I have two histograms.
int Hist1[10] = {1,4,3,5,2,5,4,6,3,2};
int Hist1[10] = {1,4,3,15,12,15,4,6,3,2};
Hist1's distribution is of type multi-modal;
Hist2's distribution is of type uni-modal with single prominent peak.
My questions are
Is there any way that i could determine the type of distribution programmatically?
How to quantify whether these two histograms are similar/dissimilar?
Thanks
Raj,
I posted a C function in your other question ( automatically compare two series -Dissimilarity test ) that will compute divergence between two sets of similar data. It's actually intended to tell you how closely real data matches predicted data but I suspect you could use it for your purpose.
Basically, the smaller the error, the more similar the two sets are.
These are just guesses, but I would try fitting each distribution as a gaussian distribution and use something like the R-squared value to determine if the distribution is uni-modal or not.
As to the similarity between the two distributions, I would try doing an autocorrelation and using the peak positive value in the autocorrelation as a similarity measure. These ideas are pretty rough, but hopefully they give you some ideas.
For #2, you could calculate their cross-correlation (so long as the buckets themselves can be sorted). That would give you a rough estimation of what "similarity".
Comparison of Histograms (For Use in Cloud Modeling).
(That's an MS .doc file.)
There are a variety of software packages that will "fit" your distributions to known discrete distributions for you - Minitab, STATA, R, etc. A reference to fitting distributions in R is here. I wouldn't advise programming this from scratch.
Regarding distribution comparisons, if neither distribution fits a known distribution (Poisson, Binomial, etc.), then you need to use non-parametric methods described here.
Related
I´ve a question regarding k-means clustering. We have a dataset with 120,000 observations and need to compute a k-means cluster solution with R. The problem is that k-means usually use Euclidean Distance. Our dataset consists of 3 continous variables, 11 ordinal (Likert 0-5) (i think it would be okay to handle them like continous) and 5 binary variables. Do you have any suggestion for a distance measure that we can use for our k-means approach with regards to the "large" dataset? We stick to k-means, so I really hope one of you has a good idea.
Cheers,
Martin
One approach would be to normalize the features and then just use the 11-dimensional
Euclidean Distance. Cast the binary values to 0/1 (Well, it's R, so it does that anyway) and go from there.
I don't see an immediate problem with this method other than k-means in 11 dimensions will definitely be hard to interpret. You could try to use a dimensionality reduction technique and hopefully make the k-means output easier to read, but you know way more about the data set than we ever could, so our ability to help you is limited.
You can certainly encode there binary variables as 0,1 too.
It is a best practise in statistics to not treat likert scale variables as numeric, because of that uneven distribution.
But I don't you will get meaningful k-means clusters. That algorithm is all about computing means. That makes sense on continuous variables. Discrete variables usually lack "resolution" for this to work well. Three mean then degrades to a "frequency" and then the data should be handled very differently.
Do not choose the problem by the hammer. Maybe your data is not a nail; and even if you'd like to make it with kmeans, it won't solve your problem... Instead, formulate your problem, then choose the right tool. So given your data, what is a good cluster? Until you have an equation that measures this, handing the data won't solve anything.
Encoding the variables to binary will not solve the underlying problem. Rather, it will only aid in increasing the data dimensionality, an added burden. It's best practice in statistics to not alter the original data to any other form like continuous to categorical or vice versa. However, if you are doing so, i.e. the data conversion then it must be in sync with the question to solve as well as you must provide valid justification.
Continuing further, as others have stated, try to reduce the dimensionality of the dataset first. Check for issues like, missing values, outliers, zero variance, principal component analysis (continuous variables), correspondence analysis (for categorical variables) etc. This can help you reduce the dimensionality. After all, data preprocessing tasks constitute 80% of analysis.
Regarding the distance measure for mixed data type, you do understand the mean in k will work only for continuous variable. So, I do not understand the logic of using the algorithm k-means for mixed datatypes?
Consider choosing other algorithm like k-modes. k-modes is an extension of k-means. Instead of distances it uses dissimilarities (that is, quantification of the total mismatches between two objects: the smaller this number, the more similar the two objects). And instead of means, it uses modes. A mode is a vector of elements that minimizes the dissimilarities between the vector itself and each object of the data.
Mixture models can be used to cluster mixed data.
You can use the R package VarSelLCM which models, within each cluster, the continuous variables by Gaussian distributions and the ordinal/binary variables.
Moreover, missing values can be managed by the model at hand.
A tutorial is available at: http://varsellcm.r-forge.r-project.org/
I have very little programming experience, but I'm working on a statistics project and would like to generate an unequal probability sample where the inclusion probability of a unit is based on its size (PPS).
Basically, I have two datasets:
ds1 lists US states and the parameter I'm trying to estimate
ds2 has the population size of each state.
My questions:
I want to use R to select a random sample from the first dataset using inclusion probabilities based on the population of each state (second dataset).
Also is there any way to use R to calculate these Generalized Unequal Probability Estimator formulas?
Also just a note on the formulas: pi_i is inclusion probability and pi_ij is joint inclusion probability.
There is a package for the same in R - pps and the documentation is here.
Also, there is another package called survey with a bit of documentation here.
I'm not sure of the difference between the two and haven't used them myself. Hope this is what you're looking for.
Yes, that's called weighted sampling. Simply set the weight to the size of the state, strictly you don't even need to normalize them by 1/sum(sizes) although it's always good practice to. There are tons of duplicate posts on SO showing how to do weighted sampling.
The only tiny complication is that you need to do a join() of the datasets ds1, ds2. Show us what code you've tried if it's causing problems. Recommend you use either dplyr or data.table.
Your second question should be asked as a separate question, and is offtopic on SO, or at least won't get a great response - best to ask statistical questions at sister site CrossValidated
I am interested in frequency distributions that are not normally distributed.
If I have a frequency distributions table which is not normally distributed.
Is there a function or package that will identify the type of distribution for me?
You can use the fitdistr function (library MASS i think) and check for yourself if you find a 'fitting' distribution. However i suggest that you plot the function first and see how it looks like. This approach is generally not recommended as you always can use different parameters to fit a distribution and thus confuse one distribution with another. If you have found a suited distribution you should test it against data.
Edit: For instance a normal distribution may look like a poisson distribution. Fitting is in my oppinion only useful if you have enough random variables. Otherwise just draw variables from your data if you need to
You can always try to test whether a distribution is adequate for your data with QQ plot. If you have data that is dynamic, I would suggest that you use ECDF (Empirical Cumulative Distribution Function) which will give you more precise distributions as your data grows. You can use ECDF in R with the ecdf() function.
I have a set of data and I'd like to know whether this data set has a logistic distribution.
When I made a histogram of my data set (see the histogram on http://imageshack.us/photo/my-images/593/histogram.png/) it seems to have a logistic distribution, but to be sure I'd like to test for a logistic distribution in R. So my question is: Is there a way to test your data for a logistic distribution and how do you do this?
Additional information: The data set consists of 8544 items. The data are horizontal distances in km between 2 geographical points.
Thanks for your attention
Sander
In R you can use the ks.test or chisq.test functions (and probably others) to test against a hypothesized distribution. Note that these tests (and others) are all rule out tests, a non-significant result does not guarentee that the data come from the given distribution, just that you cannot rule it out. Also note that with a sample size of 8544 these tests are likely to be way overpowered, meaning that it will have power to find slight meaningless differences and you are likely to reject the null hypothesis even though it is "close enough". Also the fact that you decided on a distribution based on looking at the data first could bias results.
Another approach that may give you a better feel for if a logistic distribution is "close enough" rather than exactly is to use the vis.test function in the TeachingDemos package (be sure to read the paper referenced in the help page to understand the test and what assumptions you are making).
Most importantly is understanding the science that leads to the data, does a logistic distribution make sense scientifically? what other distributions could be reasonble? Also understand what question(s) you are trying to answer with the data and what is the effect on those answers of the distribution (e.g. the CLT will let you use the normal to answer some questions, but not others, using a normal distribution even though the data comes from a logistic or something similar).
I am looking into time series data compression at the moment.
The idea is to fit a curve on a time series of n points so that the maximum deviation on any of the points is not greater than a given threshold. In other words, none of the values that the curve takes at the points where the time series is defined, should be "further away" than a certain threshold from the actual values.
Till now I have found out how to do nonlinear regression using the least squares estimation method in R (nls function) and other languages, but I haven't found any packages that implement nonlinear regression with the L-infinity norm.
I have found literature on the subject:
http://www.jstor.org/discover/10.2307/2006101?uid=3737864&uid=2&uid=4&sid=21100693651721
or
http://www.dtic.mil/dtic/tr/fulltext/u2/a080454.pdf
I could try to implement this in R for instance, but I first looking to see if this hasn't already been done and that I could maybe reuse it.
I have found a solution that I don't believe to be "very scientific": I use nonlinear least squares regression to find the starting values of the parameters which I subsequently use as starting points in the R "optim" function that minimizes the maximum deviation of the curve from the actual points.
Any help would be appreciated. The idea is to be able to find out if this type of curve-fitting is possible on a given time series sequence and to determine the parameters that allow it.
I hope there are other people that have already encountered this problem out there and that could help me ;)
Thank you.