I want to transform my excel solver model into a model in R. I need to find 3 sets of coordinates which minimizes the distance to the 5 other given coordinates. I've made a program which calculates a distance matrix which outputs the minimal distance from each input to the given coordinates. I want to minimize this function by changing the input. Id est, I want to find the coordinates such that the sum of minimal distances are minimized. I tried several methods to do so, see the code below (Yes my distance matrix function might be somewhat cluncky, but this is because I had to reduce the input to 1 variable in order to run some algorithms such as nloprt (would get warnings otherwise). I've also seen some other questions (such as GRG Non-Linear Least Squares (Optimization)) but they did not change/improve the solution.
# First half of p describes x coordinates, second half the y coordinates # yes thats cluncky
p<-c(2,4,6,5,3,2) # initial points
x_given <- c(2,2.5,4,4,5)
y_given <- c(9,5,7,1,2)
f <- function(Coordinates){
# Predining
Term_1 <- NULL
Term_2 <- NULL
x <- NULL
Distance <- NULL
min_prob <- NULL
l <- length(Coordinates)
l2 <- length(x_given)
half_length <- l/2
s <- l2*half_length
Distance_Matrix <- matrix(c(rep(1,s)), nrow=half_length)
# Creating the distance matrix
for (k in 1:half_length){
for (i in 1:l2){
Term_1[i] <- (Coordinates[k]-x_given[i])^2
Term_2[i] <- (Coordinates[k+half_length]-y_given[i])^2
Distance[i] <- sqrt(Term_1[i]+Term_2[i])
Distance_Matrix[k,i] <- Distance[i]
}
}
d <- Distance_Matrix
# Find the minimum in each row, thats what we want to obtain ánd minimize
for (l in 1:nrow(d)){
min_prob[l] <- min(d[l,])
}
som<-sum(min_prob)
return(som)
}
# Minimise
sol<-optim(p,f)
x<-sol$par[1:3]
y<-sol$par[4:6]
plot(x_given,y_given)
points(x,y,pch=19)
The solution however is clearly not that optimal. I've tried to use the nloptr function, but I'm not sure which algorithm to use. Which algorithm can I use or can I use/program another function which solves this problem? Thanks in advance (and sorry for the detailed long question)
Look at the output of optim. It reached the iteration limit and had not yet converged.
> optim(p, f)
$`par`
[1] 2.501441 5.002441 5.003209 5.001237 1.995857 2.000265
$value
[1] 0.009927249
$counts
function gradient
501 NA
$convergence
[1] 1
$message
NULL
Although the result is not that different you will need to increase the number of iterations to get convergence. If that is still unacceptable then try different starting values.
> optim(p, f, control = list(maxit = 1000))
$`par`
[1] 2.502806 4.999866 5.000000 5.003009 1.999112 2.000000
$value
[1] 0.005012449
$counts
function gradient
755 NA
$convergence
[1] 0
$message
NULL
Related
I want to discount a number of cashflows and find the ''fair'' interest rate of a financial instrument. That is, I want to set the interest rate in such a way that
$$P=\sum_i^T e^{-ri}c_i$$, where P is some value. As a toy example, I came up with this:
value <- c(1,2,3,4,5,6,7,8,9,10)
discounted_value <- c()
for(i in 1:10){
discounted_value[i] <- value[i]*exp(-r*i)
}
should_be_equal_to <- 50
I want to find a r such that the sum of the vector of discounted_values is equal to 50 (should_be_equal_to). Does anyone have an idea how to solve this? I prefer not to use a number of manual grids for r.
It's like a least square problem. You define a function that calculates the difference between the sum of your predicted value for a given r :
fn <- function(value,r) {
delta = 50 - sum(value*exp(-r*seq_along(value)))
abs(delta)
}
Then you optimize this function, providing boundaries:
optim(par = 0.5,fn=fn, value=value,method="Brent",lower=0,upper=1)
$par
[1] 0.01369665
$value
[1] 2.240363e-07
$counts
function gradient
NA NA
$convergence
[1] 0
$message
NULL
And you can try the optimized parameter:
r = 0.01369665
discounted_value = value*exp(-r*seq_along(value))
sum(discounted_value)
[1] 50
Is there a way in R to generate random coordinates with a minimum distance between them?
E.g. what I'd like to avoid
x <- c(0,3.9,4.1,8)
y <- c(1,4.1,3.9,7)
plot(x~y)
This is a classical problem from stochastic geometry. Completely random points in space where the number of points falling in disjoint regions are independent of each other corresponds to a homogeneous Poisson point process (in this case in R^2, but could be in almost any space).
An important feature is that the total number of points has to be random before you can have independence of the counts of points in disjoint regions.
For the Poisson process points can be arbitrarily close together. If you define a process by sampling the Poisson process until you don't have any points that are too close together you have the so-called Gibbs Hardcore process. This has been studied a lot in the literature and there are different ways to simulate it. The R package spatstat has functions to do this. rHardcore is a perfect sampler, but if you want a high intensity of points and a big hard core distance it may not terminate in finite time... The distribution can be obtained as the limit of a Markov chain and rmh.default lets you run a Markov chain with a given Gibbs model as its invariant distribution. This finishes in finite time but only gives a realisation of an approximate distribution.
In rmh.default you can also simulate conditional on a fixed number of points. Note that when you sample in a finite box there is of course an upper limit to how many points you can fit with a given hard core radius, and the closer you are to this limit the more problematic it becomes to sample correctly from the distribution.
Example:
library(spatstat)
beta <- 100; R = 0.1
win <- square(1) # Unit square for simulation
X1 <- rHardcore(beta, R, W = win) # Exact sampling -- beware it may run forever for some par.!
plot(X1, main = paste("Exact sim. of hardcore model; beta =", beta, "and R =", R))
minnndist(X1) # Observed min. nearest neighbour dist.
#> [1] 0.102402
Approximate simulation
model <- rmhmodel(cif="hardcore", par = list(beta=beta, hc=R), w = win)
X2 <- rmh(model)
#> Checking arguments..determining simulation windows...Starting simulation.
#> Initial state...Ready to simulate. Generating proposal points...Running Metropolis-Hastings.
plot(X2, main = paste("Approx. sim. of hardcore model; beta =", beta, "and R =", R))
minnndist(X2) # Observed min. nearest neighbour dist.
#> [1] 0.1005433
Approximate simulation conditional on number of points
X3 <- rmh(model, control = rmhcontrol(p=1), start = list(n.start = 42))
#> Checking arguments..determining simulation windows...Starting simulation.
#> Initial state...Ready to simulate. Generating proposal points...Running Metropolis-Hastings.
plot(X3, main = paste("Approx. sim. given n =", 42))
minnndist(X3) # Observed min. nearest neighbour dist.
#> [1] 0.1018068
OK, how about this? You just generate random number pairs without restriction and then remove the onces which are too close. This could be a great start for that:
minimumDistancePairs <- function(x, y, minDistance){
i <- 1
repeat{
distance <- sqrt((x-x[i])^2 + (y-y[i])^2) < minDistance # pythagorean theorem
distance[i] <- FALSE # distance to oneself is always zero
if(any(distance)) { # if too close to any other point
x <- x[-i] # remove element from x
y <- y[-i] # and remove element from y
} else { # otherwise...
i = i + 1 # repeat the procedure with the next element
}
if (i > length(x)) break
}
data.frame(x,y)
}
minimumDistancePairs(
c(0,3.9,4.1,8)
, c(1,4.1,3.9,7)
, 1
)
will lead to
x y
1 0.0 1.0
2 4.1 3.9
3 8.0 7.0
Be aware, though, of the fact that these are not random numbers anymore (however you solve problem).
You can use rejection sapling https://en.wikipedia.org/wiki/Rejection_sampling
The principle is simple: you resample until you data verify the condition.
> set.seed(1)
>
> x <- rnorm(2)
> y <- rnorm(2)
> (x[1]-x[2])^2+(y[1]-y[2])^2
[1] 6.565578
> while((x[1]-x[2])^2+(y[1]-y[2])^2 > 1) {
+ x <- rnorm(2)
+ y <- rnorm(2)
+ }
> (x[1]-x[2])^2+(y[1]-y[2])^2
[1] 0.9733252
>
The following is a naive hit-and-miss approach which for some choices of parameters (which were left unspecified in the question) works well. If performance becomes an issue, you could experiment with the package gpuR which has a GPU-accelerated distance matrix calculation.
rand.separated <- function(n,x0,x1,y0,y1,d,trials = 1000){
for(i in 1:trials){
nums <- cbind(runif(n,x0,x1),runif(n,y0,y1))
if(min(dist(nums)) >= d) return(nums)
}
return(NA) #no luck
}
This repeatedly draws samples of size n in [x0,x1]x[y0,y1] and then throws the sample away if it doesn't satisfy. As a safety, trials guards against an infinite loop. If solutions are hard to find or n is large you might need to increase or decrease trials.
For example:
> set.seed(2018)
> nums <- rand.separated(25,0,10,0,10,0.2)
> plot(nums)
runs almost instantly and produces:
Im not sure what you are asking.
if you want random coordinates here.
c(
runif(1,max=y[1],min=x[1]),
runif(1,max=y[2],min=x[2]),
runif(1,min=y[3],max=x[3]),
runif(1,min=y[4],max=x[4])
)
Here's my setup
obs1<-c(1,1,1)
obs2<-c(0,1,2)
obs3<-c(0,0,3)
absoluteError<-function(obs,x){
return(sum(abs(obs-x)))
}
Example:
> absoluteError(obs2,1)
[1] 2
For a random vector of observations, I'd like to find a minimizer, x, which minimizes the absolute error between the observation values and a vector of all x. For instance, clearly the argument that minimizes absoluteError(obs1,x) is x=1 because this results in an error of 0. How do I find a minimizer for a random vector of observations? I'd imagine this is a linear programming problem, but I've never implemented one in R before.
The median of obs is a minimizer for the absolute error. The following is a sketch of how one might try proving this:
Let the median of a set of n observations, obs, be m. Call the absolute error between obs and m f(obs,m).
Case n is odd:
Consider f(obs,m+delta) where delta is some non zero number. Suppose delta is positive - then there are (n-1)/2 +1 observations whose error is delta more than f(obs,m). The remaining (n-1)/2 observations' error is at most delta less than f(obs,m). So f(obs,m+delta)-f(obs,m)>=delta. (The same argument can be made if delta is negative.) So the median is the only minimizer in this case. Thus f(obs,m+delta)>f(obs,m) for any non zero delta so m is a minimizer for f.
Case n is even:
Basically the same logic as above, except in this case any number between the two inner most numbers in the set will be a minimizer.
I am not sure this answer is correct, and even if it is I am not sure this is what you want. Nevertheless, I am taking a stab at it.
I think you are talking about 'Least absolute deviations', a form of regression that differs from 'Least Squares'.
If so, I found this R code for solving Least absolute deviations regression:
fabs=function(beta0,x,y){
b0=beta0[1]
b1=beta0[2]
n=length(x)
llh=0
for(i in 1:n){
r2=(y[i]-b0-b1*x[i])
llh=llh + abs(r2)
}
llh
}
g=optim(c(1,1),fabs,x=x,y=y)
I found the code here:
http://www.stat.colostate.edu/~meyer/hw12ans.pdf
Assuming you are talking about Least absolute deviations, you might not be interested in the above code if you want a solution in R from scratch rather than a solution that uses optim.
The above code is for a regression line with an intercept and one slope. I modified the code as follows to handle a regression with just an intercept:
y <- c(1,1,1)
x <- 1:length(y)
fabs=function(beta0,x,y){
b0=beta0[1]
b1=0
n=length(x)
llh=0
for(i in 1:n){
r2=(y[i]-b0-b1*x[i])
llh=llh + abs(r2)
}
llh
}
# The commands to get the estimator
g = optim(c(1),fabs,x=x,y=y, method='Brent', lower = (min(y)-5), upper = (max(y)+5))
g
I was not familiar with (i.e., had not heard of) Least absolute deviations until tonight. So, hopefully my modifications are fairly reasonable.
With y <- c(1,1,1) the parameter estimate is 1 (which I think you said is the correct answer):
$par
[1] 1
$value
[1] 1.332268e-15
$counts
function gradient
NA NA
$convergence
[1] 0
$message
NULL
With y <- c(0,1,2) the parameter estimate is 1:
$par
[1] 1
$value
[1] 2
$counts
function gradient
NA NA
$convergence
[1] 0
$message
NULL
With y <- c(0,0,3) the parameter estimate is 0 (which you said is the correct answer):
$par
[1] 8.613159e-10
$value
[1] 3
$counts
function gradient
NA NA
$convergence
[1] 0
$message
NULL
If you want R code from scratch, there is additional R code in the file at the link above which might be helpful.
Alternatively, perhaps it might be possible to extract the relevant code from the source file.
Alternatively, perhaps someone else can provide the desired code (and correct any errors on my part) in the next 24 hours.
If you come up with code from scratch please post it as an answer as I would love to see it myself.
lad=function(x,y){
SAD = function(beta, x, y) {
return(sum(abs(y - (beta[1] + beta[2] * x))))
}
d=lm(y~x)
ans1 = optim(par=c(d$coefficients[1], d$coefficients[2]),method = "Nelder-Mead",fn=SAD, x=x, y=y)
coe=setNames(ans1$par,c("(Intercept)",substitute(x)))
fitted=setNames(ans1$par[1]+ans1$par[2]*x,c(1:length(x)))
res=setNames(y-fitted,c(1:length(x)))
results = list(coefficients=coe, fitted.values=fitted, residuals=res)
class(results)="lad"
return(results)
}
I would like to generate correlated variables specified by a correlation matrix.
First I generate the correlation matrix:
require(psych)
require(Matrix)
cor.table <- matrix( sample( c(0.9,-0.9) , 2500 , prob = c( 0.8 , 0.2 ) , repl = TRUE ) , 50 , 50 )
k=1
while (k<=length(cor.table[1,])){
cor.table[1,k]<-0.55
k=k+1
}
k=1
while (k<=length(cor.table[,1])){
cor.table[k,1]<-0.55
k=k+1
}
ind<-lower.tri(cor.table)
cor.table[ind]<-t(cor.table)[ind]
diag(cor.table) <- 1
This correlation matrix is not consistent, therefore, eigenvalue decomposition is impossible.
TO make it consistent I use nearPD:
c<-nearPD(cor.table)
Once this is done I generate the correlated variables:
fit<-principal(c, nfactors=50,rotate="none")
fit$loadings
loadings<-matrix(fit$loadings[1:50, 1:50],nrow=50,ncol=50,byrow=F)
loadings
cases <- t(replicate(50, rnorm(10)) )
multivar <- loadings %*% cases
T_multivar <- t(multivar)
var<-as.data.frame(T_multivar)
cor(var)
However the resulting correlations are far from anything that I specified initially.
Is it not possible to create such correlations or am I doing something wrong?
UPDATE from Greg Snow's comment it became clear that the problem is that my initial correlation matrix is unreasonable.
The question then is how can I make the matrix reasonable. The goal is:
each of the 49 variables should correlate >.5 with the first variable.
~40 of the variables should have a high >.8 correlation with each other
the remaining ~9 variables should have a low or negative correlation with each other.
Is this whole requirement impossible ?
Try using the mvrnorm function from the MASS package rather than trying to construct the variables yourself.
**Edit
Here is a matrix that is positive definite (so it works as a correlation matrix) and comes close to your criteria, you can tweak the values from there (all the Eigen values need to be positive, so you can see how changing a number affects things):
cor.mat <- matrix(0.2,nrow=50, ncol=50)
cor.mat[1,] <- cor.mat[,1] <- 0.55
cor.mat[2:41,2:41] <- 0.9
cor.mat[42:50, 42:50] <- 0.25
diag(cor.mat) <- 1
eigen(cor.mat)$values
Some numerical experimentation based on your specifications above suggests that the generated matrix will never (what never? well, hardly ever ...) be positive definite, but it also doesn't look far from PD with these values (making lcor below negative will almost certainly make things worse ...)
rmat <- function(n=49,nhcor=40,hcor=0.8,lcor=0) {
m <- matrix(lcor,n,n) ## fill matrix with 'lcor'
## select high-cor variables
hcorpos <- sample(n,size=nhcor,replace=FALSE)
## make all of these highly correlated
m[hcorpos,hcorpos] <- hcor
## compute min real part of eigenvalues
min(Re(eigen(m,only.values=TRUE)$values))
}
set.seed(101)
r <- replicate(1000,rmat())
## NEVER pos definite
max(r)
## [1] -1.069413e-15
par(las=1,bty="l")
png("eighist.png")
hist(log10(abs(r)),breaks=50,col="gray",main="")
dev.off()
I have defined a distance function as follow
jaccard.rules.dist <- function(x,y) ({
# implements feature distance. Feature "Airline" gets a different treatment, the rest
# are booleans coded as 1/0. Airline column distance = 0 if same airline, 1 otherwise
# the rest of the atributes' distance is cero iff both are 1, 1 otherwise
airline.column <- which(colnames(x)=="Aerolinea")
xmod <- x
ymod <-y
xmod[airline.column] <-ifelse(x[airline.column]==y[airline.column],1,0)
ymod[airline.column] <-1 # if they are the same, they are both ones, else they are different
andval <- sum(xmod&ymod)
orval <- sum(xmod|ymod)
return (1-andval/orval)
})
which modifies a little bit jaccard distance for dataframes of the form
t <- data.frame(Aerolinea=c("A","B","C","A"),atr2=c(1,1,0,0),atr3=c(0,0,0,1))
Now, I would like to perform some k-means clustering on my dataset, using the distance just defined. If I try to use the function kmeans, there is no way to specify my distance function. I tried the to use hclust, which accepts a distanca matrix, which I calculated as follows
distmat <- matrix(nrow=nrow(t),ncol=nrow(t))
for (i in 1:nrow(t))
for (j in i:nrow(t))
distmat[j,i] <- jaccard.rules.dist(t[j,],t[i,])
distmat <- as.dist(distmat)
and then invoked hclust
hclust(distmat)
Error in if (is.na(n) || n > 65536L) stop("size cannot be NA nor exceed 65536") :
missing value where TRUE/FALSE needed
what am i doing wrong? is there another way to do clustering that just accepts an arbitrary distance function as its input?
thanks in advance.
I think distmat (from your code) has to be a distance structure (which is different from a matrix). Try this instead:
require(proxy)
d <- dist(t, jaccard.rules.dist)
clust <- hclust(d=d)
clust#centers
[,1] [,2]
[1,] 0.044128322 -0.039518142
[2,] -0.986798495 0.975132418
[3,] -0.006441892 0.001099211
[4,] 1.487829642 1.000431146