Suppose I have 2 data sets
The first - well-separated Gaussian spheres of different sizes
The second - well-separated Gaussian spheres of the same size
Can someone explain to me why the first data set will yield higher results in k-means? How do the "different sizes" of the data affect the result?
And in addition to that, does a high value of r in fuzzy clustering lead to overfitting?
Thanks
Related
I am trying to use the KMeans clustering from faiss on a human pose dataset of body joints. I have 16 body parts so a dimension of 32. The joints are scaled in a range between 0 and 1. My dataset consists of ~ 900.000 instances. As mentioned by faiss (faiss_FAQ):
As a rule of thumb there is no consistent improvement of the k-means quantizer beyond 20 iterations and 1000 * k training points
Applying this to my problem I randomly select 50000 instances for training. As I want to check for a number of clusters k between 1 and 30.
Now to my "problem":
The inertia is increasing directly as the number of cluster increases (n_cluster on the x-axis):
I tried varying the number of iterations, the number of redos, verbose and spherical, but the results stay the same or get worse. I do not think that it is a problem of my implementation; I tested it on a small example with 2D data and very clear clusters and it worked.
Is it that the data is just bad clustered or is there another problem/mistake I have missed? Maybe the scaling of the values between 0 and 1? Should I try another approach?
I found my mistake. I had to increase the parameter max_points_per_centroid. As I have so many data points it sampled a sub-batch for the fit. For a larger number of clusters this sub-batch is larger. See FAQ of faiss:
max_points_per_centroid * k: there are too many points, making k-means unnecessarily slow. Then the training set is sampled
The larger subbatch of course has a larger inertia as there are more points in total.
I have a bundle of high-dimensional data and the instances are labeled as outliers or not. I am looking to get some insights around where these outliers reside within the data. I seek to answer questions like:
Are the outliers spread far apart from each other? Or are they clustered together?
Are the outliers lying 'in-between' clusters of good data? Or are they on the 'edge' boundaries of the data?
If outliers are clustered together, how do these cluster densities compare with clusters of good data?
'Where' are the outliers?
What kind of techniques will let me find these insights? If the data was 2 or 3-dimensional, I can easily plot the data and just look at it. But I can't do it high-dimensional data.
Analyzing the Statistical Properties of Outliers
First of all, if you can choose to focus on specific features. For
example, if you know a featues is subject to high variation, you can
draw a box plot. You can also draw a 2D graph if you want to focus on
2 features. THis shows how much the labelled outliers vary.
Next, there's a metric called a Z-score, which basically says how
many standard devations a point varies compared to the mean. The
Z-score is signed, meaning if a point is below the mean, the Z-score
will be negative. This can be used to analyze all the features of the
dataset. You can find the threshold value in your labelled dataset for which all the points above that threshold are labelled outliers
Lastly, we can find the interquartile range and similarly filter
based on it. The IQR is simply the difference between the 75
percentile and 25 percentile. You can also use this similarly to Z-score.
Using these techniques, we can analyze some of the statistical properties of the outliers.
If you also want to analyze the clusters, you can adapt the DBSCAN algorithm to your problem. This algorithm clusters data based on densities, so it will be easy to apply the techniques to outliers.
Is there any available DM script that can compare two images and know the difference?
I mean the script can compare two or more images, and it can determine the similarity of two images, for example the 95% area of one image is same as another image, then the similarity of these two images is 95%.
The script can compare brightness and contrast distribution of images.
Thanks,
This question is a bit ill-defined, as "similarity" between images depends a lot on what you want.
If by "95% of the area is the same" you mean that 95% of the pixels are of identical value in images A & B, you can simply create a mask and sum() it to count the number of pixels, i.e.:
sum( abs(A-B)==0 ? 1 : 0 )
However, this will utterly fail if the images A & B are shifted with respect to each other even by a single pixel. It will also fail, if A & B are of same contrast but different absolute value.
I guess the intended question was to find similarity of two images in a fuzzy way.
For these, one way is to do crosscorrelation. DM has this function. Like this,
image xcorr= CrossCorrelate(ref,img)
From xcorr, the peak position gives x- and y- shift between the two, the peak intensity gives "similarity" of the two.
If you know there is no shift between the two, you can just do the sum and multiplication,
number similarity1=sum(img1*img2)
Another way to do similarity is calculate Euclidian distance of the two:
number similarity2=sqrt(sum((img1-img2)**2)).
"similarity2" calculates the "pure" similarity. "similarity1" is the pure similarity plus the mean intensity of img1 and img2. The difference is essentially this,
(a-b)**2=a**2+b**2-2*a*b.
The left term is "similarity2", the last term on the right is the "crosscorrelation" or "similarity1".
I think "similarity1" is called cross-correlation, "similarity2" is called correlation coefficient.
In example comparing two diffraction patterns, if you want to compute the degree of similarity, use "similarity2". If you want to compute the degree of similarity plus a certain character of the diffraction pattern, use "similarity1".
All,
I am doing Bayesian modeling using rjags. However, when the number of observation is larger than 1000. The graph size is too big.
More specifically, I am doing a Bayesian ranking problem. Traditionally, one observation means one X[i, 1:N]-Y[i] pair, where X[i, 1:N] means the i-th item is represented by a N-size predictor vector, and Y[i] is a response. The objective is to minimize the point-wise error of predicted values,for example, least square error.
A ranking problem is different. Since we more care about the order, we use a pair-wise 1-0 indicator to represent the order between Y[i] and Y[j], for example, when Y[i]>Y[j], I(i,j)=1; otherwise I(i,j)=0. We treat this 1-0 indicator as an observation. Therefore, assuming we have K items: Y[1:K], the number of indicator is 0.5*K*(K-1). Hence when K is increased from 500 to 5000, the number of observations is very large, i.e. from 500^2 to 5000^2. The garph size of the rjags model is large too, for example graph size > 500,000. And the log-posterior will be very small.
And it takes a long time to complete the training. I think the consumed time is >40 hours. It is not practical for me to do further experiment. Therefore, do you have any idea to speed up the rjags. I heard that the RStan is faster than Rjags. Any one who has similar experience?
I have a set of vectors in multidimensional space (may be several thousands of dimensions). In this space, I can calculate distance between 2 vectors (as a cosine of the angle between them, if it matters). What I want is to visualize these vectors keeping the distance. That is, if vector a is closer to vector b than to vector c in multidimensional space, it also must be closer to it on 2-dimensional plot. Is there any kind of diagram that can clearly depict it?
I don't think so. Imagine any twodimensional picture of a tetrahedron. There is no way of depicting the four vertices in two dimensions with equal distances from each other. So you will have a hard time trying to depict more than three n-dimensional vectors in 2 dimensions conserving their mutual distances.
(But right now I can't think of a rigorous proof.)
Update:
Ok, second idea, maybe it's dumb: If you try and find clusters of closer associated objects/texts, then calculate the center or mean vector of each cluster. Then you can reduce the problem space. At first find a 2D composition of the clusters that preserves their relative distances. Then insert the primary vectors, only accounting for their relative distances within a cluster and their distance to the center of to two or three closest clusters.
This approach will be ok for a large number of vectors. But it will not be accurate in that there always will be somewhat similar vectors ending up at distant places.