I have a list of numbers. Below are some basic statistics:
N > 1000
Max: 9.24
Min: 0.00955
Mean: 1.84932
Median: 0.97696
It seems that the data is right skewed, i.e. many small numbers and a few very large numbers.
I want to find a distribution to generalize these numbers. I think Normal distribution, Gamma distribution, and Laplace distribution all look possible. How do I determine which distribution is the best?
I have to say that I usually do it in the same way you did it, by plotting the data I seeing its shape.
When being more accurate, and only for the normal distribution, I perform the Shapiro Wilk test for normality, which at least will tell me that the null hypotesis was not proven, which means that it was not possible to prove that the date does not follow a normal distribution. Usually, this is more than acceptable in scientific environments.
I know there exists equivalent tests for Laplace and Gamma distributions, although still in newly research like this. Instead, there are many sites that offer the Shapiro Wilk test online, like this one.
With all positive values and the mean being about double the median, your data are definitely skewed right. You can rule out both normal and Laplace because both are symmetric and can go negative.
Scope out some of the many fine alternatives at the Wikipedia distributions page. Make a histogram of your data and check it for similarities in shape to those distributions. Exponentials, log normals, chi-squares, and the gamma family could all give numeric results such as the ones you described, but without knowing anything about the variance/std deviation, whether your data are unimodal or multimodal, or where the mode(s) are, we can only make guesses about a very large pool of possibilities.
Related
I am learning statistics, and have some basic yet core questions on SD:
s = sample size
n = total number of observations
xi = ith observation
μ = arithmetic mean of all observations
σ = the usual definition of SD, i.e. ((1/(n-1))*sum([(xi-μ)**2 for xi in s])**(1/2) in Python lingo
f = frequency of an observation value
I do understand that (1/n)*sum([xi-μ for xi in s]) would be useless (= 0), but would not (1/n)*sum([abs(xi-μ) for xi in s]) have been a measure of variation?
Why stop at power of 1 or 2? Would ((1/(n-1))*sum([abs((xi-μ)**3) for xi in s])**(1/3) or ((1/(n-1))*sum([(xi-μ)**4 for xi in s])**(1/4) and so on have made any sense?
My notion of squaring is that it 'amplifies' the measure of variation from the arithmetic mean while the simple absolute difference is somewhat a linear scale notionally. Would it not amplify it even more if I cubed it (and made absolute value of course) or quad it?
I do agree computationally cubes and quads would have been more expensive. But with the same argument, the absolute values would have been less expensive... So why squares?
Why is the Normal Distribution like it is, i.e. f = (1/(σ*math.sqrt(2*pi)))*e**((-1/2)*((xi-μ)/σ))?
What impact would it have on the normal distribution formula above if I calculated SD as described in (1) and (2) above?
Is it only a matter of our 'getting used to the squares', it could well have been linear, cubed or quad, and we would have trained our minds likewise?
(I may not have been 100% accurate in my number of opening and closing brackets above, but you will get the idea.)
So, if you are looking for an index of dispersion, you actually don't have to use the standard deviation. You can indeed report mean absolute deviation, the summary statistic you suggested. You merely need to be aware of how each summary statistic behaves, for example the SD assigns more weight to outlying variables. You should also consider how each one can be interpreted. For example, with a normal distribution, we know how much of the distribution lies between ±2SD from the mean. For some discussion of mean absolute deviation (and other measures of average absolute deviation, such as the median average deviation) and their uses see here.
Beyond its use as a measure of spread though, SD is related to variance and this is related to some of the other reasons it's popular, because the variance has some nice mathematical properties. A mathematician or statistician would be able to provide a more informed answer here, but squared difference is a smooth function and is differentiable everywhere, allowing one to analytically identify a minimum, which helps when fitting functions to data using least squares estimation. For more detail and for a comparison with least absolute deviations see here. Another major area where variance shines is that it can be easily decomposed and summed, which is useful for example in ANOVA and regression models generally. See here for a discussion.
As to your questions about raising to higher powers, they actually do have uses in statistics! In general, the mean (which is related to average absolute mean), the variance (related to standard deviation), skewness (related to the third power) and kurtosis (related to the fourth power) are all related to the moments of a distribution. Taking differences raised to those powers and standardizing them provides useful information about the shape of a distribution. The video I linked provides some easy intuition.
For some other answers and a larger discussion of why SD is so popular, See here.
Regarding the relationship of sigma and the normal distribution, sigma is simply a parameter that stretches the standard normal distribution, just like the mean changes its location. This is simply a result of the way the standard normal distribution (a normal distribution with mean=0 and SD=variance=1) is mathematically defined, and note that all normal distributions can be derived from the standard normal distribution. This answer illustrates this. Now, you can parameterize a normal distribution in other ways as well, but I believe you do need to provide sigma, whether using the SD or precisions. I don't think you can even parametrize a normal distribution using just the mean and the mean absolute difference. Now, a deeper question is why normal distributions are so incredibly useful in representing widely different phenomena and crop up everywhere. I think this is related to the Central Limit Theorem, but I do not understand the proofs of the theorem well enough to comment further.
I am trying to generate a model that uses several physico-chemical properties of a molecule (incl. number of atoms, number of rings, volume, etc.) to predict a numeric value Y. I would like to use PLS Regression, and I understand that standardization is very important here. I am programming in Python, using scikit-learn. The type and range for the features varies. Some are int64 while other are float. Some features generally have small (positive or negative) values, while other have very large value. I have tried using various scalers (e.g. standard scaler, normalize, minmax scaler, etc.). Yet, the R2/Q2 are still low. I have a few questions:
Is it possible that by scaling, some of the very important features lose their significance, and thus contribute less to explaining the variance of the response variable?
If yes, if I identify some important features (by expert knowledge), is it OK to scale other features but those? Or scale the important features only?
Some of the features, although not always correlated, have values that are in a similar range (e.g. 100-400), compared to others (e.g. -1 to 10). Is it possible to scale only a specific group of features that are within the same range?
The whole idea of scaling is to make models more robust to analysis on features space. For example, if you have 2 features as 5 Kg and 5000 gm, we know both are same, but for some algorithm, which are sensitive to metric space such as KNN, PCA etc, they will be more weighted towards second features, so scaling must be done for these algos.
Now coming to your question,
Scaling doesn't effect the significance of features. As i explained above, it helps in better analysis of data.
No, you should not do, reason explained above.
If you want to include domain knowledge in your model, you can use it as prior information. In short, for linear model, this is same as regularization. It has very good features. if you think, you have many useless-features, you can use L1 regularization, which creates sparse effect on features space, which is nothing but assign 0 weight to useless features. Here is the link for more-info.
One more point, some method such as tree based model doesn't need scaling, In last, it mostly depend on the model, you choose.
Lose significance? Yes. Contribute less? No.
No, it's not OK. It's either all or nothing.
No. The idea of scaling is not to decrease / increase significance / effect of a variable. It's to transform all variables to a common scale that can be interpreted.
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 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.
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.