This article explains how to calculate bigmin and litmax (big minimum and little max).
I have a hard time understanding the phrasing of the step 4 to see how it would translate to pseudocode, C or python.
Step 4
A horizontal division we know we need to calculate the Latitude value closest to the division line of yn. Take the most significant bit’s of both min and max up to where they first differ, yn, and call it y[1..m] we know that the Latitude value just above the division line will be binary coded y[1..m] 0111..., and y[1..m] 1000... which translates to our Latitude value for LitMax, and BigMin.
Since the most significant bit that differed in our example was y4, our LitMax and BigMin Latitude values equals 0111 and 1000.
In a vertical division we just reverse this to the x bits and Longitude.
I have no problem translating z order index from coord and vice versa.
I'm just interested in calculating bigmin and litmax since they allow a big speed up. I've searched and I can't find good details on those particular bitmask operations (the answer here How to use Morton Order(z order curve) in range search? doesn't really cover it, and neither does the linked dynamodb article).
Take a look to the page 76 decision tables in the original article http://hermanntropf.de/media/multidimensionalrangequery.pdf The ideas behind the Bigmin Calculation for Z-Indexing are described a bit more elaborately in the US patent US7321890B2 which is about using hilbert order instead of Z-order. See section 2: "BIGMIN Solution for Z-Indexing" which has been included to ease the description as much of the ideas in the patent is in common with the application to Hilbert indexing. https://patents.google.com/patent/US7321890B2/en?assignee=hermann+tropf&oq=hermann+tropf. Once realized for Bigmin, just take a look at the Litmax Decision Table in the original article. I hope this may help.
Related
RSME calculates how close the predicted value is compared to the actual value, but in a point cloud, there are 2 things that I am confused about:
How do we know which point corresponds to which point, to be subtracted from?
Point clouds are 3-dimensional since it has xyz values, but how do people turn those 3 values to one RSME value?
First of all, it's RMSE, not RSME. It stands for Root Mean Square Error:
https://en.wikipedia.org/wiki/Root-mean-square_deviation
With 3D coordinates you can compare component wise, or however else you choose to define a distance measure. Then you plug this into the RMSE formula. Essentially this means comparing an expected value to your observed value.
As for the point correspondence - this depends on the algorithm of choice. Probably one of the most famous examples is ICP:
https://de.wikipedia.org/wiki/Iterative_Closest_Point_Algorithm
In a nutshell for every point of one cloud, the closest point of the other cloud is determined. Then an error measure is calculated and lastly points are transformed. This is done an arbitrary number of times, depending on the desired precision.
Since I strongly suspect that you are indeed looking for ICP, here is the description as to how they are put together:
https://en.wikipedia.org/wiki/Iterative_closest_point
Other than that you will have to do some reading yourself.
Problem
I want to find
The first root
The first local minimum/maximum
of a black-box function in a given range.
The function has following properties:
It's continuous and differentiable.
It's combination of constant and periodic functions. All periods are known.
(It's better if it can be done with weaker assumptions)
What is the fastest way to get the root and the extremum?
Do I need more assumptions or bounds of the function?
What I've tried
I know I can use root-finding algorithm. What I don't know is how to find the first root efficiently.
It needs to be fast enough so that it can run within a few miliseconds with precision of 1.0 and range of 1.0e+8, which is the problem.
Since the range could be quite large and it should be precise enough, I can't brute-force it by checking all the possible subranges.
I considered bisection method, but it's too slow to find the first root if the function has only one big root in the range, as every subrange should be checked.
It's preferable if the solution is in java, but any similar language is fine.
Background
I want to calculate when arbitrary celestial object reaches certain height.
It's a configuration-defined virtual object, so I can't assume anything about the object.
It's not easy to get either analytical solution or simple approximation because various coordinates are involved.
I decided to find a numerical solution for this.
For a general black box function, this can't really be done. Any root finding algorithm on a black box function can't guarantee that it has found all the roots or any particular root, even if the function is continuous and differentiable.
The property of being periodic gives a bit more hope, but you can still have periodic functions with infinitely many roots in a bounded domain. Given that your function relates to celestial objects, this isn't likely to happen. Assuming your periodic functions are sinusoidal, I believe you can get away with checking subranges on the order of one-quarter of the shortest period (out of all the periodic components).
Maybe try Brent's Method on the shortest quarter period subranges?
Another approach would be to apply your root finding algorithm iteratively. If your range is (a, b), then apply your algorithm to that range to find a root at say c < b. Then apply your algorithm to the range (a, c) to find a root in that range. Continue until no more roots are found. The last root you found is a good candidate for your minimum root.
Black box function for any range? You cannot even be sure it has the continuous domain over that range. What kind of solutions are you looking for? Natural numbers, integers, real numbers, complex? These are all the question that greatly impact the answer.
So 1st thing should be determining what kind of number you accept as the result.
Second is having some kind of protection against limes of function that will try to explode your calculations as it goes for plus or minus infinity.
Since we are touching the limes topics you could have your solution edge towards zero and look like a solution but never touch 0 and become a solution. This depends on your margin of error, how close something has to be to be considered ok, it's good enough.
I think for this your SIMPLEST TO IMPLEMENT bet for real number solutions (I assume those) is to take an interval and this divide and conquer algorithm:
Take lower and upper border and middle value (or approx middle value for infinity decimals border/borders)
Try to calculate solution with all 3 and have some kind of protection against infinities
remember all 3 values in an array with results from them (3 pair of values)
remember the current best value (one its closest to solution) in seperate variable (a pair of value and result for that value)
STEP FORWARD - repeat above with 1st -2nd value range and 2nd -3rd value range
have a new pair of value and result to be closest to solution.
clear the old value-result pairs, replace them with new ones gotten from this iteration while remembering the best value solution pair (total)
Repeat above for how precise you wish to get and look at that memory explode with each iteration, keep in mind you are gonna to have exponential growth of values there. It can be further improved if you lets say take one interval and go as deep as you wanna, remember best value-result pair and then delete all other memory and go for next interval and dig deep.
How big should epsilon be when checking if dot product is close to 0?
I am working on raytracing project, and i need to check if the dot product is 0. But
that will probably never happen, so I want to take it as 0 if its value is in a small
area [-eps, +eps], but I am not sure how big should eps be?
Thanks
Since you describe this as part of a ray-tracing project, your accuracy needed is likely dictated by the "world coordinates" of a scene, or perhaps even the screen coordinates to which those are translated. Either of these would give an acceptable target for absolute error in your calculation.
It might be possible to backtrack from that to the accuracy required in the intermediate calculations you are doing, such as forming an inner product which is supposed "theoretically" to be zero. For example, you might be trying to find a shortest path (reflected light) between two smooth bodies, and the disappearance of the inner product (perpendicularity) gives the location of a point.
In a case like that the inner product may be a quadratic in the unknowns (location of a point) that you seek. It's possible that the unknowns form a "double root" (zero of multiplicity 2), making the location of that root extra sensitive to the computation of the inner product being zero.
For such cases you would want to get roughly twice the number of digits "zero" in the inner product as needed in the accuracy of the location. Essentially the inner product changes very slowly with location in the neighborhood of a double root.
But your application might not be so sensitive; analysis of the algorithm involved is necessary to give you a good answer. As a general rule I do the inner product in double precision to get answers that may be reliable as far as single precision, but this may be too costly if the ray-tracing is to be done in real time.
There is no definitive answer. I use two approaches.
If you are only concerned by floating point error then you can use a pretty small value, comparable to the smallest floating number that the compiler can handle. In c/c++ you can use the definitions provided in float.h, such as DBL_MIN to check for these numbers. I'd use a small multiple of the number, e.g., 10. * DBL_MIN as the value for eps.
If the problem is not floating point math rounding error, then I use a value that is small (say 1%) compared to the modulus of the smallest vector.
I have a linear regression equation from school , which gives a value between 1 and -1 indicative of whether or not a set of data points are close enough to a linear function
and the equation given here
http://people.hofstra.edu/stefan_waner/realworld/calctopic1/regression.html
under best fit of a line. I would like to use these to do simple gesture detection based on a point in 3-space (x,y,z) - forward, back, left, right, up, down. First I would see if they fall on a line in 2 of the 3 dimensions, then I would see if that line's slope approached zero or infinity.
Is this fast enough for functional gesture recognition? If not, could someone propose an alternative algorithm?
If I've understood your question correctly then (1) the calculation you describe here would probably be plenty fast enough, (2) it may not actually do what you want, and (3) the stuff that'll be slow in an actual implementation would lie elsewhere.
So, I think you're proposing to do this. (1) Identify the positions of ... something ... (the user's hand, perhaps) in three-dimensional space, at several successive times. (2) For (say) each of {x,y} and {x,z}, look at those two coordinates of each point, compute the correlation coefficient (which is what your formula describes) and see whether it's close to +-1. (3) If both correlation coefficients are close to +-1 then the points lie approximately on a straight line; calculate the gradient of that line (using a formula similar to that of the correlation coefficient). (4) If the gradients are both very close to 0 or +- infinity, then your line is approximately parallel to one axis, which is the case you're trying to recognize.
1: Is it fast enough? You might perhaps be sampling at 50 frames per second or thereabouts, and your gestures might take a second to execute. So you'll have somewhere on the order of 50 positions. So, the total number of arithmetic operations you'll need is maybe a few hundred (including a modest number of square roots). In the worst case, you might be doing this in emulated floating-point on a slow ARM processor or something; in that case, each arithmetic operation might take a couple of hundred cycles, so the whole thing might be 100k cycles, which for a really slow processor running at 100MHz would be about a millisecond. You're not going to have any problem with the time taken to do this calculation.
2: Is it the right thing? It's not clear that it's the right calculation. For instance, suppose your user's hand moves back and forth rapidly several times along the x-axis; that will give you a positive result; is that what you want? Suppose the user attempts the gesture you want but moves at slightly the wrong angle; you may get a negative result. Suppose they move exactly along the x-axis for a bit and then along the y-axis for a bit; then the projections onto the {x,y}, {x,z} and {y,z} planes will all pass your test. These all seem like results you might not want.
3: Is it where the real cost will lie? This all assumes you've already got (x,y,z) coordinates. Getting those is probably going to be more expensive than processing them. For instance, if you have a camera-based system of some kind then there'll be some nontrivial image processing for every frame. Or perhaps you're integrating up data from accelerometers (which, by the way, is likely to give nasty inaccurate position results); the chances are that you're doing some filtering and other calculations to get position data. I bet that the cost of performing a calculation like this one will be substantially less than the cost of getting the coordinates in the first place.
Okay, so this is a straight math question and I read up on meta that those need to be written to sound like programming questions. I'll do my best...
So I have graph made in flot that shows the network usage (in bytes/sec) for the user. The data is 4 minutes apart when there is activity, and otherwise set at the start of the usage range (let's say day 1) and the end of the range (day 7). The data is coming from a CGI script I have no control over, so I'm fairly limited in what I can provide the user.
I never took trig or calculus, so I'm pretty much in over my head. What I want is for the user to have the option to click any point on the graph and see their bandwidth usage for that moment. Since the lines between real data points are drawn straight, this can be done by getting the points before and after where the user has clicked and finding the y-interval.
It took me weeks to finally get a helpful math person to explain this to me. Everyone else has insisted on trying to teach me Riemann sum techniques and all sorts of other heavy stuff that not only is confusing to me, doesn't seem necessary for the problem.
But I also want the user to be able to highlight the graph from two arbitrary points on the y-axis (time) to get the amount of network usage total during that range. I know this would be inaccurate, but I need it to be the right inaccurate using a solid equation.
I thought this was the area under the line, but experiments with much simpler graphs makes this seem just far too high. I figured out I could take the distance from y2 - y1 and multiply it by x2 - x1 and then divide by two to get the area of the graph below the line like a triangle, but again, the numbers seemed to high. (maybe they are just big numbers and I don't get this math stuff at all).
So what I need, if anyone would be really awesome enough to provide it before this question is closed down for being too pure-math, is either the name of the concept I should be researching or the equation itself. Or the bad news that I do need advanced math to get an accurate result.
I am not bad at math, just as a last note, I just am not familiar with math beyond 10th grade and so I need some place to start. All the math sites seem to keep it too simple or way over my paygrade.
If I understood correctly what you're asking (and that is somewhat doubtful), you should find what you seek in these links:
Linear interpolation
(calculating the value of the point in between)
Trapezoidal rule
(calculating the area below the "curve")
*****Edit, so we can get this over :) without much ado:*****
So I have graph made in flot that shows the network usage (in bytes/sec) for the user. The data is 4 minutes apart when there is activity, and otherwise set at the start of the usage range (let's say day 1) and the end of the range (day 7). The data is coming from a CGI script I have no control over, so I'm fairly limited in what I can provide the user.
What is a "flot" ?
Okey, so you have speed on y axis [in bytes/sec]; and time on x axis in [sec], right?
That means, that if you're flotting (I'm bored, yes :) speed over time, in linear segments, interpolating at some particular point in time you'll get speed at that particular point in time.
If you wish to calculate how much bandwidth you've spend, you need to determine the area beneath that curve. The area from point "a" to point "b" will determine the spended bandwidth in [bytes] in that time period.
It took me weeks to finally get a helpful math person to explain this to me. Everyone else has insisted on trying to teach me Riemann sum techniques and all sorts of other heavy stuff that not only is confusing to me, doesn't seem necessary for the problem.
In the immortal words of Snoopy: "Good grief !"
But I also want the user to be able to highlight the graph from two arbitrary points on the y-axis (time) to get the amount of network usage total during that range. I know this would be inaccurate, but I need it to be the right inaccurate using a solid equation.
It would not be inaccurate.
It would be actually perfectly accurate (well, apart from roundoff error in bytes :), since you're using linear interpolation on linear segments.
I thought this was the area under the line, but experiments with much simpler graphs makes this seem just far too high. I figured out I could take the distance from y2 - y1 and multiply it by x2 - x1 and then divide by two to get the area of the graph below the line like a triangle, but again, the numbers seemed to high. (maybe they are just big numbers and I don't get this math stuff at all).
"like a triangle" --> should be "like a trapezoid"
If you do deltax*(y2-y1)/2 you will get the area, yes (this works only for linear segments). This is the basis principle of trapezoidal rule.
If you're uncertain about what you're calculating use dimensional analysis: speed is in bytes/sec, time is in sec, bandwidth is in bytes. Multiplying speed*time=bandwidth, and so on.
What I want is for the user to have
the option to click any point on the
graph and see their bandwidth usage
for that moment. Since the lines
between real data points are drawn
straight, this can be done by getting
the points before and after where the
user has clicked and finding the
y-interval.
Yes, that's a good way to find that instantaneous value. When you report that value back, it's in the same units as the y-axis, so that means bytes/sec, right?
I don't know how rapidly the rate changes between points, but it's even simpler if you simply pick the closest point and report its value. You simplify your problem without sacrificing too much accuracy.
I thought this was the area under the
line, but experiments with much
simpler graphs makes this seem just
far too high. I figured out I could
take the distance from y2 - y1 and
multiply it by x2 - x1 and then divide
by two to get the area of the graph
below the line like a triangle, but
again, the numbers seemed to high.
(maybe they are just big numbers and I
don't get this math stuff at all).
To calculate the total bytes over a given time interval, you should find the index closest to the starting and ending point and multiply the value of y by the spacing of your x-points and add them all together. That will give you the total # of bytes consumed during that time interval, but there's one more wrinkle you might have forgotten.
You said that the points come in "4 minutes apart", and your y-axis is in bytes/second. Remember that units matter. Your area is the sum of bytes/second times a spacing in minutes. To make the units come out right you have to multiply by 60 seconds/minute to get the final value of bytes that you want.
If that "too high" value is still off, consider units again. It's 1024 bytes per kbyte, and 1024*1024 bytes per MB. Check the units of the values you're checking the calculation against.
UPDATE:
No wonder you're having problems. Your original question CLEARLY stated bytes/sec. Even this question is imprecise and confusing. How did you arrive at "amount of data" at a given time stamp? Are those the total bits transferred since the last time stamp? If yes, simply add the values between the start and end of the interval you want and convert to the units convenient for you.
The network usage total is not in bytes (kilo-, mega-, whatever) per second. It would be in just straight bytes (or kilo-, or whatever).
For example, 2 megabytes per second over an interval of 10 seconds would be 20 megabytes total. It would not be 20 megabytes per second.
Or do you perhaps want average bytes per second over an interval?
This would be a lot easier for you if you would accept that there is well-established terminology for the concepts that you are having trouble expressing concisely or accurately, and that these mathematical terms have been around far longer than you. Since you've clearly gone through most of the trouble of understanding the concepts, you might as well break down and start calling them by their proper names.
That said:
There are 2 obvious ways to graph bandwidth, and two ways you might be getting the bandwidth data from the server. First, there's the cumulative usage function, which for any time is simply the total amount of data transferred since the start of the measurement. If you plot this function, you get a graph that never decreases (since you can't un-download something). The units of the values of this function will be bytes or kB or something like that.
What users are typically interested is in the instantaneous usage function, which is an indicator of how much bandwidth you are using right now. This is what users typically want to see. In mathematical terms, this is the derivative of the cumulative function. This derivative can take on any value from 0 (you aren't downloading) to the rated speed of your network link (indicating that you're pushing as much data as possible through your connection). The units of this function are bytes per second, or something related like Mbps (megabits per second).
You can approximate the instantaneous bandwidth with the average data usage over the past few seconds. This is computed as
(number of bytes transferred)
-----------------------------------------------------------------
(number of seconds that elapsed while transferring those bytes)
Generally speaking, the smaller the time interval, the more accurate the approximation. For simplicity's sake, you usually want to compute this as "number of bytes transferred since last report" divided by "number of seconds since last report".
As an example, if the server is giving you a report every 4 minutes of "total number of bytes transferred today", then it is giving you the cumulative function and you need to approximate the derivative. The instantaneous bandwidth usage rate you can report to users is:
(total transferred as of now) - (total as of 4 minutes ago) bytes
-----------------------------------------------------------
4*60 seconds
If the server is giving you reports of the form "number of bytes transferred since last report", then you can directly report this to users and plot that data relative to time. On the other hand, if the user (or you) is concerned about a quota on total bytes transferred per day, then you will need to transform the (approximately) instantaneous data you have into the cumulative data. This process, known as computing the integral, is the opposite of computing the derivative, and is in some ways conceptually simpler. If you've kept track of each of the reports from the server and the timestamp, then for each time, the value you plot is the total of all the reports that came in before that time. If you're doing this in realtime, then every time you get a new report, the graph jumps up by the amount in that report.
I am not bad at math, ... I just am not familiar with math beyond 10th grade
This is like saying "I'm not bad at programming, I have no trouble with ifs and loops but I never got around to writing more than one function."
I would suggest you enrol in a maths class of some kind. An understanding of matrices and the basics of calculus gives you an appreciation of many things, and can be useful in all sorts of areas. You'll be able to understand more of Wikipedia articles and SO answers - and questions!
If you can't afford that, try to find some lecture videos or something.
Everyone else has insisted on trying to teach me Riemann sum techniques
I can't see why. You don't need them for this - though if you had learned them, I expect you would find it easier to come up with a solution. You see, Riemann sums attempt to give you a "familiar" notion of area. The sort of area you (hopefully) learned years ago.
Getting the area below your usage graph between two points will tell you (approximately) how much was used over that period.
How do you find the area of a floor plan? You break it up into rectangles and triangles, find the area of each, and add them together. You can do the same thing with your graph, basically. Someone has worked out a simple way of doing this called the trapezoidal rule. It's just a matter of choosing how to divide your graph into strips, and in your case this is easy: just use the data points themselves as dividers. (You'll also need to work out the value of the graph at the left and right ends of the region selected by the user, using linear interpolation.)
If there's anything I've said that isn't clear to you (as there may well be), please leave a comment.