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I have a program which calculates probability values
(p-values),
but it is entering a very large negative number into the
exp function
exp(-626294.830) which evaluates to zero instead of the very small
positive number that it should be.
How can I get this to evaluate as a very small floating point number?
I have tried
Math::BigFloat,
bignum, and
bigrat
but all have failed.
Wolfram Alpha says that exp(-626294.830) is 4.08589×10^-271997... zero is a pretty close approximation to that ;-) Although you've edited and removed the context from your question, do you really need to work with such tiny numbers, or perhaps there is some way you could optimize your algorithm or scale your numbers?
Anyway, you are correct that code like Math::BigFloat->new("-626294.830")->bexp seems to take quite some time, even with the support of use Math::BigFloat lib => 'GMP';.
The only alternative I can offer at the moment is Math::Prime::Util::GMP's expreal, although you need to specify a precision to it.
use Math::Prime::Util::GMP qw/expreal/;
use Math::BigFloat;
my $e = Math::BigFloat->new(expreal(-626294.830,272000));
print $e->bnstr,"\n";
__END__
4.086e-271997
But on my machine, even that still takes ~20s to run, which brings us back to the question of potential optimization in other places.
Floating point numbers do not have infinite precision. Assuming the number is represented as an IEEE 754 double, we have 52 bits for a fraction, 11 bits for the exponent, and one bit for the sign. Due to the way exponents are encoded, the smallest positive number that can be represented is 2^-1022.
If we look at your number e^-626294.830, we can do a change of base and see that it equals 2^(log_2 e · -626294.830) = 2^-903552.445, which is significantly smaller than 2^-1022. Approximating your number as zero is therefore correct.
Instead of calculating this value using arbitrary-precision numerics, you are likely better off solving the necessary equations by hand, then coding this in a way that does not require extreme precision. For example, it is unlikely that you need the exact value of e^-626294.830, but perhaps just the magnitude. Then, you can calculate the logarithm instead of using exp().
I am trying to turn off denormal number support in matlab, so that basically any two computations that would result in a denormal number would instead just result in zero (DAZ, FTZ)
I've researched several sites include the one below, but I haven't found anything about doing this.
http://blogs.mathworks.com/cleve/2014/07/21/floating-point-denormals-insignificant-but-controversial-2/
I've never heard of such an option in Matlab. It would likely require deep manipulation of a lot of the floating-point math, effectively requiring a new datatype to be supported if this were to be an easily toggle-able option in Matlab. You could write your own mex C code to do this (more here and here) for an individual function.
And of course you can get something like this with one line of Matlab – here's an example:
a = [1e-300 1e-310 1e-310];
b = [1e-301 1e-311 1e-310];
x = a-b;
x(abs(x(:)) < realmin(class(x))) = 0;
where realmin is the smallest normalized floating-point number. However, the floating point math is still performed using the extended denormal/subnormal values in a. It's just the output that's clipped to zero.
Unless you're doing this for fun an experimentation, or possibly running code on an embedded platform, I'd really recommend against disabling denormals as a form of optimization. Instead, focus on why your values are so small and how you might rescale your problem to avoid the issue entirely.
Is it a good practice to manually set numbers with large negative exponential like 1e-300 to zero to avoid underflow in Matlab?
If not how can we avoid harm of underflow when implementing functions like log(1+exp(x))?
Typically, you get trouble when adding very large and very small values, because this can lead to a high relative error. Get rid of this summation (1+exp(x)), it quickly runs out of the range of double values when x is large.
log(1+exp(x))
log(1+1/exp(x))*exp(x))
log(1+1/exp(x))+log(exp(x))
log(1+1/exp(x))+x
An alternative is the use of vpa:
log(1+exp(vpa(10^6)))
very slow, but you get a result with the configured precision.
I never saw a case where manually setting small values to zero was a good solution, typically comparing with a tolerance is better.
I know matlab has a built in pdist function that will calculate pairwise distances. However, my matrix is so large that its 60000 by 300 and matlab runs out of memory.
This question is a follow up on Matlab euclidean pairwise square distance function.
Is there any workaround for this computational inefficiency. I tried manually coding the pairwise distance calculations and it usually takes a full day to run (sometimes 6 to 7 hours).
Any help is greatly appreciated!
Well, I couldn't resist playing around. I created a Matlab mex C file called pdistc that implements pairwise Euclidean distance for single and double precision. On my machine using Matlab R2012b and R2015a it's 20–25% faster than pdist(and the underlying pdistmex helper function) for large inputs (e.g., 60,000-by-300).
As has been pointed out, this problem is fundamentally bounded by memory and you're asking for a lot of it. My mex C code uses minimal memory beyond that needed for the output. In comparing its memory usage to that of pdist, it looks like the two are virtually the same. In other words, pdist is not using lots of extra memory. Your memory problem is likely in the memory used up before calling pdist (can you use clear to remove any large arrays?) or simply because you're trying to solve a big computational problem on tiny hardware.
So, my pdistc function likely won't be able to save you memory overall, but you may be able to use another feature I built in. You can calculate chunks of your overall pairwise distance vector. Something like this:
m = 6e3;
n = 3e2;
X = rand(m,n);
sz = m*(m-1)/2;
for i = 1:m:sz-m
D = pdistc(X', i, i+m); % mex C function, X is transposed relative to pdist
... % Process chunk of pairwise distances
end
This is considerably slower (10 times or so) and this part of my C code is not optimized well, but it will allow much less memory use – assuming that you don't need the entire array at one time. Note that you could do the same thing much more efficiently with pdist (or pdistc) by creating a loop where you passed in subsets of X directly, rather than all of it.
If you have a 64-bit Intel Mac, you won't need to compile as I've included the .mexmaci64 binary, but otherwise you'll need to figure out how to compile the code for your machine. I can't help you with that. It's possible that you may not be able to get it to compile or that there will be compatibility issues that you'll need to solve by editing the code yourself. It's also possible that there are bugs and the code will crash Matlab. Also, note that you may get slightly different outputs relative to pdist with differences between the two in the range of machine epsilon (eps). pdist may or may not do fancy things to avoid overflows for large inputs and other numeric issues, but be aware that my code does not.
Additionally, I created a simple pure Matlab implementation. It is massively slower than the mex code, but still faster than a naïve implementation or the code found in pdist.
All of the files can be found here. The ZIP archive includes all of the files. It's BSD licensed. Feel free to optimize (I tried BLAS calls and OpenMP in the C code to no avail – maybe some pointer magic or GPU/OpenCL could further speed it up). I hope that it can be helpful to you or someone else.
On my system the following is the fastest (Even faster than the C code pdistc by #horchler):
function [ mD ] = CalcDistMtx ( mX )
vSsqX = sum(mX .^ 2);
mD = sqrt(bsxfun(#plus, vSsqX.', vSsqX) - (2 * (mX.' * mX)));
end
You'll need a very well tuned C code to beat this, I think.
Update
Since MATLAB R2016b MATLAB supports implicit broadcasting without the use of bsxfun().
Hence the code can be written:
function [ mD ] = CalcDistMtx ( mX )
vSsqX = sum(mX .^ 2, 1);
mD = sqrt(vSsqX.'+ vSsqX - (2 * (mX.' * mX)));
end
A generalization is given in my Calculate Distance Matrix project.
P. S.
Using MATLAB's pdist for comparison: squareform(pdist(mX.')) is equivalent to CalcDistMtx(mX).
Namely the input should be transposed.
Computers are not infinitely large, or infinitely fast. People think that they have a lot of memory, a fast CPU, so they just create larger and larger problems, and then eventually wonder why their problem runs slowly. The fact is, this is NOT computational inefficiency. It is JUST an overloaded CPU.
As Oli points out in a comment, there are something like 2e9 values to compute, even assuming you only compute the upper or lower half of the distance matrix. (6e4^2/2 is approximately 2e9.) This will require roughly 16 gigabytes of RAM to store, assuming that only ONE copy of the array is created in memory. If your code is sloppy, you might easily double or triple that. As soon as you go into virtual memory, things get much slower.
Wanting a big problem to run fast is not enough. To really help you, we need to know how much RAM is available. Is this a virtual memory issue? Are you using 64 bit MATLAB, on a CPU that can handle all the needed RAM?
I am clustering a large set of points. Throughout the iterations, I want to avoid re-computing cluster properties if the assigned points are the same as the previous iteration. Each cluster keeps the IDs of its points. I don't want to compare them element wise, comparing the sum of the ID vector is risky (a small ID can be compensated with a large one), may be I should compare the sum of squares? Is there a hashing method in Matlab which I can use with confidence?
Example data:
a=[2,13,14,18,19,21,23,24,25,27]
b=[6,79,82,85,89,111,113,123,127,129]
c=[3,9,59,91,99,101,110,119,120,682]
d=[11,57,74,83,86,90,92,102,103,104]
So the problem is that if I just check the sum, it could be that cluster d for example, looses points 11,103 and gets 9,105. Then I would mistakenly think that there has been no change in the cluster.
This is one of those (very common) situations where the more we know about your data and application the better we are able to help. In the absence of better information than you provide, and in the spirit of exposing the weakness of answers such as this in that absence, here are a couple of suggestions you might reject.
One appropriate data structure for set operations is a bit-set, that is a set of length equal to the cardinality of the underlying universe of things in which each bit is set on or off according to the things membership of the (sub-set). You could implement this in Matlab in at least two ways:
a) (easy, but possibly consuming too much space): define a matrix with as many columns as there are points in your data, and one row for each cluster. Set the (cluster, point) value to true if point is a member of cluster. Set operations are then defined by vector operations. I don't have a clue about the relative (time) efficiency of setdiff versus rowA==rowB.
b) (more difficult): actually represent the clusters by bit sets. You'll have to use Matlab's bit-twiddling capabilities of course, but the pain might be worth the gain. Suppose that your universe comprises 1024 points, then you'll need an array of 16 uint64 values to represent the bit set for each cluster. The presence of, say, point 563 in a cluster requires that you set, for the bit set representing that cluster, bit 563 (which is probably bit 51 in the 9th element of the set) to 1.
And perhaps I should have started by writing that I don't think that this is a hashing sort of a problem, it's a set sort of a problem. Yeah, you could use a hash but then you'll have to program around the limitations of using a screwdriver on a nail (choose your preferred analogy).
If I understand correctly, to hash the ID's I would recommend using the matlab Java interface to use the Java hashing algorithms
http://docs.oracle.com/javase/1.4.2/docs/api/java/security/MessageDigest.html
You'll do something like:
hash = java.security.MessageDigest.getInstance('SHA');
Hope this helps.
I found the function
DataHash on FEX it is quiet fast for vectors and the strcmp on the keys is a lot faster than I expected.