I have been researching and trying to figure this one out to no avail. I have found many ways not to solve this...
The gist of the problem: I am looking for a method to calculate the deviance from an original path traveled by way of GPS coordinates. I have multiple csv files that contain latitude, longitude, and UTC time. I have created KML files from this information for a visual viewing of the deviance and now would like to put a value on this deviation. I ahve chosen a route as a reference and would like to measure the other routes against the reference route. There are multiple routes each having it's own reference route, each of which has many runs. No two runs are the same, and some of the routes deviate more than the next. I cannot use time, only lat and lon since the runs were completed over many weeks of data collection.
What I have tried thus far:
Haversine and Equirectangular formulas (looping through and measuring point to point).
Outcome: The coordinates only line up for a short period of time and the difference in the number of points varies greatly.
Area under each curve: was going to find the difference of the two routes by this method.
Outcome: Really unsure how to proceed, nor find equations suitable for this calculation.
There were a couple more feeble attempts, but have been working on this for a few weeks now, with not much to show for and still unsure on how to proceed.
Any help or ideas would be greatly appreciated.
Possible solution 1: Instead of calculating the "sideways" deviation between the two routes, just compare the respective arc lengths (Matlab: arclength).
Possible solution 2: To compare two routes, each going from the same start A to the same end point B: Draw a straight line between A and B, place a number of equidistant points along AB, and then average the perpendicular distance from these points on AB to the paths you want to compare. The absolute difference between the cumulative deviations from the straight-line reference is your deviation.
Possible solution 3: Calculate the arc length of each route. Place a number of equidistant points along each route. Average the distance between these points.
Both solution 2 and 3 will depend on the number of points you place, but with a higher number of points, the average deviation will converge. Note that these solutions are both related to calculating the area under each curve.
Related
I have calculated the essential matrix using the 5 point algorithm. I'm not sure how to integrate it with ransac so it gives me a better outcome.
Here is the source code. https://github.com/lunzhang/openar/blob/master/src/utils/5point/computeEssential.js
Currently, I was thinking about computing the essential matrix for 5 random points then convert the essential matrix to fundamental and see the error threshold using this equation x'Fx = 0. But then I'm not sure, what to do after.
How do I know which points to set as outliners? If the errors too big, do I set them as outliners right away? Could it be possible that one point could produce different essential matrices depending on what the other 4 points are?
Well, here is a short explanation, in pseudo-code, of how you can integrate this with ransac. Basically, all Ransac does is compute your model (here the Essential) using a subset of the data, and then sees if the rest of data "is happy" with that result. It keeps the result for which a highest portion of the dataset "is happy".
highest_number_of_happy_points=-1;
best_estimated_essential_matrix=Identity;
for iter=1 to max_iter_number:
n_pts=get_n_random_pts(P);//get a subset of n points from the set of points P. You can use 5, but you can also use more.
E=compute_essential(n_pts);
number_of_happy_points=0;
for pt in P:
//we want to know if pt is happy with the computed E
err=cost_function(pt,E);//for example x^TFx as you propose, or X^TEX with the essential.
if(err<some_threshold):
number_of_happy_points+=1;
if(number_of_happy_points>highest_number_of_happy_points):
highest_number_of_happy_points=number_of_happy_points;
best_estimated_essential_matrix=E;
This should do the trick. Usually, you set some_threshold experimentally to a low value. There are of course more sophisticated Ransacs, you can easily find them by googling.
Your idea of using x^TFx is fine in my opinion.
Once this Ransac completes, you will have best_estimated_essential_matrix. The outliers are those that have a x^TFx value that is greater than your optional threshold.
To answer your final question, yes, a point could produce a different matrix given 4 different points, because their spatial configuration is different (you can have degenerate situations). In an ideal settings this wouldn't be the case, but we always have noise, matching errors and so on, so what happens in the end is that the equations you obtain with 5 points wont produce the exact same results as for 5 other points.
Hope this helps.
What would b the best way to implement a simple shape-matching algorithm to match a plot interpolated from just 8 points (x, y) against a database of similar plots (> 12 000 entries), each plot having >100 nodes. The database has 6 categories of plots (signals measured under 6 different conditions), and the main aim is to find the right category (so for every category there's around 2000 plots to compare against).
The 8-node plot would represent actual data from measurement, but for now I am simulating this by selecting a random plot from the database, then 8 points from it, then smearing it using gaussian random number generator.
What would be the best way to implement non-linear least-squares to compare the shape of the 8-node plot against each plot from the database? Are there any c++ libraries you know of that could help with this?
Is it necessary to find the actual formula (f(x)) of the 8-node plot to use it with least squares, or will it be sufficient to use interpolation in requested points, such as interpolation from the gsl library?
You can certainly use least squares without knowing the actual formula. If all of your plots are measured at the same x value, then this is easy -- you simply compute the sum in the normal way:
where y_i is a point in your 8-node plot, sigma_i is the error on the point and Y(x_i) is the value of the plot from the database at the same x position as y_i. You can see why this is trivial if all your plots are measured at the same x value.
If they're not, you can get Y(x_i) either by fitting the plot from the database with some function (if you know it) or by interpolating between the points (if you don't know it). The simplest interpolation is just to connect the points with straight lines and find the value of the straight lines at the x_i that you want. Other interpolations might do better.
In my field, we use ROOT for these kind of things. However, scipy has a great collections of functions, and it might be easier to get started with -- if you don't mind using Python.
One major problem you could have would be that the two plots are not independent. Wikipedia suggests McNemar's test in this case.
Another problem you could have is that you don't have much information in your test plot, so your results will be affected greatly by statistical fluctuations. In other words, if you only have 8 test points and two plots match, how will you know if the underlying functions are really the same, or if the 8 points simply jumped around (inside their error bars) in such a way that it looks like the plot from the database -- purely by chance! ... I'm afraid you won't really know. So the plots that test well will include false positives (low purity), and some of the plots that don't happen to test well were probably actually good matches (low efficiency).
To solve that, you would need to either use a test plot with more points or else bring in other information. If you can throw away plots from the database that you know can't match for other reasons, that will help a lot.
100 periods have been collected from a 3 dimensional periodic signal. The wavelength slightly varies. The noise of the wavelength follows Gaussian distribution with zero mean. A good estimate of the wavelength is known, that is not an issue here. The noise of the amplitude may not be Gaussian and may be contaminated with outliers.
How can I compute a single period that approximates 'best' all of the collected 100 periods?
Time-series, ARMA, ARIMA, Kalman Filter, autoregression and autocorrelation seem to be keywords here.
UPDATE 1: I have no idea how time-series models work. Are they prepared for varying wavelengths? Can they handle non-smooth true signals? If a time-series model is fitted, can I compute a 'best estimate' for a single period? How?
UPDATE 2: A related question is this. Speed is not an issue in my case. Processing is done off-line, after all periods have been collected.
Origin of the problem: I am measuring acceleration during human steps at 200 Hz. After that I am trying to double integrate the data to get the vertical displacement of the center of gravity. Of course the noise introduces a HUGE error when you integrate twice. I would like to exploit periodicity to reduce this noise. Here is a crude graph of the actual data (y: acceleration in g, x: time in second) of 6 steps corresponding to 3 periods (1 left and 1 right step is a period):
My interest is now purely theoretical, as http://jap.physiology.org/content/39/1/174.abstract gives a pretty good recipe what to do.
We have used wavelets for noise suppression with similar signal measured from cows during walking.
I'm don't think the noise is so much of a problem here and the biggest peaks represent actual changes in the acceleration during walking.
I suppose that the angle of the leg and thus accelerometer changes during your experiment and you need to account for that in order to calculate the distance i.e you need to know what is the orientation of the accelerometer in each time step. See e.g this technical note for one to account for angle.
If you need get accurate measures of the position the best solution would be to get an accelerometer with a magnetometer, which also measures orientation. Something like this should work: http://www.sparkfun.com/products/10321.
EDIT: I have looked into this a bit more in the last few days because a similar project is in my to do list as well... We have not used gyros in the past, but we are doing so in the next project.
The inaccuracy in the positioning doesn't come from the white noise, but from the inaccuracy and drift of the gyro. And the error then accumulates very quickly due to the double integration. Intersense has a product called Navshoe, that addresses this problem by zeroing the error after each step (see this paper). And this is a good introduction to inertial navigation.
Periodic signal without noise has the following property:
f(a) = f(a+k), where k is the wavelength.
Next bit of information that is needed is that your signal is composed of separate samples. Every bit of information you've collected are based on samples, which are values of f() function. From 100 samples, you can get the mean value:
1/n * sum(s_i), where i is in range [0..n-1] and n = 100.
This needs to be done for every dimension of your data. If you use 3d data, it will be applied 3 times. Result would be (x,y,z) points. You can find value of s_i from the periodic signal equation simply by doing
s_i(a).x = f(a+k*i).x
s_i(a).y = f(a+k*i).y
s_i(a).z = f(a+k*i).z
If the wavelength is not accurate, this will give you additional source of error or you'll need to adjust it to match the real wavelength of each period. Since
k*i = k+k+...+k
if the wavelength varies, you'll need to use
k_1+k_2+k_3+...+k_i
instead of k*i.
Unfortunately with errors in wavelength, there will be big problems keeping this k_1..k_i chain in sync with the actual data. You'd actually need to know how to regognize the starting position of each period from your actual data. Possibly need to mark them by hand.
Now, all the mean values you calculated would be functions like this:
m(a) :: R->(x,y,z)
Now this is a curve in 3d space. More complex error models will be left as an excersize for the reader.
If you have a copy of Curve Fitting Toolbox, localized regression might be a good choice.
Curve Fitting Toolbox supports both lowess and loess localized regression models for curve and curve fitting.
There is an option for robust localized regression
The following blog post shows how to use cross validation to estimate an optimzal spaning parameter for a localized regression model, as well as techniques to estimate confidence intervals using a bootstrap.
http://blogs.mathworks.com/loren/2011/01/13/data-driven-fitting/
I'm trying to detect how well an input vector fits a given cluster centre. I can find the best match quite easily (the centre with the minimum euclidean distance to the input vector is the best), however, I now need to work how good a match that is.
To do this I need to find the spread (standard deviation?) of the vectors which build up the centroid, then see if the distance from my input vector to the centre is less than the spread. If it's more than the spread than I should be able to say that I have no clusters to fit it (given that the best doesn't fit the input vector well).
I'm not sure how to find the spread per cluster. I have all the centre vectors, and all the training vectors are labelled with their closest cluster, I just can't quite fathom exactly what I need to do to get the spread.
I hope that's clear? If not I'll try to reword it!
TIA
Ian
Use the distance function and calculate the distance from your center point to each labeled point, then figure out the mean of those distances. That should give you the standard deviation.
If you switch to using a different algorithm, such as Mixture of Gaussians, you get the spread (e.g., std. deviation) as part of the model (clustering result).
http://home.deib.polimi.it/matteucc/Clustering/tutorial_html/mixture.html
http://en.wikipedia.org/wiki/Mixture_model
As the user drags their stylus across the tablet, you receive a series of coordinates. You want to approximate the pen's path with a smooth line, trailing only a few sample points behind it. How would you do this?
In other words, how would you render a nice smooth responsive line as a user draws it with their tablet? Simply connecting the dots with straight lines is not good enough. Real drawing programs do a much better job of curving the line, no matter how close or far the sample points are. Some even let you give them a number to indicate the amount of smoothing to be done, accounting for jittery pens and hands. Where can I learn to do this stuff?
I know this is an old question but I had the same problem and I came with 2 different solutions:
The first approach is use two resolutions: One , when the user is inserting the path points connecting them with straight lines. Two , when the user finish the stroke delete the lines and draw the spline over. That should be smoother than the straight lines.
The second approach it is to smooth the new points with a weighted mean of the sampled points. So each time you get a new point [x1,y1] instead of painting it directly, you take the previous points [x2,y2] and create a new intermediate point with the weighted mean of the two points. The pseudocode could be something like:
newPoint = [x1,y1];
oldPoint = [x2,y2];
point2Paint = [(x1*0.3) + (x2*0.7), (y1*0.3) + (y2*0.7)];
oldPoint= newPoint;
Being 0.7 and 0.3 the coefficients for the weighted mean ( You can change them to get your desired smoothing :)
I hope this would help
UPDATE Dec 13: Here it is an article explaining different drawing methods, there are good concepts that can be applied (edge smoothing, bezier curves, smooth joints)
http://perfectionkills.com/exploring-canvas-drawing-techniques
I never had to implement these (only for academic purposes), but you may want to take a look at wikipedia's interpolation article.
Extracted from the article:
interpolation is a method of constructing new data points within the range of a discrete set of known data points.
In engineering and science one often has a number of data points, as obtained by sampling or experimentation, and tries to construct a function which closely fits those data points. This is called curve fitting or regression analysis. Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points.
Hope it helps.