Fast Algorithm for computing percentiles to remove outliers

Answer 1

Divide the interval between minimum and maximum of your data into (say) 1000 bins and calculate a histogram. Then build partial sums and see where they first exceed 5000 or 95000.

Answer 2

Not an expert, but my memory suggests:

• to determine percentile points exactly you need to sort and count
• taking a sample from the data and calculating the percentile values sounds like a good plan for decent approximation if you can get a good sample
• if not, as suggested by Henrik, you can avoid the full sort if you do the buckets and count them

Answer 3

You could estimate your percentiles from just a part of your dataset, like the first few thousand points.

The Glivenko–Cantelli theorem ensures that this would be a fairly good estimate, if you can assume your data points to be independent.

Answer 4

There are a couple basic approaches I can think of. First is to compute the range (by finding the highest and lowest values), project each element to a percentile ((x - min) / range) and throw out any that evaluate to lower than .05 or higher than .95.

The second is to compute the mean and standard deviation. A span of 2 standard deviations from the mean (in both directions) will enclose 95% of a normally-distributed sample space, meaning your outliers would be in the <2.5 and >97.5 percentiles. Calculating the mean of a series is linear, as is the standard dev (square root of the sum of the difference of each element and the mean). Then, subtract 2 sigmas from the mean, and add 2 sigmas to the mean, and you've got your outlier limits.

Both of these will compute in roughly linear time; the first one requires two passes, the second one takes three (once you have your limits you still have to discard the outliers). Since this is a list-based operation, I do not think you will find anything with logarithmic or constant complexity; any further performance gains would require either optimizing the iteration and calculation, or introducing error by performing the calculations on a sub-sample (such as every third element).

Answer 5

I used to identify outliers by calculating the standard deviation. Everything with a distance more as 2 (or 3) times the standard deviation from the avarage is an outlier. 2 times = about 95%.

Since your are calculating the avarage, its also very easy to calculate the standard deviation is very fast.

You could also use only a subset of your data to calculate the numbers.

Answer 6

The histogram solution from Henrik will work. You can also use a selection algorithm to efficiently find the k largest or smallest elements in an array of n elements in O(n). To use this for the 95th percentile set k=0.05n and find the k largest elements.

Reference:

http://en.wikipedia.org/wiki/Selection_algorithm#Selecting_k_smallest_or_largest_elements

Answer 7

According to it's creator a SoftHeap can be used to:

compute exact or approximate medians and percentiles optimally. It is also useful for approximate sorting...

Answer 8

A good general answer to your problem seems to be RANSAC. Given a model, and some noisy data, the algorithm efficiently recovers the parameters of the model.
You will have to chose a simple model that can map your data. Anything smooth should be fine. Let say a mixture of few gaussians. RANSAC will set the parameters of your model and estimate a set of inliners at the same time. Then throw away whatever doesn't fit the model properly.

Answer 9

One set of data of 100k elements takes almost no time to sort, so I assume you have to do this repeatedly. If the data set is the same set just updated slightly, you're best off building a tree (O(N log N)) and then removing and adding new points as they come in (O(K log N) where K is the number of points changed). Otherwise, the kth largest element solution already mentioned gives you O(N) for each dataset.

Answer 10

You could filter out 2 or 3 standard deviation even if the data is not normally distributed; at least, it will be done in a consistent manner, that should be important.

As you remove the outliers, the std dev will change, you could do this in a loop until the change in std dev is minimal. Whether or not you want to do this depends upon why are you manipulating the data this way. There are major reservations by some statisticians to removing outliers. But some remove the outliers to prove that the data is fairly normally distributed.