Certain sensors are to trigger a signal based on the rate of change of the value rather than a threshold.
For instance, heat detectors in fire alarms are supposed to trigger an alarm quicker if the rate of temperature rise is higher: A temperature rise of 1K/min should trigger an alarm after 30 minutes, a rise of 5K/min after 5 minutes and a rise of 30K/min after 30 seconds.
I am wondering how this is implemented in embedded systems, where resources are scares. Is there a clever data structure to minimize the data stored?
The naive approach would be to measure the temperature every 5 seconds or so and keep the data for 30 minutes. On these data one can calculate change rates over arbitrary time windows. But this requires a lot of memory.
I thought about small windows (e.g. 10 seconds) for which min and max are stored, but this would not save much memory.
From a mathematical point of view, the examples you have described can be greatly simplified:
1K/min for 30 mins equals a total change of 30K
5K/min for 5 mins equals a total change of 25K
Obviously there is some adjustment to be made because you have picked round numbers for the example, but it sounds like what you care about is having a single threshold for the total change. This makes sense because taking the integral of a differential results in just a delta.
However, if we disregard the numeric example and just focus on your original question then here are some answers:
First, it has already been mentioned in the comments that one byte every five seconds for half an hour is really not very much memory at all for almost any modern microcontroller, as long as you are able to keep your main RAM turned on between samples, which you usually can.
If however you need to discard the contents of RAM between samples to preserve battery life, then a simpler method is just to calculate one differential at a time.
In your example you want to have a much higher sample rate (every 5 seconds) than the time you wish to calculate the delta over (eg: 30 mins). You can reduce your storage needs to a single data point if you make your sample rate equal to your delta period. The single previous value could be stored in a small battery retained memory (eg: backup registers on STM32).
Obviously if you choose this approach you will have to compromise between accuracy and latency, but maybe 30 seconds would be a suitable timebase for your temperature alarm example.
You can also set several thresholds of K/sec, and then allocate counters to count how many consecutive times the each threshold has been exceeded. This requires only one extra integer per threshold.
In signal processing terms, the procedure you want to perform is:
Apply a low-pass filter to smooth quick variations in the temperature
Take the derivative of its output
The cut-off frequency of the filter would be set according to the time frame. There are 2 ways to do this.
You could apply a FIR (finite impulse response) filter, which is a weighted moving average over the time frame of interest. Naively, this requires a lot of memory, but it's not bad if you do a multi-stage decimation first to reduce your sample rate. It ends up being a little complicated, but you have fine control over the response.
You could apply in IIR (Infinite impulse response) filter, which utilizes feedback of the output. The exponential moving average is the simplest example of this. These filters require far less memory -- only a few samples' worth, but your control over the precise shape of the response is limited. A classic example like the Butterworth filter would probably be great for your application, though.
Related
Assume I have huge set of data about a system idle time.
Day 1 - 5 mins
Day 2 - 3 mins
Day 3 - 7 mins
...
Day 'n' - 'k' mins
We can assume that even though the idletime is random, the pattern repeats.
Using this as a training data, is it possible for me to identify the idle time behavior of the system. With that, can a abnormality be predicted
Which algorithm would best suit for this purpose
I tried to fit in regression, but it can just answer me " What is the expected idle time today "
But what I want to do is. When the idle time goes away from the pattern, it has to be detected.
Edit:
Or does it make sense to predict for the current day only. i.e Today the expected idle time is 'x' mins. Tomorrow it may differ
I would try a Fourier Transformation and have a look if your system behaves in a periodic way (this would mean there are some peaks in the frequency domain).
Than get rid of the frequencies with low values and use the rest to predict the system behavior in the future.
If the real behavior differs a lot from the prediction that is what you want to detect.
wikipedia: Fast Fourier Transformation
Scenario:
Suppose we have infinite cache memory size. Caching is just limited by timeout, value of this timeout is half an hour. Cache is initially empty.
Problem:
We have 50,000 distinct request. Our system is querying, randomly, at the rate of 15 request/second i.e. 27,000 request in half an hour . What kind of curve or average value of cache hit rate could we expect for first 5 hours?
Note: This scenario is fixed. I need an approach to find out hit rate. If you think tag is wrong, please suggest appropriate tag.
I think you're right and this is a math question (certainly not a programming
problem).
One approach is to consider the extremes -- what is the hit rate for the
first query when the the system starts running? For the second query?
After one second? After 10? After a minute? And what is the likelyhood
that any random query will be found in the cache once the system has been
running a long time?
These are few specific values, and together they give you a curve.
I don't think great numeric precision is necessary; the long-term average
and the shape of the curve is more interesting.
There's a large set of objects. Set is dynamic: objects can be added or deleted any time. Let's call the total number of objects N.
Each object has two properties: mass (M) and time (T) of last update.
Every X minutes a small batch of those should be selected for processing, which updates their T to current time. Total M of all objects in a batch is limited: not more than L.
I am looking to solve three tasks here:
find a next batch object picking algorithm;
introduce object classes: simple, priority (granted fit into at least each n-th batch) and frequent (fit into each batch);
forecast system capacity exhaust (time to add next server = increase L).
What kind of model best describes such a system?
The whole thing is about a service that processes the "objects" in time intervals. Each object should be "measured" each N hours. N can vary in a range. X is fixed.
Objects are added/deleted by humans. N grows exponentially, rather slow, with some spikes caused by publications. Of course forecast can't be precise, just some estimate. M varies from 0 to 1E7 with exponential distribution, most are closer to 0.
I see there can be several strategies here:
A. full throttle - pack each batch as much as close to 100%. As N grows, average interval a particular object gets a hit will grow.
B. equal temperament :) - try to keep an average interval around some value. A batch fill level will be growing from some low level. When it reaches closer to 100% – time to get more servers.
C. - ?
Here is a pretty complete design for your problem.
Your question does not optimally match your description of the system this is for. So I'll assume that the description is accurate.
When you schedule a measurement you should pass an object, a first time it can be measured, and when you want the measurement to happen by. The object should have a weight attribute and a measured method. When the measurement happens, the measured method will be called, and the difference between your classes is whether, and with what parameters, they will reschedule themselves.
Internally you will need a couple of priority queues. See http://en.wikipedia.org/wiki/Heap_(data_structure) for details on how to implement one.
The first queue is by time the measurement can happen, all of the objects that can't be measured yet. Every time you schedule a batch you will use that to find all of the new measurements that can happen.
The second queue is of measurements that are ready to go now, and is organized by which scheduling period they should happen by, and then weight. I would make them both ascending. You can schedule a batch by pulling items off of that queue until you've got enough to send off.
Now you need to know how much to put in each batch. Given the system that you have described, a spike of events can be put in manually, but over time you'd like those spikes to smooth out. Therefore I would recommend option B, equal temperament. So to do this, as you put each object into the "ready now" queue, you can calculate its "average work weight" as its weight divided by the number of periods until it is supposed to happen. Store that with the object, and keep a running total of what run rate you should be at. Every period I would suggest that you keep adding to the batch until one of three conditions has been met:
You run out of objects.
You hit your maximum batch capacity.
You exceed 1.1 times your running total of your average work weight. The extra 10% is because it is better to use a bit more capacity now than to run out of capacity later.
And finally, capacity planning.
For this you need to use some heuristic. Here is a reasonable one which may need some tweaking for your system. Maintain an array of your past 10 measurements of running total of average work weight. Maintain an "exponentially damped average of your high water mark." Do that by updating each time according to the formula:
average_high_water_mark
= 0.95 * average_high_water_mark
+ 0.5 * max(last 10 running work weight)
If average_high_water_mark ever gets within, say, 2 servers of your maximum capacity, then add more servers. (The idea is that a server should be able to die without leaving you hosed.)
I think answer A is good. Bin packing is to maximize or minimize and you have only one batch. Sort the objects by m and n.
I am using codahale metrics for monitoring purposes. Lets say there is a spike in latency at some point and later there are no values reported due to attribute that there are no traffic, the value in the graph stays as is(I am using a histogram). At times it gives a notion that the spike remains and we might need to address it, but it actually means that no values are reported after that and hence the graph doesn't decay. Am I missing any config parameter in this case or is the behaviour expected?
The way we update the metrics is
metrics.processingTime.update(processingTime);
So, when there is no traffic, we don't update this metric.
I know that the histogram takes into consideration datapoints from the past (for an irregular period of time) in order to display a statistical image of the data.
When there are no new datapoints, only the outlier is taken into consideration and averaged on and on.
The meters have the same behavior, displaying the data through moving averages of 1,5,15 minutes.
The solution in the histogram case is to use HDRhistogram and flush it periodically.
Background
Echo Nest have a rate limited API. A given application (identified in requests using an API key) can make up to 120 REST calls a minute. The service response includes an estimate of the total number of calls made in the last minute; repeated abuse of the API (exceeding the limit) may cause the API key to be revoked.
When used from a single machine (a web server providing a service to clients) it is easy to control access - the server has full knowledge of the history of requests and can regulate itself correctly.
But I am working on a program where distributed, independent clients make requests in parallel.
In such a case it is much less clear what an optimal solution would be. And in general the problem appears to be undecidable - if over 120 clients, all with no previous history, make an initial request at the same time, then the rate will be exceeded.
But since this is a personal project, and client use is expected to be sporadic (bursty), and my projects have never been hugely successful, that is not expected to be a huge problem. A more likely problem is that there are times when a smaller number of clients want to make many requests as quickly as possible (for example, a client may need, exceptionally, to make several thousand requests when starting for the first time - it is possible two clients would start at around the same time, so they must cooperate to share the available bandwidth).
Given all the above, what are suitable algorithms for the clients so that they rate-limit appropriately? Note that limited cooperation is possible because the API returns the total number of requests in the last minute for all clients.
Current Solution
My current solution (when the question was written - a better approach is given as an answer) is quite simple. Each client has a record of the time the last call was made and the number of calls made in the last minute, as reported by the API, on that call.
If the number of calls is less than 60 (half the limit) the client does not throttle. This allows for fast bursts of small numbers of requests.
Otherwise (ie when there are more previous requests) the client calculates the limiting rate it would need to work at (ie period = 60 / (120 - number of previous requests)) and then waits until the gap between the previous call and the current time exceeds that period (in seconds; 60 seconds in a minute; 120 max requests per minute). This effectively throttles the rate so that, if it were acting alone, it would not exceed the limit.
But the above has problems. If you think it through carefully you'll see that for large numbers of requests a single client oscillates and does not reach maximum throughput (this is partly because of the "initial burst" which will suddenly "fall outside the window" and partly because the algorithm does not make full use of its history). And multiple clients will cooperate to an extent, but I doubt that it is optimal.
Better Solutions
I can imagine a better solution that uses the full local history of the client and models other clients with, say, a Hidden Markov Model. So each client would use the API report to model the other (unknown) clients and adjust its rate accordingly.
I can also imagine an algorithm for a single client that progressively transitions from unlimited behaviour for small bursts to optimal, limited behaviour for many requests without introducing oscillations.
Do such approaches exist? Can anyone provide an implementation or reference? Can anyone think of better heuristics?
I imagine this is a known problem somewhere. In what field? Queuing theory?
I also guess (see comments earlier) that there is no optimal solution and that there may be some lore / tradition / accepted heuristic that works well in practice. I would love to know what... At the moment I am struggling to identify a similar problem in known network protocols (I imagine Perlman would have some beautiful solution if so).
I am also interested (to a lesser degree, for future reference if the program becomes popular) in a solution that requires a central server to aid collaboration.
Disclaimer
This question is not intended to be criticism of Echo Nest at all; their service and conditions of use are great. But the more I think about how best to use this, the more complex/interesting it becomes...
Also, each client has a local cache used to avoid repeating calls.
Updates
Possibly relevant paper.
The above worked, but was very noisy, and the code was a mess. I am now using a simpler approach:
Make a call
From the response, note the limit and count
Calculate
barrier = now() + 60 / max(1, (limit - count))**greedy
On the next call, wait until barrier
The idea is quite simple: that you should wait some length of time proportional to how few requests are left in that minute. For example, if count is 39 and limit is 40 then you wait an entire minute. But if count is zero then you can make a request soon. The greedy parameter is a trade-off - when greater than 1 the "first" calls are made more quickly, but you are more likely hit the limit and end up waiting for 60s.
The performance of this is similar to the approach above, and it's much more robust. It is particularly good when clients are "bursty" as the approach above gets confused trying to estimate linear rates, while this will happily let a client "steal" a few rapid requests when demand is low.
Code here.
After some experimenting, it seems that the most important thing is getting as good an estimate as possible for the upper limit of the current connection rates.
Each client can track their own (local) connection rate using a queue of timestamps. A timestamp is added to the queue on each connection and timestamps older than a minute are discarded. The "long term" (over a minute) average rate is then found from the first and last timestamps and the number of entries (minus one). The "short term" (instantaneous) rate can be found from the times of the last two requests. The upper limit is the maximum of these two values.
Each client can also estimate the external connection rate (from the other clients). The "long term" rate can be found from the number of "used" connections in the last minute, as reported by the server, corrected by the number of local connections (from the queue mentioned above). The "short term" rate can be estimated from the "used" number since the previous request (minus one, for the local connection), scaled by the time difference. Again, the upper limit (maximum of these two values) is used.
Each client computes these two rates (local and external) and then adds them to estimate the upper limit to the total rate of connections to the server. This value is compared with the target rate band, which is currently set to between 80% and 90% of the maximum (0.8 to 0.9 * 120 per minute).
From the difference between the estimated and target rates, each client modifies their own connection rate. This is done by taking the previous delta (time between the last connection and the one before) and scaling it by 1.1 (if the rate exceeds the target) or 0.9 (if the rate is lower than the target). The client then refuses to make a new connection until that scaled delta has passed (by sleeping if a new connected is requested).
Finally, nothing above forces all clients to equally share the bandwidth. So I add an additional 10% to the local rate estimate. This has the effect of preferentially over-estimating the rate for clients that have high rates, which makes them more likely to reduce their rate. In this way the "greedy" clients have a slightly stronger pressure to reduce consumption which, over the long term, appears to be sufficient to keep the distribution of resources balanced.
The important insights are:
By taking the maximum of "long term" and "short term" estimates the system is conservative (and more stable) when additional clients start up.
No client knows the total number of clients (unless it is zero or one), but all clients run the same code so can "trust" each other.
Given the above, you can't make "exact" calculations about what rate to use, but you can make a "constant" correction (in this case, +/- 10% factor) depending on the global rate.
The adjustment to the client connection frequency is made to the delta between the last two connection (adjusting based on the average over the whole minute is too slow and leads to oscillations).
Balanced consumption can be achieved by penalising the greedy clients slightly.
In (limited) experiments this works fairly well (even in the worst case of multiple clients starting at once). The main drawbacks are: (1) it doesn't allow for an initial "burst" (which would improve throughput if the server has few clients and a client has only a few requests); (2) the system does still oscillate over ~ a minute (see below); (3) handling a larger number of clients (in the worst case, eg if they all start at once) requires a larger gain (eg 20% correction instead of 10%) which tends to make the system less stable.
The "used" amount reported by the (test) server, plotted against time (Unix epoch). This is for four clients (coloured), all trying to consume as much data as possible.
The oscillations come from the usual source - corrections lag signal. They are damped by (1) using the upper limit of the rates (predicting long term rate from instantaneous value) and (2) using a target band. This is why an answer informed by someone who understand control theory would be appreciated...
It's not clear to me that estimating local and external rates separately is important (they may help if the short term rate for one is high while the long-term rate for the other is high), but I doubt removing it will improve things.
In conclusion: this is all pretty much as I expected, for this kind of approach. It kind-of works, but because it's a simple feedback-based approach it's only stable within a limited range of parameters. I don't know what alternatives might be possible.
Since you're using the Echonest API, why don't you take advantage of the rate limit headers that are returned with every API call?
In general you get 120 requests per minute. There are three headers that can help you self-regulate your API consumption:
X-Ratelimit-Used
X-Ratelimit-Remaining
X-Ratelimit-Limit
**(Notice the lower-case 'ell' in 'Ratelimit'--the documentation makes you think it should be capitalized, but in practice it is lower case.)
These counts account for calls made by other processes using your API key.
Pretty neat, huh? Well, I'm afraid there is a rub...
That 120-request-per-minute is really an upper bound. You can't count on it. The documentation states that value can fluctuate according to system load. I've seen it as low as 40ish in some calls I've made, and have in some cases seen it go below zero (I really hope that was a bug in the echonest API!)
One approach you can take is to slow things down once utilization (used divided by limit) reaches a certain threshold. Keep in mind though, that on the next call your limit may have been adjusted download significantly enough that 'used' is greater than 'limit'.
This works well up until a point. Since the Echonest doesn't adjust the limit in a predictable mannar, it is hard to avoid 400s in practice.
Here are some links that I've found helpful:
http://blog.echonest.com/post/15242456852/managing-your-api-rate-limit
http://developer.echonest.com/docs/v4/#rate-limits