Hello
Assume I have the set of numbers I want a quick to calculate some measure of uniformity.
I know the variance is the most obvious answer but i am afraid the complexity of naive algorithm is too high
Anyone have any suggestions?
"Intuitive" algorithms for calculating variance usually suffer one or both of the following:
Use two loops (one for calculating the mean, the other for the variance)
Are not numerically stable
A good algorithm, with only one loop and numerically stable is due to D. Knuth (as always).
From Wikipedia:
n = 0
mean = 0
M2 = 0
def calculate_online_variance(x):
n = n + 1
delta = x - mean
mean = mean + delta/n
M2 = M2 + delta*(x - mean) # This expression uses the new value of mean
variance_n = M2/n
variance = M2/(n - 1) #note on the first pass with n=1 this will fail (should return Inf)
return variance
You should invoke calculate_online_variance(x) for each point, and it returns the variance calculated so far.
Related
I'm trying to calculate the average of a value that is changing, and I would like to do so without storing all the previous values in an array and iterating over them.
I found this formula
avg = avg + (value - avg) / n
where n is the number of changes to value.
TL;DR
My question is if this formula is identical to the normal way of
calculating an average (which it seems to be when I compare them), or
if it might give different results under certain circumstances?
I'm not sure what the correct name of this formula is - I've seen "running average, "rolling average", "moving average", etc. The results of it seem to be exactly the same as storing each historical value, summing them up and dividing by n - i.e. a "normal average".
What's confusing is that people sometimes call this formula a "moving average", which in my mind sounds more like you're using a subset of the historical values to calculate an average. Others say it's an exponential moving average (see comment by Julia on OP).
Is this formula is identical to the normal way of calculating an average?
With infinite precision, this formula does indeed compute the sum of the first n samples if avg is set equal to 0 at the start.
It is clearly true when n=1 because the average of 1 sample works out as:
avg' = avg + (value - avg) / n
= 0 + (value - 0) / 1
= value
For larger values, assume it is true for n-1 (i.e. avg=(x[1]+..+x[n-1])/(n-1) ).
Then:
avg' = avg + (x[n] - avg) / n
= (n-1)*avg/n + x[n]/n
= (x[1]+...+x[n-1])/n + x[n]/n
= (x[1]+...+x[n])/n
So the new value of avg is also equal to the average of the first n samples.
Is this a moving average?
Normally by "moving average" people are referring to a simple moving average.
This formula is actually known as a cumulative moving average.
I'm currently utilizing an online variance algorithm to calculate the variance for a given sequence. This works nicely, and also gives good numerical stability and overflow resistance, at the cost of some speed, which is fine. My question is, does an algorithm exist that will be faster than this if the sample mean is already known, while having similar stability and resistance to overflow (hence not something like a naive variance calculation).
The current online variance calculation is a single-pass algorithm with both divisions and multiplications in the main loop (which is what impacts speed). From wikipedia:
def online_variance(data):
n = 0
mean = 0
M2 = 0
for x in data:
n = n + 1
delta = x - mean
mean = mean + delta/n
M2 = M2 + delta*(x - mean)
variance = M2/(n - 1)
return variance
The thing that causes a naive variance calculation to go unstable is the fact that you separately sum the X (to get mean(x)) and the X^2 values and then take the difference
var = mean(x^2) - (mean(x))^2
But since the definition of variance is
var = mean((x - mean(x))^2)
You can just evaluate that and it will be as fast as it can be. When you don't know the mean, you have to compute it first for stability, or use the "naive" formulation that goes through the data only once at the expense of numerical stability.
EDIT
Now that you have given the "original" code, it's easy to be better (faster). As you correctly point out, the division in the inner loop is slowing you down. Try this one for comparison:
def newVariance(data, mean):
n = 0
M2 = 0
for x in data:
n = n + 1
delta = x - mean
M2 = M2 + delta * delta
variance = M2 / (n - 1)
return variance
Note - this looks a lot like the two_pass_variance algorithm from Wikipedia, except that you don't need the first pass to compute the mean since you say it is already known.
I'm trying to calculate the absolute deviation of a vector online, that is, as each item in the vector is received, without using the entire vector. The absolute deviation is the sum of the absolute difference between each item in a vector and the mean:
I know that the variance of a vector can be calculated in such a manner. Variance is similar to absolute deviation, but each difference is squared:
The online algorithm for variance is as follows:
n = 0
mean = 0
M2 = 0
def calculate_online_variance(x):
n = n + 1
delta = x - mean
mean = mean + delta/n
M2 = M2 + delta*(x - mean) # This expression uses the new value of mean
variance_n = M2/n
return variance_n
Is there such an algorithm for calculating absolute deviance? I cannot formulate a recursive definition myself, but wiser heads may prevail!
As the absolute deviation between x and the mean can be defined as the square root of the squared difference, the adaptation is trivial if you are happy with a consistent but biased estimate (meaning the limit to infinity is the expected value) :
n = 0
mean = 0
M2 = 0
def calculate_online_avg_abs_dev(x):
n = n + 1
delta = x - mean
mean = mean + delta/n
M2 = M2 + sqrt(delta*(x - mean))
avg_abs_dev_n = M2/n
This is for the case of the average absolute deviation. Normally the mad is used (median absolute deviation), which is impossible to program recursively. but the average absolute deviation is as useful in most cases. When we're talking about hundreds of values from close-to-normal distributions, both values are very close.
If you just want the sum of the absolute devations, life is even simpler: just return M2.
Be aware of the fact that BOTH the algorithm you gave and the trivial adaptation for the absolute deviation are slightly biased.
A simulation in R to prove the algorithm works this way :
The red line is the true value, the black line is the progressive value following the algorithm outlined above.
Code :
calculate_online_abs_dev <- function(x,n){
M2=0
mean=0
out <- numeric(n)
for(i in 1:n) {
delta <- x[i] - mean
mean <- mean + delta/i
M2 = M2 + sqrt(delta*(x[i] - mean))
out[i] <- M2/i
}
return(out)
}
set.seed(2010)
x <- rnorm(100)
Abs_Dev <- calculate_online_abs_dev(x,length(x))
True_Val <- sapply(1:length(x),function(i)sum(abs(x[1:i]-mean(x[1:i])))/i)
plot(1:length(x),Abs_Dev,type="l",xlab="number of values",lwd=2)
lines(1:length(x),True_Val,col="red",lty=2,lwd=2)
legend("bottomright",lty=c(1,2),col=c("black","red"),
legend=c("Online Calc","True Value"))
I don't think it's possible.
In the formula for variance it is possible to separate the x and x2 terms, so that it suffices to keep track of those sums (and n). In the formula for absolute deviation this is not possible.
I think the best one can do (apart from keeping the whole vector and calculating the absolute deviation on demand) is keep a sorted list of elements. This is O(log(n)) for each new element, but after you add an element the cost of recalculating the absolute deviation is O(log(n)). This may or may not be worthwhile, depending on your application.
The formula for variance that you give is ONE of the many that are possible (I can think of three distinct ways to do that computation) although I have not verified that yours is correct. It looks reasonably close to what I recall.
The problem is that absolute value is actually more "nonlinear" in a sense than is the sum of squares of the deviations. This prevents you from being able to do that calculation in a recursive form in a loop, at least not without retaining all of the previous values of x. You must compute the overall mean in advance for that sum.
Edit: I see that beta agrees with me. IF you saved all of the previous data points, in a sorted list, you could then efficiently compute the updated desired deviation. But this is counter to the spirit of your request.
I've been trying to implement a modular exponentiator recently. I'm writing the code in VHDL, but I'm looking for advice of a more algorithmic nature. The main component of the modular exponentiator is a modular multiplier which I also have to implement myself. I haven't had any problems with the multiplication algorithm- it's just adding and shifting and I've done a good job of figuring out what all of my variables mean so that I can multiply in a pretty reasonable amount of time.
The problem that I'm having is with implementing the modulus operation in the multiplier. I know that performing repeated subtractions will work, but it will also be slow. I found out that I could shift the modulus to effectively subtract large multiples of the modulus but I think there might still be better ways to do this. The algorithm that I'm using works something like this (weird pseudocode follows):
result,modulus : integer (n bits) (previously defined)
shiftcount : integer (initialized to zero)
while( (modulus<result) and (modulus(n-1) != 1) ){
modulus = modulus << 1
shiftcount++
}
for(i=shiftcount;i>=0;i--){
if(modulus<result){result = result-modulus}
if(i!=0){modulus = modulus >> 1}
}
So...is this a good algorithm, or at least a good place to start? Wikipedia doesn't really discuss algorithms for implementing the modulo operation, and whenever I try to search elsewhere I find really interesting but incredibly complicated (and often unrelated) research papers and publications. If there's an obvious way to implement this that I'm not seeing, I'd really appreciate some feedback.
I'm not sure what you're calculating there to be honest. You talk about modulo operation, but usually a modulo operation is between two numbers a and b, and its result is the remainder of dividing a by b. Where is the a and b in your pseudocode...?
Anyway, maybe this'll help: a mod b = a - floor(a / b) * b.
I don't know if this is faster or not, it depends on whether or not you can do division and multiplication faster than a lot of subtractions.
Another way to speed up the subtraction approach is to use binary search. If you want a mod b, you need to subtract b from a until a is smaller than b. So basically you need to find k such that:
a - k*b < b, k is min
One way to find this k is a linear search:
k = 0;
while ( a - k*b >= b )
++k;
return a - k*b;
But you can also binary search it (only ran a few tests but it worked on all of them):
k = 0;
left = 0, right = a
while ( left < right )
{
m = (left + right) / 2;
if ( a - m*b >= b )
left = m + 1;
else
right = m;
}
return a - left*b;
I'm guessing the binary search solution will be the fastest when dealing with big numbers.
If you want to calculate a mod b and only a is a big number (you can store b on a primitive data type), you can do it even faster:
for each digit p of a do
mod = (mod * 10 + p) % b
return mod
This works because we can write a as a_n*10^n + a_(n-1)*10^(n-1) + ... + a_1*10^0 = (((a_n * 10 + a_(n-1)) * 10 + a_(n-2)) * 10 + ...
I think the binary search is what you're looking for though.
There are many ways to do it in O(log n) time for n bits; you can do it with multiplication and you don't have to iterate 1 bit at a time. For example,
a mod b = a - floor((a * r)/2^n) * b
where
r = 2^n / b
is precomputed because typically you're using the same b many times. If not, use the standard superconverging polynomial iteration method for reciprocal (iterate 2x - bx^2 in fixed point).
Choose n according to the range you need the result (for many algorithms like modulo exponentiation it doesn't have to be 0..b).
(Many decades ago I thought I saw a trick to avoid 2 multiplications in a row... Update: I think it's Montgomery Multiplication (see REDC algorithm). I take it back, REDC does the same work as the simpler algorithm above. Not sure why REDC was ever invented... Maybe slightly lower latency due to using the low-order result into the chained multiplication, instead of the higher-order result?)
Of course if you have a lot of memory, you can just precompute all the 2^n mod b partial sums for n = log2(b)..log2(a). Many software implementations do this.
If you're using shift-and-add for the multiplication (which is by no means the fastest way) you can do the modulo operation after each addition step. If the sum is greater than the modulus you then subtract the modulus. If you can predict the overflow, you can do the addition and subtraction at the same time. Doing the modulo at each step will also reduce the overall size of your multiplier (same length as input rather than double).
The shifting of the modulus you're doing is getting you most of the way towards a full division algorithm (modulo is just taking the remainder).
EDIT Here is my implementation in Python:
def mod_mul(a,b,m):
result = 0
a = a % m
b = b % m
while (b>0):
if (b&1)!=0:
result += a
if result >= m: result -= m
a = a << 1
if a>=m: a-= m
b = b>>1
return result
This is just modular multiplication (result = a*b mod m). The modulo operations at the top are not needed, but serve as a reminder that the algorithm assumes a and b are less than m.
Of course for modular exponentiation you'll have an outer loop that does this entire operation at each step doing either squaring or multiplication. But I think you knew that.
For modulo itself, I'm not sure. For modulo as part of the larger modular exponential operation, did you look up Montgomery multiplication as mentioned in the wikipedia page on modular exponentiation? It's been a while since I've looked into this type of algorithm, but from what I recall, it's commonly used in fast modular exponentiation.
edit: for what it's worth, your modulo algorithm seems ok at first glance. You're basically doing division which is a repeated subtraction algorithm.
That test (modulus(n-1) != 1) //a bit test?
-seems redundant combined with (modulus<result).
Designing for hardware implementation i would be conscious of the smaller/greater than tests implying more logic (subtraction) than bitwise operations and branching on zero.
If we can do bitwise tests easily, this could be quick:
m=msb_of(modulus)
while( result>0 )
{
r=msb_of(result) //countdown from prev msb onto result
shift=r-m //countdown from r onto modulus or
//unroll the small subtraction
takeoff=(modulus<<(shift)) //or integrate this into count of shift
result=result-takeoff; //necessary subtraction
if(shift!=0 && result<0)
{ result=result+(takeoff>>1); }
} //endwhile
if(result==0) { return result }
else { return result+takeoff }
(code untested may contain gotchas)
result is repetively decremented by modulus shifted to match at most significant bits.
After each subtraction: result has a ~50/50 chance of loosing more than 1 msb. It also has ~50/50 chance of going negative,
addition of half what was subtracted will always put it into positive again. > it should be put back in positive if shift was not=0
The working loop exits when result is underrun and 'shift' was 0.
Given positive integers b, c, m where (b < m) is True it is to find a positive integer e such that
(b**e % m == c) is True
where ** is exponentiation (e.g. in Ruby, Python or ^ in some other languages) and % is modulo operation. What is the most effective algorithm (with the lowest big-O complexity) to solve it?
Example:
Given b=5; c=8; m=13 this algorithm must find e=7 because 5**7%13 = 8
From the % operator I'm assuming that you are working with integers.
You are trying to solve the Discrete Logarithm problem. A reasonable algorithm is Baby step, giant step, although there are many others, none of which are particularly fast.
The difficulty of finding a fast solution to the discrete logarithm problem is a fundamental part of some popular cryptographic algorithms, so if you find a better solution than any of those on Wikipedia please let me know!
This isn't a simple problem at all. It is called calculating the discrete logarithm and it is the inverse operation to a modular exponentation.
There is no efficient algorithm known. That is, if N denotes the number of bits in m, all known algorithms run in O(2^(N^C)) where C>0.
Python 3 Solution:
Thankfully, SymPy has implemented this for you!
SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python.
This is the documentation on the discrete_log function. Use this to import it:
from sympy.ntheory import discrete_log
Their example computes \log_7(15) (mod 41):
>>> discrete_log(41, 15, 7)
3
Because of the (state-of-the-art, mind you) algorithms it employs to solve it, you'll get O(\sqrt{n}) on most inputs you try. It's considerably faster when your prime modulus has the property where p - 1 factors into a lot of small primes.
Consider a prime on the order of 100 bits: (~ 2^{100}). With \sqrt{n} complexity, that's still 2^{50} iterations. That being said, don't reinvent the wheel. This does a pretty good job. I might also add that it was almost 4x times more memory efficient than Mathematica's MultiplicativeOrder function when I ran with large-ish inputs (44 MiB vs. 173 MiB).
Since a duplicate of this question was asked under the Python tag, here is a Python implementation of baby step, giant step, which, as #MarkBeyers points out, is a reasonable approach (as long as the modulus isn't too large):
def baby_steps_giant_steps(a,b,p,N = None):
if not N: N = 1 + int(math.sqrt(p))
#initialize baby_steps table
baby_steps = {}
baby_step = 1
for r in range(N+1):
baby_steps[baby_step] = r
baby_step = baby_step * a % p
#now take the giant steps
giant_stride = pow(a,(p-2)*N,p)
giant_step = b
for q in range(N+1):
if giant_step in baby_steps:
return q*N + baby_steps[giant_step]
else:
giant_step = giant_step * giant_stride % p
return "No Match"
In the above implementation, an explicit N can be passed to fish for a small exponent even if p is cryptographically large. It will find the exponent as long as the exponent is smaller than N**2. When N is omitted, the exponent will always be found, but not necessarily in your lifetime or with your machine's memory if p is too large.
For example, if
p = 70606432933607
a = 100001
b = 54696545758787
then 'pow(a,b,p)' evaluates to 67385023448517
and
>>> baby_steps_giant_steps(a,67385023448517,p)
54696545758787
This took about 5 seconds on my machine. For the exponent and the modulus of those sizes, I estimate (based on timing experiments) that brute force would have taken several months.
Discrete logarithm is a hard problem
Computing discrete logarithms is believed to be difficult. No
efficient general method for computing discrete logarithms on
conventional computers is known.
I will add here a simple bruteforce algorithm which tries every possible value from 1 to m and outputs a solution if it was found. Note that there may be more than one solution to the problem or zero solutions at all. This algorithm will return you the smallest possible value or -1 if it does not exist.
def bruteLog(b, c, m):
s = 1
for i in xrange(m):
s = (s * b) % m
if s == c:
return i + 1
return -1
print bruteLog(5, 8, 13)
and here you can see that 3 is in fact the solution:
print 5**3 % 13
There is a better algorithm, but because it is often asked to be implemented in programming competitions, I will just give you a link to explanation.
as said the general problem is hard. however a prcatical way to find e if and only if you know e is going to be small (like in your example) would be just to try each e from 1.
btw e==3 is the first solution to your example, and you can obviously find that in 3 steps, compare to solving the non discrete version, and naively looking for integer solutions i.e.
e = log(c + n*m)/log(b) where n is a non-negative integer
which finds e==3 in 9 steps