 0.00984281 102.331 106.324 20.4943 72.0499 151.310 20.0025
3: 640.87434: 98.8571 0.0108578 101.193 106.850 23.2101 72.2779 152.977 19.8476
4: 623.76939: 99.0151 0.0109923 101.882 107.031 23.5743 72.0464 153.266 19.5296
5: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5785 72.0456 153.270 19.5260
6: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5786 72.0456 153.270 19.5260
7: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5786 72.0456 153.270 19.5260
Nonlinear regression model
model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2)
data: Gauss2
b1 b2 b3 b4 b5 b6 b7 b8
99.01833 0.01099 101.88023 107.03096 23.57858 72.04559 153.27010 19.52597
residual sum-of-squares: 1248
Algorithm "port", convergence message: both X-convergence and relative convergence (5)
4683.131 : 98.0000 0.0105 103.0000 105.0000 20.0000 73.0000 150.0000 20.0000
1381.816 : 99.06960234 0.01095555 100.60411638 106.47040277 22.97605594 71.93900304 152.51045630 20.55444025
1247.944 : 99.01184313 0.01099244 101.92671701 107.06144403 23.60363847 71.96538483 153.30523084 19.50014812
1247.528 : 99.01836275 0.01099502 101.87990857 107.03121551 23.57917927 72.04543633 153.27053360 19.52563078
1247.528 : 99.01832878 0.01099495 101.88021963 107.03095945 23.57859459 72.04558762 153.27010974 19.52596578
Nonlinear regression model
model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2)
data: Gauss2
b1 b2 b3 b4 b5 b6 b7 b8
99.01833 0.01099 101.88022 107.03096 23.57859 72.04559 153.27011 19.52597
residual sum-of-squares: 1248
Number of iterations to convergence: 4
Achieved convergence tolerance: 3.023e-06
0: 2341.5654: 98.0000 0.0105000 103.000 105.000 20.0000 73.0000 150.000 20.0000
1: 1638.6179: 98.1294 0.0103405 103.082 105.233 20.3072 73.1871 150.192 20.5139
2: 1051.9250: 98.2969 0.0105585 102.705 105.674 21.0771 72.6786 150.907 21.1880
3: 652.33375: 98.5643 0.0108103 101.472 106.680 23.0275 71.7470 152.677 20.0730
4: 623.77872: 99.0146 0.0109922 101.888 107.033 23.5753 72.0218 153.275 19.5309
5: 623.76411: 99.0183 0.0109949 101.880 107.031 23.5786 72.0457 153.270 19.5259
6: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5786 72.0456 153.270 19.5260
7: 623.76410: 99.0183 0.0109949 101.880 107.031 23.5786 72.0456 153.270 19.5260
Nonlinear regression model
model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2)
data: Gauss2
b1 b2 b3 b4 b5 b6 b7 b8
99.01833 0.01099 101.88023 107.03096 23.57858 72.04559 153.27010 19.52597
residual sum-of-squares: 1248
Algorithm "port", convergence message: both X-convergence and relative convergence (5)
8866.014 : 0.00900 106.00000 18.00000 151.00000 18.00000 95.96598 105.75578 74.22943
1530.342 : 0.01049287 106.44248073 22.07937985 152.06773601 20.39981995 97.99785791 101.80779903 71.49880586
1248.21 : 0.01097101 107.00572248 23.51555256 153.24897146 19.57794034 98.96295874 101.87954364 71.99181084
1247.528 : 0.01099465 107.03066676 23.57755988 153.26937113 19.52603325 99.01791651 101.88068859 72.04622122
1247.528 : 0.01099494 107.03094632 23.57856542 153.27008950 19.52599140 99.01833202 101.88023739 72.04558783
1247.528 : 0.01099495 107.03095508 23.57858371 153.27010169 19.52597272 99.01832832 101.88022541 72.04558961
Nonlinear regression model
model: y ~ cbind(exp(-b2 * x), exp(-(x - b4)^2/b5^2), exp(-(x - b7)^2/b8^2))
data: Gauss2
b2 b4 b5 b7 b8 .lin1 .lin2 .lin3
0.01099 107.03096 23.57858 153.27010 19.52597 99.01833 101.88023 72.04559
residual sum-of-squares: 1248
Number of iterations to convergence: 5
Achieved convergence tolerance: 2.795e-07
4284.417 : 0.01050 105.00000 20.00000 150.00000 20.00000 98.93239 104.69502 75.76338
1364.82 : 0.01094259 106.42733492 22.89080469 152.39077575 20.54810931 98.91377303 101.74111247 71.71053368
1247.775 : 0.01099405 107.05255932 23.59552897 153.30814069 19.50747579 99.01575775 101.88267666 72.02700682
1247.528 : 0.01099498 107.03121621 23.57908192 153.27051044 19.52553428 99.01827563 101.87995094 72.04552691
1247.528 : 0.01099495 107.03095858 23.57859303 153.27010901 19.52596820 99.01832968 101.88022108 72.04558683
Nonlinear regression model
model: y ~ cbind(exp(-b2 * x), exp(-(x - b4)^2/b5^2), exp(-(x - b7)^2/b8^2))
data: Gauss2
b2 b4 b5 b7 b8 .lin1 .lin2 .lin3
0.01099 107.03096 23.57859 153.27011 19.52597 99.01833 101.88022 72.04559
residual sum-of-squares: 1248
Number of iterations to convergence: 4
Achieved convergence tolerance: 7.344e-06
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