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I'm interested in developing a semi-autonomous RC lawnmower.
That is, the operator would decide when to stop, turn, etc., but could request "slightly overlap previous cut" and the mower would automatically do so. (Having operated high-end RC mowers at trade shows, this is the tedious part. Overcoming that, plus the high cost -- which I believe is possible -- would make a commercial success.)
This feature would require accurate horizontal positioning. I have investigated ultrasonic, laser, optical, and GPS. Each has its problems in this application. (I'll resist the temptation to go off on these tangents here.)
So... my question...
I know GPS horizontal accuracy is only 3-4m. Not good enough, but:
I don't need to know where I am on the planet. I only need to know where I am relative to where I was a minute ago.
So, my question is, is the inaccuracy consistent in the short term? if so, I think it would work for me. If it varies wildly by +- 1.5m from one second to the next, then it will not work.
I have tried to find this information but have had no success (possibly because of the ubiquity of other GPS-accuracy discussion), so I appreciate any guidance.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Edit ~~~~~~~~~~~~~~~~~~~~~~
It's looking to me like GPS is not just skewed but granular. I'd be interested in hearing from anyone who can give better insight into this, but for now I'm going to explore other options.
I realized that even though my intended application is "outdoor", this question is technically in the field of "indoor positioning systems" so I am adding that tag.
My latest thinking is to have 3 "intelligent" high-dB ultrasonic (US) speaker units. The mower emits RF requests for a tone from each speaker in rapid sequence, measuring the time it takes to "hear" each unit's response, thereby calculating distance to each of these fixed point and using trilateration to get position. if the fixed-point speakers are 300' away from the mower, the mower may have moved several feet between the 1st and 3rd response, so this would have to be allowed for in the software. If it is possible to differentiate 3 different US frequencies, they could be requested/received "simultaneously". Though you still run into issues when you're close to one fixed unit and far from another. So some software correction may still be necessary. If we can assume the mower is moving in a straight line, this isn't too complicated.
Another variation is the mower does not request the tones. The fixed units send RF "here comes tone from unit A" etc., and the mower unit just monitors both RF info and US tones. This may simplify things somewhat, but it seems it really requires the ability to determine which speaker a tone is coming from.
This seems like the kind of thing you could (and should) measure empirically. Just set a GPS of your liking down in the middle of a field on a clear day and wait an hour. Then come back and see what you find.
Because I'm in a city, I can't run out and do this for you. However, I found a paper entitled iGeoTrans – A novel iOS application for GPS positioning in geosciences.
That includes this figure which duplicates the test I propose. You'll note that both the iPhone4 and Garmin eTrex10 perform pretty poorly versus the accuracy you say you need.
But the authors do some Math Magic™ to reduce the uncertainty in the position, presumably by using some kind of averaging. That gets them to a 3.53m RMSE measure.
If you have real-time differential GPS, you can do better. But this requires relatively expensive hardware and software.
Even aside from the above, you have the potential issue of GPS reflection and multipath error. What if your mower has to go under a deck, or thick trees, or near the wall of a house? These common yard features will likely break the assumptions needed to make a good averaging algorithm work and even frustrate attempts at DGPS by blocking critical signals.
To my mind, this seems like a computer vision problem. And not just because that'll give you more accurate row overlaps... you definitely don't want to run over a dog!
In my opinion a standard GPS is no way accurate enough for this application. A typical consumer grade receiver that I have used has a position accuracy defined as a CEP of 2.5 metres. This means that for a stationary receiver in a "perfect" sky view environment over time 50% of the position fixes will lie within a circle with a radius of 2.5 metres. If you look at the position that the receiver reports it appears to wander at random around the true position sometimes moving a number of metres away from its true location. When I have monitored the position data from a number of stationary units that I have used they could appear to be moving at speeds of up to 0.5 metres per second. In your application this would mean that the lawnmower could be out of position by some not insignificant distance (with disastrous consequences for your prized flowerbeds).
There is a way that this can be done, as has been proved by the tractor manufacturers who can position the seed drills and agricultural sprayers to millimetre accuracy. These systems use Differential GPS where there is a fixed reference station positioned in the neighbourhood of the tractor being controlled. This reference station transmits error corrections to the mobile unit allowing it to correct its reported position to a high degree of accuracy. Unfortunately this sort of positioning system is very expensive.
We've all poked fun at the 'X minutes remaining' dialog which seems to be too simplistic, but how can we improve it?
Effectively, the input is the set of download speeds up to the current time, and we need to use this to estimate the completion time, perhaps with an indication of certainty, like '20-25 mins remaining' using some Y% confidence interval.
Code that did this could be put in a little library and used in projects all over, so is it really that difficult? How would you do it? What weighting would you give to previous download speeds?
Or is there some open source code already out there?
Edit: Summarising:
Improve estimated completion time via better algo/filter etc.
Provide interval instead of single time ('1h45-2h30 mins'), or just limit the precision ('about 2 hours').
Indicate when progress has stalled - although if progress consistently stalls and then continues, we should be able to deal with that. Perhaps 'about 2 hours, currently stalled'
More generally, I think you are looking for a way to give an instant mesure of the transfer speed, which is generally obtained by an average over a small period.
The problem is generally that in order to be reactive, the period is usually extremely small, which leads to the yoyo effect.
I would propose a very simple scheme, let's model it.
Think of a curve speed (y) over time (x).
the Instant Speed, is no more than reading y for the current x (x0).
the Average Speed, is no more than Integral(f(x), x in [x0-T,x0]) / T
the scheme I propose is to apply a filter, to give more weight to the last moments, while still taking into account the past moments.
It can be easily implement as g(x,x0,T) = 2 * (x - x0) + 2T which is a simple triangle of surface T.
And now you can compute Integral(f(x)*g(x,x0,T), x in [x0-T,x0]) / T, which should work because both functions are always positive.
Of course you could have a different g as long as it's always positive in the given interval and that its integral on the interval is T (so that its own average is exactly 1).
The advantage of this method is that because you give more weight to immediate events, you can remain pretty reactive even if you consider larger time intervals (so that the average is more precise, and less susceptible to hiccups).
Also, what I have rarely seen but think would provide more precise estimates would be to correlate the time used for computing the average to the estimated remaining time:
if I download a 5ko file, it's going to be loaded in an instant, no need to estimate
if I download a 15 Mo file, it's going to take between 2 minutes roughly, so I would like estimates say... every 5 seconds ?
if I download a 1.5 Go file, it's going to take... well around 200 minutes (with the same speed)... which is to say 3h20m... perhaps that an estimates every minute would be sufficient ?
So, the longer the download is going to take, the less reactive I need to be, and the more I can average out. In general, I would say that a window could cover 2% of the total time (perhaps except for the few first estimates, because people appreciate immediate feedback). Also, indicating progress by whole % at a time is sufficient. If the task is long, I was prepared to wait anyway.
I wonder, would a state estimation technique produce good results here? Something like a Kalman Filter?
Basically you predict the future by looking at your current model, and change the model at each time step to reflect the changes to the real world. I think this kind of technique is used for estimating the time left on your laptop battery, which can also vary according to use, age of battery, etc'.
see http://en.wikipedia.org/wiki/Kalman_filter for a more in depth description of the algorithm.
The filter also gives a variance measure, which could be used to indicate your confidence of the estimate (allthough, as was mentioned by other answers, it might not be the best idea to show this to the end user)
Does anyone know if this is actually used somewhere for download (or file copy) estimation?
Don't confuse your users by providing more information than they need. I'm thinking of the confidence interval. Skip it.
Internet download times are highly variable. The microwave interferes with WiFi. Usage varies by time of day, day of week, holidays, and releases of new exciting games. The server may be heavily loaded right now. If you carry your laptop to cafe, the results will be different than at home. So, you probably can't rely on historical data to predict the future of download speeds.
If you can't accurately estimate the time remaining, then don't lie to your user by offering such an estimate.
If you know how much data must be downloaded, you can provide % completed progress.
If you don't know at all, provide a "heartbeat" - a piece of moving UI that shows the user that things are working, even through you don't know how long remains.
Improving the estimated time itself: Intuitively, I would guess that the speed of the net connection is a series of random values around some temporary mean speed - things tick along at one speed, then suddenly slow or speed up.
One option, then, could be to weight the previous set of speeds by some exponential, so that the most recent values get the strongest weighting. That way, as the previous mean speed moves further into the past, its effect on the current mean reduces.
However, if the speed randomly fluctuates, it might be worth flattening the top of the exponential (e.g. by using a Gaussian filter), to avoid too much fluctuation.
So in sum, I'm thinking of measuring the standard deviation (perhaps limited to the last N minutes) and using that to generate a Gaussian filter which is applied to the inputs, and then limiting the quoted precision using the standard deviation.
How, though, would you limit the standard deviation calculation to the last N minutes? How do you know how long to use?
Alternatively, there are pattern recognition possibilities to detect if we've hit a stable speed.
I've considered this off and on, myself. I the answer starts with being conservative when computing the current (and thus, future) transfer rate, and includes averaging over longer periods, to get more stable estimates. Perhaps low-pass filtering the time that is displayed, so that one doesn't get jumps between 2 minutes and 2 days.
I don't think a confidence interval is going to be helpful. Most people wouldn't be able to interpret it, and it would just be displaying more stuff that is a guess.
We've all poked fun at the 'X minutes remaining' dialog which seems to be too simplistic, but how can we improve it?
Effectively, the input is the set of download speeds up to the current time, and we need to use this to estimate the completion time, perhaps with an indication of certainty, like '20-25 mins remaining' using some Y% confidence interval.
Code that did this could be put in a little library and used in projects all over, so is it really that difficult? How would you do it? What weighting would you give to previous download speeds?
Or is there some open source code already out there?
Edit: Summarising:
Improve estimated completion time via better algo/filter etc.
Provide interval instead of single time ('1h45-2h30 mins'), or just limit the precision ('about 2 hours').
Indicate when progress has stalled - although if progress consistently stalls and then continues, we should be able to deal with that. Perhaps 'about 2 hours, currently stalled'
More generally, I think you are looking for a way to give an instant mesure of the transfer speed, which is generally obtained by an average over a small period.
The problem is generally that in order to be reactive, the period is usually extremely small, which leads to the yoyo effect.
I would propose a very simple scheme, let's model it.
Think of a curve speed (y) over time (x).
the Instant Speed, is no more than reading y for the current x (x0).
the Average Speed, is no more than Integral(f(x), x in [x0-T,x0]) / T
the scheme I propose is to apply a filter, to give more weight to the last moments, while still taking into account the past moments.
It can be easily implement as g(x,x0,T) = 2 * (x - x0) + 2T which is a simple triangle of surface T.
And now you can compute Integral(f(x)*g(x,x0,T), x in [x0-T,x0]) / T, which should work because both functions are always positive.
Of course you could have a different g as long as it's always positive in the given interval and that its integral on the interval is T (so that its own average is exactly 1).
The advantage of this method is that because you give more weight to immediate events, you can remain pretty reactive even if you consider larger time intervals (so that the average is more precise, and less susceptible to hiccups).
Also, what I have rarely seen but think would provide more precise estimates would be to correlate the time used for computing the average to the estimated remaining time:
if I download a 5ko file, it's going to be loaded in an instant, no need to estimate
if I download a 15 Mo file, it's going to take between 2 minutes roughly, so I would like estimates say... every 5 seconds ?
if I download a 1.5 Go file, it's going to take... well around 200 minutes (with the same speed)... which is to say 3h20m... perhaps that an estimates every minute would be sufficient ?
So, the longer the download is going to take, the less reactive I need to be, and the more I can average out. In general, I would say that a window could cover 2% of the total time (perhaps except for the few first estimates, because people appreciate immediate feedback). Also, indicating progress by whole % at a time is sufficient. If the task is long, I was prepared to wait anyway.
I wonder, would a state estimation technique produce good results here? Something like a Kalman Filter?
Basically you predict the future by looking at your current model, and change the model at each time step to reflect the changes to the real world. I think this kind of technique is used for estimating the time left on your laptop battery, which can also vary according to use, age of battery, etc'.
see http://en.wikipedia.org/wiki/Kalman_filter for a more in depth description of the algorithm.
The filter also gives a variance measure, which could be used to indicate your confidence of the estimate (allthough, as was mentioned by other answers, it might not be the best idea to show this to the end user)
Does anyone know if this is actually used somewhere for download (or file copy) estimation?
Don't confuse your users by providing more information than they need. I'm thinking of the confidence interval. Skip it.
Internet download times are highly variable. The microwave interferes with WiFi. Usage varies by time of day, day of week, holidays, and releases of new exciting games. The server may be heavily loaded right now. If you carry your laptop to cafe, the results will be different than at home. So, you probably can't rely on historical data to predict the future of download speeds.
If you can't accurately estimate the time remaining, then don't lie to your user by offering such an estimate.
If you know how much data must be downloaded, you can provide % completed progress.
If you don't know at all, provide a "heartbeat" - a piece of moving UI that shows the user that things are working, even through you don't know how long remains.
Improving the estimated time itself: Intuitively, I would guess that the speed of the net connection is a series of random values around some temporary mean speed - things tick along at one speed, then suddenly slow or speed up.
One option, then, could be to weight the previous set of speeds by some exponential, so that the most recent values get the strongest weighting. That way, as the previous mean speed moves further into the past, its effect on the current mean reduces.
However, if the speed randomly fluctuates, it might be worth flattening the top of the exponential (e.g. by using a Gaussian filter), to avoid too much fluctuation.
So in sum, I'm thinking of measuring the standard deviation (perhaps limited to the last N minutes) and using that to generate a Gaussian filter which is applied to the inputs, and then limiting the quoted precision using the standard deviation.
How, though, would you limit the standard deviation calculation to the last N minutes? How do you know how long to use?
Alternatively, there are pattern recognition possibilities to detect if we've hit a stable speed.
I've considered this off and on, myself. I the answer starts with being conservative when computing the current (and thus, future) transfer rate, and includes averaging over longer periods, to get more stable estimates. Perhaps low-pass filtering the time that is displayed, so that one doesn't get jumps between 2 minutes and 2 days.
I don't think a confidence interval is going to be helpful. Most people wouldn't be able to interpret it, and it would just be displaying more stuff that is a guess.
A delay will always occur between a user action and an application response.
It is well known that the lower the response delay, the greater the feeling of the application responding instantaneously. It is also commonly known that a delay of up to 100ms is generally not perceivable. But what about a delay of 110ms?
What is the shortest application response delay that can be perceived?
I'm interested in any solid evidence, general thoughts and opinions.
The 100 ms threshold was established over 30 yrs ago. See:
Card, S. K., Robertson, G. G., and Mackinlay, J. D. (1991). The information visualizer: An information workspace. Proc. ACM CHI'91 Conf. (New Orleans, LA, 28 April-2 May), 181-188.
Miller, R. B. (1968). Response time in man-computer conversational transactions. Proc. AFIPS Fall Joint Computer Conference Vol. 33, 267-277.
Myers, B. A. (1985). The importance of percent-done progress indicators for computer-human interfaces. Proc. ACM CHI'85 Conf. (San Francisco, CA, 14-18 April), 11-17.
What I remember learning was that any latency of more than 1/10th of a second (100ms) for the appearance of letters after typing them begins to negatively impact productivity (you instinctively slow down, less sure you have typed correctly, for example), but that below that level of latency productivity is essentially flat.
Given that description, it's possible that a latency of less than 100ms might be perceivable as not being instantaneous (for example, trained baseball umpires can probably resolve the order of two events even closer together than 100ms), but it is fast enough to be considered an immediate response for feedback, as far as effects on productivity. A latency of 100ms and greater is definitely perceivable, even if it's still reasonably fast.
That's for visual feedback that a specific input has been received. Then there'd be a standard of responsiveness in a requested operation. If you click on a form button, getting visual feedback of that click (eg. the button displays a "depressed" look) within 100ms is still ideal, but after that you expect something else to happen. If nothing happens within a second or two, as others have said, you really wonder if it took the click or ignored it, thus the standard of displaying some sort of "working..." indicator when an operation might take more than a second before showing a clear effect (eg. waiting for a new window to pop up).
New research as of January, 2014:
http://newsoffice.mit.edu/2014/in-the-blink-of-an-eye-0116
...a team of neuroscientists from MIT has found that the human brain
can process entire images that the eye sees for as little as 13
milliseconds...That speed is far faster than the 100 milliseconds
suggested by previous studies...
At the San Francisco Opera house, we routinely setup precise delay setting for each of our speakers. We can detect 5 millisecond changes in delay times to our speakers. When you make such subtle changes, you change where the sound sources from. Often times we want sound to sound as if it's coming from someplace other than were the speakers are. Precise delay adjustments make this possible. Sound delays of 15 milliseconds are very obvious even to untrained ears because it radically shifts where the sound sources from. A simple test is to prove this is to play sound through multiple speakers, and have the subject close their eyes and point to where the sound is coming from. Now make a slight change in the delay time to one of the speakers of just a few milliseconds, and have the person point again to where the sound is coming from. Making changes in delay times is acoustically very similar to moving the actual speakers.
I don't think anecdotes or opinions are really valid for answers here. This question touches on the psychology of user experience and the sub-conscious mind. The human brain is powerful and fast and mere milliseconds do count and are registered. I am no expert but I know there is much science behind e.g. what Matt Jacobsen mentioned. Check out Google's study here http://services.google.com/fh/files/blogs/google_delayexp.pdf for an idea of how much it can affect site traffic.
Here's another study by Akami - 2 second response time
http://www.akamai.com/html/about/press/releases/2009/press_091409.html (From https://ux.stackexchange.com/questions/5529/once-apon-a-time-there-was-a-10-seconds-to-load-a-page-rule-what-is-it-nowa )
Does anyone have any other studies to share?
Persistence of vision is around 100ms so it should be a reasonable visual feedback delay. 110ms should make no difference, as it is an approximate value. In practice you won't notice a delay below 200ms.
Out of my memory, studies have shown that users lose patience and retry an operation after around 2s of inactivity (in the absence of feedback), e.g. clicking on a confirm or action button. So plan on using some kind of animation if the action takes longer than 1s.
I worked on an application that had a explicit business goal of being blindingly fast, and we had a max allowed server time of 150ms for processing a full web page.
No solid evidence but for our own application, we allow a maximum of one second between a user action and feedback. If it does take longer, a "waiting box" should be shown.
A user should see "something" happening within a second of causing an action.
Use the dual of test for visual spatial resolution ( two parallel black bars, with an equal width, and an equal gap between them. Reduce angular subtense until they appear to be one line, ie scale down or simply move away. The point at which it seems to merge into one line shows the threshold).
Use function gen to blink an LED on for an interval, then off, then on, then off --- same time delay each interval, but repeat the pattern while gradually decreasing that delay, thus same as above, but time in place of space.
Imagine an oscilloscope image like so:
_________/^d^\_d_/^d^\_________
I note that at 41 ms interval, I perceive one longer blink only, but at 42 ms, I just perceive it as extremely rapid double blink. Thus, threshold is ~42ms. Probably varies depending on person, age, condition etc.
This is close to 24 fps, which is probably why cinema works at that presentation rate.
Reaction time to see something, then decide to react, say by clicking mouse etc, is longer much longer again. Thus, it's not surprising that experiments requiring a reaction response to measure yield a longer time, but that longer delay wasn't what you were asking for, and the above experiment is easy and illuminating!
But note also -- smoothly moving animations require the visual cortex to work harder, delaying visual comprehension. This delay is 'hidden' from perception, so longer delays (several hundred ms) can be 'hidden' by just providing something thats difficult to see because moving.
The effect that hides it is called Chronostasis. Basically, glancing somewhere 'new' requires the visual cortex to work harder to 'de-render' / 'recognise' the scene. This takes a remarkably long time, during which your consciousness is essentially 'paused'.
Once looking at a mostly-constant scene, only changes need this processing, so smaller/faster changes are possible and your perceptual experience resumes, and faster/smaller movements are detectable.
The detection of changes visually is processed basically on your retina. Your eyes also have a natural 'bandpass' response -- stare unblinkingly at anything for sufficient time, and at sufficient distance for saccades to be unable to change the image much, and you will find your visual feed fading out to 'grey'. This is what gives us our 'white balance', and is somewhat similar to the automatic gain control on analogue radio/tv.
The point is, that your eyes themselves have a time constant to respond, but this is actually dependant on the strength of the stimulus. (brightness of the LED, for our case).
Too bright, and the ability of your retinal cells to 'relax' back from the brightness, ie, respond to the 'sudden dark', is compromised.
The effect which keeps you seeing bright things after the light has stopped is called 'persistence of vision', and old cathode-ray picture tubes more or less depend heavily on it for them to work at all.
This is the one that's usually 100 ms or so, but it's not a 'sharp' interval -- more like a exponential roll-off, and again -- changes duration depending on how bright the stimulus is relative to how dark-adjusted (ie, sensitive) the eye is at that moment.
For duller, faster changes, especially changes outside your fovea, you will perceive even higher rates easily. Eg, flickering lights. Those outer parts of your retina (most of the area, actually) are adapted to detecting movement, and bringing it to your attention. So it makes sense that although lacking spatial resolution, they have greater time resolution / shorter response rate.
But this also means animating things usually requires even finer time steps, otherwise 'jumpiness' is perceptible, mostly due to that faster response.
Note all the scaling/sliding full screen animations iOS uses -- these essentially exploit chronostasis to hide technically unavoidable loading delays, giving the perception that those products respond instantly and smoothly at all times.
So, show something different within 42 ms -> instant response.
Keep animating otherwise useless hard-to-see-properly visuals continuously at high frame rates, then stop suddenly when done -> hides the delay so long as enough is visually busy, and the delay isn't too long. (probably 250ms is pushing the friendship).
This also seems to tee up with other's perceptions of input lag, for example : http://danluu.com/input-lag/
100ms is totally wrong. You can prove this yourself using your fingers, a desk, and a watch with visible seconds. Synchronising to the watch's seconds, drum out beats on the desk continuously such that 16 beats are drummed out every second. I chose 16 because it is natural to drum out multiples of two, so it's like four strong beats with three weak beats in between. Adjacent beats are clearly discernible by their sound. The beats are separated by about 60ms, so even 60 ms is actually still too high. Therefore the threshold is way below 100ms, especially if sound is involved.
For instance, a drum app or a keyboard app needs a delay of more like 30ms, or else it gets really annoying, because you hear the sound coming from the physical button / pad / key well before the sound comes out of the speakers. Software like ASIO and jack were made specifically to deal with this issue, so no excuses. If your drum app has a 100ms delay, I will hate you.
The situation for VoIP and high powered gaming is actually worse, because you need to react to events in real time, and in music, at least you get to plan ahead at least a little. For an average human reaction time of 200ms, a further 100ms delay is an enormous penalty. It noticeably changes the conversational flow of VoIP. In gaming, 200ms reaction time is generous, especially if the players have a lot of practice.
For a reasonably current scholarly article, try out How Much Faster is Fast Enough? User Perception of
Latency & Latency Improvements in Direct and Indirect Touch (PDF). While the main focus was on JND (Just Noticeable Difference) of delay, there is some good background on on absolute delay perception and they also acknowledge and account for 60Hz monitors (16.7 ms repaint times) in their second experiment.
I am a cognitive neuroscientist who studies visual perception and cognition.
The paper by Mary Potter mentioned above regards the minimum time required to categorize a visual stimulus. However, understand that this is under laboratory conditions in the absence of any other visual stimuli, which certainly would not be the case in the real world user experience.
The typical benchmark for a stimulus-response / input-stimulus interaction, that is, the average amount of time for an individuals minimum reaction speed or input-response detection is ~200ms. to be certain there is no detectable difference, this threshold could be lowered to around 100ms. Below this threshold, the temporal dynamics of your cognitive processes take longer to compute the event than the event itself, so there is nearly no chance of any ability to detect or differentiate it. You could go lower to say 50 ms, but it really wouldn't be necessary. 10 ms and you've gone into the territory of overkill.
For web applications 200ms is considered as unnoticable delay, while 500ms is acceptable.
I'm writing an app that lets a diabetic user enter his/her "blood glucose" readings, and then charts them on a graph over time from left to right. Since the blood readings will be done only several times a day, an algorithm would be handy to:
a) fill in the gaps on the graph between readings (curves would be more realistic than jerky lines) and allow a more accurate "blood glucose level" daily average
b) roughly predict what will happen in the future (if the user eats nothing that will affect his blood levels)
I suck at calculus. I'm hoping someone here knows a library for this stuff? I'm hoping someone knows of an algorithm that has been tailored for this specific problem already (e.g.: where someone has compared it to real data from diabetics)
Disclaimer: I am very aware that any such algorithm would vary wildly depending on the user. I'm just looking to improve on straight angular lines. Regardless of the diabetic, there is a limit to the rate that blood sugars can rise and fall.
I'm using Javascript, but as it's just math, I could port it from C, Java or whatever.
Blood sugar behavior is very complicated. It is affected by
Current blood sugar (complicated by the possible presence of ketones if the patient is hyperglycemic)
recent food out to several hours depending on the type and how much
recent fast acting insulin (with variety and patient dependent reaction profiles between 45 minutes and two hours long. Oh, and delivery mechanism)
long-acting insulin out past 12 hours (again patient and variety dependent)
activity levels
stress levels
illness
basal insulin rate if the patient wears a pump
ad nauseum
Very hard problem. Any heuristic---any heuristic---you chose would be highly misleading. So short answer:
Don't do it.
This comes, in part, from having compared a diabetic's 24-hour continous glucose log with the ~10 finger pricks taken during the same time. I.e. my suggestion is data driven.
Edit: Evidently I didn't make myself clear.
You can't even get close.
Nothing you can do with finger prick data can be remotely reliable.
Connecting the dots with any lines (even straight segments) is just plain wrong. It doesn't reflect reality. Not even a little bit.
I'm an experimental particle physicist. Complicated data sets are what I do. There is a diabetic in my life (did you guess?). This matters to me.
But I've seen the high frequency data logs, side-by-side with a log of the days finger-pricks, exercise, food, and insulin.
If you could get every-fifteen-minutes data, I'd say go ahead and use a spline. It won't be dangerously misleading. But, if you have 6-10 measurements across the day, you know nothing.
Good news: continuous monitoring is coming down in price. It's out of the lab and available with some pumps even now.
For those who aren't familiar with this: compliant diabetic patients do (results of extremely unscientific polling) 4-6+ glucose tests a day as a matter of course, and several additional ones in the 1-2 hours following any unexpected excursion (they get physical symptoms that allow them to detect severe excursions).
This serves to give the patient a rough idea of how they are doing at controlling their glucose levels, but they also go to a lab to get a Hemoglobin A1C drawn every quarter (or so). The A1C result is dependent mostly on their average blood glucose.
I've talked to people who clocked in a 80-110 (quite favorable numbers) four times a day for months, and got back an A1C suggesting an average above 150 (not desirable at all). Presumable they were going high in the night. And I've heard similar stories from people who we probably going low---very low---in their sleep.
The lesson is:
Finger prick readings have their place, but don't try to extrapolate them to times not well sampled.
If you want to do just a straight fit of the data to make things easier to view then something like what Charlie Martin recommended would likely work well. However, as noted by dmckee this data really wouldn't mean anything.
What you are trying to do is actually more in line with pharmacokenetics which is an entire scientific study in and of itself. In this case I'm not even sure it would entirely apply except in the case of Type I Diabetes as most of what I know about pharamcokenetics only applies drug studies, but if something is being produced by the body then you are likely looking at entirely different types of analysis. If you are interested in the subject then there are quite a few book previews on Google Books if you do a search for "pharmacokienetics" but due to the nature of the subject they are very math heavy and assume that you have an understanding of chemistry and biology as well.
okay, you're going to be looking for some fitted curve. The thing with that is that for n points there are fit polynomials up to order... n-1 I think. It's been a while. Yep. by golly, I'm right. The common thing when you have lots of points and don't wants a complicated function (which you don't) is to use a least-squares approximation.
probably the best thing is to look for a canned routine you can use; these exist in most stats packages. Give us a little more detail on the environment you want and we might be able to point you more closely to something suitable.
This is most likely not going to work but Artificial Neural Networks may, and i repeat may be able to get something out of a good data set. By good, i mean like weeks or months of continuous recording, and even then i wouldn't trust the data set unless i had very good reason to. I also don't think you'll get predictive data out of it, but it may depend on how you implement it. Overall if you were to do this it would seem to be more of a hobby thing to see if it even even come close, like "oh neat i got a neural network to within X amount of accuracy". Again, i must stress, don't use this in any sort of production situations or anywhere where it could possibly hurt or kill someone!