I have a complicated theoretical Probability Density Function (PDF) that I define in mathematica and that depends on some parameters that I need to estimate from comparison with real data. From a big simulation done on a cluster and not my laptop I have acquired a lot of events (over 10^9).
The way I understand things, given that I know what the PDF is I 'just' need to sum the probability that those events appear for a given set of parameters and maximise this quantity by adjusting the parameters.
However, given the number of events I would rather work with something less computer-time consuming and work for example with something easily generated like an histogram of my data. But then how would my log-likelihood estimator work?
Thanks a lot for your answers!
I am running an analysis of several thousand (e.g., 10,000) text documents. I have computed TF-IDF weights and have a matrix with pairwise cosine similarities. I want to treat the documents as a graph to analyze various properties (e.g., the path length separating groups of documents) and to visualize the connections as a network.
The problem is that there are too many similarities. Most are too small to be meaningful. I see many people dealing with this problem by dropping all similarities below a particular threshold, e.g., similarities below 0.5.
However, 0.5 (or 0.6, or 0.7, etc.) is an arbitrary threshold, and I'm looking for techniques that are more objective or systematic to get rid of tiny similarities.
I'm open to many different strategies. For example, is there a different alternative to tf-idf that would make most of the small similarities 0? Other methods to keep only significant similarities?
In short, take the average cosine value of an initial clustering or even all of the initial sentences and accept or reject clusters based on something akin to the following.
One way to look at the problem is to try and develop a score based on a distance from the mean similarity (1.5 standard deviations (86th percentile if the data were normal) tends to mark an outlier with 3 (99.9th percentile) being an extreme outlier), taking the high end for good measure. I cannot remember where, but this idea has had traction in other forums and formed the basis for my similarity.
Keep in mind that the data is not likely to be normally distributed.
average(cosine_similarities)+alpha*standard_deviation(cosine_similarities)
In order to obtain alpha, you could use the Wu Palmer score or another score as described by NLTK. Strong similarities with Wu Palmer should lead to a larger range of acceptance while lower Wu Palmer scores should lead to a more strict acceptance. Therefore, taking 1-Wu Palmer score would be adviseable. You can even use this method for LSA or LDA groups. To be even more strict and take things close to 1.5 or more standard deviations, you could even try 1+Wu Palmer (the cream of the crop), re-find the ultimate K,find the new score, cluster, and repeat.
Beware though, this would mean finding the Wu Palmer of all relevant words and is quite a large computational problem. Also, 10000 documents is peanuts compared to most algorithms. The smallest I have seen for tweets was 15,000 and the 20 news groups set was 20,000 documents. I am pretty sure Alchemy API uses something akin to the 20 news groups set. They definitely use senti-wordnet.
The basic equation is not really mine so feel free to dig around for it.
Another thing to keep in mind is that the calculation is time intensive. It may be a good idea to use a student t value for estimating the expected value/mean wu-palmer score of SOV pairings and especially good if you try to take the entire sentence. Commons Math3 for java/scala includes the distribution as does scipy for python and R should already have something as well.
Xbar +/- tsub(alpha/2)*sample_std/sqrt(sample_size)
Note: There is another option with this weight. You could use an algorithm that adds or subtracts from this threshold until achieving the best result. This would likely not be related solely to the cosine importance but possibly to an inflection point or gap as with Tibshirani's gap statistic.
I have a time-series of weekly usage data and I'm going to attempt to use some statistics to segment the population. Skewness and Kurtosis to may allow me to describe the time-series and group the people in different ways. But I also notice some appear to have saw-tooth patterns, or bimodal patterns, then I don't think these two aforementioned statistics will describe them well. Distance from the mean would tell me who has continual steady usage vs. unpredictable usage.
What descriptive statistics are commonly used for time-series data?
Thanks,
The periodogram and the autocorrelation function are two common sources of information
used to analyse and model time series. You can use this information to compare the series.
In the periodogram you can detect the frequencies at which the estimated spectral density is the highest. This will tell you which series are dominated by cycles of the same frequency.
The autocorrelation function (the time domain counterpart of the periodogram) and the partial autocorrelation function can similarly be used to compare and group the series. Those series with significant autocorrelations at the same lag orders could be grouped together.
You may need to transform the series in order to discern some of this information, for example taking differences to render the data stationary. Alternatively you can select an ARIMA model for each series and compare the characteristics of each model (those characteristics will be pretty much the same as those observed in the autocorrelation functions).
I'm trying to find confidence intervals for the means of various variables in a database using SPSS, and I've run into a spot of trouble.
The data is weighted, because each of the people who was surveyed represents a different portion of the overall population. For example, one young man in our sample might represent 28000 young men in the general population. The problem is that SPSS seems to think that the young man's database entries each represent 28000 measurements when they actually just represent one, and this makes SPSS think we have much more data than we actually do. As a result SPSS is giving very very low standard error estimates and very very narrow confidence intervals.
I've tried fixing this by dividing every weight value by the mean weight. This gives plausible figures and an average weight of 1, but I'm not sure the resulting numbers are actually correct.
Is my approach sound? If not, what should I try?
I've been using the Explore command to find mean and standard error (among other things), in case it matters.
You do need to scale weights to the actual sample size, but only the procedures in the Complex Samples option are designed to account for sampling weights properly. The regular weight variable in Statistics is treated as a frequency weight.
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.