Matlab, is there any way to manipulate random variable - statistics
In Maple, there is some feature that allows you to calculate the pdf of a function of a random variable. For example, if X is exponentially distributed, and you want to know the distribution of X^2, then there is a function that will do that for you.
My question is , is there a functionality in matlab that allows you to do so? I have looked through the matlab's guide, but I didn't see it.
The Statistics toolbox includes many probability distributions for you to choose from, both parametric and non-parametric distributions. For each it provides functions for PDF, CDF, fitting, random number generation, etc..
I suggest you start with the "Distribution Fitting app": dfittool.
EDIT:
In addition, MuPAD has support for a number of distributions, which you can manipulate symbolically. Example:
The function intlib::changevar might be of interest here, though it seems intended for integrals...
Also, if you're interested in getting the values of the PMF, or discrete PDF, then, given x some RV with some distribution,
my_pmf = hist(x)/sum(x);
So try,
doc hist
Related
How do I check data gaussianity with excel?
Well, I have a data set (length 20) and I want to check its gaussianity or non-gaussianity. I'm an somewhat advanced user of matlab so I already know (it doesn't fit a gaussian) but I have to prove it with excel and I have never used its statistical tools very much. What's the best way to do it? EDIT: I had several ideas, but none of those appear to be practical in excel. First, I thought it would be a fitting tool (something like matlab's histfit) but I didn't find it. Second, I thought I could say my data is approximate gaussian if the deciles of my data set are the approximate the same of the gaussian distribution with mean=dataSetMean and variance=dataSetVariance, but excel doesn't have any of those.
Interpolation technique for weirdly spaced point data
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.
Run Haskell benchmarks on inputs of varying size
Often I’d like to compare the runtime performance of multiple implementations of the same function. For individual inputs, criterion is a good tool. But what is an easy way to plot the performance of the code over varying input size, e.g. to see the algorithmic complexity? Ideally, I pass the library a value of type Benchmarkable r => [(String, Int -> r)], i.e. a list of size-dependent benchmarks, and the library will automatically find a the sensible input range for each value and creates a nice plot from it.
e.g. to see the algorithmic complexity? There's a package for that: http://hackage.haskell.org/package/complexity However, I typically use QuickCheck to drive testing at random data sizes, then plot the result.
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.
Create CDF for Anderson Darling test for Octave forge Statistics package function
I am using Octave and I would like to use the anderson_darling_test from the Octave forge Statistics package to test if two vectors of data are drawn from the same statistical distribution. Furthermore, the reference distribution is unlikely to be "normal". This reference distribution will be the known distribution and taken from the help for the above function " 'If you are selecting from a known distribution, convert your values into CDF values for the distribution and use "uniform'. " My question therefore is: how would I convert my data values into CDF values for the reference distribution? Some background information for the problem: I have a vector of raw data values from which I extract the cyclic component (this will be the reference distribution); I then wish to compare this cyclic component with the raw data itself to see if the raw data is essentially cyclic in nature. If the the null hypothesis that the two are the same can be rejected I will then know that most of the movement in the raw data is not due to cyclic influences but is due to either trend or just noise.
If your data has a specific distribution, for instance beta(3,3) then p = betacdf(x, 3, 3) will be uniform by the definition of a CDF. If you want to transform it to a normal, you can just call the inverse CDF function x=norminv(p,0,1) on the uniform p. Once transformed, use your favorite test. I'm not sure I understand your data, but you might consider using a Kolmogorov-Smirnov test instead, which is a nonparametric test of distributional equality.
Your approach is misguided in multiple ways. Several points: The Anderson-Darling test implemented in Octave forge is a one-sample test: it requires one vector of data and a reference distribution. The distribution should be known - not come from data. While you quote the help-file correctly about using a CDF and the "uniform" option for a distribution that is not built in, you are ignoring the next sentence of the same help file: Do not use "uniform" if the distribution parameters are estimated from the data itself, as this sharply biases the A^2 statistic toward smaller values. So, don't do it. Even if you found or wrote a function implementing a proper two-sample Anderson-Darling or Kolmogorov-Smirnov test, you would still be left with a couple of problems: Your samples (the data and the cyclic part estimated from the data) are not independent, and these tests assume independence. Given your description, I assume there is some sort of time predictor involved. So even if the distributions would coincide, that does not mean they coincide at the same time-points, because comparing distributions collapses over the time. The distribution of cyclic trend + error would not expected to be the same as the distribution of the cyclic trend alone. Suppose the trend is sin(t). Then it never will go above 1. Now add a normally distributed random error term with standard deviation 0.1 (small, so that the trend is dominant). Obviously you could get values well above 1. We do not have enough information to figure out the proper thing to do, and it is not really a programming question anyway. Look up time series theory - separating cyclic components is a major topic there. But many reasonable analyses will probably be based on the residuals: (observed value - predicted from cyclic component). You will still have to be careful about auto-correlation and other complexities, but at least it will be a move in the right direction.