hyper.fit: Top level function that attempts to fit a hyperplane to...
In asgr/hyper.fit: N-Dimensional Hyperplane Fitting with Errors
hyper.fit R Documentation
Top level function that attempts to fit a hyperplane to provided data.
Description
Top level fitting function that uses downhill searches (optim/LaplaceApproximation) or MCMC (LaplacesDemon) to search out the best fitting parameters for a hyperplane (minimum a 1D line for 2D data), including the intrinsic scatter as part of the fit.
Usage
hyper.fit(X, covarray, vars, parm, parm.coord, parm.beta, parm.scat, parm.errorscale=1,
vert.axis, weights, k.vec, prior, itermax = 1e4, coord.type = 'alpha',
scat.type = 'vert.axis', algo.func = 'optim', algo.method = 'default',
Specs = list(Grid=seq(-0.1,0.1, len=5), dparm=NULL, CPUs=1, Packages=NULL, Dyn.libs=NULL),
doerrorscale = FALSE, ...)
Arguments
X
A position matrix with N (number of data points) rows by d (number of dimensions) columns.
covarray
A dxdxN array containing the full covariance (d=dimensions, N=number of dxd matrices in the array stack). The makecovarray2d and makecovarray3d are convenience functions that make populating 2x2xN and 3x3xN arrays easier for a novice user.
vars
A variance matrix with the N (numver of data points) rows by d (number of dimensions) columns. In effect this is the diagonal elements of covarray where all other terms are zero. If covarray is also provided that input argument is used instead.
parm
Vector of all initial parameters. This should be a concatenation of c(parm.coord, parm.beta, parm.scat). If doerrorscale=TRUE then there should be an extrat element to give c(parm.coord, parm.beta, parm.scat, parm.errorscale). See the parm.coord, parm.beta, parm.scat and parm.errorscale arguments below for more information on what these different quantities describe.
parm.coord
Vector of initial coord parameters. These are either angles that produce the vectors that predict the vert.axis dimension (coord.type="theta"), the gradients of these (coord.type="alpha") or they are the vector elements normal to the hyperplane (coord.type="normvec"). Depending on the argument used, either beta along the vertical axis is generated (coord.type="alpha"/theta") or no beta is required (coord.type="normvec").
parm.beta
Initial value of beta. This is either specified as the absolute distance from the origin to the hyperplane or as the intersection of the hyperplane on the vert.axis dimension being predicted.
parm.scat
Initial value of the intrinsic scatter. This is either specified as the scatter orthogonal to the hyperplane or as the scatter along the vert.axis dimension being predicted.
parm.errorscale
If doerrorscale=TRUE, this argument is the initial errorscale (default is 1). See the doerrorscale argument below for more details.
vert.axis
Which axis should the hyperplane equation be formulated for. This must be a number which specifies the column of position matrix X to be defined by the hyperplane. If missing, then the projection dimension is assumed to be the last column of the X matrix.
weights
Vector of multiplicative weights for each row of the X data matrix. i.e. if this is 2 then it is equivalent to having two identical data points with weights equal to 1. Should be either of length 1 (in which case elements are repeated as required) or the same length as the number of rows in the data matrix X.
k.vec
A vector defining the direction of an exponential sampling distribution in the data. The length is the scaling "a" of the generative exponent (i.e., exp(a*x)), and it points in the direction of *increasing* density (see example below). If provided, k.vec must be the same length as the dimensions of the data. k.vec has the most noticeable effect on the beta offset parameter. It is correcting for the phenomenom Astronomers often call Eddington bias. For discussion on the use of k.vec see Appendix B of Robotham & Obreschkow.
prior
A function that will take in parm and return a logged-likelihood that will be added to the data-model log-likelihood to give the true log-posterior. By default this will add 0, which means it is strictly an improper prior over infinite domain. The user will need to be careful about the ordering and parameter type when passing in parm to the prior function (especially when this is not provided, and the initial parameter guess is made internally). For clarity the prior function provided will be passed the parm vector with the exact ordering and coorindate types as seen in the output parm, so it is probably a good idea to run once without prior set, check the parm output, and build the prior function as appropriate.
itermax
The maximum iterations to use for either the LaplaceApproximation function or LaplacesDemon function.
coord.type
This specifies whether the fit should be done in terms of the normal vector to the hyperplane (coord.type="normvec") gradients defined to produce values along the vert.axis dimension (coord.type="alpha") or by the values of the angles that form the gradients (coord.type="theta"). "alpha" is the default since it is the most common means of parameterising hyperplane fits in the astronomy literature.
scat.type
This specifies whether the intrinsic scatter should be defined orthogonal to the plane (orth) or along the vert.axis of interest (vert.axis).
algo.func
If algo.func="optim" (default) hyper.fit will optimise using the R base optim function. If algo.func="LA" will optimise using the LaplaceApproximation function. If algo.func="LD" will optimise using the LaplacesDemon function. For both algo.func="LA" and algo.func="LD" the LaplacesDemon package will be used (see http://www.bayesian-inference.com/software).
algo.method
Specifies the method argument of optim function when using algo.func="optim" (if not specified hyper.fit will use "Nelder-Mead" for optim). Specifies Method argument of LaplaceApproximation function when using algo.func="LA" (if not specified hyper.fit will use "NM" for LaplaceApproximation). Specifies Algorithm argument of LaplacesDemon function when using algo.func="LD" (if not specified hyper.fit will use "GG" (i.e. Griddy Gibbs) for LaplacesDemon, but consider also using "CHARM"). When using algo.func="LD" the user can also specify further options via the Specs argument below.
Specs
Inputs to pass to the LaplacesDemon function. Default Specs is for the default GG (Griddy Gibbs) algorithm (see algo.method above). If using CHARM, then you probably want to set Specs=list(alpha.star = 0.44).
doerrorscale
If FALSE then the provided covariance are treated as it. If TRUE then the likelihood function is also allowed to rescale all errors by a uniform multiplicative value.
...
Further arguments to be passed to optim, LaplaceApproximation or LaplacesDemon depending on the option used for algo.func.
Details
Setting doerrorscale to TRUE allows for stable solutions when errors are overestimated, e.g. when the intrinsic scatter is equal to zero but the data is more clustered around the optimal likelihood plane than expected from the data covariance array. See Examples below for a 2D scenario where this is helpful.
algo.func="LD" also returns the probability that the generative model has exactly zero intrinsic scatter (zeroscatprob in the output). The other available functions (algo.func="optim" and algo.func="LA") can find exact solutions equal to zero since they are strictly mode finding (i.e. maximum likelihood) routines. algo.func="LD" is MCMC based, so naturally returns a mean/expectation which *must* have a finite positive value for the intrinsic scatter (it's not allowed to travel below zero). The zeroscatprob output works better with a Metropolis type scheme, e.g. algo.method='CHARM', Specs=list(alpha.star = 0.44) works well for many cases.
Value
The function returns a multi-component list containing:
parm
Vector of the main paramter fit outputs specified as set by the coord.type and scat.type options.
parm.vert.axis
Vector of the main parameter fit outputs specified strictly along the last column dimension of X (for both the intrinsic scatter and the beta offset). The order of the columns in X therefore affects the contents of this vector. This output assume a coord.type="alpha" and scat.type="vert.axis".
fit
The direct output of the specified algo.func. So either the natural return from optim (class type "optim"), LaplaceApproximation (class type "laplace") or LaplacesDemon (class type "demonoid").
sigcor
If algo.func="optim" or algo.func="LA" then this list element contains the unbiased population estimator for the intrinsic scatter corrected via the sampbias2popunbias output of hyper.sigcor, i.e. it uses the full correction from hyper.sigcor (element 3 of the correction vector generated). If algo.func="LD" then this list element contains the unbiased population estimator for the intrinsic scatter corrected via the bias2unbias output of hyper.sigcor, i.e. it uses the standard deviation bias correction from hyper.sigcor (element 1 of the correction vector generated). See hyper.sigcor for details on the different outputs and the convergence tests the demonstrate why these different correction methods are desirable.
parm.covar
The covariance matrix for parm. Only for algo.func="optim" and algo.func="LA".
zeroscatprob
The fraction of samples for the intrinsic scatter which are at *exactly* zero, which provides a guideline probability for the intrinsic scatter being truly zero, rather than the expectation which will always be a finite amount above zero. Only for algo.func="LD". See Details above.
hyper.plane
Object class of type hyper.plane.param, as output by hyper.covert. This standardises the final fit in the users requested coord.type and scat.type, and allows easy conversion to other systems by using the class dependent convert function on it.
N
The number of rows of matrix X.
dims
The number of columns of matrix X.
X
The input matrix X.
covarray
The covarray used for fitting.
weights
The weights used for fitting.
call
The actual call used to run hyper.fit.
args
The arguments used to run hyper.fit.
LL
A list containing elements sum (the total log-likelihodd), val (the individual log likelihoods) and sig (the effective sigma offsets). See hyper.like for more information on these outputs.
func
A function that can take the input X and output the target vert.axis.
predict.vert.axis
Vector of vert.axis predictions.
Author(s)
Aaron Robotham and Danail Obreschkow
References
Robotham, A.S.G., & Obreschkow, D., PASA, in press
See Also
hyper.basic, hyper.convert, hyper.fit-data, hyper.fit, hyper.plot, hyper.sigcor, hyper.summary
Examples
#### A very simple 2D example ####
#Make the simple data:
simpledata=cbind(x=1:10,y=c(12.2, 14.2, 15.9, 18.0, 20.1, 22.1, 23.9, 26.0, 27.9, 30.1))
simpfit=hyper.fit(simpledata)
summary(simpfit)
plot(simpfit)
#Increase the scatter:
simpledata2=cbind(x=1:10,y=c(11.6, 13.7, 15.5, 18.2, 21.2, 21.5, 23.6, 25.6, 27.9, 30.1))
simpfit2=hyper.fit(simpledata2)
summary(simpfit2)
plot(simpfit2)
#Assuming the error in each y data point is the same sy=0.5, we no longer need any
#component of intrinsic scatter to explain the data:
simpledata2err=cbind(sx=0, sy=rep(0.5, length(simpledata2[, 1])))
simpfit2werr=hyper.fit(simpledata2, vars=simpledata2err)
summary(simpfit2werr)
plot(simpfit2werr)
#We can fit for 6 different combinations of coordinate system:
print(hyper.fit(simpledata, coord.type='theta', scat.type='orth')$parm)
print(hyper.fit(simpledata, coord.type='alpha', scat.type='orth')$parm)
print(hyper.fit(simpledata, coord.type='normvec', scat.type='orth')$parm)
print(hyper.fit(simpledata, coord.type='theta', scat.type='vert.axis')$parm)
print(hyper.fit(simpledata, coord.type='alpha', scat.type='vert.axis')$parm)
print(hyper.fit(simpledata, coord.type='normvec', scat.type='vert.axis')$parm)
#These all describe the same hyperplane (or line in this case). We can convert between
#systems by using the hyper.convert utility function:
fit4normvert=hyper.fit(simpledata, coord.type='normvec', scat.type='vert.axis')$parm
hyper.convert(fit4normvert, in.coord.type='normvec', out.coord.type='theta',
in.scat.type='vert.axis', out.scat.type='orth')$parm
#### Simple Example in hyper.fit paper ####
#Fit with no error:
xval = c(-1.22, -0.78, 0.44, 1.01, 1.22)
yval = c(-0.15, 0.49, 1.17, 0.72, 1.22)
fitnoerror=hyper.fit(cbind(xval, yval))
plot(fitnoerror)
#Fit with independent x and y error:
xerr = c(0.12, 0.14, 0.20, 0.07, 0.06)
yerr = c(0.13, 0.03, 0.07, 0.11, 0.08)
fitwitherror=hyper.fit(cbind(xval, yval), vars=cbind(xerr, yerr)^2)
plot(fitwitherror)
#Fit with correlated x and y error:
xycor = c(0.90, -0.40, -0.25, 0.00, -0.20)
fitwitherrorandcor=hyper.fit(cbind(xval, yval), covarray=makecovarray2d(xerr, yerr, xycor))
plot(fitwitherrorandcor)
#### A 2D example with fitting a line ####
#Setup the initial data:
set.seed(650)
sampN=200
initscat=3
randatax=runif(sampN, -100, 100)
randatay=rnorm(sampN, sd=initscat)
sx=runif(sampN, 0, 10); sy=runif(sampN, 0, 10)
mockvararray=makecovarray2d(sx, sy, corxy=0)
errxy={}
for(i in 1:sampN){
rancovmat=ranrotcovmat2d(mockvararray[,,i])
errxy=rbind(errxy, mvrnorm(1, mu=c(0, 0), Sigma=rancovmat))
mockvararray[,,i]=rancovmat
}
randatax=randatax+errxy[,1]
randatay=randatay+errxy[,2]
#Rotate the data to an arbitrary angle theta:
ang=30
mock=rotdata2d(randatax, randatay, theta=ang)
xerrang={}; yerrang={}; corxyang={}
for(i in 1:sampN){
covmatrot=rotcovmat(mockvararray[,,i], theta=ang)
xerrang=c(xerrang, sqrt(covmatrot[1,1])); yerrang=c(yerrang, sqrt(covmatrot[2,2]))
corxyang=c(corxyang, covmatrot[1,2]/(xerrang[i]*yerrang[i]))
}
corxyang[xerrang==0 & yerrang==0]=0
mock=data.frame(x=mock[,1], y=mock[,2], sx=xerrang, sy=yerrang, corxy=corxyang)
#Do the fit:
X=cbind(mock$x, mock$y)
covarray=makecovarray2d(mock$sx, mock$sy, mock$corxy)
fitline=hyper.fit(X=X, covarray=covarray, coord.type='theta')
summary(fitline)
plot(fitline, trans=0.2, asp=1)
#We can add increasingly strenuous priors on theta (which becomes much like fixing theta):
fitline_p1=hyper.fit(X=X, covarray=covarray, coord.type='theta',
prior=function(parm){dnorm(parm[1],mean=40,sd=1,log=TRUE)})
plot(fitline_p1, trans=0.2, asp=1)
fitline_p2=hyper.fit(X=X, covarray=covarray, coord.type='theta',
prior=function(parm){dnorm(parm[1],mean=40,sd=0.01,log=TRUE)})
plot(fitline_p2, trans=0.2, asp=1)
#We can test to see if the errors are compatable with the intrinsic scatter:
fitlineerrscale=hyper.fit(X=X, covarray=covarray, coord.type='theta', doerrorscale=TRUE)
summary(fitlineerrscale)
plot(fitline, parm.errorscale=fitlineerrscale$parm['errorscale'], trans=0.2, asp=1)
#Within errors the errorscale parameter is 1, i.e. the errors are realistic, which we know
#they should be a priori since we made them ourselves.
#### A 2D example with exponential sampling & fitting a line ####
#Setup the initial data:
set.seed(650)
#The effect of an exponential density function along y is to offset the Gaussian mean by
#0.5 times the factor 'a' in exp(a*x), i.e.:
normfac=dnorm(0,sd=1.1)/(dnorm(10*1.1^2,sd=1.1)*exp(10*10*1.1^2))
magplot(seq(5,15,by=0.01), normfac*dnorm(seq(5,15, by=0.01), sd=1.1)*exp(10*seq(5,15,
by=0.01)), type='l')
abline(v=10*1.1^2,lty=2)
#The above will not be correctly normalised to form a true PDF, but the shift in the mean
#is clear, and it doesn't alter the standard deviation at all:
points(seq(5,15,by=0.1), dnorm(seq(5,15, by=0.1), mean=10*1.1^2, sd=1.1),col='red')
#Applying the same principal to our random data we apply the offset due to our exponential
#generative slope in y:
set.seed(650)
sampN=200
vert.scat=10
sampexp=0.1
ang=30
randatax=runif(200,-100,100)
randatay=randatax*tan(ang*pi/180)+rnorm(sampN, mean=sampexp*vert.scat^2, sd=vert.scat)
sx=runif(sampN, 0, 10); sy=runif(sampN, 0, 10)
mockvararray=makecovarray2d(sx, sy, corxy=0)
errxy={}
for(i in 1:sampN){
rancovmat=ranrotcovmat2d(mockvararray[,,i])
errxy=rbind(errxy, mvrnorm(1, mu=c(0, sampexp*sy[i]^2), Sigma=rancovmat))
mockvararray[,,i]=rancovmat
}
randatax=randatax+errxy[,1]
randatay=randatay+errxy[,2]
sx=sqrt(mockvararray[1,1,]); sy=sqrt(mockvararray[2,2,]); corxy=mockvararray[1,2,]/(sx*sy)
mock=data.frame(x=randatax, y=randatay, sx=sx, sy=sy, corxy=corxy)
#Do the fit. Notice that the second element of k.vec has the positive sign, i.e. we are moving
#data that has been shifted positively by the positive exponential slope in y back to where it
#would exist without the slope (i.e. if it had an equal chance of being scattered in both
#directions, rather than being preferentially offset in the direction away from denser data).
#This dense -> less-dense shift i s known as Eddington bias in astronomy, and is common in all
#power-law distributions that have intrinsic scatter (e.g. Schechter LF and dark matter HMF).
X=cbind(mock$x, mock$y)
covarray=makecovarray2d(mock$sx, mock$sy, mock$corxy)
fitlineexp=hyper.fit(X=X, covarray=covarray, coord.type='theta', k.vec=c(0,sampexp),
scat.type='vert.axis')
summary(fitlineexp)
plot(fitlineexp, k.vec=c(0,sampexp))
#If we ignore the k.vec when calculating the plotting sigma values you can see it has
#a significant effect:
plot(fitlineexp, trans=0.2, asp=1)
#Compare this to not including the known exponential slope:
fitlinenoexp=hyper.fit(X=X, covarray=covarray, coord.type='theta', k.vec=c(0,0),
scat.type='vert.axis')
summary(fitlinenoexp)
plot(fitlinenoexp, trans=0.2, asp=1)
#The theta and intrinsic scatter are similar, but the offset is shifted significantly
#away from zero.
#### A 3D example with fitting a plane ####
#Setup the initial data:
set.seed(650)
sampN=200
initscat=3
randatax=runif(sampN, -100, 100)
randatay=runif(sampN, -100, 100)
randataz=rnorm(sampN, sd=initscat)
sx=runif(sampN, 0, 5); sy=runif(sampN, 0, 5); sz=runif(sampN, 0, 5)
mockvararray=makecovarray3d(sx, sy, sz, corxy=0, corxz=0, coryz=0)
errxyz={}
for(i in 1:sampN){
rancovmat=ranrotcovmat3d(mockvararray[,,i])
errxyz=rbind(errxyz,mvrnorm(1, mu=c(0, 0, 0), Sigma=rancovmat))
mockvararray[,,i]=rancovmat
}
randatax=randatax+errxyz[,1]
randatay=randatay+errxyz[,2]
randataz=randataz+errxyz[,3]
sx=sqrt(mockvararray[1,1,]); sy=sqrt(mockvararray[2,2,]); sz=sqrt(mockvararray[3,3,])
corxy=mockvararray[1,2,]/(sx*sy); corxz=mockvararray[1,3,]/(sx*sz)
coryz=mockvararray[2,3,]/(sy*sz)
#Rotate the data to an arbitrary angle theta/phi:
desiredxtozang=10
desiredytozang=40
ang=c(desiredxtozang*cos(desiredytozang*pi/180), desiredytozang)
newxyz=rotdata3d(randatax, randatay, randataz, theta=ang[1], dim='y')
newxyz=rotdata3d(newxyz[,1], newxyz[,2], newxyz[,3], theta=ang[2], dim='x')
mockplane=data.frame(x=newxyz[,1], y=newxyz[,2], z=newxyz[,3])
xerrang={};yerrang={};zerrang={}
corxyang={};corxzang={};coryzang={}
for(i in 1:sampN){
newcovmatrot=rotcovmat(makecovmat3d(sx=sx[i], sy=sy[i], sz=sz[i], corxy=corxy[i],
corxz=corxz[i], coryz=coryz[i]), theta=ang[1], dim='y')
newcovmatrot=rotcovmat(newcovmatrot, theta=ang[2], dim='x')
xerrang=c(xerrang, sqrt(newcovmatrot[1,1]))
yerrang=c(yerrang, sqrt(newcovmatrot[2,2]))
zerrang=c(zerrang, sqrt(newcovmatrot[3,3]))
corxyang=c(corxyang, newcovmatrot[1,2]/(xerrang[i]*yerrang[i]))
corxzang=c(corxzang, newcovmatrot[1,3]/(xerrang[i]*zerrang[i]))
coryzang=c(coryzang, newcovmatrot[2,3]/(yerrang[i]*zerrang[i]))
}
corxyang[xerrang==0 & yerrang==0]=0
corxzang[xerrang==0 & zerrang==0]=0
coryzang[yerrang==0 & zerrang==0]=0
mockplane=data.frame(x=mockplane$x, y=mockplane$y, z=mockplane$z, sx=xerrang, sy=yerrang,
sz=zerrang, corxy=corxyang, corxz=corxzang, coryz=coryzang)
X=cbind(mockplane$x, mockplane$y, mockplane$z)
covarray=makecovarray3d(mockplane$sx, mockplane$sy, mockplane$sz, mockplane$corxy,
mockplane$corxz, mockplane$coryz)
fitplane=hyper.fit(X=X, covarray=covarray, coord.type='theta', scat.type='orth')
summary(fitplane)
plot(fitplane)
#### Example using the data from Hogg 2010 ####
#Example using the data from Hogg 2010: http://arxiv.org/pdf/1008.4686v1.pdf
#Load data:
data(hogg)
#Fit:
fithogg=hyper.fit(X=cbind(hogg$x, hogg$y), covarray=makecovarray2d(hogg$x_err, hogg$y_err,
hogg$corxy), coord.type='theta', scat.type='orth')
summary(fithogg)
plot(fithogg, trans=0.2)
#We now do exercise 17 of Hogg 2010 using trimmed data:
hoggtrim=hogg[-3,]
fithoggtrim=hyper.fit(X=cbind(hoggtrim$x, hoggtrim$y), covarray=makecovarray2d(hoggtrim$x_err,
hoggtrim$y_err, hoggtrim$corxy), coord.type='theta', scat.type='orth', algo.func='LA')
summary(fithoggtrim)
plot(fithoggtrim, trans=0.2)
#We can get more info from looking at the Summary1 output of the LaplaceApproximation:
print(fithoggtrim$fit$Summary1)
#MCMC (exercise 18):
fithoggtrimMCMC=hyper.fit(X=cbind(hoggtrim$x, hoggtrim$y), covarray=
makecovarray2d(hoggtrim$x_err, hoggtrim$y_err, hoggtrim$corxy), coord.type='theta',
scat.type='orth', algo.func='LD', algo.method='CHARM', Specs=list(alpha.star = 0.44))
summary(fithoggtrimMCMC)
#We can get additional info from looking at the Summary1 output of the LaplacesDemon:
print(fithoggtrimMCMC$fit$Summary2)
magplot(density(fithoggtrimMCMC$fit$Posterior2[,3]), xlab='Intrinsic Scatter',
ylab='Probability Density')
abline(v=quantile(fithoggtrimMCMC$fit$Posterior2[,3], c(0.95,0.99)), lty=2)
#### Example using 'real' data with intrinsic scatter ####
data(intrin)
fitintrin=hyper.fit(X=cbind(intrin$x, intrin$y), vars=cbind(intrin$x_err,
intrin$y_err)^2, coord.type='theta', scat.type='orth', algo.func='LA')
summary(fitintrin)
plot(fitintrin, trans=0.1, pch='.', asp=1)
fitintrincor=hyper.fit(X=cbind(intrin$x, intrin$y), covarray=makecovarray2d(intrin$x_err,
intrin$y_err, intrin$corxy), coord.type='theta', scat.type='orth', algo.func='LA')
summary(fitintrincor)
plot(fitintrincor, trans=0.1, pch='.', asp=1)
#### Example using flaring trumpet data ####
data(trumpet)
fittrumpet=hyper.fit(X=cbind(trumpet$x, trumpet$y), covarray=makecovarray2d(trumpet$x_err,
trumpet$y_err, trumpet$corxy), coord.type='normvec', algo.func='LA')
summary(fittrumpet)
plot(fittrumpet, trans=0.1, pch='.', asp=1)
#The best fit solution has a scat.orth very close to 0, so it is worth considering if the
#data should truly have 0 intrinsic scatter.
#To find the likelihood of zero intrinsic scatter we will need to run LaplacesDemon. The
#following will take a minute or so to run. Note we change the Specs for the
#default Griddy Gibbs sampler.
set.seed(650)
fittrumpetMCMC = hyper.fit(X=cbind(trumpet$x, trumpet$y), covarray=makecovarray2d(
trumpet$x_err, trumpet$y_err, trumpet$corxy), coord.type='normvec', algo.func='LD',
itermax=1e4, Specs = list(Grid=seq(-0.001,0.001, len=5), dparm=NULL, CPUs=1,
Packages=NULL, Dyn.libs=NULL))
#Assuming the user has specified the same initial seed we should find that the data
#has exactly zero intrinsic scatter with ~47% likelihood:
print(fittrumpetMCMC$zeroscatprob)
#We can also make an assessment of whether the data has even less scatter than expected
#given the expected errors:
set.seed(650)
fittrumpetMCMCerrscale=hyper.fit(X=cbind(trumpet$x, trumpet$y), covarray=makecovarray2d(
trumpet$x_err, trumpet$y_err, trumpet$corxy), itermax=1e4, coord.type='normvec', algo.func='LD',
algo.method='CHARM', Specs=list(alpha.star = 0.44), doerrorscale=TRUE)
#Assuming the user has specified the same initial seed we should find that the data
#has exactly zero intrinsic scatter with ~69% likelihood:
print(fittrumpetMCMCerrscale$zeroscatprob)
#### Example using 6dFGS Fundamental Plane data ####
data(FP6dFGS)
#First we try the fit without using any weights:
fitFP6dFGS=hyper.fit(FP6dFGS[,c('logIe_J', 'logsigma', 'logRe_J')],
vars=FP6dFGS[,c('logIe_J_err', 'logsigma_err', 'logRe_J_err')]^2, coord.type='alpha',
scat.type='vert.axis')
summary(fitFP6dFGS)
plot(fitFP6dFGS, doellipse=FALSE, alpha=0.5)
#Next we add the censoring weights provided by C. Magoulas:
fitFP6dFGSw=hyper.fit(FP6dFGS[,c('logIe_J', 'logsigma', 'logRe_J')],
vars=FP6dFGS[,c('logIe_J_err', 'logsigma_err', 'logRe_J_err')]^2, weights=FP6dFGS[,'weights'],
coord.type='alpha', scat.type='vert.axis')
summary(fitFP6dFGSw)
plot(fitFP6dFGSw, doellipse=FALSE, alpha=0.5)
#It is interesting to note the scatter orthogonal to the plane for the fundmental plane:
print(hyper.convert(coord=fitFP6dFGSw$parm[1:2], beta=fitFP6dFGSw$parm[3],
scat=fitFP6dFGSw$parm[4], in.scat.type='vert.axis', out.scat.type='orth',
in.coord.type='alpha'))
#### Example using GAMA mass-size relation data ####
data(GAMAsmVsize)
fitGAMAsmVsize=hyper.fit(GAMAsmVsize[,c('logmstar', 'rekpc')],
vars=GAMAsmVsize[,c('logmstar_err', 'rekpc_err')]^2, weights=GAMAsmVsize[,'weights'],
coord.type='alpha', scat.type='vert.axis')
summary(fitGAMAsmVsize)
#We turn the ellipse plotting off to speed things up:
plot(fitGAMAsmVsize, doellipse=FALSE, unlog='x')
#This is obviously a poor fit since the y data has a non-linear dependence on x. Let's try
#using the logged y-axis and converted errors:
fitGAMAsmVsizelogre=hyper.fit(GAMAsmVsize[,c('logmstar', 'logrekpc')],
vars=GAMAsmVsize[,c('logmstar_err', 'logrekpc_err')]^2, weights=GAMAsmVsize[,'weights'],
coord.type='alpha', scat.type='vert.axis')
summary(fitGAMAsmVsizelogre)
#We turn the ellipse plotting off to speed things up:
plot(fitGAMAsmVsizelogre, doellipse=FALSE, unlog='xy')
#We can compare to a fit with no errors used:
fitGAMAsmVsizelogrenoerr=hyper.fit(GAMAsmVsize[,c('logmstar', 'logrekpc')],
weights=GAMAsmVsize[,'weights'], coord.type='alpha', scat.type='vert.axis')
summary(fitGAMAsmVsizelogrenoerr)
#We turn the ellipse plotting off to speed things up:
plot(fitGAMAsmVsizelogrenoerr, doellipse=FALSE, unlog='xy')
### Example using Tully-Fisher relation data ###
data(TFR)
TFRfit=hyper.fit(X=TFR[,c('logv','M_K')],vars=TFR[,c('logv_err','M_K_err')]^2)
plot(TFRfit, xlim=c(1.7,2.5), ylim=c(-19,-26))
### Mase-Angular Momentum-Bulge/Total ###
data(MJB)
MJBfit=hyper.fit(X=MJB[,c('logM','logj','B.T')], covarray=makecovarray3d(MJB$logM_err,
MJB$logj_err, MJB$B.T_err, MJB$corMJ, 0, 0))
plot(MJBfit)
asgr/hyper.fit documentation built on June 8, 2024, 12:38 a.m.
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| hyper.fit | R Documentation |
Top level function that attempts to fit a hyperplane to provided data.
Description
Top level fitting function that uses downhill searches (optim/LaplaceApproximation) or MCMC (LaplacesDemon) to search out the best fitting parameters for a hyperplane (minimum a 1D line for 2D data), including the intrinsic scatter as part of the fit.
Usage
hyper.fit(X, covarray, vars, parm, parm.coord, parm.beta, parm.scat, parm.errorscale=1,
vert.axis, weights, k.vec, prior, itermax = 1e4, coord.type = 'alpha',
scat.type = 'vert.axis', algo.func = 'optim', algo.method = 'default',
Specs = list(Grid=seq(-0.1,0.1, len=5), dparm=NULL, CPUs=1, Packages=NULL, Dyn.libs=NULL),
doerrorscale = FALSE, ...)
Arguments
X |
A position matrix with N (number of data points) rows by d (number of dimensions) columns. |
covarray |
A dxdxN array containing the full covariance (d=dimensions, N=number of dxd matrices in the array stack). The makecovarray2d and makecovarray3d are convenience functions that make populating 2x2xN and 3x3xN arrays easier for a novice user. |
vars |
A variance matrix with the N (numver of data points) rows by d (number of dimensions) columns. In effect this is the diagonal elements of covarray where all other terms are zero. If covarray is also provided that input argument is used instead. |
parm |
Vector of all initial parameters. This should be a concatenation of c(parm.coord, parm.beta, parm.scat). If doerrorscale=TRUE then there should be an extrat element to give c(parm.coord, parm.beta, parm.scat, parm.errorscale). See the parm.coord, parm.beta, parm.scat and parm.errorscale arguments below for more information on what these different quantities describe. |
parm.coord |
Vector of initial coord parameters. These are either angles that produce the vectors that predict the vert.axis dimension (coord.type="theta"), the gradients of these (coord.type="alpha") or they are the vector elements normal to the hyperplane (coord.type="normvec"). Depending on the argument used, either beta along the vertical axis is generated (coord.type="alpha"/theta") or no beta is required (coord.type="normvec"). |
parm.beta |
Initial value of beta. This is either specified as the absolute distance from the origin to the hyperplane or as the intersection of the hyperplane on the vert.axis dimension being predicted. |
parm.scat |
Initial value of the intrinsic scatter. This is either specified as the scatter orthogonal to the hyperplane or as the scatter along the vert.axis dimension being predicted. |
parm.errorscale |
If doerrorscale=TRUE, this argument is the initial errorscale (default is 1). See the doerrorscale argument below for more details. |
vert.axis |
Which axis should the hyperplane equation be formulated for. This must be a number which specifies the column of position matrix X to be defined by the hyperplane. If missing, then the projection dimension is assumed to be the last column of the X matrix. |
weights |
Vector of multiplicative weights for each row of the X data matrix. i.e. if this is 2 then it is equivalent to having two identical data points with weights equal to 1. Should be either of length 1 (in which case elements are repeated as required) or the same length as the number of rows in the data matrix X. |
k.vec |
A vector defining the direction of an exponential sampling distribution in the data. The length is the scaling "a" of the generative exponent (i.e., exp(a*x)), and it points in the direction of *increasing* density (see example below). If provided, k.vec must be the same length as the dimensions of the data. k.vec has the most noticeable effect on the beta offset parameter. It is correcting for the phenomenom Astronomers often call Eddington bias. For discussion on the use of k.vec see Appendix B of Robotham & Obreschkow. |
prior |
A function that will take in parm and return a logged-likelihood that will be added to the data-model log-likelihood to give the true log-posterior. By default this will add 0, which means it is strictly an improper prior over infinite domain. The user will need to be careful about the ordering and parameter type when passing in parm to the prior function (especially when this is not provided, and the initial parameter guess is made internally). For clarity the prior function provided will be passed the parm vector with the exact ordering and coorindate types as seen in the output parm, so it is probably a good idea to run once without prior set, check the parm output, and build the prior function as appropriate. |
itermax |
The maximum iterations to use for either the LaplaceApproximation function or LaplacesDemon function. |
coord.type |
This specifies whether the fit should be done in terms of the normal vector to the hyperplane (coord.type="normvec") gradients defined to produce values along the vert.axis dimension (coord.type="alpha") or by the values of the angles that form the gradients (coord.type="theta"). "alpha" is the default since it is the most common means of parameterising hyperplane fits in the astronomy literature. |
scat.type |
This specifies whether the intrinsic scatter should be defined orthogonal to the plane (orth) or along the vert.axis of interest (vert.axis). |
algo.func |
If algo.func="optim" (default) hyper.fit will optimise using the R base |
algo.method |
Specifies the method argument of |
Specs |
Inputs to pass to the LaplacesDemon function. Default Specs is for the default GG (Griddy Gibbs) algorithm (see algo.method above). If using CHARM, then you probably want to set Specs=list(alpha.star = 0.44). |
doerrorscale |
If FALSE then the provided covariance are treated as it. If TRUE then the likelihood function is also allowed to rescale all errors by a uniform multiplicative value. |
... |
Further arguments to be passed to |
Details
Setting doerrorscale to TRUE allows for stable solutions when errors are overestimated, e.g. when the intrinsic scatter is equal to zero but the data is more clustered around the optimal likelihood plane than expected from the data covariance array. See Examples below for a 2D scenario where this is helpful.
algo.func="LD" also returns the probability that the generative model has exactly zero intrinsic scatter (zeroscatprob in the output). The other available functions (algo.func="optim" and algo.func="LA") can find exact solutions equal to zero since they are strictly mode finding (i.e. maximum likelihood) routines. algo.func="LD" is MCMC based, so naturally returns a mean/expectation which *must* have a finite positive value for the intrinsic scatter (it's not allowed to travel below zero). The zeroscatprob output works better with a Metropolis type scheme, e.g. algo.method='CHARM', Specs=list(alpha.star = 0.44) works well for many cases.
Value
The function returns a multi-component list containing:
parm |
Vector of the main paramter fit outputs specified as set by the coord.type and scat.type options. |
parm.vert.axis |
Vector of the main parameter fit outputs specified strictly along the last column dimension of X (for both the intrinsic scatter and the beta offset). The order of the columns in X therefore affects the contents of this vector. This output assume a coord.type="alpha" and scat.type="vert.axis". |
fit |
The direct output of the specified algo.func. So either the natural return from optim (class type "optim"), LaplaceApproximation (class type "laplace") or LaplacesDemon (class type "demonoid"). |
sigcor |
If algo.func="optim" or algo.func="LA" then this list element contains the unbiased population estimator for the intrinsic scatter corrected via the sampbias2popunbias output of |
parm.covar |
The covariance matrix for parm. Only for algo.func="optim" and algo.func="LA". |
zeroscatprob |
The fraction of samples for the intrinsic scatter which are at *exactly* zero, which provides a guideline probability for the intrinsic scatter being truly zero, rather than the expectation which will always be a finite amount above zero. Only for algo.func="LD". See Details above. |
hyper.plane |
Object class of type hyper.plane.param, as output by hyper.covert. This standardises the final fit in the users requested coord.type and scat.type, and allows easy conversion to other systems by using the class dependent convert function on it. |
N |
The number of rows of matrix X. |
dims |
The number of columns of matrix X. |
X |
The input matrix X. |
covarray |
The covarray used for fitting. |
weights |
The weights used for fitting. |
call |
The actual call used to run hyper.fit. |
args |
The arguments used to run hyper.fit. |
LL |
A list containing elements sum (the total log-likelihodd), val (the individual log likelihoods) and sig (the effective sigma offsets). See hyper.like for more information on these outputs. |
func |
A function that can take the input X and output the target vert.axis. |
predict.vert.axis |
Vector of vert.axis predictions. |
Author(s)
Aaron Robotham and Danail Obreschkow
References
Robotham, A.S.G., & Obreschkow, D., PASA, in press
See Also
hyper.basic, hyper.convert, hyper.fit-data, hyper.fit, hyper.plot, hyper.sigcor, hyper.summary
Examples
#### A very simple 2D example ####
#Make the simple data:
simpledata=cbind(x=1:10,y=c(12.2, 14.2, 15.9, 18.0, 20.1, 22.1, 23.9, 26.0, 27.9, 30.1))
simpfit=hyper.fit(simpledata)
summary(simpfit)
plot(simpfit)
#Increase the scatter:
simpledata2=cbind(x=1:10,y=c(11.6, 13.7, 15.5, 18.2, 21.2, 21.5, 23.6, 25.6, 27.9, 30.1))
simpfit2=hyper.fit(simpledata2)
summary(simpfit2)
plot(simpfit2)
#Assuming the error in each y data point is the same sy=0.5, we no longer need any
#component of intrinsic scatter to explain the data:
simpledata2err=cbind(sx=0, sy=rep(0.5, length(simpledata2[, 1])))
simpfit2werr=hyper.fit(simpledata2, vars=simpledata2err)
summary(simpfit2werr)
plot(simpfit2werr)
#We can fit for 6 different combinations of coordinate system:
print(hyper.fit(simpledata, coord.type='theta', scat.type='orth')$parm)
print(hyper.fit(simpledata, coord.type='alpha', scat.type='orth')$parm)
print(hyper.fit(simpledata, coord.type='normvec', scat.type='orth')$parm)
print(hyper.fit(simpledata, coord.type='theta', scat.type='vert.axis')$parm)
print(hyper.fit(simpledata, coord.type='alpha', scat.type='vert.axis')$parm)
print(hyper.fit(simpledata, coord.type='normvec', scat.type='vert.axis')$parm)
#These all describe the same hyperplane (or line in this case). We can convert between
#systems by using the hyper.convert utility function:
fit4normvert=hyper.fit(simpledata, coord.type='normvec', scat.type='vert.axis')$parm
hyper.convert(fit4normvert, in.coord.type='normvec', out.coord.type='theta',
in.scat.type='vert.axis', out.scat.type='orth')$parm
#### Simple Example in hyper.fit paper ####
#Fit with no error:
xval = c(-1.22, -0.78, 0.44, 1.01, 1.22)
yval = c(-0.15, 0.49, 1.17, 0.72, 1.22)
fitnoerror=hyper.fit(cbind(xval, yval))
plot(fitnoerror)
#Fit with independent x and y error:
xerr = c(0.12, 0.14, 0.20, 0.07, 0.06)
yerr = c(0.13, 0.03, 0.07, 0.11, 0.08)
fitwitherror=hyper.fit(cbind(xval, yval), vars=cbind(xerr, yerr)^2)
plot(fitwitherror)
#Fit with correlated x and y error:
xycor = c(0.90, -0.40, -0.25, 0.00, -0.20)
fitwitherrorandcor=hyper.fit(cbind(xval, yval), covarray=makecovarray2d(xerr, yerr, xycor))
plot(fitwitherrorandcor)
#### A 2D example with fitting a line ####
#Setup the initial data:
set.seed(650)
sampN=200
initscat=3
randatax=runif(sampN, -100, 100)
randatay=rnorm(sampN, sd=initscat)
sx=runif(sampN, 0, 10); sy=runif(sampN, 0, 10)
mockvararray=makecovarray2d(sx, sy, corxy=0)
errxy={}
for(i in 1:sampN){
rancovmat=ranrotcovmat2d(mockvararray[,,i])
errxy=rbind(errxy, mvrnorm(1, mu=c(0, 0), Sigma=rancovmat))
mockvararray[,,i]=rancovmat
}
randatax=randatax+errxy[,1]
randatay=randatay+errxy[,2]
#Rotate the data to an arbitrary angle theta:
ang=30
mock=rotdata2d(randatax, randatay, theta=ang)
xerrang={}; yerrang={}; corxyang={}
for(i in 1:sampN){
covmatrot=rotcovmat(mockvararray[,,i], theta=ang)
xerrang=c(xerrang, sqrt(covmatrot[1,1])); yerrang=c(yerrang, sqrt(covmatrot[2,2]))
corxyang=c(corxyang, covmatrot[1,2]/(xerrang[i]*yerrang[i]))
}
corxyang[xerrang==0 & yerrang==0]=0
mock=data.frame(x=mock[,1], y=mock[,2], sx=xerrang, sy=yerrang, corxy=corxyang)
#Do the fit:
X=cbind(mock$x, mock$y)
covarray=makecovarray2d(mock$sx, mock$sy, mock$corxy)
fitline=hyper.fit(X=X, covarray=covarray, coord.type='theta')
summary(fitline)
plot(fitline, trans=0.2, asp=1)
#We can add increasingly strenuous priors on theta (which becomes much like fixing theta):
fitline_p1=hyper.fit(X=X, covarray=covarray, coord.type='theta',
prior=function(parm){dnorm(parm[1],mean=40,sd=1,log=TRUE)})
plot(fitline_p1, trans=0.2, asp=1)
fitline_p2=hyper.fit(X=X, covarray=covarray, coord.type='theta',
prior=function(parm){dnorm(parm[1],mean=40,sd=0.01,log=TRUE)})
plot(fitline_p2, trans=0.2, asp=1)
#We can test to see if the errors are compatable with the intrinsic scatter:
fitlineerrscale=hyper.fit(X=X, covarray=covarray, coord.type='theta', doerrorscale=TRUE)
summary(fitlineerrscale)
plot(fitline, parm.errorscale=fitlineerrscale$parm['errorscale'], trans=0.2, asp=1)
#Within errors the errorscale parameter is 1, i.e. the errors are realistic, which we know
#they should be a priori since we made them ourselves.
#### A 2D example with exponential sampling & fitting a line ####
#Setup the initial data:
set.seed(650)
#The effect of an exponential density function along y is to offset the Gaussian mean by
#0.5 times the factor 'a' in exp(a*x), i.e.:
normfac=dnorm(0,sd=1.1)/(dnorm(10*1.1^2,sd=1.1)*exp(10*10*1.1^2))
magplot(seq(5,15,by=0.01), normfac*dnorm(seq(5,15, by=0.01), sd=1.1)*exp(10*seq(5,15,
by=0.01)), type='l')
abline(v=10*1.1^2,lty=2)
#The above will not be correctly normalised to form a true PDF, but the shift in the mean
#is clear, and it doesn't alter the standard deviation at all:
points(seq(5,15,by=0.1), dnorm(seq(5,15, by=0.1), mean=10*1.1^2, sd=1.1),col='red')
#Applying the same principal to our random data we apply the offset due to our exponential
#generative slope in y:
set.seed(650)
sampN=200
vert.scat=10
sampexp=0.1
ang=30
randatax=runif(200,-100,100)
randatay=randatax*tan(ang*pi/180)+rnorm(sampN, mean=sampexp*vert.scat^2, sd=vert.scat)
sx=runif(sampN, 0, 10); sy=runif(sampN, 0, 10)
mockvararray=makecovarray2d(sx, sy, corxy=0)
errxy={}
for(i in 1:sampN){
rancovmat=ranrotcovmat2d(mockvararray[,,i])
errxy=rbind(errxy, mvrnorm(1, mu=c(0, sampexp*sy[i]^2), Sigma=rancovmat))
mockvararray[,,i]=rancovmat
}
randatax=randatax+errxy[,1]
randatay=randatay+errxy[,2]
sx=sqrt(mockvararray[1,1,]); sy=sqrt(mockvararray[2,2,]); corxy=mockvararray[1,2,]/(sx*sy)
mock=data.frame(x=randatax, y=randatay, sx=sx, sy=sy, corxy=corxy)
#Do the fit. Notice that the second element of k.vec has the positive sign, i.e. we are moving
#data that has been shifted positively by the positive exponential slope in y back to where it
#would exist without the slope (i.e. if it had an equal chance of being scattered in both
#directions, rather than being preferentially offset in the direction away from denser data).
#This dense -> less-dense shift i s known as Eddington bias in astronomy, and is common in all
#power-law distributions that have intrinsic scatter (e.g. Schechter LF and dark matter HMF).
X=cbind(mock$x, mock$y)
covarray=makecovarray2d(mock$sx, mock$sy, mock$corxy)
fitlineexp=hyper.fit(X=X, covarray=covarray, coord.type='theta', k.vec=c(0,sampexp),
scat.type='vert.axis')
summary(fitlineexp)
plot(fitlineexp, k.vec=c(0,sampexp))
#If we ignore the k.vec when calculating the plotting sigma values you can see it has
#a significant effect:
plot(fitlineexp, trans=0.2, asp=1)
#Compare this to not including the known exponential slope:
fitlinenoexp=hyper.fit(X=X, covarray=covarray, coord.type='theta', k.vec=c(0,0),
scat.type='vert.axis')
summary(fitlinenoexp)
plot(fitlinenoexp, trans=0.2, asp=1)
#The theta and intrinsic scatter are similar, but the offset is shifted significantly
#away from zero.
#### A 3D example with fitting a plane ####
#Setup the initial data:
set.seed(650)
sampN=200
initscat=3
randatax=runif(sampN, -100, 100)
randatay=runif(sampN, -100, 100)
randataz=rnorm(sampN, sd=initscat)
sx=runif(sampN, 0, 5); sy=runif(sampN, 0, 5); sz=runif(sampN, 0, 5)
mockvararray=makecovarray3d(sx, sy, sz, corxy=0, corxz=0, coryz=0)
errxyz={}
for(i in 1:sampN){
rancovmat=ranrotcovmat3d(mockvararray[,,i])
errxyz=rbind(errxyz,mvrnorm(1, mu=c(0, 0, 0), Sigma=rancovmat))
mockvararray[,,i]=rancovmat
}
randatax=randatax+errxyz[,1]
randatay=randatay+errxyz[,2]
randataz=randataz+errxyz[,3]
sx=sqrt(mockvararray[1,1,]); sy=sqrt(mockvararray[2,2,]); sz=sqrt(mockvararray[3,3,])
corxy=mockvararray[1,2,]/(sx*sy); corxz=mockvararray[1,3,]/(sx*sz)
coryz=mockvararray[2,3,]/(sy*sz)
#Rotate the data to an arbitrary angle theta/phi:
desiredxtozang=10
desiredytozang=40
ang=c(desiredxtozang*cos(desiredytozang*pi/180), desiredytozang)
newxyz=rotdata3d(randatax, randatay, randataz, theta=ang[1], dim='y')
newxyz=rotdata3d(newxyz[,1], newxyz[,2], newxyz[,3], theta=ang[2], dim='x')
mockplane=data.frame(x=newxyz[,1], y=newxyz[,2], z=newxyz[,3])
xerrang={};yerrang={};zerrang={}
corxyang={};corxzang={};coryzang={}
for(i in 1:sampN){
newcovmatrot=rotcovmat(makecovmat3d(sx=sx[i], sy=sy[i], sz=sz[i], corxy=corxy[i],
corxz=corxz[i], coryz=coryz[i]), theta=ang[1], dim='y')
newcovmatrot=rotcovmat(newcovmatrot, theta=ang[2], dim='x')
xerrang=c(xerrang, sqrt(newcovmatrot[1,1]))
yerrang=c(yerrang, sqrt(newcovmatrot[2,2]))
zerrang=c(zerrang, sqrt(newcovmatrot[3,3]))
corxyang=c(corxyang, newcovmatrot[1,2]/(xerrang[i]*yerrang[i]))
corxzang=c(corxzang, newcovmatrot[1,3]/(xerrang[i]*zerrang[i]))
coryzang=c(coryzang, newcovmatrot[2,3]/(yerrang[i]*zerrang[i]))
}
corxyang[xerrang==0 & yerrang==0]=0
corxzang[xerrang==0 & zerrang==0]=0
coryzang[yerrang==0 & zerrang==0]=0
mockplane=data.frame(x=mockplane$x, y=mockplane$y, z=mockplane$z, sx=xerrang, sy=yerrang,
sz=zerrang, corxy=corxyang, corxz=corxzang, coryz=coryzang)
X=cbind(mockplane$x, mockplane$y, mockplane$z)
covarray=makecovarray3d(mockplane$sx, mockplane$sy, mockplane$sz, mockplane$corxy,
mockplane$corxz, mockplane$coryz)
fitplane=hyper.fit(X=X, covarray=covarray, coord.type='theta', scat.type='orth')
summary(fitplane)
plot(fitplane)
#### Example using the data from Hogg 2010 ####
#Example using the data from Hogg 2010: http://arxiv.org/pdf/1008.4686v1.pdf
#Load data:
data(hogg)
#Fit:
fithogg=hyper.fit(X=cbind(hogg$x, hogg$y), covarray=makecovarray2d(hogg$x_err, hogg$y_err,
hogg$corxy), coord.type='theta', scat.type='orth')
summary(fithogg)
plot(fithogg, trans=0.2)
#We now do exercise 17 of Hogg 2010 using trimmed data:
hoggtrim=hogg[-3,]
fithoggtrim=hyper.fit(X=cbind(hoggtrim$x, hoggtrim$y), covarray=makecovarray2d(hoggtrim$x_err,
hoggtrim$y_err, hoggtrim$corxy), coord.type='theta', scat.type='orth', algo.func='LA')
summary(fithoggtrim)
plot(fithoggtrim, trans=0.2)
#We can get more info from looking at the Summary1 output of the LaplaceApproximation:
print(fithoggtrim$fit$Summary1)
#MCMC (exercise 18):
fithoggtrimMCMC=hyper.fit(X=cbind(hoggtrim$x, hoggtrim$y), covarray=
makecovarray2d(hoggtrim$x_err, hoggtrim$y_err, hoggtrim$corxy), coord.type='theta',
scat.type='orth', algo.func='LD', algo.method='CHARM', Specs=list(alpha.star = 0.44))
summary(fithoggtrimMCMC)
#We can get additional info from looking at the Summary1 output of the LaplacesDemon:
print(fithoggtrimMCMC$fit$Summary2)
magplot(density(fithoggtrimMCMC$fit$Posterior2[,3]), xlab='Intrinsic Scatter',
ylab='Probability Density')
abline(v=quantile(fithoggtrimMCMC$fit$Posterior2[,3], c(0.95,0.99)), lty=2)
#### Example using 'real' data with intrinsic scatter ####
data(intrin)
fitintrin=hyper.fit(X=cbind(intrin$x, intrin$y), vars=cbind(intrin$x_err,
intrin$y_err)^2, coord.type='theta', scat.type='orth', algo.func='LA')
summary(fitintrin)
plot(fitintrin, trans=0.1, pch='.', asp=1)
fitintrincor=hyper.fit(X=cbind(intrin$x, intrin$y), covarray=makecovarray2d(intrin$x_err,
intrin$y_err, intrin$corxy), coord.type='theta', scat.type='orth', algo.func='LA')
summary(fitintrincor)
plot(fitintrincor, trans=0.1, pch='.', asp=1)
#### Example using flaring trumpet data ####
data(trumpet)
fittrumpet=hyper.fit(X=cbind(trumpet$x, trumpet$y), covarray=makecovarray2d(trumpet$x_err,
trumpet$y_err, trumpet$corxy), coord.type='normvec', algo.func='LA')
summary(fittrumpet)
plot(fittrumpet, trans=0.1, pch='.', asp=1)
#The best fit solution has a scat.orth very close to 0, so it is worth considering if the
#data should truly have 0 intrinsic scatter.
#To find the likelihood of zero intrinsic scatter we will need to run LaplacesDemon. The
#following will take a minute or so to run. Note we change the Specs for the
#default Griddy Gibbs sampler.
set.seed(650)
fittrumpetMCMC = hyper.fit(X=cbind(trumpet$x, trumpet$y), covarray=makecovarray2d(
trumpet$x_err, trumpet$y_err, trumpet$corxy), coord.type='normvec', algo.func='LD',
itermax=1e4, Specs = list(Grid=seq(-0.001,0.001, len=5), dparm=NULL, CPUs=1,
Packages=NULL, Dyn.libs=NULL))
#Assuming the user has specified the same initial seed we should find that the data
#has exactly zero intrinsic scatter with ~47% likelihood:
print(fittrumpetMCMC$zeroscatprob)
#We can also make an assessment of whether the data has even less scatter than expected
#given the expected errors:
set.seed(650)
fittrumpetMCMCerrscale=hyper.fit(X=cbind(trumpet$x, trumpet$y), covarray=makecovarray2d(
trumpet$x_err, trumpet$y_err, trumpet$corxy), itermax=1e4, coord.type='normvec', algo.func='LD',
algo.method='CHARM', Specs=list(alpha.star = 0.44), doerrorscale=TRUE)
#Assuming the user has specified the same initial seed we should find that the data
#has exactly zero intrinsic scatter with ~69% likelihood:
print(fittrumpetMCMCerrscale$zeroscatprob)
#### Example using 6dFGS Fundamental Plane data ####
data(FP6dFGS)
#First we try the fit without using any weights:
fitFP6dFGS=hyper.fit(FP6dFGS[,c('logIe_J', 'logsigma', 'logRe_J')],
vars=FP6dFGS[,c('logIe_J_err', 'logsigma_err', 'logRe_J_err')]^2, coord.type='alpha',
scat.type='vert.axis')
summary(fitFP6dFGS)
plot(fitFP6dFGS, doellipse=FALSE, alpha=0.5)
#Next we add the censoring weights provided by C. Magoulas:
fitFP6dFGSw=hyper.fit(FP6dFGS[,c('logIe_J', 'logsigma', 'logRe_J')],
vars=FP6dFGS[,c('logIe_J_err', 'logsigma_err', 'logRe_J_err')]^2, weights=FP6dFGS[,'weights'],
coord.type='alpha', scat.type='vert.axis')
summary(fitFP6dFGSw)
plot(fitFP6dFGSw, doellipse=FALSE, alpha=0.5)
#It is interesting to note the scatter orthogonal to the plane for the fundmental plane:
print(hyper.convert(coord=fitFP6dFGSw$parm[1:2], beta=fitFP6dFGSw$parm[3],
scat=fitFP6dFGSw$parm[4], in.scat.type='vert.axis', out.scat.type='orth',
in.coord.type='alpha'))
#### Example using GAMA mass-size relation data ####
data(GAMAsmVsize)
fitGAMAsmVsize=hyper.fit(GAMAsmVsize[,c('logmstar', 'rekpc')],
vars=GAMAsmVsize[,c('logmstar_err', 'rekpc_err')]^2, weights=GAMAsmVsize[,'weights'],
coord.type='alpha', scat.type='vert.axis')
summary(fitGAMAsmVsize)
#We turn the ellipse plotting off to speed things up:
plot(fitGAMAsmVsize, doellipse=FALSE, unlog='x')
#This is obviously a poor fit since the y data has a non-linear dependence on x. Let's try
#using the logged y-axis and converted errors:
fitGAMAsmVsizelogre=hyper.fit(GAMAsmVsize[,c('logmstar', 'logrekpc')],
vars=GAMAsmVsize[,c('logmstar_err', 'logrekpc_err')]^2, weights=GAMAsmVsize[,'weights'],
coord.type='alpha', scat.type='vert.axis')
summary(fitGAMAsmVsizelogre)
#We turn the ellipse plotting off to speed things up:
plot(fitGAMAsmVsizelogre, doellipse=FALSE, unlog='xy')
#We can compare to a fit with no errors used:
fitGAMAsmVsizelogrenoerr=hyper.fit(GAMAsmVsize[,c('logmstar', 'logrekpc')],
weights=GAMAsmVsize[,'weights'], coord.type='alpha', scat.type='vert.axis')
summary(fitGAMAsmVsizelogrenoerr)
#We turn the ellipse plotting off to speed things up:
plot(fitGAMAsmVsizelogrenoerr, doellipse=FALSE, unlog='xy')
### Example using Tully-Fisher relation data ###
data(TFR)
TFRfit=hyper.fit(X=TFR[,c('logv','M_K')],vars=TFR[,c('logv_err','M_K_err')]^2)
plot(TFRfit, xlim=c(1.7,2.5), ylim=c(-19,-26))
### Mase-Angular Momentum-Bulge/Total ###
data(MJB)
MJBfit=hyper.fit(X=MJB[,c('logM','logj','B.T')], covarray=makecovarray3d(MJB$logM_err,
MJB$logj_err, MJB$B.T_err, MJB$corMJ, 0, 0))
plot(MJBfit)
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