Interpolation technique for weirdly spaced point data - geospatial
I have a spatial dataset that consists of a large number of point measurements (n=10^4) that were taken along regular grid lines (500m x 500m) and some arbitrary lines and blocks in between. Single measurements taken with a spacing of about 0.3-1.0m (varying) along these lines (see example showing every 10th point).
The data can be assumed to be normally distributed but shows a strong small-scale variability in some regions. And there is some trend with elevation (r=0.5) that can easily be removed.
Regardless of the coding platform, I'm looking for a good or "the optimal" way to interpolate these points to a regular 25 x 25m grid over the entire area of interest (5000 x 7000m). I know about the wide range of kriging techniques but I wondered if somebody has a specific idea on how to handle the "oversampling along lines" with rather large gaps between the lines.
Thank you for any advice!
Leo
Kriging technique does not perform well when the points to interpolate are taken on a regular grid, because it is necessary to have a wide range of different inter-points distances in order to well estimate the covariance model.
Your case is a bit particular... The oversampling over the lines is not a problem at all. The main problem is the big holes you have in your grid. If think that these holes will create problems whatever the interpolation technique you use.
However it is difficult to predict a priori if kriging will behave well. I advise you to try it anyway.
Kriging is only suited for interpolating. You cannot extrapolate with kriging metamodel, so that you won't be able to predict values in the bottom left part of your figure for example (because you have no point here).
To perform kriging, I advise you to use the following tools (depending the languages you're more familiar with):
DiceKriging package in R (the one I use preferably)
fields package in R (which is more specialized on spatial fields)
DACE toolbox in matlab
Bonus: a link to a reference book about kriging which is available online: http://www.gaussianprocess.org/
PS: This type of question is more statistics oriented than programming and may be better suited to the stats.stackexchange.com website.
Related
How do I analyze the change in the relationship between two variables?
I'm working on a simple project in which I'm trying to describe the relationship between two positively correlated variables and determine if that relationship is changing over time, and if so, to what degree. I feel like this is something people probably do pretty often, but maybe I'm just not using the correct terminology because google isn't helping me very much. I've plotted the variables on a scatter plot and know how to determine the correlation coefficient and plot a linear regression. I thought this may be a good first step because the linear regression tells me what I can expect y to be for a given x value. This means I can quantify how "far away" each data point is from the regression line (I think this is called the squared error?). Now I'd like to see what the error looks like for each data point over time. For example, if I have 100 data points and the most recent 20 are much farther away from where the regression line/function says it should be, maybe I could say that the relationship between the variables is showing signs of changing? Does that make any sense at all or am I way off base? I have a suspicion that there is a much simpler way to do this and/or that I'm going about it in the wrong way. I'd appreciate any guidance you can offer!
I can suggest two strands of literature that study changing relationships over time. Typing these names into google should provide you with a large number of references so I'll stick to more concise descriptions. (1) Structural break modelling. As the name suggest, this assumes that there has been a sudden change in parameters (e.g. a correlation coefficient). This is applicable if there has been a policy change, change in measurement device, etc. The estimation approach is indeed very close to the procedure you suggest. Namely, you would estimate the squared error (or some other measure of fit) on the full sample and the two sub-samples (before and after break). If the gains in fit are large when dividing the sample, then you would favour the model with the break and use different coefficients before and after the structural change. (2) Time-varying coefficient models. This approach is more subtle as coefficients will now evolve more slowly over time. These changes can originate from the time evolution of some observed variables or they can be modeled through some unobserved latent process. In the latter case the estimation typically involves the use of state-space models (and thus the Kalman filter or some more advanced filtering techniques). I hope this helps!
Finding powerlines in LIDAR point clouds with RANSAC
I'm trying to find powerlines in LIDAR points clouds with skimage.measures ransac() function. This is my very first time meddling with these modules in python so bear with me.
So far all I knew how to do reliably was filtering low or 'ground' points from the cloud to reduce the number of points to deal with.
def filter_Z(las, threshold):
filtered = laspy.create(point_format = las.header.point_format, file_version = las.header.version)
filtered.points = las.points[las.Z > las.Z.min() + threshold]
print(f'original size: {len(las.points)}')
print(f'filtered size: {len(filtered.points)}')
filtered.write('filtered_points2.las')
return filtered
The threshold is something I put in by hand since in the las files I worked with are some nasty outliers that prevent me from dynamically calculating it.
The filtered point cloud, or one of them atleast looks like this:
Note the evil red outliers on top, maybe they're birds or something. Along with them are trees and roofs of buildings. If anyone wants to take a look at the .las files, let me know. I can't put a wetransfer link in the body of the question.
A top down view:
I've looked into it as much as I could, and found the skimage.measure module and the ransac function that comes with it. I played around a bit to get a feel for it and currently I'm stumped on how to continue.
def ransac_linefit_sklearn(points):
model_robust, inliers = ransac(points, LineModelND, min_samples=2, residual_threshold=1000, max_trials=1000)
return model_robust, inliers
The result is quite predictable (I ran ransac on a 2D view of the cloud just to make it a bit easier on the pc)
Using this doesn't really yield any good results in examples like the one I posted. The vegetation clusters have too many points and the line is fitted through it because it has the highest point density.
I tried DBSCAN() to cluster up the points but it didn't work. I also attempted OPTICS() but as I write it still hasn't finished running.
From what I've read on various articles, the best course of action would be to cluster up the points and perform RANSAC on each individual cluster to find lines, but I'm not really sure on how to do that or what clustering method to use in situations like these.
One thing I'm also curious about doing is just filtering out the big blobs of trees that mess with model fititng.
Inadequacy of RANSAC RANSAC works best whenever your data fits a mono-modal distribution around your model. In the case of this point cloud, it works best whenever there is only one line with outliers, but there are at least 5 lines when viewed birds-eye. Check out this older SO post that discusses your problem. Francesco's response suggests an iterative RANSAC based approach. Octrees and SVD Colleagues worked on a similar problem in my previous job. I am not fluent in the approach, but I know enough to provide some hints. Their approach resembled Francesco's suggestion. They partitioned the point-cloud into octrees and calculated the singular value decomposition (SVD) within each partition. The three resulting singular values will correspond to the geometric distribution of the data. If the first singular value is significantly greater than the other two, then the points are line-like. If the first and second singular values are significantly greater than the other, then the points are plane-like If all three values are of similar magnitude, then the data is just a "glob" of points. They used these rules iteratively to rule out which points were most likely NOT part of the lines. Literature If you want to look into published methods, maybe this paper is a good starting point. Power lines are modeled as hyperbolic functions.
I need a function that describes a set of sequences of zeros and ones?
I have multiple sets with a variable number of sequences. Each sequence is made of 64 numbers that are either 0 or 1 like so: Set A sequence 1: 0,0,0,0,0,0,1,1,0,0,0,0,1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0 sequence 2: 0,0,0,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0 sequence 3: 0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0 ... Set B sequence1: 0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1 sequence2: 0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,0 ... I would like to find a mathematical function that describes all possible sequences in the set, maybe even predict more and that does not contain the sequences in the other sets. I need this because I am trying to recognize different gestures in a mobile app based on the cells in a grid that have been touched (1 touch/ 0 no touch). The sets represent each gesture and the sequences a limited sample of variations in each gesture. Ideally the function describing the sequences in a set would allow me to test user touches against it to determine which set/gesture is part of. I searched for a solution, either using Excel or Mathematica, but being very ignorant about both and mathematics in general I am looking for the direction of an expert. Suggestions for basic documentation on the subject is also welcome.
It looks as if you are trying to treat what is essentially 2D data in 1D. For example, let s1 represent the first sequence in set A in your question. Then the command ArrayPlot[Partition[s1, 8]] produces this picture: The other sequences in the same set produce similar plots. One of the sequences from the second set produces, in response to the same operations, the picture: I don't know what sort of mathematical function you would like to define to describe these pictures, but I'm not sure that you need to if your objective is to recognise user gestures. You could do something much simpler, such as calculate the 'average' picture for each of your gestures. One way to do this would be to calculate the average value for each of the 64 pixels in each of the pictures. Perhaps there are 6 sequences in your set A describing gesture A. Sum the sequences element-by-element. You will now have a sequence with values ranging from 0 to 6. Divide each element by 6. Now each element represents a sort of probability that a new gesture, one you are trying to recognise, will touch that pixel. Repeat this for all the sets of sequences representing your set of gestures. To recognise a user gesture, simply compute the difference between the sequence representing the gesture and each of the sequences representing the 'average' gestures. The smallest (absolute) difference will direct you to the gesture the user made. I don't expect that this will be entirely foolproof, it may well result in some user gestures being ambiguous or not recognisable, and you may want to try something more sophisticated. But I think this approach is simple and probably adequate to get you started.
In Mathematica the following expression will enumerate all the possible combinations of {0,1} of length 64.
Tuples[{1, 0}, {64}]
But there are 2^62 or 18446744073709551616 of them, so I'm not sure what use that will be to you.
Maybe you just wanted the unique sequences contained in each set, in that case all you need is the Mathematica Union[] function applied to the set. If you have a the sets grouped together in a list in Mathematica, say mySets, then you can apply the Union operator to every set in the list my using the map operator.
Union/#mySets
If you want to do some type of prediction a little more information might be useful.
Thanks you for the clarifications.
Machine Learning
The task you want to solve falls under the disciplines known by a variety of names, but probably most commonly as Machine Learning or Pattern Recognition and if you know which examples represent the same gestures, your case would be known as supervised learning.
Question: In your case do you know which gesture each example represents ?
You have a series of examples for which you know a label ( the form of gesture it is ) from which you want to train a model and use that model to label an unseen example to one of a finite set of classes. In your case, one of a number of gestures. This is typically known as classification.
Learning Resources
There is a very extensive background of research on this topic, but a popular introduction to the subject is machine learning by Christopher Bishop.
Stanford have a series of machine learning video lectures Standford ML available on the web.
Accuracy
You might want to consider how you will determine the accuracy of your system at predicting the type of gesture for an unseen example. Typically you train the model using some of your examples and then test its performance using examples the model has not seen. The two of the most common methods used to do this are 10 fold Cross Validation or repeated 50/50 holdout. Having a measure of accuracy enables you to compare one method against another to see which is superior.
Have you thought about what level of accuracy you require in your task, is 70% accuracy enough, 85%, 99% or better?
Machine learning methods are typically quite sensitive to the specific type of data you have and the amount of examples you have to train the system with, the more examples, generally the better the performance.
You could try the method suggested above and compare it against a variety of well proven methods, amongst which would be Random Forests, support vector machines and Neural Networks. All of which and many more are available to download in a variety of free toolboxes.
Toolboxes
Mathematica is a wonderful system, is infinitely flexible and my favourite environment, but out of the box it doesn't have a great deal of support for machine learning.
I suspect you will make a great deal of progress more quickly by using a custom toolbox designed for machine learning. Two of the most popular free toolboxes are WEKA and R both support more than 50 different methods for solving your task along with methods for measuring the accuracy of the solutions.
With just a little data reformatting, you can convert your gestures to a simple file format called ARFF, load them into WEKA or R and experiment with dozens of different algorithms to see how each performs on your data. The explorer tool in WEKA is definitely the easiest to use, requiring little more than a few mouse clicks and typing some parameters to get started.
Once you have an idea of how well the established methods perform on your data you have a good starting point to compare a customised approach against should they fail to meet your criteria.
Handwritten Digit Recognition
Your problem is similar to a very well researched machine learning problem known as hand written digit recognition. The methods that work well on this public data set of handwritten digits are likely to work well on your gestures.
k-means with ellipsoids
I have n points in R^3 that I want to cover with k ellipsoids or cylinders (I don't really care; whichever is easier). I want to approximately minimize the union of the volumes. Let's say n is tens of thousands and k is a handful. Development time (i.e. simplicity) is more important than runtime. Obviously I can run k-means and use perfect balls for my ellipsoids. Or I can run k-means, then use minimum enclosing ellipsoids per cluster rather than covering with balls, though in the worst case that's no better. I've seen talk of handling anisotropy with k-means but the links I saw seemed to think I had a tensor in hand; I don't, I just know the data will be a union of ellipsoids. Any suggestions? [Edit: There's a couple votes for fitting a mixture of multivariate Gaussians, which seems like a viable thing to try. Firing up an EM code to do that won't minimize the volume of the union, but of course k-means doesn't minimize volume either.]
So you likely know k-means is NP-hard, and this problem is even more general (harder). Because you want to do ellipsoids it might make a lot of sense to fit a mixture of k multivariate gaussian distributions. You would probably want to try and find a maximum likelihood solution, which is a non-convex optimization, but at least it's easy to formulate and there is likely code available. Other than that you're likely to have to write your own heuristic search algorithm from scratch, this is just a huge undertaking.
I did something similar with multi-variate gaussians using this method. The authors use kurtosis as the split measure, and I found it to be a satisfactory method for my application, clustering points obtained from a laser range finder (i.e. computer vision).
If the ellipsoids can overlap a lot, then methods like k-means that try to assign points to single clusters won't work very well. Part of each ellipsoid has to fit the surface of your object, but the rest may be inside it, don't-cares. That is, covering algorithms seem to me quite different from clustering / splitting algorithms; unions are not splits. Gaussian mixtures with lots of overlaps ? No idea, but see the picture and code on Numerical Recipes p. 845. Coverings are hard even in 2d, see find-near-minimal-covering-set-of-discs-on-a-2-d-plane.
What are the efficient and accurate algorithms to exclude outliers from a set of data?
I have set of 200 data rows(implies a small set of data). I want to carry out some statistical analysis, but before that I want to exclude outliers. What are the potential algos for the purpose? Accuracy is a matter of concern. I am very new to Stats, so need help in very basic algos.
Overall, the thing that makes a question like this hard is that there is no rigorous definition of an outlier. I would actually recommend against using a certain number of standard deviations as the cutoff for the following reasons: A few outliers can have a huge impact on your estimate of standard deviation, as standard deviation is not a robust statistic. The interpretation of standard deviation depends hugely on the distribution of your data. If your data is normally distributed then 3 standard deviations is a lot, but if it's, for example, log-normally distributed, then 3 standard deviations is not a lot. There are a few good ways to proceed: Keep all the data, and just use robust statistics (median instead of mean, Wilcoxon test instead of T-test, etc.). Probably good if your dataset is large. Trim or Winsorize your data. Trimming means removing the top and bottom x%. Winsorizing means setting the top and bottom x% to the xth and 1-xth percentile value respectively. If you have a small dataset, you could just plot your data and examine it manually for implausible values. If your data looks reasonably close to normally distributed (no heavy tails and roughly symmetric), then use the median absolute deviation instead of the standard deviation as your test statistic and filter to 3 or 4 median absolute deviations away from the median.
Start by plotting the leverage of the outliers and then go for some good ol' interocular trauma (aka look at the scatterplot). Lots of statistical packages have outlier/residual diagnostics, but I prefer Cook's D. You can calculate it by hand if you'd like using this formula from mtsu.edu (original link is dead, this is sourced from archive.org).
You may have heard the expression 'six sigma'. This refers to plus and minus 3 sigma (ie, standard deviations) around the mean. Anything outside the 'six sigma' range could be treated as an outlier. On reflection, I think 'six sigma' is too wide. This article describes how it amounts to "3.4 defective parts per million opportunities." It seems like a pretty stringent requirement for certification purposes. Only you can decide if it suits you.
Depending on your data and its meaning, you might want to look into RANSAC (random sample consensus). This is widely used in computer vision, and generally gives excellent results when trying to fit data with lots of outliers to a model. And it's very simple to conceptualize and explain. On the other hand, it's non deterministic, which may cause problems depending on the application.
Compute the standard deviation on the set, and exclude everything outside of the first, second or third standard deviation.
Here is how I would go about it in SQL Server The query below will get the average weight from a fictional Scale table holding a single weigh-in for each person while not permitting those who are overly fat or thin to throw off the more realistic average: select w.Gender, Avg(w.Weight) as AvgWeight from ScaleData w join ( select d.Gender, Avg(d.Weight) as AvgWeight, 2*STDDEVP(d.Weight) StdDeviation from ScaleData d group by d.Gender ) d on w.Gender = d.Gender and w.Weight between d.AvgWeight-d.StdDeviation and d.AvgWeight+d.StdDeviation group by w.Gender There may be a better way to go about this, but it works and works well. If you have come across another more efficient solution, I’d love to hear about it. NOTE: the above removes the top and bottom 5% of outliers out of the picture for purpose of the Average. You can adjust the number of outliers removed by adjusting the 2* in the 2*STDDEVP as per: http://en.wikipedia.org/wiki/Standard_deviation
If you want to just analyse it, say you want to compute the correlation with another variable, its ok to exclude outliers. But if you want to model / predict, it is not always best to exclude them straightaway. Try to treat it with methods such as capping or if you suspect the outliers contain information/pattern, then replace it with missing, and model/predict it. I have written some examples of how you can go about this here using R.