I'm trying to find a name for my problem, so I don't have to re-invent wheel when coding an algorithm which solves it...
I have say 2,000 binary (row) vectors and I need to pick 500 from them. In the picked sample I do column sums and I want my sample to be as close as possible to a pre-defined distribution of the column sums. I'll be working with 20 to 60 columns.
A tiny example:
Out of the vectors:
110
010
011
110
100
I need to pick 2 to get column sums 2, 1, 0. The solution (exact in this case) would be
110
100
My ideas so far
one could maybe call this a binary multidimensional knapsack, but I did not find any algos for that
Linear Programming could help, but I'd need some step by step explanation as I got no experience with it
as exact solution is not always feasible, something like simulated annealing brute force could work well
a hacky way using constraint solvers comes to mind - first set the constraints tight and gradually loosen them until some solution is found - given that CSP should be much faster than ILP...?
My concrete, practical (if the approximation guarantee works out for you) suggestion would be to apply the maximum entropy method (in Chapter 7 of Boyd and Vandenberghe's book Convex Optimization; you can probably find several implementations with your favorite search engine) to find the maximum entropy probability distribution on row indexes such that (1) no row index is more likely than 1/500 (2) the expected value of the row vector chosen is 1/500th of the predefined distribution. Given this distribution, choose each row independently with probability 500 times its distribution likelihood, which will give you 500 rows on average. If you need exactly 500, repeat until you get exactly 500 (shouldn't take too many tries due to concentration bounds).
Firstly I will make some assumptions regarding this problem:
Regardless whether the column sum of the selected solution is over or under the target, it weighs the same.
The sum of the first, second, and third column are equally weighted in the solution (i.e. If there's a solution whereas the first column sum is off by 1, and another where the third column sum is off by 1, the solution are equally good).
The closest problem I can think of this problem is the Subset sum problem, which itself can be thought of a special case of Knapsack problem.
However both of these problem are NP-Complete. This means there are no polynomial time algorithm that can solve them, even though it is easy to verify the solution.
If I were you the two most arguably efficient solution of this problem are linear programming and machine learning.
Depending on how many columns you are optimising in this problem, with linear programming you can control how much finely tuned you want the solution, in exchange of time. You should read up on this, because this is fairly simple and efficient.
With Machine learning, you need a lot of data sets (the set of vectors and the set of solutions). You don't even need to specify what you want, a lot of machine learning algorithms can generally deduce what you want them to optimise based on your data set.
Both solution has pros and cons, you should decide which one to use yourself based on the circumstances and problem set.
This definitely can be modeled as (integer!) linear program (many problems can). Once you have it, you can use a program such as lpsolve to solve it.
We model vector i is selected as x_i which can be 0 or 1.
Then for each column c, we have a constraint:
sum of all (x_i * value of i in column c) = target for column c
Taking your example, in lp_solve this could look like:
min: ;
+x1 +x4 +x5 >= 2;
+x1 +x4 +x5 <= 2;
+x1 +x2 +x3 +x4 <= 1;
+x1 +x2 +x3 +x4 >= 1;
+x3 <= 0;
+x3 >= 0;
bin x1, x2, x3, x4, x5;
If you are fine with a heuristic based search approach, here is one.
Go over the list and find the minimum squared sum of the digit wise difference between each bit string and the goal. For example, if we are looking for 2, 1, 0, and we are scoring 0, 1, 0, we would do it in the following way:
Take the digit wise difference:
2, 0, 1
Square the digit wise difference:
4, 0, 1
Sum:
5
As a side note, squaring the difference when scoring is a common method when doing heuristic search. In your case, it makes sense because bit strings that have a 1 in as the first digit are a lot more interesting to us. In your case this simple algorithm would pick first 110, then 100, which would is the best solution.
In any case, there are some optimizations that could be made to this, I will post them here if this kind of approach is what you are looking for, but this is the core of the algorithm.
You have a given target binary vector. You want to select M vectors out of N that have the closest sum to the target. Let's say you use the eucilidean distance to measure if a selection is better than another.
If you want an exact sum, have a look at the k-sum problem which is a generalization of the 3SUM problem. The problem is harder than the subset sum problem, because you want an exact number of elements to add to a target value. There is a solution in O(N^(M/2)). lg N), but that means more than 2000^250 * 7.6 > 10^826 operations in your case (in the favorable case where vectors operations have a cost of 1).
First conclusion: do not try to get an exact result unless your vectors have some characteristics that may reduce the complexity.
Here's a hill climbing approach:
sort the vectors by number of 1's: 111... first, 000... last;
use the polynomial time approximate algorithm for the subset sum;
you have an approximate solution with K elements. Because of the order of elements (the big ones come first), K should be a little as possible:
if K >= M, you take the M first vectors of the solution and that's probably near the best you can do.
if K < M, you can remove the first vector and try to replace it with 2 or more vectors from the rest of the N vectors, using the same technique, until you have M vectors. To sumarize: split the big vectors into smaller ones until you reach the correct number of vectors.
Here's a proof of concept with numbers, in Python:
import random
def distance(x, y):
return abs(x-y)
def show(ls):
if len(ls) < 10:
return str(ls)
else:
return ", ".join(map(str, ls[:5]+("...",)+ls[-5:]))
def find(is_xs, target):
# see https://en.wikipedia.org/wiki/Subset_sum_problem#Pseudo-polynomial_time_dynamic_programming_solution
S = [(0, ())] # we store indices along with values to get the path
for i, x in is_xs:
T = [(x + t, js + (i,)) for t, js in S]
U = sorted(S + T)
y, ks = U[0]
S = [(y, ks)]
for z, ls in U:
if z == target: # use the euclidean distance here if you want an approximation
return ls
if z != y and z < target:
y, ks = z, ls
S.append((z, ls))
ls = S[-1][1] # take the closest element to target
return ls
N = 2000
M = 500
target = 1000
xs = [random.randint(0, 10) for _ in range(N)]
print ("Take {} numbers out of {} to make a sum of {}", M, xs, target)
xs = sorted(xs, reverse = True)
is_xs = list(enumerate(xs))
print ("Sorted numbers: {}".format(show(tuple(is_xs))))
ls = find(is_xs, target)
print("FIRST TRY: {} elements ({}) -> {}".format(len(ls), show(ls), sum(x for i, x in is_xs if i in ls)))
splits = 0
while len(ls) < M:
first_x = xs[ls[0]]
js_ys = [(i, x) for i, x in is_xs if i not in ls and x != first_x]
replace = find(js_ys, first_x)
splits += 1
if len(replace) < 2 or len(replace) + len(ls) - 1 > M or sum(xs[i] for i in replace) != first_x:
print("Give up: can't replace {}.\nAdd the lowest elements.")
ls += tuple([i for i, x in is_xs if i not in ls][len(ls)-M:])
break
print ("Replace {} (={}) by {} (={})".format(ls[:1], first_x, replace, sum(xs[i] for i in replace)))
ls = tuple(sorted(ls[1:] + replace)) # use a heap?
print("{} elements ({}) -> {}".format(len(ls), show(ls), sum(x for i, x in is_xs if i in ls)))
print("AFTER {} splits, {} -> {}".format(splits, ls, sum(x for i, x in is_xs if i in ls)))
The result is obviously not guaranteed to be optimal.
Remarks:
Complexity: find has a polynomial time complexity (see the Wikipedia page) and is called at most M^2 times, hence the complexity remains polynomial. In practice, the process is reasonably fast (split calls have a small target).
Vectors: to ensure that you reach the target with the minimum of elements, you can improve the order of element. Your target is (t_1, ..., t_c): if you sort the t_js from max to min, you get the more importants columns first. You can sort the vectors: by number of 1s and then by the presence of a 1 in the most important columns. E.g. target = 4 8 6 => 1 1 1 > 0 1 1 > 1 1 0 > 1 0 1 > 0 1 0 > 0 0 1 > 1 0 0 > 0 0 0.
find (Vectors) if the current sum exceed the target in all the columns, then you're not connecting to the target (any vector you add to the current sum will bring you farther from the target): don't add the sum to S (z >= target case for numbers).
I propose a simple ad hoc algorithm, which, broadly speaking, is a kind of gradient descent algorithm. It seems to work relatively well for input vectors which have a distribution of 1s “similar” to the target sum vector, and probably also for all “nice” input vectors, as defined in a comment of yours. The solution is not exact, but the approximation seems good.
The distance between the sum vector of the output vectors and the target vector is taken to be Euclidean. To minimize it means minimizing the sum of the square differences off sum vector and target vector (the square root is not needed because it is monotonic). The algorithm does not guarantee to yield the sample that minimizes the distance from the target, but anyway makes a serious attempt at doing so, by always moving in some locally optimal direction.
The algorithm can be split into 3 parts.
First of all the first M candidate output vectors out of the N input vectors (e.g., N=2000, M=500) are put in a list, and the remaining vectors are put in another.
Then "approximately optimal" swaps between vectors in the two lists are done, until either the distance would not decrease any more, or a predefined maximum number of iterations is reached. An approximately optimal swap is one where removing the first vector from the list of output vectors causes a maximal decrease or minimal increase of the distance, and then, after the removal of the first vector, adding the second vector to the same list causes a maximal decrease of the distance. The whole swap is avoided if the net result is not a decrease of the distance.
Then, as a last phase, "optimal" swaps are done, again stopping on no decrease in distance or maximum number of iterations reached. Optimal swaps cause a maximal decrease of the distance, without requiring the removal of the first vector to be optimal in itself. To find an optimal swap all vector pairs have to be checked. This phase is much more expensive, being O(M(N-M)), while the previous "approximate" phase is O(M+(N-M))=O(N). Luckily, when entering this phase, most of the work has already been done by the previous phase.
from typing import List, Tuple
def get_sample(vects: List[Tuple[int]], target: Tuple[int], n_out: int,
max_approx_swaps: int = None, max_optimal_swaps: int = None,
verbose: bool = False) -> List[Tuple[int]]:
"""
Get a sample of the input vectors having a sum close to the target vector.
Closeness is measured in Euclidean metrics. The output is not guaranteed to be
optimal (minimum square distance from target), but a serious attempt is made.
The max_* parameters can be used to avoid too long execution times,
tune them to your needs by setting verbose to True, or leave them None (∞).
:param vects: the list of vectors (tuples) with the same number of "columns"
:param target: the target vector, with the same number of "columns"
:param n_out: the requested sample size
:param max_approx_swaps: the max number of approximately optimal vector swaps,
None means unlimited (default: None)
:param max_optimal_swaps: the max number of optimal vector swaps,
None means unlimited (default: None)
:param verbose: print some info if True (default: False)
:return: the sample of n_out vectors having a sum close to the target vector
"""
def square_distance(v1, v2):
return sum((e1 - e2) ** 2 for e1, e2 in zip(v1, v2))
n_vec = len(vects)
assert n_vec > 0
assert n_out > 0
n_rem = n_vec - n_out
assert n_rem > 0
output = vects[:n_out]
remain = vects[n_out:]
n_col = len(vects[0])
assert n_col == len(target) > 0
sumvect = (0,) * n_col
for outvect in output:
sumvect = tuple(map(int.__add__, sumvect, outvect))
sqdist = square_distance(sumvect, target)
if verbose:
print(f"sqdist = {sqdist:4} after"
f" picking the first {n_out} vectors out of {n_vec}")
if max_approx_swaps is None:
max_approx_swaps = sqdist
n_approx_swaps = 0
while sqdist and n_approx_swaps < max_approx_swaps:
# find the best vect to subtract (the square distance MAY increase)
sqdist_0 = None
index_0 = None
sumvect_0 = None
for index in range(n_out):
tmp_sumvect = tuple(map(int.__sub__, sumvect, output[index]))
tmp_sqdist = square_distance(tmp_sumvect, target)
if sqdist_0 is None or sqdist_0 > tmp_sqdist:
sqdist_0 = tmp_sqdist
index_0 = index
sumvect_0 = tmp_sumvect
# find the best vect to add,
# but only if there is a net decrease of the square distance
sqdist_1 = sqdist
index_1 = None
sumvect_1 = None
for index in range(n_rem):
tmp_sumvect = tuple(map(int.__add__, sumvect_0, remain[index]))
tmp_sqdist = square_distance(tmp_sumvect, target)
if sqdist_1 > tmp_sqdist:
sqdist_1 = tmp_sqdist
index_1 = index
sumvect_1 = tmp_sumvect
if sumvect_1:
tmp = output[index_0]
output[index_0] = remain[index_1]
remain[index_1] = tmp
sqdist = sqdist_1
sumvect = sumvect_1
n_approx_swaps += 1
else:
break
if verbose:
print(f"sqdist = {sqdist:4} after {n_approx_swaps}"
f" approximately optimal swap{'s'[n_approx_swaps == 1:]}")
diffvect = tuple(map(int.__sub__, sumvect, target))
if max_optimal_swaps is None:
max_optimal_swaps = sqdist
n_optimal_swaps = 0
while sqdist and n_optimal_swaps < max_optimal_swaps:
# find the best pair to swap,
# but only if the square distance decreases
best_sqdist = sqdist
best_diffvect = diffvect
best_pair = None
for i0 in range(M):
tmp_diffvect = tuple(map(int.__sub__, diffvect, output[i0]))
for i1 in range(n_rem):
new_diffvect = tuple(map(int.__add__, tmp_diffvect, remain[i1]))
new_sqdist = sum(d * d for d in new_diffvect)
if best_sqdist > new_sqdist:
best_sqdist = new_sqdist
best_diffvect = new_diffvect
best_pair = (i0, i1)
if best_pair:
tmp = output[best_pair[0]]
output[best_pair[0]] = remain[best_pair[1]]
remain[best_pair[1]] = tmp
sqdist = best_sqdist
diffvect = best_diffvect
n_optimal_swaps += 1
else:
break
if verbose:
print(f"sqdist = {sqdist:4} after {n_optimal_swaps}"
f" optimal swap{'s'[n_optimal_swaps == 1:]}")
return output
from random import randrange
C = 30 # number of columns
N = 2000 # total number of vectors
M = 500 # number of output vectors
F = 0.9 # fill factor of the target sum vector
T = int(M * F) # maximum value + 1 that can be appear in the target sum vector
A = 10000 # maximum number of approximately optimal swaps, may be None (∞)
B = 10 # maximum number of optimal swaps, may be None (unlimited)
target = tuple(randrange(T) for _ in range(C))
vects = [tuple(int(randrange(M) < t) for t in target) for _ in range(N)]
sample = get_sample(vects, target, M, A, B, True)
Typical output:
sqdist = 2639 after picking the first 500 vectors out of 2000
sqdist = 9 after 27 approximately optimal swaps
sqdist = 1 after 4 optimal swaps
P.S.: As it stands, this algorithm is not limited to binary input vectors, integer vectors would work too. Intuitively I suspect that the quality of the optimization could suffer, though. I suspect that this algorithm is more appropriate for binary vectors.
P.P.S.: Execution times with your kind of data are probably acceptable with standard CPython, but get better (like a couple of seconds, almost a factor of 10) with PyPy. To handle bigger sets of data, the algorithm would have to be translated to C or some other language, which should not be difficult at all.
I am trying to implement a roulette wheel selection. I have understood this algorithm:
Calculate the sum S of all chromosome fitnesses in population
Generate a random number, r, from interval (0,S)
Loop through the population and sum fitnesses from 0 till S, this
is the partial sum, call it P.
When the P > S: stop and return the corresponding chromosome.
What I don't understand is how this corresponds to doing this instead: Roulette wheel selection algorithm
(the answer with 44 votes). This makes sense to me, but not the one above.
The following is done using the sum
def choose_parent_using_RWS(genes, S, points):
P = randint(0, int(S))
for x in genes:
P += evaluate(x, points)
if P > S:
return x
return genes[-1]
the following is done by normalizing between 0 and 1
def choose_parent_using_RWS(genes, S, points):
P = randint(0, int(S))/S
for x in genes:
P += evaluate(x, points)/S
if P > S/S:
return x
return genes[-1]
In the answer with 44 votes, the range has been normalised between 0 to 1 which is easier to understand but requires extra steps for calculations.
You can implement the approach you mentioned. In that while calculating the sum, each individual chromosome adds its own value, so when a random number is generated between 0 and S we assume that if r is between 2 numbers whose range is equal to the above mentioned value, it is chosen with the probability proportional to its fitness value. The bigger the value the more is the probability of r to come in its range.
For example, lets say that a chromosome having a fitness of 23 (assumption) is the 5th chromosome when you iterate and the total sum S is 130. The sum of the first 4 chromosomes is, lets say, 54. So if random r is between 55 and 77 (both inclusive), this chromosome is chosen.
After normalisation, 55/130 ~= 0.423 and 77/130 ~= 0.5923 is the range a random number r2 (between 0 and 1) should fall for this chromosome to be selected.
I have small question and I will be very happy if you can give me a solution or any idea for solution of probability distribution of the following idea:
I have a random variable x which follows exponntial distribution with parameter lambda1,I have one more variable y which follows exponential distribution with parameter lambda2. z is a discrete value, how can I define the probability distribution of k in the following formula ?
k=z-x-y
Thank you so much
Ok, lets start with rewriting formula a bit:
k = z-x-y = -(x-y) + z = - (x + y + -z)
That parts in the parentheses looks manageable. Let's start with x+y. For random variable x and y if one wants to find out their sum, answer is PDFs convolution.
q = x+y
PDF(q) = S PDFx(q-t) PDFy(t) dt
where S denotes integration. For x and y being exponential, the convolution integral is known and equal to expression here when lambdas are different, or to Gamma(2,lambda) when lambdas are equal, Gamma being Gamma distribution.
If z is some constant discrete value, then we could express it as continuous RV with PDF
PDF(t) = 𝛿(t+z)
where 𝛿 is Delta function, and we take into account that peak would be at -z as expected. It is normalized, so integral over t is eqaul to 1. It could be easily extended to discrete RV, as sum of 𝛿-functions at those values, multiplied by probabilities such that sum of them is equal to 1.
Again, we have sum of two RV, with known PDFs, and solution is convolution, which is easy to compute due to property of 𝛿-function. So final PDF of x + y + -z would be
PDF(q+z) dq
where PDF is taken from sum expression from Exponential distribution wiki, of Gamma distribution from Gamma wiki.
You just have to negate, and that's it
I'm currently trying to optimize some MATLAB/Octave code by means of an algorithmic change, but can't figure out how to deal with some randomness here. Suppose that I have a vector V of integers, with each element representing a count of some things, photons in my case. Now I want to randomly pick some amount of those "things" and create a new vector of the same size, but with the counts adjusted.
Here's how I do this at the moment:
function W = photonfilter(V, eff)
% W = photonfilter(V, eff)
% Randomly takes photons from V according to the given efficiency.
%
% Args:
% V: Input vector containing the number of emitted photons in each
% timeslot (one element is one timeslot). The elements are rounded
% to integers before processing.
% eff: Filter efficiency. On the average, every 1/eff photon will be
% taken. This value must be in the range 0 < eff <= 1.
% W: Output row vector with the same length as V and containing the number
% of received photons in each timeslot.
%
% WARNING: This function operates on a photon-by-photon basis in that it
% constructs a vector with one element per photon. The storage requirements
% therefore directly depend on sum(V), not only on the length of V.
% Round V and make it flat.
Ntot = length(V);
V = round(V);
V = V(:);
% Initialize the photon-based vector, so that each element contains
% the original index of the photon.
idxV = zeros(1, sum(V), 'uint32');
iout = 1;
for i = 1:Ntot
N = V(i);
idxV(iout:iout+N-1) = i;
iout = iout + N;
end;
% Take random photons.
idxV = idxV(randperm(length(idxV)));
idxV = idxV(1:round(length(idxV)*eff));
% Generate the output vector by placing the remaining photons back
% into their timeslots.
[W, trash] = hist(idxV, 1:Ntot);
This is a rather straightforward implementation of the description above. But it has an obvious performance drawback: The function creates a vector (idxV) containing one element per single photon. So if my V has only 1000 elements but an average count of 10000 per element, the internal vector will have 10 million elements making the function slow and heavy.
What I'd like to achieve now is not to directly optimize this code, but to use some other kind of algorithm which immediately calculates the new counts without giving each photon some kind of "identity". This must be possible somehow, but I just can't figure out how to do it.
Requirements:
The output vector W must have the same number of elements as the input vector V.
W(i) must be an integer and bounded by 0 <= W(i) <= V(i).
The expected value of sum(W) must be sum(V)*eff.
The algorithm must somehow implement this "random picking" of photons, i.e. there should not be some deterministic part like "run through V dividing all counts by the stepsize and propagating the remainders", as the whole point of this function is to bring randomness into the system.
An explicit loop over V is allowed if unavoidable, but a vectorized approach is preferable.
Any ideas how to implement something like this? A solution using only a random vector and then some trickery with probabilities and rounding would be ideal, but I haven't had any success with that so far.
Thanks! Best regards, Philipp
The method you employ to compute W is called Monte Carlo method. And indeed there can be some optimizations. Once of such is instead of calculating indices of photons, let's imagine a set of bins. Each bin has some probability and the sum of all bins' probabilities adds up to 1. We divide the segment [0, 1] into parts whose lengths are proportional to the probabilities of the bins. Now for every random number within [0, 1) that we generate we can quickly find the bin that it belongs to. Finally, we count numbers in the bins to obtain the final result. The code below illustrates the idea.
% Population size (number of photons).
N = 1000000;
% Sample size, size of V and W as well.
% For convenience of plotting, V and W are of the same size, but
% the algorithm doesn't enforce this constraint.
M = 10000;
% Number of Monte Carlo iterations, greater numbers give better quality.
K = 100000;
% Generate population of counts, use gaussian distribution to test the method.
% If implemented correctly histograms should have the same shape eventually.
V = hist(randn(1, N), M);
P = cumsum(V / sum(V));
% For every generated random value find its bin and then count the bins.
% Finally we normalize counts by the ration of N / K.
W = hist(lookup(P, rand(1, K)), M) * N / K;
% Compare distribution plots, they should be the same.
hold on;
plot(W, '+r');
plot(V, '*b');
pause
Based on the answer from Alexander Solovets, this is how the code now looks:
function W = photonfilter(V, eff, impl=1)
Ntot = length(V);
V = V(:);
if impl == 0
% Original "straightforward" solution.
V = round(V);
idxV = zeros(1, sum(V), 'uint32');
iout = 1;
for i = 1:Ntot
N = V(i);
idxV(iout:iout+N-1) = i;
iout = iout + N;
end;
idxV = idxV(randperm(length(idxV)));
idxV = idxV(1:round(length(idxV)*eff));
[W, trash] = hist(idxV, 1:Ntot);
else
% Monte Carlo approach.
Nphot = sum(V);
P = cumsum(V / Nphot);
W = hist(lookup(P, rand(1, round(Nphot * eff))), 0:Ntot-1);
end;
The results are quite comparable, as long as eff if not too close to 1 (with eff=1, the original solution yields W=V while the Monte Carlo approach still has some randomness, thereby violating the upper bound constraints).
Test in the interactive Octave shell:
octave:1> T=linspace(0,10*pi,10000);
octave:2> V=100*(1+sin(T));
octave:3> W1=photonfilter(V, 0.1, 0);
octave:4> W2=photonfilter(V, 0.1, 1);
octave:5> plot(T,V,T,W1,T,W2);
octave:6> legend('V','Random picking','Monte Carlo')
octave:7> sum(W1)
ans = 100000
octave:8> sum(W2)
ans = 100000
Plot:
I have a matrix z (3 x 20000). Consider each row as a random variable and each column as one simulation. I wrote the following function in R using apply command to find the empirical cumulative distribution function (EMP.CDF) in 3 dimensions. This k-variate empirical CDF was explained on the page 2 of this pdf, under the section of "The Multivariate ECDF".
EMP.CDF=function(z) {
# z is a matrix (3 x 20000) and each row is a realization of a random variable
q1=z[1,];q2=z[2,];q3=z[3,]
# qi = the realization of the ith random variable, i=1,2,3
# Now I am going to evaluate the empirical cumulative distribution function at
# each column of z
# Given each column, the function should return an empirical
# cumulative probability.
d=apply(z,2, function(x) sum(q1<=x[1] & q2<=x[2] & q3<=x[3])/(length(q1)))
return(d)}
> z=matrix(0,3,20000)
> z[1,]=runif(20000,1,2)
> z[2,]=runif(20000,3,5)
> z[3,]=runif(20000,7,9)
> system.time(EMP.CDF(z))
user system elapsed
30.18 0.01 30.39
In above code k=3. Is there any way I can vectorize the above function to reduce the system time?
A 3-dimensional cumulative distribution function is a function of 3 variables.
If you estimate it on a grid, it could be represented as a 3-dimensional array,
but it would be imprecise and huge (your function returns a 1-dimensional array,
so it is not what it is computing).
Given a point x, just compute the proportion of points all of whose coordinates are less than those of x.
z <- matrix(runif(60000), 3, 20000)
emp.cdf <- function(z)
function(x) mean( apply( z <= x, 2, all ) )
emp.cdf(z)( c(.5,.5,.5) ) # Approximately 1/8
The following reproduces the plots in the document you cite:
n <- 10
z <- matrix(runif(2*n), 2, n)
f <- emp.cdf(z)
g <- function(u,v) f(c(u,v))
persp( outer( sort(z[1,]), sort(z[2,]), Vectorize(g) ) )
x <- seq(0,1,length=100)
persp( outer( x, x, Vectorize(g) ) )
If you want to evaluate the cumulative probability distribution on the initial points,
you can just use apply (if you wanted to evaluate it on a grid, you could use expand.grid to build it).
n <- 100
z <- matrix(runif(3*n), 3, n)
f <- emp.cdf(z)
p <- apply( z, 2, f )
But this algorithm is quadratic: there are n probabilities to compute,
and for each of them, we examine all 3*n coordinates.
For your 20,000 points, that will take a while.
You can use a divide-and-conquer approach
to speed up the computations,
but it is not straightforward:
pick up a point at random,
use it to separate the space into 8 octants,
recursively compute the number of points in each octant;
you can then use the resulting tree
to compute the probability at any point,
by examining only a fraction of the points.
This is not unlike the preprocessing step
used to compute the k-nearest neighbours,
or to speed up n-body simulations.