Imagine, you have 100-1000 images that look like the following
What is the best algorithm to identify this pattern uniquely, even if it's rotated
or zoomed
or even shifted and/or partly cropped?
What you are trying to solve here is Cluster identification problem. The 100-1000 images you describe in your question are all large cluster of unlabeled dataset. There exist multiple Cluster Identification algorithms which will be perfect in your case such as k-means algorithm, k-modes algorithm or k-Nearest Neighbor algorithm.
Basically how data clustering works is that they statistically categorize similar clusters based on multiple similarity features like the cluster's size, density, distance, shape, etc. into classes such that there forms a group of similar and dissimilar clusters. Using the clustering algorithm your machine can learn to recognize patterns by observing as many dataset you intend to feed it.
Now, when you zoom the image or rotate/crop the image you just increase the noise in your dataset. Noises makes the data clustering process more tedious but it is doable. You can refer to this paper if you want to learn more about data clustering algorithms.
Related
I could use some advice on methods in R to determine the optimal number of clusters and later on describe the clusters with different statistical criteria. I’m new to R with basic knowledge about the statistical foundations of cluster analysis.
Methods to determine the number of clusters: In the literature one common method to do so is the so called "Elbow-criterion" which compares the Sum of Squared Differences (SSD) for different cluster solutions. Therefore the SSD is plotted against the numbers of Cluster in the analysis and an optimal number of clusters is determined by identifying the “elbow” in the plot (e.g. here: https://en.wikipedia.org/wiki/File:DataClustering_ElbowCriterion.JPG)
This method is a first approach to get a subjective impression. Therefore I’d like to implement it in R. The information on the internet on this is sparse. There is one good example here: http://www.mattpeeples.net/kmeans.html where the author also did an interesting iterative approach to see if the elbow is somehow stable after several repetitions of the clustering process (nevertheless it is for partitioning cluster methods not for hierarchical).
Other methods in Literature comprise the so called “stopping rules”. MILLIGAN & COOPER compared 30 of these stopping rules in their paper “An examination of procedures for determining the number of clusters in a data set” (available here: http://link.springer.com/article/10.1007%2FBF02294245) finding that the Stopping Rule from Calinski and Harabasz provided the best results in a Monte Carlo evaluation. Information on implementing this in R is even sparser.
So if anyone has ever implemented this or another Stopping rule (or other method) some advice would be very helpful.
Statistically describe the clusters:For describing the clusters I thought of using the mean and some sort of Variance Criterion. My data is on agricultural land-use and shows the production numbers of different crops per Municipality. My aim is to find similar patterns of land-use in my dataset.
I produced a script for a subset of objects to do a first test-run. It looks like this (explanations on the steps within the script, sources below).
#Clusteranalysis agriculture
#Load data
agriculture <-read.table ("C:\\Users\\etc...", header=T,sep=";")
attach(agriculture)
#Define Dataframe to work with
df<-data.frame(agriculture)
#Define a Subset of objects to first test the script
a<-df[1,]
b<-df[2,]
c<-df[3,]
d<-df[4,]
e<-df[5,]
f<-df[6,]
g<-df[7,]
h<-df[8,]
i<-df[9,]
j<-df[10,]
k<-df[11,]
#Bind the objects
aTOk<-rbind(a,b,c,d,e,f,g,h,i,j,k)
#Calculate euclidian distances including only the columns 4 to 24
dist.euklid<-dist(aTOk[,4:24],method="euclidean",diag=TRUE,upper=FALSE, p=2)
print(dist.euklid)
#Cluster with Ward
cluster.ward<-hclust(dist.euklid,method="ward")
#Plot the dendogramm. define Labels with labels=df$Geocode didn't work
plot(cluster.ward, hang = -0.01, cex = 0.7)
#here are missing methods to determine the optimal number of clusters
#Calculate different solutions with different number of clusters
n.cluster<-sapply(2:5, function(n.cluster)table(cutree(cluster.ward,n.cluster)))
n.cluster
#Show the objects within clusters for the three cluster solution
three.cluster<-cutree(cluster.ward,3)
sapply(unique(three.cluster), function(g)aTOk$Geocode[three.cluster==g])
#Calculate some statistics to describe the clusters
three.cluster.median<-aggregate(aTOk[,4:24],list(three.cluster),median)
three.cluster.median
three.cluster.min<-aggregate(aTOk[,4:24],list(three.cluster),min)
three.cluster.min
three.cluster.max<-aggregate(aTOk[,4:24],list(three.cluster),max)
three.cluster.max
#Summary statistics for one variable
three.cluster.summary<-aggregate(aTOk[,4],list(three.cluster),summary)
three.cluster.summary
detach(agriculture)
Sources:
http://www.r-tutor.com/gpu-computing/clustering/distance-matrix
How to apply a hierarchical or k-means cluster analysis using R?
http://statistics.berkeley.edu/classes/s133/Cluster2a.html
The elbow criterion as your links indicated is for k-means. Also the cluster mean is obviously related to k-means, and is not appropriate for linkage clustering (in particular not for single-linkage, see single-link-effect).
Your question title however mentions hierarchical clustering, and so does your code?
Note that the elbow criterion does not choose the optimal number of clusters. It chooses the optimal number of k-means clusters. If you use a different clustering method, it may need a different number of clusters.
There is no such thing as the objectively best clustering. Thus, there also is no objectively best number of clusters. There is a rule of thumb for k-means that chooses a (maybe best) tradeoff between number of clusters and minimizing the target function (because increasing the number of clusters always can improve the target function); but that is mostly to counter a deficit of k-means. It is by no means objective.
Cluster analysis in itself is not an objective task. A clustering may be mathematically good, but useless. A clustering may score much worse mathematically, but it may provide you insight to your data that cannot be measured mathematically.
This is a very late answer and probably not useful for the asker anymore - but maybe for others. Check out the package NbClust. It contains 26 indices that give you a recommended number of clusters (and you can also choose your type of clustering). You can run it in such a way that you get the results for all the indices and then you can basically go with the number of clusters recommended by most indices. And yes, I think the basic statistics are the best way to describe clusters.
You can also try the R-NN Curves method.
http://rguha.net/writing/pres/rnn.pdf
K means Clustering is highly sensitive to the scale of data e.g. for a person's age and salary, if not normalized, K means would consider salary more important variable for clustering rather than age, which you do not want. So before applying the Clustering Algorithm, it is always a good practice to normalize the scale of data, bring them to the same level and then apply the CA.
I'm working with a set of co-ordinates, and want to dynamically (I have many sets that need to go through this process) understand how many distinct groups there are within the data. My approach was to apply k-means to investigate whether it would find the centroids and I could go from there.
When plotting some data with 6 distinct clusters (visually) the k-means algorithm continues to ignore two significant clusters while putting many centroids into another.
See image below:
Red are the co-ordinate data points and blue are centroids that k-means has provided. In this specific case I've gone for 15 (arbitrary), but it still doesn't recognise those patches of data on the right hand side, rather putting a mid point between them while putting in 8 in the cluster in the top right.
Admittedly there are slightly more data points in the top right, but not by much.
I'm using the standard k-means algorithm in R and just feeding in x and y co-ordinates. I've tried standardising the data, but this doesn't make any difference.
Any thoughts on why this is, or other potential methodologies that could be applied to try and dynamically understand the number of distinct clusters there are in the data?
You could try with Self-organizing map:
this is a clustering algorithm based on Neural Networks which create a discretized representation of the input space of the training samples, called a map, and is, therefore, a method to do dimensionality reduction (SOM).
This algorithm is very good for clustering also because does not require a priori selection of the number of clusters (in k-mean you need to choose k, here no). In your case, it hopefully finds automatically the optimal number of cluster, and you can actually visualize it.
You can find a very nice python package called somoclu which has got this algorithm implemented and an easy way to visualize the result. Else you can go with R. Here you can find a blog post with a tutorial, and Cran package manual for SOM.
K-means is a randomized algorithm and it will get stuck in local minima.
Because of these problems, it is common to run k-means several times, and keep the result with least squares, I.e., the best of the local minima found.
For my thesis assignment I need to perform a cluster analysis on a high dimensional data set containing purchase data from a retail store (+1000 dimensions). Because traditional clustering algorithms are not well suited for high dimensions (and dimension reduction is not really an option), I would like to try algorithms specifically developed for high dimensional data(e.g. ProClus).
Here however, my problem starts.
I have no clue what value I should use for parameter d. Can anyone help me?
This is just one of the many limitations of ProClus.
The parameter is the average dimensionality of your cluster. It assumes there is a linear cluster somewhere in your data. This likely will not hold for purchase data, but you can try. For sparse data such as purchases, I would rather focus on frequent itemset mining.
There is no universal clustering algorithm. Any clustering algorithm will come with a variety of parameters that you need to experiment with.
For cluster analysis it is essential that you somehow can visualize or analyze the result, to be able to find out if and how well the method worked.
I want to cluster binary vectors (millions of them) into k clusters.I am using hamming distance for finding the nearest neighbors to initial clusters (which is very slow as well). I think K-means clustering does not really fit here. The problem is in calculating mean of the nearest neighbors (which are binary vectors) to some initial cluster center, to update the centroid.
A second option is to use K-medoids in which the new cluster center is chosen from one of the nearest neighbors ( the one which is closest to all neighbors for a particular cluster center). But finding that is another problem because numbers of nearest neighbors are also quite large.
Can someone please guide me?
It is possible to do k-means with clustering with binary feature vectors. The paper called TopSig I co-authored has the details. The centroids are calculated by taking the most frequently occurring bit in each dimension. The TopSig paper applied this to document clustering where we had binary feature vectors created by random projection of sparse high dimensional bag-of-words feature vectors. There is an implementation in java at http://ktree.sf.net. We are currently working on a C++ version but it is very early code which is still messy, and probably contains bugs, but you can find it at http://github.com/cmdevries/LMW-tree. If you have any questions, please feel free to contact me at chris#de-vries.id.au.
If you are wanting to cluster a lot of binary vectors there are also more scalable tree based clustering algorithms of K-tree, TSVQ and EM-tree. For more details related to these algorithms you can see a paper I have recently submitted for peer review that is not yet published relating to the EM-tree.
Indeed k-means is not too appropriate here, because the means won't be reasonable on binary data.
Why do you need exactly k clusters? This will likely mean that some vectors won't fit to their clusters very well.
Some stuff you could look into for clustering: minhash, locality sensitive hashing.
I could use some advice on methods in R to determine the optimal number of clusters and later on describe the clusters with different statistical criteria. I’m new to R with basic knowledge about the statistical foundations of cluster analysis.
Methods to determine the number of clusters: In the literature one common method to do so is the so called "Elbow-criterion" which compares the Sum of Squared Differences (SSD) for different cluster solutions. Therefore the SSD is plotted against the numbers of Cluster in the analysis and an optimal number of clusters is determined by identifying the “elbow” in the plot (e.g. here: https://en.wikipedia.org/wiki/File:DataClustering_ElbowCriterion.JPG)
This method is a first approach to get a subjective impression. Therefore I’d like to implement it in R. The information on the internet on this is sparse. There is one good example here: http://www.mattpeeples.net/kmeans.html where the author also did an interesting iterative approach to see if the elbow is somehow stable after several repetitions of the clustering process (nevertheless it is for partitioning cluster methods not for hierarchical).
Other methods in Literature comprise the so called “stopping rules”. MILLIGAN & COOPER compared 30 of these stopping rules in their paper “An examination of procedures for determining the number of clusters in a data set” (available here: http://link.springer.com/article/10.1007%2FBF02294245) finding that the Stopping Rule from Calinski and Harabasz provided the best results in a Monte Carlo evaluation. Information on implementing this in R is even sparser.
So if anyone has ever implemented this or another Stopping rule (or other method) some advice would be very helpful.
Statistically describe the clusters:For describing the clusters I thought of using the mean and some sort of Variance Criterion. My data is on agricultural land-use and shows the production numbers of different crops per Municipality. My aim is to find similar patterns of land-use in my dataset.
I produced a script for a subset of objects to do a first test-run. It looks like this (explanations on the steps within the script, sources below).
#Clusteranalysis agriculture
#Load data
agriculture <-read.table ("C:\\Users\\etc...", header=T,sep=";")
attach(agriculture)
#Define Dataframe to work with
df<-data.frame(agriculture)
#Define a Subset of objects to first test the script
a<-df[1,]
b<-df[2,]
c<-df[3,]
d<-df[4,]
e<-df[5,]
f<-df[6,]
g<-df[7,]
h<-df[8,]
i<-df[9,]
j<-df[10,]
k<-df[11,]
#Bind the objects
aTOk<-rbind(a,b,c,d,e,f,g,h,i,j,k)
#Calculate euclidian distances including only the columns 4 to 24
dist.euklid<-dist(aTOk[,4:24],method="euclidean",diag=TRUE,upper=FALSE, p=2)
print(dist.euklid)
#Cluster with Ward
cluster.ward<-hclust(dist.euklid,method="ward")
#Plot the dendogramm. define Labels with labels=df$Geocode didn't work
plot(cluster.ward, hang = -0.01, cex = 0.7)
#here are missing methods to determine the optimal number of clusters
#Calculate different solutions with different number of clusters
n.cluster<-sapply(2:5, function(n.cluster)table(cutree(cluster.ward,n.cluster)))
n.cluster
#Show the objects within clusters for the three cluster solution
three.cluster<-cutree(cluster.ward,3)
sapply(unique(three.cluster), function(g)aTOk$Geocode[three.cluster==g])
#Calculate some statistics to describe the clusters
three.cluster.median<-aggregate(aTOk[,4:24],list(three.cluster),median)
three.cluster.median
three.cluster.min<-aggregate(aTOk[,4:24],list(three.cluster),min)
three.cluster.min
three.cluster.max<-aggregate(aTOk[,4:24],list(three.cluster),max)
three.cluster.max
#Summary statistics for one variable
three.cluster.summary<-aggregate(aTOk[,4],list(three.cluster),summary)
three.cluster.summary
detach(agriculture)
Sources:
http://www.r-tutor.com/gpu-computing/clustering/distance-matrix
How to apply a hierarchical or k-means cluster analysis using R?
http://statistics.berkeley.edu/classes/s133/Cluster2a.html
The elbow criterion as your links indicated is for k-means. Also the cluster mean is obviously related to k-means, and is not appropriate for linkage clustering (in particular not for single-linkage, see single-link-effect).
Your question title however mentions hierarchical clustering, and so does your code?
Note that the elbow criterion does not choose the optimal number of clusters. It chooses the optimal number of k-means clusters. If you use a different clustering method, it may need a different number of clusters.
There is no such thing as the objectively best clustering. Thus, there also is no objectively best number of clusters. There is a rule of thumb for k-means that chooses a (maybe best) tradeoff between number of clusters and minimizing the target function (because increasing the number of clusters always can improve the target function); but that is mostly to counter a deficit of k-means. It is by no means objective.
Cluster analysis in itself is not an objective task. A clustering may be mathematically good, but useless. A clustering may score much worse mathematically, but it may provide you insight to your data that cannot be measured mathematically.
This is a very late answer and probably not useful for the asker anymore - but maybe for others. Check out the package NbClust. It contains 26 indices that give you a recommended number of clusters (and you can also choose your type of clustering). You can run it in such a way that you get the results for all the indices and then you can basically go with the number of clusters recommended by most indices. And yes, I think the basic statistics are the best way to describe clusters.
You can also try the R-NN Curves method.
http://rguha.net/writing/pres/rnn.pdf
K means Clustering is highly sensitive to the scale of data e.g. for a person's age and salary, if not normalized, K means would consider salary more important variable for clustering rather than age, which you do not want. So before applying the Clustering Algorithm, it is always a good practice to normalize the scale of data, bring them to the same level and then apply the CA.