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I'm having some difficulties with fitting a discrete distribution function (I'm specifically using the negative binomial distribution). Here's my setting: I have a source of incoming items, each with an unknown lifetime. Everyday, some expire (a big portion in the first day, some more in the second day, etc.). For an existing source of incoming items (source is older than 180 days), I've managed to model the lifespan of a new item with the negative binomial distribution to an acceptable error (using MLE - Maximum Likelihood Estimation).
My problem starts with new sources of incoming items. I want to estimate their items' lifetime distribution after a short time (say, after 5-7 days). When I try to apply the MLE, I get significantly lower means (i.e. 3 instead of 30). I assume it's because the MLE can't understand the last day's (7th day) mass is actually the 1-CDF(6) (cumulative distribution function of the previous 6 days) and actually contains living items as well.
Is there a good approach to fit the discrete distribution only based on the early data values and the sum of the mass of the other values? I could write some optimization function for it and only give weight to the 6 previous days, but I feel it will give me sub-optimal performance.
I'm ok with theory explanation, but if you can address specific functions or libraries, I can work in Matlab, R, Python and C#.
The problem you have encountered is called "censored" data. Essentially you at a certain only that the lifetime of some items is greater than (now minus start time). Your guess about how to correct the likelihood function is pointing in the right direction. I think censored data are usually considered in texts about survival analysis. The Wikipedia article [1] has some brief remarks about censored data that might help too.
There is a package for survival analysis in R named 'survival'. There may be other R packages. Dunno about packages for other systems.
[1] http://en.wikipedia.org/wiki/Survival_analysis
I'm attempting to estimate the total amount of results for app engine queries that will return large amounts of results.
In order to do this, I assigned a random floating point number between 0 and 1 to every entity. Then I executed the query for which I wanted to estimate the total results with the following 3 settings:
* I ordered by the random numbers that I had assigned in ascending order
* I set the offset to 1000
* I fetched only one entity
I then plugged the entities's random value that I had assigned for this purpose into the following equation to estimate the total results (since I used 1000 as the offset above, the value of OFFSET would be 1000 in this case):
1 / RANDOM * OFFSET
The idea is that since each entity has a random number assigned to it, and I am sorting by that random number, the entity's random number assignment should be proportionate to the beginning and end of the results with respect to its offset (in this case, 1000).
The problem I am having is that the results I am getting are giving me low estimates. And the estimates are lower, the lower the offset. I had anticipated that the lower the offset that I used, the less accurate the estimate should be, but I thought that the margin of error would be both above and below the actual number of results.
Below is a chart demonstrating what I am talking about. As you can see, the predictions get more consistent (accurate) as the offset increases from 1000 to 5000. But then the predictions predictably follow a 4 part polynomial. (y = -5E-15x4 + 7E-10x3 - 3E-05x2 + 0.3781x + 51608).
Am I making a mistake here, or does the standard python random number generator not distribute numbers evenly enough for this purpose?
Thanks!
Edit:
It turns out that this problem is due to my mistake. In another part of the program, I was grabbing entities from the beginning of the series, doing an operation, then re-assigning the random number. This resulted in a denser distribution of random numbers towards the end.
I did a little more digging into this concept, fixed the problem, and tried it again on a different query (so the number of results are different from above). I found that this idea can be used to estimate the total results for a query. One thing of note is that the "error" is very similar for offsets that are close by. When I did a scatter chart in excel, I expected the accuracy of the predictions at each offset to "cloud". Meaning that offsets at the very begging would produce a larger, less dense cloud that would converge to a very tiny, dense could around the actual value as the offsets got larger. This is not what happened as you can see below in the cart of how far off the predictions were at each offset. Where I thought there would be a cloud of dots, there is a line instead.
This is a chart of the maximum after each offset. For example the maximum error for any offset after 10000 was less than 1%:
When using GAE it makes a lot more sense not to try to do large amounts work on reads - it's built and optimized for very fast requests turnarounds. In this case it's actually more efficent to maintain a count of your results as and when you create the entities.
If you have a standard query, this is fairly easy - just use a sharded counter when creating the entities. You can seed this using a map reduce job to get the initial count.
If you have queries that might be dynamic, this is more difficult. If you know the range of possible queries that you might perform, you'd want to create a counter for each query that might run.
If the range of possible queries is infinite, you might want to think of aggregating counters or using them in more creative ways.
If you tell us the query you're trying to run, there might be someone who has a better idea.
Some quick thought:
Have you tried Datastore Statistics API? It may provide a fast and accurate results if you won't update your entities set very frequently.
http://code.google.com/appengine/docs/python/datastore/stats.html
[EDIT1.]
I did some math things, I think the estimate method you purposed here, could be rephrased as an "Order statistic" problem.
http://en.wikipedia.org/wiki/Order_statistic#The_order_statistics_of_the_uniform_distribution
For example:
If the actual entities number is 60000, the question equals to "what's the probability that your 1000th [2000th, 3000th, .... ] sample falling in the interval [l,u]; therefore, the estimated total entities number based on this sample, will have an acceptable error to 60000."
If the acceptable error is 5%, the interval [l, u] will be [0.015873015873015872, 0.017543859649122806]
I think the probability won't be very large.
This doesn't directly deal with the calculations aspect of your question, but would using the count attribute of a query object work for you? Or have you tried that out and it's not suitable? As per the docs, it's only slightly faster than retrieving all of the data, but on the plus side it would give you the actual number of results.
http://code.google.com/appengine/docs/python/datastore/queryclass.html#Query_count
I am looking for an algorithm that determines percentiles for live data capture.
For example, consider the development of a server application.
The server might have response times as follows:
17 ms
33 ms
52 ms
60 ms
55 ms
etc.
It is useful to report the 90th percentile response time, 80th percentile response time, etc.
The naive algorithm is to insert each response time into a list. When statistics are requested, sort the list and get the values at the proper positions.
Memory usages scales linearly with the number of requests.
Is there an algorithm that yields "approximate" percentile statistics given limited memory usage? For example, let's say I want to solve this problem in a way that I process millions of requests but only want to use say one kilobyte of memory for percentile tracking (discarding the tracking for old requests is not an option since the percentiles are supposed to be for all requests).
Also require that there is no a priori knowledge of the distribution. For example, I do not want to specify any ranges of buckets ahead of time.
If you want to keep the memory usage constant as you get more and more data, then you're going to have to resample that data somehow. That implies that you must apply some sort of rebinning scheme. You can wait until you acquire a certain amount of raw inputs before beginning the rebinning, but you cannot avoid it entirely.
So your question is really asking "what's the best way of dynamically binning my data"? There are lots of approaches, but if you want to minimise your assumptions about the range or distribution of values you may receive, then a simple approach is to average over buckets of fixed size k, with logarithmically distributed widths. For example, lets say you want to hold 1000 values in memory at any one time. Pick a size for k, say 100. Pick your minimum resolution, say 1ms. Then
The first bucket deals with values between 0-1ms (width=1ms)
Second bucket: 1-3ms (w=2ms)
Third bucket: 3-7ms (w=4ms)
Fourth bucket: 7-15ms (w=8ms)
...
Tenth bucket: 511-1023ms (w=512ms)
This type of log-scaled approach is similar to the chunking systems used in hash table algorithms, used by some filesystems and memory allocation algorithms. It works well when your data has a large dynamic range.
As new values come in, you can choose how you want to resample, depending on your requirements. For example, you could track a moving average, use a first-in-first-out, or some other more sophisticated method. See the Kademlia algorithm for one approach (used by Bittorrent).
Ultimately, rebinning must lose you some information. Your choices regarding the binning will determine the specifics of what information is lost. Another way of saying this is that the constant size memory store implies a trade-off between dynamic range and the sampling fidelity; how you make that trade-off is up to you, but like any sampling problem, there's no getting around this basic fact.
If you're really interested in the pros and cons, then no answer on this forum can hope to be sufficient. You should look into sampling theory. There's a huge amount of research on this topic available.
For what it's worth, I suspect that your server times will have a relatively small dynamic range, so a more relaxed scaling to allow higher sampling of common values may provide more accurate results.
Edit: To answer your comment, here's an example of a simple binning algorithm.
You store 1000 values, in 10 bins. Each bin therefore holds 100 values. Assume each bin is implemented as a dynamic array (a 'list', in Perl or Python terms).
When a new value comes in:
Determine which bin it should be stored in, based on the bin limits you've chosen.
If the bin is not full, append the value to the bin list.
If the bin is full, remove the value at the top of the bin list, and append the new value to the bottom of the bin list. This means old values are thrown away over time.
To find the 90th percentile, sort bin 10. The 90th percentile is the first value in the sorted list (element 900/1000).
If you don't like throwing away old values, then you can implement some alternative scheme to use instead. For example, when a bin becomes full (reaches 100 values, in my example), you could take the average of the oldest 50 elements (i.e. the first 50 in the list), discard those elements, and then append the new average element to the bin, leaving you with a bin of 51 elements that now has space to hold 49 new values. This is a simple example of rebinning.
Another example of rebinning is downsampling; throwing away every 5th value in a sorted list, for example.
I hope this concrete example helps. The key point to take away is that there are lots of ways of achieving a constant memory aging algorithm; only you can decide what is satisfactory given your requirements.
I've once published a blog post on this topic. The blog is now defunct but the article is included in full below.
The basic idea is to reduce the requirement for an exact calculation in favor of "95% percent of responses take 500ms-600ms or less" (for all exact percentiles of 500ms-600ms).
As we’ve recently started feeling that response times of one of our webapps got worse, we decided to spend some time tweaking the apps’ performance. As a first step, we wanted to get a thorough understanding of current response times. For performance evaluations, using minimum, maximum or average response times is a bad idea: “The ‘average’ is the evil of performance optimization and often as helpful as ‘average patient temperature in the hospital'” (MySQL Performance Blog). Instead, performance tuners should be looking at the percentile: “A percentile is the value of a variable below which a certain percent of observations fall” (Wikipedia). In other words: the 95th percentile is the time in which 95% of requests finished. Therefore, a performance goals related to the percentile could be similar to “The 95th percentile should be lower than 800 ms”. Setting such performance goals is one thing, but efficiently tracking them for a live system is another one.
I’ve spent quite some time looking for existing implementations of percentile calculations (e.g. here or here). All of them required storing response times for each and every request and calculate the percentile on demand or adding new response times in order. This was not what I wanted. I was hoping for a solution that would allow memory and CPU efficient live statistics for hundreds of thousands of requests. Storing response times for hundreds of thousands of requests and calculating the percentile on demand does neither sound CPU nor memory efficient.
Such a solution as I was hoping for simply seems not to exist. On second thought, I came up with another idea: For the type of performance evaluation I was looking for, it’s not necessary to get the exact percentile. An approximate answer like “the 95th percentile is between 850ms and 900ms” would totally suffice. Lowering the requirements this way makes an implementation extremely easy, especially if upper and lower borders for the possible results are known. For example, I’m not interested in response times higher than several seconds – they are extremely bad anyway, regardless of being 10 seconds or 15 seconds.
So here is the idea behind the implementation:
Define any random number of response time buckets (e.g. 0-100ms, 100-200ms, 200-400ms, 400-800ms, 800-1200ms, …)
Count number of responses and number of response each bucket (For a response time of 360ms, increment the counter for the 200ms – 400ms bucket)
Estimate the n-th percentile by summing counter for buckets until the sum exceeds n percent of the total
It’s that simple. And here is the code.
Some highlights:
public void increment(final int millis) {
final int i = index(millis);
if (i < _limits.length) {
_counts[i]++;
}
_total++;
}
public int estimatePercentile(final double percentile) {
if (percentile < 0.0 || percentile > 100.0) {
throw new IllegalArgumentException("percentile must be between 0.0 and 100.0, was " + percentile);
}
for (final Percentile p : this) {
if (percentile - p.getPercentage() <= 0.0001) {
return p.getLimit();
}
}
return Integer.MAX_VALUE;
}
This approach only requires two int values (= 8 byte) per bucket, allowing to track 128 buckets with 1K of memory. More than sufficient for analysing response times of a web application using a granularity of 50ms). Additionally, for the sake of performance, I’ve intentionally implemented this without any synchronization(e.g. using AtomicIntegers), knowing that some increments might get lost.
By the way, using Google Charts and 60 percentile counters, I was able to create a nice graph out of one hour of collected response times:
I believe there are many good approximate algorithms for this problem. A good first-cut approach is to simply use a fixed-size array (say 1K worth of data). Fix some probability p. For each request, with probability p, write its response time into the array (replacing the oldest time in there). Since the array is a subsampling of the live stream and since subsampling preserves the distribution, doing the statistics on that array will give you an approximation of the statistics of the full, live stream.
This approach has several advantages: it requires no a-priori information, and it's easy to code. You can build it quickly and experimentally determine, for your particular server, at what point growing the buffer has only a negligible effect on the answer. That is the point where the approximation is sufficiently precise.
If you find that you need too much memory to give you statistics that are precise enough, then you'll have to dig further. Good keywords are: "stream computing", "stream statistics", and of course "percentiles". You can also try "ire and curses"'s approach.
(It's been quite some time since this question was asked, but I'd like to point out a few related research papers)
There has been a significant amount of research on approximate percentiles of data streams in the past few years. A few interesting papers with full algorithm definitions:
A fast algorithm for approximate quantiles in high speed data streams
Space-and time-efficient deterministic algorithms for biased quantiles over data streams
Effective computation of biased quantiles over data streams
All of these papers propose algorithms with sub-linear space complexity for the computation of approximate percentiles over a data stream.
Try the simple algorithm defined in the paper “Sequential Procedure for Simultaneous Estimation of Several Percentiles” (Raatikainen). It’s fast, requires 2*m+3 markers (for m percentiles) and tends to an accurate approximation quickly.
Use a dynamic array T[] of large integers or something where T[n] counts the numer of times the response time was n milliseconds. If you really are doing statistics on a server application then possibly 250 ms response times are your absolute limit anyway. So your 1 KB holds one 32 bits integer for every ms between 0 and 250, and you have some room to spare for an overflow bin.
If you want something with more bins, go with 8 bit numbers for 1000 bins, and the moment a counter would overflow (i.e. 256th request at that response time) you shift the bits in all bins down by 1. (effectively halving the value in all bins). This means you disregard all bins that capture less than 1/127th of the delays that the most visited bin catches.
If you really, really need a set of specific bins I'd suggest using the first day of requests to come up with a reasonable fixed set of bins. Anything dynamic would be quite dangerous in a live, performance sensitive application. If you choose that path you'd better know what your doing, or one day you're going to get called out of bed to explain why your statistics tracker is suddenly eating 90% CPU and 75% memory on the production server.
As for additional statistics: For mean and variance there are some nice recursive algorithms that take up very little memory. These two statistics can be usefull enough in themselves for a lot of distributions because the central limit theorem states that distributions that that arise from a sufficiently large number of independent variables approach the normal distribution (which is fully defined by mean and variance) you can use one of the normality tests on the last N (where N sufficiently large but constrained by your memory requirements) to monitor wether the assumption of normality still holds.
#thkala started off with some literature citations. Let me extend that.
Implementations
T-digest from the 2019 Dunning paper has reference implementation in Java, and ports on that page to Python, Go, Javascript, C++, Scala, C, Clojure, C#, Kotlin, and C++ port by facebook, and a further rust port of that C++ port
Spark implements "GK01" from the 2001 Greenwald/Khanna paper for approximate quantiles
Beam: org.apache.beam.sdk.transforms.ApproximateQuantiles has approximate quantiles
Java: Guava:com.google.common.math.Quantiles implements exact quantiles, thus taking more memory
Rust: quantiles crate has implementations for the 2001 GK algorithm "GK01", and the 2005 CKMS algorithm. (caution: I found the CKMS implementation slow - issue)
C++: boost quantiles has some code, but I didn't understand it.
I did some profiling of the options in Rust [link] for up to 100M items, and found GK01 the best, T-digest the second, and "keep 1% top values in priority queue" the third.
Literature
2001: Space-efficient online computation of quantile summaries (by Greenwald, Khanna). Implemented in Rust: quantiles::greenwald_khanna.
2004: Medians and beyond: new aggregation techniques for sensor networks (by Shrivastava, Buragohain, Agrawal, Suri). Introduces "q-digests", used for fixed-universe data.
2005: Effective computation of biased quantiles over data streams (by Cormode, Korn, Muthukrishnan, Srivastava)... Implemented in Rust: quantiles::ckms which notes that the IEEE presentation is correct but the self-published one has flaws. With carefully crafted data, space can grow linearly with input size. "Biased" means it focuses on P90/P95/P99 rather than all the percentiles).
2006: Space-and time-efficient deterministic algorithms for biased quantiles over data streams (by Cormode, Korn, Muthukrishnan, Srivastava)... improved space bound over 2005 paper
2007: A fast algorithm for approximate quantiles in high speed data streams (by Zhang, Wang). Claims 60-300x speedup over GK. The 2020 literature review below says this has state-of-the-art space upper bound.
2019 Computing extremely accurate quantiles using t-digests (by Dunning, Ertl). Introduces t-digests, O(log n) space, O(1) updates, O(1) final calculation. It's neat feature is you can build partial digests (e.g. one per day) and merge them into months, then merge months into years. This is what the big query engines use.
2020 A survey of approximate quantile computation on large-scale data (technical report) (by Chen, Zhang).
2021 The t-digest: Efficient estimates of distributions - an approachable wrap-up paper about t-digests.
Cheap hack for P99 of <10M values: just store top 1% in a priority queue!
This'll sound stupid, but if I want to calculate the P99 of 10M float64s, I just created a priority queue with 100k float32s (takes 400kB). This takes only 4x as much space as "GK01" and is much faster. For 5M or fewer items, it takes less space than GK01!!
struct TopValues {
values: std::collections::BinaryHeap<std::cmp::Reverse<ordered_float::NotNan<f32>>>,
}
impl TopValues {
fn new(count: usize) -> Self {
let capacity = std::cmp::max(count / 100, 1);
let values = std::collections::BinaryHeap::with_capacity(capacity);
TopValues { values }
}
fn render(&mut self) -> String {
let p99 = self.values.peek().unwrap().0;
let max = self.values.drain().min().unwrap().0;
format!("TopValues, p99={:.4}, max={:.4}", p99, max)
}
fn insert(&mut self, value: f64) {
let value = value as f32;
let value = std::cmp::Reverse(unsafe { ordered_float::NotNan::new_unchecked(value) });
if self.values.len() < self.values.capacity() {
self.values.push(value);
} else if self.values.peek().unwrap().0 < value.0 {
self.values.pop();
self.values.push(value);
} else {
}
}
}
You can try the following structure:
Take on input n, ie. n = 100.
We'll keep an array of ranges [min, max] sorted by min with count.
Insertion of value x – binary search for min range for x. If not found take preceeding range (where min < x). If value belongs to range (x <= max) increment count. Otherwise insert new range with [min = x, max = x, count = 1].
If number of ranges hits 2*n – collapse/merge array into n (half) by taking min from odd and max from even entries, summing their count.
To get ie. p95 walk from the end summing the count until next addition would hit threshold sum >= 95%, take p95 = min + (max - min) * partial.
It will settle on dynamic ranges of measurements. n can be modified to trade accuracy for memory (to lesser extent cpu). If you make values more discrete, ie. by rounding to 0.01 before insertion – it'll stabilise on ranges sooner.
You could improve accuracy by not assuming that each range holds uniformly distributed entries, ie. something cheap like sum of values which will give you avg = sum / count, it would help to read closer p95 value from range where it sits.
You can also rotate them, ie. after m = 1 000 000 entries start filling new array and take p95 as weighted sum on count in array (if array B has 10% of count of A, then it contributes 10% to p95 value).
What's the rationale behind the formula used in the hive_trend_mapper.py program of this Hadoop tutorial on calculating Wikipedia trends?
There are actually two components: a monthly trend and a daily trend. I'm going to focus on the daily trend, but similar questions apply to the monthly one.
In the daily trend, pageviews is an array of number of page views per day for this topic, one element per day, and total_pageviews is the sum of this array:
# pageviews for most recent day
y2 = pageviews[-1]
# pageviews for previous day
y1 = pageviews[-2]
# Simple baseline trend algorithm
slope = y2 - y1
trend = slope * log(1.0 +int(total_pageviews))
error = 1.0/sqrt(int(total_pageviews))
return trend, error
I know what it's doing superficially: it just looks at the change over the past day (slope), and scales this up to the log of 1+total_pageviews (log(1)==0, so this scaling factor is non-negative). It can be seen as treating the month's total pageviews as a weight, but tempered as it grows - this way, the total pageviews stop making a difference for things that are "popular enough," but at the same time big changes on insignificant don't get weighed as much.
But why do this? Why do we want to discount things that were initially unpopular? Shouldn't big deltas matter more for items that have a low constant popularity, and less for items that are already popular (for which the big deltas might fall well within a fraction of a standard deviation)? As a strawman, why not simply take y2-y1 and be done with it?
And what would the error be useful for? The tutorial doesn't really use it meaningfully again. Then again, it doesn't tell us how trend is used either - this is what's plotted in the end product, correct?
Where can I read up for a (preferably introductory) background on the theory here? Is there a name for this madness? Is this a textbook formula somewhere?
Thanks in advance for any answers (or discussion!).
As the in-line comment goes, this is a simple "baseline trend algorithm",
which basically means before you compare the trends of two different pages, you have to establish
a baseline. In many cases, the mean value is used, it's straightforward if you
plot the pageviews against the time axis. This method is widely used in monitoring
water quality, air pollutants, etc. to detect any significant changes w.r.t the baseline.
In OP's case, the slope of pageviews is weighted by the log of totalpageviews.
This sorta uses the totalpageviews as a baseline correction for the slope. As Simon put it, this puts a balance
between two pages with very different totalpageviews.
For exmaple, A has a slope 500 over 1000,000 total pageviews, B is 1000 over 1,000.
A log basically means 1000,000 is ONLY twice more important than 1,000 (rather than 1000 times).
If you only consider the slope, A is less popular than B.
But with a weight, now the measure of popularity of A is the same as B. I think it is quite intuitive:
though A's pageviews is only 500 pageviews, but that's because it's saturating, you still gotta give it enough credit.
As for the error, I believe it comes from the (relative) standard error, which has a factor 1/sqrt(n), where
n is the number of data points. In the code, the error is equal to (1/sqrt(n))*(1/sqrt(mean)).
It roughly translates into : the more data points, the more accurate the trend. I don't see
it is an exact math formula, just a brute trend analysis algorithm, anyway the relative
value is more important in this context.
In summary, I believe it's just an empirical formula. More advanced topics can be found in some biostatistics textbooks (very similar to monitoring the breakout of a flu or the like.)
The code implements statistics (in this case the "baseline trend"), you should educate yourself on that and everything becomes clearer. Wikibooks has a good instroduction.
The algorithm takes into account that new pages are by definition more unpopular than existing ones (because - for example - they are linked from relatively few other places) and suggests that those new pages will grow in popularity over time.
error is the error margin the system expects for its prognoses. The higher error is, the more unlikely the trend will continue as expected.
The reason for moderating the measure by the volume of clicks is not to penalise popular pages but to make sure that you can compare large and small changes with a single measure. If you just use y2 - y1 you will only ever see the click changes on large volume pages. What this is trying to express is "significant" change. 1000 clicks change if you attract 100 clicks is really significant. 1000 click change if you attract 100,000 is less so. What this formula is trying to do is make both of these visible.
Try it out at a few different scales in Excel, you'll get a good view of how it operates.
Hope that helps.
another way to look at it is this:
suppose your page and my page are made at same day, and ur page gets total views about ten million, and mine about 1 million till some point. then suppose the slope at some point is a million for me, and 0.5 million for you. if u just use slope, then i win, but ur page already had more views per day at that point, urs were having 5 million, and mine 1 million, so that a million on mine still makes it 2 million, and urs is 5.5 million for that day. so may be this scaling concept is to try to adjust the results to show that ur page is also good as a trend setter, and its slope is less but it already was more popular, but the scaling is only a log factor, so doesnt seem too problematic to me.
What is an algorithm to compare multiple sets of numbers against a target set to determine which ones are the most "similar"?
One use of this algorithm would be to compare today's hourly weather forecast against historical weather recordings to find a day that had similar weather.
The similarity of two sets is a bit subjective, so the algorithm really just needs to diferentiate between good matches and bad matches. We have a lot of historical data, so I would like to try to narrow down the amount of days the users need to look through by automatically throwing out sets that aren't close and trying to put the "best" matches at the top of the list.
Edit:
Ideally the result of the algorithm would be comparable to results using different data sets. For example using the mean square error as suggested by Niles produces pretty good results, but the numbers generated when comparing the temperature can not be compared to numbers generated with other data such as Wind Speed or Precipitation because the scale of the data is different. Some of the non-weather data being is very large, so the mean square error algorithm generates numbers in the hundreds of thousands compared to the tens or hundreds that is generated by using temperature.
I think the mean square error metric might work for applications such as weather compares. It's easy to calculate and gives numbers that do make sense.
Since your want to compare measurements over time you can just leave out missing values from the calculation.
For values that are not time-bound or even unsorted, multi-dimensional scatter data it's a bit more difficult. Choosing a good distance metric becomes part of the art of analysing such data.
Use the pearson correlation coefficient. I figured out how to calculate it in an SQL query which can be found here: http://vanheusden.com/misc/pearson.php
In finance they use Beta to measure the correlation of 2 series of numbers. EG, Beta could answer the question "Over the last year, how much would the price of IBM go up on a day that the price of the S&P 500 index went up 5%?" It deals with the percentage of the move, so the 2 series can have different scales.
In my example, the Beta is Covariance(IBM, S&P 500) / Variance(S&P 500).
Wikipedia has pages explaining Covariance, Variance, and Beta: http://en.wikipedia.org/wiki/Beta_(finance)
Look at statistical sites. I think you are looking for correlation.
As an example, I'll assume you're measuring temp, wind, and precip. We'll call these items "features". So valid values might be:
Temp: -50 to 100F (I'm in Minnesota, USA)
Wind: 0 to 120 Miles/hr (not sure if this is realistic but bear with me)
Precip: 0 to 100
Start by normalizing your data. Temp has a range of 150 units, Wind 120 units, and Precip 100 units. Multiply your wind units by 1.25 and Precip by 1.5 to make them roughly the same "scale" as your temp. You can get fancy here and make rules that weigh one feature as more valuable than others. In this example, wind might have a huge range but usually stays in a smaller range so you want to weigh it less to prevent it from skewing your results.
Now, imagine each measurement as a point in multi-dimensional space. This example measures 3d space (temp, wind, precip). The nice thing is, if we add more features, we simply increase the dimensionality of our space but the math stays the same. Anyway, we want to find the historical points that are closest to our current point. The easiest way to do that is Euclidean distance. So measure the distance from our current point to each historical point and keep the closest matches:
for each historicalpoint
distance = sqrt(
pow(currentpoint.temp - historicalpoint.temp, 2) +
pow(currentpoint.wind - historicalpoint.wind, 2) +
pow(currentpoint.precip - historicalpoint.precip, 2))
if distance is smaller than the largest distance in our match collection
add historicalpoint to our match collection
remove the match with the largest distance from our match collection
next
This is a brute-force approach. If you have the time, you could get a lot fancier. Multi-dimensional data can be represented as trees like kd-trees or r-trees. If you have a lot of data, comparing your current observation with every historical observation would be too slow. Trees speed up your search. You might want to take a look at Data Clustering and Nearest Neighbor Search.
Cheers.
Talk to a statistician.
Seriously.
They do this type of thing for a living.
You write that the "similarity of two sets is a bit subjective", but it's not subjective at all-- it's a matter of determining the appropriate criteria for similarity for your problem domain.
This is one of those situation where you are much better off speaking to a professional than asking a bunch of programmers.
First of all, ask yourself if these are sets, or ordered collections.
I assume that these are ordered collections with duplicates. The most obvious algorithm is to select a tolerance within which numbers are considered the same, and count the number of slots where the numbers are the same under that measure.
I do have a solution implemented for this in my application, but I'm looking to see if there is something that is better or more "correct". For each historical day I do the following:
function calculate_score(historical_set, forecast_set)
{
double c = correlation(historical_set, forecast_set);
double avg_history = average(historical_set);
double avg_forecast = average(forecast_set);
double penalty = abs(avg_history - avg_forecast) / avg_forecast
return c - penalty;
}
I then sort all the results from high to low.
Since the correlation is a value from -1 to 1 that says whether the numbers fall or rise together, I then "penalize" that with the percentage difference the averages of the two sets of numbers.
A couple of times, you've mentioned that you don't know the distribution of the data, which is of course true. I mean, tomorrow there could be a day that is 150 degree F, with 2000km/hr winds, but it seems pretty unlikely.
I would argue that you have a very good idea of the distribution, since you have a long historical record. Given that, you can put everything in terms of quantiles of the historical distribution, and do something with absolute or squared difference of the quantiles on all measures. This is another normalization method, but one that accounts for the non-linearities in the data.
Normalization in any style should make all variables comparable.
As example, let's say that a day it's a windy, hot day: that might have a temp quantile of .75, and a wind quantile of .75. The .76 quantile for heat might be 1 degree away, and the one for wind might be 3kmh away.
This focus on the empirical distribution is easy to understand as well, and could be more robust than normal estimation (like Mean-square-error).
Are the two data sets ordered, or not?
If ordered, are the indices the same? equally spaced?
If the indices are common (temperatures measured on the same days (but different locations), for example, you can regress the first data set against the second,
and then test that the slope is equal to 1, and that the intercept is 0.
http://stattrek.com/AP-Statistics-4/Test-Slope.aspx?Tutorial=AP
Otherwise, you can do two regressions, of the y=values against their indices. http://en.wikipedia.org/wiki/Correlation. You'd still want to compare slopes and intercepts.
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If unordered, I think you want to look at the cumulative distribution functions
http://en.wikipedia.org/wiki/Cumulative_distribution_function
One relevant test is Kolmogorov-Smirnov:
http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test
You could also look at
Student's t-test,
http://en.wikipedia.org/wiki/Student%27s_t-test
or a Wilcoxon signed-rank test http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test
to test equality of means between the two samples.
And you could test for equality of variances with a Levene test http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm
Note: it is possible for dissimilar sets of data to have the same mean and variance -- depending on how rigorous you want to be (and how much data you have), you could consider testing for equality of higher moments, as well.
Maybe you can see your set of numbers as a vector (each number of the set being a componant of the vector).
Then you can simply use dot product to compute the similarity of 2 given vectors (i.e. set of numbers).
You might need to normalize your vectors.
More : Cosine similarity