I'm new to this site as well as new to cluster analysis, so I apologize if I violate conventions.
I've been using Cluster 3.0 to perform Hierarchical Cluster Analysis with Euclidean Distance and Average linkage. Cluster 3.0 outputs a .gtr file with a node joining a gene and their similarity score. I've noticed that the first line in the .gtr file always links a gene with another gene followed by the similarity score. But, how do I reproduce this similarity score?
In my data set, I have 8 genes and create a distance matrix where d_{ij} contains the Euclidian distance between gene i and gene j. Then I normalize the matrix by dividing each element by the max value in the matrix. To get the similarity matrix, I subtract all the elements from 1. However, my result does not use the linkage type and differs from the output similarity score.
I am mainly confused how linkages affect the similarity of the first node (the joining of the two closest genes) and how to compute the similarity score.
Thank you!
The algorithm compares clusters using some linkage method, not data points. However, in the first iteration of the algorithm each data point forms its own cluster; this means that your linkage method is actually reduced to the metric you use to measure the distance between data points (for your case Euclidean distance). For subsequent iterations, the distance between clusters will be measured according to your linkage method, which in your case is average link. For two clusters A and B, this is calculated as follows:
where d(a,b) is the Euclidean distance between the two data points. Convince yourself that when A and B contain just one data point (as in the first iteration) this equation reduces itself to d(a,b). I hope this makes things a bit more clear. If not, please provide more details of what exactly you want to do.
Related
I have a big dataset and need to calculate cosine similarities between products in the context of item-item collaborative filtering for product recommendations. As the data contains more than 50000 items and 25000 rows, I opted for using Spark and found the function columnSimilarities() which can be used on DistributedMatrix, specifically on a RowMatrix or IndexedRowMatrix.
But, there is 2 issues I'm wondering about.
1) In the documentation, it's mentioned that:
A RowMatrix is backed by an RDD of its rows, where each row is a local
vector. Since each row is represented by a local vector, the number of
columns is limited by the integer range but it should be much smaller
in practice.
As I have many products it seems that RowMatrix is not the best choice for building the similarity Matrix from my input which is a Spark Dataframe. That's why I decided to start by converting the dataframe to a CoordinateMatrix and then use toRowMatrix() because columnSimilarities() requires input parameter as RowMatrix. Meanwhile, I'm not sure of its performance..
2) I found out that:
the columnSimilarities method only returns the off diagonal entries of
the upper triangular portion of the similarity matrix.
reference
Does this mean I cannot get the similarity vectors of all the products?
So your current strategy is to compute the similarity between each item, i, and each other item. This means at best you have to compute the upper triangular of the distance matrix, I think that's (i^2 / 2) - i calculations. Then you have to sort for each of those i items.
If you are willing to trade off a little accuracy for runtime you can use approximate nearest neighbors (ANN). You might not find exactly the top NNS for an item but you will find very similar items and it will be orders of magnitude faster. No one dealing with moderately sized datasets calculates (or has the time to wait to calculate) the full set of distances.
Each ANN search method creates an index that will only generate a small set of candidates and compute distances within that subset (this is the fast part). The way the index is constructed provides different guarantees about the accuracy of the NN retrieval (this is the approximate part).
There are various ANN search libraries out there, annoy, nmslib, LSH. An accessible introduction is here: https://erikbern.com/2015/10/01/nearest-neighbors-and-vector-models-part-2-how-to-search-in-high-dimensional-spaces.html
HTH. Tim
I'm trying to implement the Bag of Features model.
Given a descriptors matrix object (representing an image) belonging to the initial dataset, compute its histogram is easy, since we already know to which cluster each descriptor vector belongs to from k-means.
But what about if we want to compute the histogram of a query matrix? The only solution that crosses my mind is to compute the distance between each vector descriptor to each of the k cluster centroids.
This can be inefficient: supposing that k=100 (so 100 centroids), then we have an query image represented through 1000 SIFT descriptors, so a matrix 1000x100.
What we have to do now is computing 1000 * 100 eucledian distances in 128 dimensions. This seems really inefficient.
How to solve this problem?
NOTE: can you suggest me some implementations where this point is explained?
NOTE: I know LSH is a solution (since we are using high-dim vectors), but I don't think that actual implementations use it.
UPDATE:
I was talking with a collegue of mine: using a hierarchical cluster approach instead of classic k-means, should speed up the process so much! Is it correct to say that if we have k centroids, with an hierarchical cluster we have to do only log(k) comparisons in order to find the closest centroid instead of k comparisons?
For a bag of features approach, you indeed need to quantize the descriptors. Yes, if you have 10000 features and 100 features that 10000*100 distances (unless you use an index here).
Compare this to comparing each of the 10000 features to each of the 10000 features of each image in your database. Does it still sound that bad?
I have a question about using a clustering method vs fitting the same data with a distribution.
Assuming that I have a dataset with 2 features (feat_A and feat_B) and let's assume that I use a clustering algorithm to divide the data in an optimal number of clusters...say 3.
My goal is to assign for each of the input data [feat_Ai,feat_Bi] a probability (or something similar) that the point belongs to cluster 1 2 3.
a. First approach with clustering:
I cluster the data in the 3 clusters and I assign to each point the probability of belonging to a cluster depending on the distance from the center of that cluster.
b. Second approach using mixture model:
I fit a mixture model or mixture distribution to the data. Data are fit to the distribution using an expectation maximization (EM) algorithm, which assigns posterior probabilities to each component density with respect to each observation. Clusters are assigned by selecting the component that maximizes the posterior probability.
In my problem I find the cluster centers (or I fit the model if approach b. is used) with a subsample of data. Then I have to assign a probability to a lot of other data... I would like to know in presence of new data which approach is better to use to still have meaningful assignments.
I would go for a clustering method for example a kmean because:
If the new data come from a distribution different from the one used to create the mixture model, the assignment could be not correct.
With new data the posterior probability changes.
The clustering method minimizes the variance of the clusters in order to find a kind of optimal separation border, the mixture model take into consideration the variance of the data to create the model (not sure that the clusters that will be formed are separated in an optimal way).
More info about the data:
Features shouldn't be assumed dependent.
Feat_A represents the duration of a physical activity Feat_B the step counts In principle we could say that with an higher duration of the activity the step counts increase, but it is not always true.
Please help me to think and if you have any other point please let me know..
I've been studying about k-means clustering, and one big thing which is not clear is what Silhouette function really tell to me?
i know it shows that what appropriate k should be detemine but i cant understand what mean of silhouette function really say to me?
i read somewhere, if the mean of silhouette is less than 0.5 your clustering is not valid.
thanks for your answers in advance.
From the definition of silhouette :
Silhouette Value
The silhouette value for each point is a measure of how similar that
point is to points in its own cluster compared to points in other
clusters, and ranges from -1 to +1.
The silhouette value for the ith point, Si, is defined as
Si = (bi-ai)/ max(ai,bi) where ai is the average distance from the ith
point to the other points in the same cluster as i, and bi is the
minimum average distance from the ith point to points in a different
cluster, minimized over clusters.
This method just compares the intra-group similarity to closest group similarity. If any data member average distance to other members of the same cluster is higher than average distance to some other cluster members, then this value is negative and clustering is not successful. On the other hand, silhuette values close to 1 indicates a successful clustering operation. 0.5 is not an exact measure for clustering.
#fatihk gave a good citation;
additionally, you may think about the Silhouette value as a degree of
how clusters overlap with each other, i.e. -1: overlap perfectly,
+1: clusters are perfectly separable;
BUT low Silhouette values for a particular algorithm does NOT mean that there are no clusters, rather it means that the algorithm used cannot separate clusters and you may consider tune your algorithm or use a different algorithm (think about K-means for concentric circles, vs DBSCAN).
There is an explicit formula associated with the elbow method to automatically determine the number of clusters. The formula tells you about the strength of the elbow(s) being detected when using the elbow method to determine the number of clusters, see here. See illustration here:
Enhanced Elbow rule
This is a Homework question. I have a huge document full of words. My challenge is to classify these words into different groups/clusters that adequately represent the words. My strategy to deal with it is using the K-Means algorithm, which as you know takes the following steps.
Generate k random means for the entire group
Create K clusters by associating each word with the nearest mean
Compute centroid of each cluster, which becomes the new mean
Repeat Step 2 and Step 3 until a certain benchmark/convergence has been reached.
Theoretically, I kind of get it, but not quite. I think at each step, I have questions that correspond to it, these are:
How do I decide on k random means, technically I could say 5, but that may not necessarily be a good random number. So is this k purely a random number or is it actually driven by heuristics such as size of the dataset, number of words involved etc
How do you associate each word with the nearest mean? Theoretically I can conclude that each word is associated by its distance to the nearest mean, hence if there are 3 means, any word that belongs to a specific cluster is dependent on which mean it has the shortest distance to. However, how is this actually computed? Between two words "group", "textword" and assume a mean word "pencil", how do I create a similarity matrix.
How do you calculate the centroid?
When you repeat step 2 and step 3, you are assuming each previous cluster as a new data set?
Lots of questions, and I am obviously not clear. If there are any resources that I can read from, it would be great. Wikipedia did not suffice :(
As you don't know exact number of clusters - I'd suggest you to use a kind of hierarchical clustering:
Imagine that all your words just a points in non-euclidean space. Use Levenshtein distance to calculate distance between words (it works great, in case, if you want to detect clusters of lexicographically similar words)
Build minimum spanning tree which contains all of your words
Remove links, which have length greater than some threshold
Linked groups of words are clusters of similar words
Here is small illustration:
P.S. you can find many papers in web, where described clustering based on building of minimal spanning tree
P.P.S. If you want to detect clusters of semantically similar words, you need some algorithms of automatic thesaurus construction
That you have to choose "k" for k-means is one of the biggest drawbacks of k-means.
However, if you use the search function here, you will find a number of questions that deal with the known heuristical approaches to choosing k. Mostly by comparing the results of running the algorithm multiple times.
As for "nearest". K-means acutally does not use distances. Some people believe it uses euclidean, other say it is squared euclidean. Technically, what k-means is interested in, is the variance. It minimizes the overall variance, by assigning each object to the cluster such that the variance is minimized. Coincidentially, the sum of squared deviations - one objects contribution to the total variance - over all dimensions is exactly the definition of squared euclidean distance. And since the square root is monotone, you can also use euclidean distance instead.
Anyway, if you want to use k-means with words, you first need to represent the words as vectors where the squared euclidean distance is meaningful. I don't think this will be easy or maybe not even possible.
About the distance: In fact, Levenshtein (or edit) distance satisfies triangle inequality. It also satisfies the rest of the necessary properties to become a metric (not all distance functions are metric functions). Therefore you can implement a clustering algorithm using this metric function, and this is the function you could use to compute your similarity matrix S:
-> S_{i,j} = d(x_i, x_j) = S_{j,i} = d(x_j, x_i)
It's worth to mention that the Damerau-Levenshtein distance doesn't satisfy the triangle inequality, so be careful with this.
About the k-means algorithm: Yes, in the basic version you must define by hand the K parameter. And the rest of the algorithm is the same for a given metric.