[edit]
x <- c(32,64,96,118,126,144,152.5,158) y <- c(99.5,104.8,108.5,100,86,64,35.3,15) plot(x,y,pch=19) xx <- seq(30,160, length=50) fit <- lm(y~poly(x,4,raw=TRUE)) summary(fit) lines(xx, predict(fit, data.frame(x=xx)), col="purple")
> summary(fit)
Call:
lm(formula = y ~ poly(x, 4, raw = TRUE))
Residuals:
1 2 3 4 5 6 7 8
0.1242 -0.6912 1.6355 1.4491 -5.1240 4.0360 -0.4692 -0.9604
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.474e+01 5.473e+01 1.366 0.265
poly(x, 4, raw = TRUE)1 1.426e+00 3.095e+00 0.461 0.676
poly(x, 4, raw = TRUE)2 -2.854e-02 5.729e-02 -0.498 0.653
poly(x, 4, raw = TRUE)3 2.878e-04 4.278e-04 0.673 0.549
poly(x, 4, raw = TRUE)4 -1.134e-06 1.113e-06 -1.018 0.384
Residual standard error: 4.04 on 3 degrees of freedom
Multiple R-squared: 0.9943, Adjusted R-squared: 0.9868
F-statistic: 131.5 on 4 and 3 DF, p-value: 0.001064
#data
val <- c(462604.2114,724545.9975,677961.1773,556621.1925,572068.4823,550408.0245,442748.9577,463050.285,485241.4509,406399.9335,394768.1661,474606.3792,408121.4988,254290.8273,234755.1933,201690.9834,
175243.2,171106.0665,228024.2613,199481.5251,141505.8969,148199.988,138756.7692,140078.0631,146512.2765,183688.7274,169084.7955,103705.1421,107350.3998,106554.8355,99301.161,105367.9611,148672.9455,
135343.5096,73087.3671,75604.4967,69973.8132,66367.3878,72191.2371,98143.1619,88506.7773,59722.086,59705.1591,54111.3165,49504.2126,50762.7774,68940.2766,67763.3592,50401.3383,50509.8696,51749.5161,
56774.814,56875.3797,65898.4131,58036.3659,45687.6945,44862.2592,41548.5696,37896.342,39405.8232,54220.8435,50035.9164,36221.5746,35067.5583,33099.0594,41550.561,33312.1392,42200.7531,36982.2894,26394.0156,
25939.9764,24163.6476,23641.9008,25495.8942,35859.1398,31292.8596,20783.2461,21233.3025,20770.302,20744.4138,19822.3956,31262.9886,28102.6368,17650.7739,17666.7051,17097.1647,16079.5593,17644.7997,26329.2951,
22879.1946,14800.0848,15000.2205,15227.2401,19307.6187,20345.1381,27219.4509,22791.573,14828.9601,26459.7318,65060.0337)
seq <- 1:length(val)
tmp <- data.frame(seq, val)
#log-normal function
fun <- function(x, mu, sd){
a <- 1/(x * sqrt(2*pi*sd))
b <- (log(x)-mu)^2 * -1
c <- 2 * sd^2
return (a * exp(b/c))
}
#fitting
library("minpack.lm")
fit <- nlsLM(val ~ a * fun(seq, b, c) + d,
start = list(a=1, b=0.01, c=100, d=1),
control = nls.lm.control(maxiter = 1000),
trace = TRUE,
data=tmp)
#plotting
r <- sort(unique(tmp$seq))
xlim <- c(1,max(r))
ylim <- c(0, max(tmp$val) + max(tmp$val) * 0.3)
plot(tmp$seq, tmp$val, cex=0.1, pch=19, xlim=xlim, ylim=ylim)
lines(r, predict(fit, data.frame(seq=r)), col="red")
tmp <- textConnection( "lv ex 2 10 3 11 4 12 5 13 6 15 7 17 8 19 9 21 10 24 11 26 12 29 13 33 14 37 15 41 16 46 17 51 18 57 19 64 20 71 21 79 22 88 23 99 24 110 25 123 26 137 27 153 28 170 29 190 30 212 31 237 32 264 33 294 34 328 35 366 36 409 37 456 38 508 39 567 40 632 41 705 42 787 43 878 44 979 45 1092 46 1218 47 1358 48 1515 49 1689 50 1884") x <- read.table(tmp, header=TRUE) close.connection(tmp) head(x) xx <- x$lv plot(x$lv,x$ex,pch=19) cc <- coef(lm(log(ex) ~ lv, data=x)) fit <- nls(ex~ exp(a + b * lv), data=x, start = list(a = cc[1], b = cc[2])) lines(xx, predict(fit, data.frame(lv=xx)), col="red") summary(fit)
y <- c(0.0941,0.2372,0.2923,0.1750,0.0863,0.0419,0.0207,0.0128,0.0142,0.0071,0.0041,0.0031,0.0032,0.0057,0.0022,0.0001) x <- 1:length(y) library(MASS) est <- fitdistr(y,"gamma")$estimate # scale=0.4495207, shape=7.1923309 fit <- nls(y ~ dgamma(x, scale=a, shape=b), start=list(a=est[1], b=est[2]), trace=T) plot(x,y) lines(x, predict(fit, data.frame(x=x)), col="red")
x <- 1:10 y <- c(2,4,6,8,7,12,14,16,18,20) lo <- loess(y~x) plot(x,y) lines(predict(lo), col='red', lwd=2)
#install.packages("distr")
library(distr)
X <- rnorm(1000,1,0.5)
Y <- rlnorm(1000,1.5,0.75)
Z <- X + Y
N <- Norm(mean=1, sd=0.5) # N is signature for normal dist
L <- Lnorm(meanlog=1.5,sdlog=0.75) # same for log-normal
conv <- convpow(L+N,1) # object of class AbscontDistribution
f.Z <- d(conv) # distribution function
hist(Z,freq=F,breaks=50, xlim=c(0,30))
z <- seq(0,50,0.01)
lines(z,f.Z(z),lty=2,col="red")
#install.packages("propagate")
library(propagate)
y <- c(0.0941,0.2372,0.2923,0.1750,0.0863,0.0419,0.0207,0.0128,0.0142,0.0071,0.0041,0.0031,0.0032,0.0057,0.0022,0.0001)
fit.result <- fitDistr(y)
fit.result
> fit.result <- fitDistr(y)
Fitting Normal distribution...Done.
Fitting Skewed-normal distribution...Done.
Fitting Generalized normal distribution...........10.........20.......Done.
Fitting Log-normal distribution...Done.
Fitting Scaled/shifted t- distribution...Done.
Fitting Logistic distribution...Done.
Fitting Uniform distribution...Done.
Fitting Triangular distribution...Done.
Fitting Trapezoidal distribution...Done.
Fitting Curvilinear Trapezoidal distribution...Done.
Fitting Generalized Trapezoidal distribution...Done.
Fitting Gamma distribution...Done.
Fitting Cauchy distribution...Done.
Fitting Laplace distribution...Done.
Fitting Gumbel distribution...Done.
Fitting Johnson SU distribution...........10.........20.........30.........40.........50
.........60.........70.........80.Done.
Fitting Johnson SB distribution...........10.........20.........30.........40.........50
.........60.........70.........80.Done.
Fitting 3P Weibull distribution...........10.........20.......Done.
Fitting 4P Beta distribution...Done.
Fitting Arcsine distribution...Done.
Fitting von Mises distribution...Done.
> fit.result
$aic
Distribution AIC
17 Johnson SB 613.7082
18 3P Weibull 624.7589
3 Generalized normal 624.7920
16 Johnson SU 626.7920
4 Log-normal 627.1804
12 Gamma 634.9023
14 Laplace 639.8102
13 Cauchy 640.2875
5 Scaled/shifted t- 642.1013
15 Gumbel 642.8420
2 Skewed-normal 646.8412
6 Logistic 647.8476
1 Normal 652.1788
9 Trapezoidal 654.7066
20 Arcsine 670.4624
19 4P Beta 705.6771
11 Generalized Trapezoidal 836.8772
21 von Mises 882.3921
7 Uniform 897.6248
8 Triangular 905.4962
10 Curvilinear Trapezoidal 911.1446
$fit
$fit$Normal
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.000717984960447366, 0.00725890392042258
residual sum-of-squares: 197174
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`Skewed-normal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.00583038510142194, 0.0106443425236555, 8.50327745471171
residual sum-of-squares: 174116
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`Generalized normal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.0818148807363509, 0.0385147298611958, -2.26477841096039
residual sum-of-squares: 119822
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`Log-normal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -4.42473398922409, 2.05820723941306
residual sum-of-squares: 129074
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`Scaled/shifted t-`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.002960556811455, 0.00543082394110837, 0.847275538361585
residual sum-of-squares: 160675
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$Logistic
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00106198716159507, 0.00448320432790849
residual sum-of-squares: 183218
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$Uniform
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00249994429141502, 0.289899967178183
residual sum-of-squares: 12634542
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$Triangular
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.58442884603643, 0.276474920439136, 0.00750016053696837
residual sum-of-squares: 13956564
reason terminated: Relative error between `par' and the solution is at most `ptol'.
$fit$Trapezoidal
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.267697695351802, 0.162169769399358, -0.118546092194162, 0.0169541854752057
residual sum-of-squares: 192315
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`Curvilinear Trapezoidal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -3.51380272100041, 0.781019642275179, 0.778518731596065
residual sum-of-squares: 15358740
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`Generalized Trapezoidal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.316014522465303, -0.0122579620659835, 0.00250001098860818, -0.0569154565231172, 16.9883350504257, -13.687268319418, 1.38700131982915
residual sum-of-squares: 3808834
reason terminated: Relative error between `par' and the solution is at most `ptol'.
$fit$Gamma
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.365709818937871, 25.9401735353146
residual sum-of-squares: 147123
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$Cauchy
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.0027034462361882, 0.00566077284092268
residual sum-of-squares: 161183
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$Laplace
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00208489337704117, 0.0115719008947262
residual sum-of-squares: 159884
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$Gumbel
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.000564496234569831, 0.00585195632924723
residual sum-of-squares: 168315
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`Johnson SU`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00238945145777224, 2.41428652017113e-08, -6.58408257452447, 0.441518979946343
residual sum-of-squares: 119822
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`Johnson SB`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00249961631424682, 0.290585603944236, 0.633747903543896, 0.35472053982219
residual sum-of-squares: 95990
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`3P Weibull`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00206926024634554, 0.60287651598625, 0.0650136696113521
residual sum-of-squares: 119755
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`4P Beta`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 243.96530522719, 244.003863012748, -0.219805373171359, 0.221019706262809
residual sum-of-squares: 456251
reason terminated: Relative error between `par' and the solution is at most `ptol'.
$fit$Arcsine
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00241801362975452, 0.295326975917706
residual sum-of-squares: 268803
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fit$`von Mises`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00235128799831663, 314.160915522474
residual sum-of-squares: 9759620
reason terminated: Number of iterations has reached `maxiter' == 1024.
$bestfit
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00249961631424682, 0.290585603944236, 0.633747903543896, 0.35472053982219
residual sum-of-squares: 95990
reason terminated: Relative error in the sum of squares is at most `ftol'.
$fitted
[1] 62.219673 20.891649 12.605071 9.187138 7.285103 6.065145 5.213740 4.585130 4.101920 3.719054 3.408463
[12] 3.151718 2.936210 2.753014 2.595632 2.459222 2.340100 2.235416 2.142929 2.060858 1.987768 1.922492
[23] 1.864074 1.811724 1.764788 1.722718 1.685056 1.651422 1.621494 1.595006 1.571740 1.551516 1.534193
[34] 1.519663 1.507852 1.498715 1.492237 1.488439 1.487372 1.489126 1.493832 1.501671 1.512883 1.527777
[45] 1.546754 1.570328 1.599166 1.634138 1.676391 1.727475 1.789524 1.865572 1.960094 2.080031 2.236881
[56] 2.451540 2.767639 3.300214 4.355798
$residuals
[1] 0.2803273 4.1083505 12.3949289 -9.1871379 5.2148973 -6.0651453 -5.2137405 -4.5851299 8.3980803 -3.7190542
[11] -3.4084626 -3.1517175 -2.9362096 -2.7530136 -2.5956321 -2.4592222 -2.3401002 10.2645843 10.3570709 -2.0608580
[21] -1.9877677 -1.9224916 -1.8640737 -1.8117243 -1.7647878 -1.7227176 -1.6850565 -1.6514218 -1.6214937 -1.5950062
[31] -1.5717396 -1.5515155 -1.5341925 -1.5196632 10.9921481 -1.4987145 -1.4922374 -1.4884391 -1.4873721 -1.4891258
[41] -1.4938321 -1.5016714 -1.5128828 -1.5277768 -1.5467535 -1.5703281 -1.5991664 10.8658625 -1.6763910 -1.7274750
[51] -1.7895244 -1.8655720 -1.9600941 -2.0800311 -2.2368812 -2.4515397 -2.7676387 -3.3002136 8.1442017
