i have an array of date=>values, like this
"2010-10-12 14:58:36" =>13.4
"2010-10-17 14:58:36" =>12
"2010-10-22 14:58:36" =>17.6
"2010-10-27 14:58:36" =>22
"2010-11-01 14:58:36" =>10
[...]
I use this date-value combination to paint an graph in javascript.
Now i like to mark those dates, who are "very special".
My problem (and Question) is, which aspect should consider to find those specific dates?
As an human, i prefer the date "2010-10-17 14:58:36", because "something" should be happens on this date, because the value on the next dates rises for 5.6 points, which is the biggest step up followed by one mor big step up. On the other hand, also the date "2010-10-27 14:58:36" is an "highlight", because this is
the top of all values and
after this date, there comes the biggest step down.
So as an human, i would be choose both dates.
My problem is: how could an algorithm look like?
I tried averages values for n dates before and after the current values, which results in an accumulation of those specifics dates at the beginning and at the end of the graph
So i tried to find the biggest percentage step up (depending on the date before), but I'm not sure, if i really find the specific dates, I'm looking for?!
How would you tackle the problem?
Thank you.
Looks like financial stocking issue :-) You are looking for Time series analysis - this is a statistical issue. I'd recommend to use R programming language to play with it (you can do complex statistical things very fast). There are tens of special packages, for sure financial one's too. Once you know what you want, you may implement the solution in any other language.
Just try to google time series analysis r.
EDIT: note that R is very powerful - I'd bet there is a tool how to use R packages from other languages.
If you have information over a timeline you could use Inerpolation.
A Polynomial interpolation will give you an approximated polynomial that goes through the points.
What's nice about this is you can then use Mathematical analysis which is easy on polynomials to find interesting points (large gradients, min-max points etc...)
Also you get an approximation of how the function behaves, so you could "future" points and see what may happen in the near future.
Of course looking into the future isn't so accurate, but forms of interpolation are used in analytic to see trends and behaviors.
And of course, it's easy to plot a polynomial, which is always nice.
This is really a question of Statistics http://en.wikipedia.org/wiki/Statistics and the context of your data and what you're looking to highlight, for example, the fact that between 12/10 and 17/10 the data moved negative 1.4 units may be more useful in some scenarios than a larger positive step change.
You need sample data, on which build up a function which can calculate an expected value for any given date; for instance averaging the values of the day before, the same week day of the previous week, of the previous month and so on. After that decide a threshold: interesting date are those for which real value is outside expected value +- threshold
Related
I’ve got a statistical/mathematical problem I’m stumped on and I was really hoping to get some help. I’m working on a research where I need to compare a weekly graph with its own history to see when in the past it was almost the same. Think of this as “finding the closest match”. The information is displayed as a line graph, but it’s readily available as raw data:
Date...................Result
08/10/18......52.5
08/07/18......60.2
08/06/18......58.5
08/05/18......55.4
08/04/18......55.2
and so on...
What I really want is the output to be a form of correlation between the current data points with the other set of 5 concurrent data points in history. So, something like:
Date range.....................Correlation
07/10/18-07/15/18....0.98
We’ll be getting a code written in Python for the software to do this automatically (so that as new data is added, it automatically runs and finds the closest set of numbers to match the current one).
Here’s where the difficulty sets in: Since numbers are on a general upward trend over time, we don’t want it to compare the absolute value (since the numbers might never really match). One suggestion has been to compare the delta (rate of change as a percentage over the previous day), or using a log scale.
I’m wondering: how do I go about this? What kind of calculation I can use to get the desired results? I’ve looked at the different kind of correlation equations, but they don’t account for the “shape” of the data, and they generally just average it out. The shape of the line chart is the important thing.
Thanks very much in advance!
I would simply divide the data of each week by their average (i.e., normalize them to an average of 1), then sum the squares of the differences of each day of each pair of weeks. This sum is what you want to minimize.
If you don't care about how much a graph oscillates relative to its mean, you can normalize also the variance. For each week, calculate mean and variance, then subtract the mean and divide by the root of the variance. Each week will have mean 0 and variance 1. Then minimize the sum of squares of differences like before.
If the normalization of data is all you can change in your workflow, just leave out the sum of squares of differences minimization part.
I'm logging temperature values in a room, saving them to the database. I'd like to be alerted when temperature rises suddenly. I can't set fixed values, because 18°C is acceptable in winter and 25°C is acceptable in summer. But if it jumps from 20°C to 25°C during, let's say, 30 minutes and stays like this for 5 minutes (to eliminate false readouts), I'd like to be informed.
My current idea is to take readouts from last 30 minutes (A) and readouts from last 5 minutes (B), calculate median of A and B and check if difference between them is less then my desired threshold.
Is this correct way to solve this or is there a better algorithm? I searched for a specific one but most of them seem overcomplicated.
Thanks!
Detecting changes in a time-series is a well-researched subject, and hundreds if not thousands of papers have been written on this subject. As you've seen many methods are quite advanced, but proved to be quite useful for many use cases. Whatever method you choose, you should evaluate it against real of simulated data, and optimize its parameters for your use case.
As you require, let me suggest a very simple method that in many cases prove to be good enough, and is quite similar to that you considered.
Basically, you have two concerns:
Detecting a monotonous change in a sampled noisy signal
Ignoring false readouts
First, note that medians are not commonly used for detecting trends. For the series (1,2,3,30,35,3,2,1) the medians of 5 consecutive terms is be (3, 3, 3, 3). It is much more common to use averages.
One common trick is to throw the extreme values before averaging (e.g. for each 7 values average only the middle 5). If many false readouts are expected - try to take measurements at a faster rate, and throw more extreme values (e.g. for each 13 values average the middle 9).
Also, you should throw away unfeasible values and replace them with the last measured value (unfeasible means out of range, or non-physical change rate).
Your idea of comparing a short-period measure with a long-period measure is a good idea, and indeed it is commonly used (e.g. in econometrics).
Quoting from "Financial Econometric Models - Some Contributions to the Field [Nicolau, 2007]:
Buy and sell signals are generated by two moving averages of the price
level: a long-period average and a short-period average. A typical
moving average trading rule prescribes a buy (sell) when the
short-period moving average crosses the long-period moving average
from below (above) (i.e. when the original time series is rising
(falling) relatively fast).
When you say "rises suddenly," mathematically you are talking about the magnitude of the derivative of the temperature signal.
There is a nice algorithm to simultaneously smooth a signal and calculate its derivative called the Savitzky–Golay filter. It's explained with examples on Wikipedia, or you can use Matlab to help you generate the convolution coefficients required. Once you have the coefficients the calculation is very simple.
I don't have a real question but I'm more like seeking for creative input for a problem.
I want to compare two (most likely unequal) Date values and calculate the ratio of their similarity. So for example if I'd compare 08.01.2013 and 10.01.2013 I would get a relative high value but between 08.01.2013 and 17.04.1998it would be really low.
But now I'm not sure how I should exactly calculate the similarity. First I was thinking about turning the Date values into Strings and then use the EditDistance on them (number of single char operations to transform one String into another). This seems like a good idea for some cases and I'll definitly implement it but I also need an appropriate calculation for something like 31.01.2013 and 02.02.2013
Why not use the difference in days between two dates as a starting point?
It is "low" for similar dates and "high" for unequal dates, then use arithmetic to obtain a "similarity ratio" which matches your requirements.
Consider a fixed reference date "early enough" in the past if you get stuck.
The edit distance can be calculated using the Levenshtein distance.
A change in the year would mean a lot more "distance" than a change in the day.
The usual way to compare days would be to calculate the distance in days or hours. To do that, you'd convert both dates in a serial day number. Microsoft offers a DateDiff() function for date comparisons and distance calculations.
I have to track if given a week full of data integers ( 40, 30, 25, 55, 5, 40, etc ) raise an alert when the deviation from the norm happens (the '5' in the above case). An extra nice thing to have would be to actually learn if 5 is a normal event for that day of the week.
Do you know an implementation in ruby that is meant for this issue? In case this is a classic problem, what's the name of the problem/algorithm?
It's a very easy thing to calculate, but you will need to tune one parameter. You want to know if any given value is X standard deviations from the mean. To figure this out, calculate the standard deviation (see Wikipedia), then compare each value's deviation abs(mean - value) from the mean to this value. If a value's deviation is say, more than two standard deviations from the mean, flag it.
Edit:
To track deviations by weekday, keep an array of integers, one for each day. Every time you encounter a deviation, increment that day's counter by one. You could also use doubles and instead maintain a percentage of deviations for that day (num_friday_deviations/num_fridays) for example.
This is often referred to as "anomaly detection" and there is a lot of work out there if you google for it. The paper Mining Deviants in Time Series Data Streams may help you with your specific needs.
From the abstract:
We present first-known algorithms for identifying deviants on massive data streams. Our algorithms monitor
streams using very small space (polylogarithmic in data
size) and are able to quickly find deviants at any instant,
as the data stream evolves over time.
http://en.wikipedia.org/wiki/Control_chart describes classical ways of doing this sort of thing. As Jonathan Feinberg commented, there are different approaches.
The name of the algorithm could be as simple as "calculate standard deviation."
http://en.wikipedia.org/wiki/Standard_deviation
However, any analysis you do should be specific to the data set. You should inspect historical data to get at the right algorithm. Standard deviation won't be a good measure at all unless your data is normally distributed. Your data might even be such that you just want to look for numbers above a certain max value... it really depends.
So, my advice to you is:
1) Google for statistics overview and read up on basic statistics.
2) Inspect any historical data you have.
3) Come up with some reasonable measure of an odd number.
4) Test your measure against your historical data and see if it highlights the numbers you think it should.
5) Repeat steps 2-4 as necessary to refine your algorithm.
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.
====
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