I have several laplacian spectra of graphs (networks) to compare, and I'm looking for a distance measure between each pair of spectrum.
I have tried Hellinger distance and euclidian distance (the n-dimensionnal euclidian distance between two vectors of eigenvalues, where n is the number of bins), which give me different distances, obviously, but also different rankings, which leads me to think that the method used is quite important in the quality of the results.
I also saw that histograms distances can be measured through other methods (correlation, chi-square, intersection, heart mover's distance, etc.)
Is there an accepted distance measure for this kind of histogram comparison problem, where relatively small features are quite important (like the peaks around eigenvalue 0.3) ?
The spectra:
Related
Problem:
To cluster the similar colour pixels in CIE LAB using K means.
I want to use CIE 94 for distance between 2 pixels
Formula of CIE94
What i read was Kmeans work in "Euclidean space" where the positional cordinates are minimised by cost function which is (sum of squared difference)
The reason of not Using kmeans in space other than euclidean is
"""algorithm is often presented as assigning objects to the nearest cluster by distance. The standard algorithm aims at minimizing the within-cluster sum of squares (WCSS) objective, and thus assigns by "least sum of squares", which is exactly equivalent to assigning by the smallest Euclidean distance. Using a different distance function other than (squared) Euclidean distance may stop the algorithm from converging""(source wiki)
So how to use distance CIE 94 in LAB SPACE for similar colour clustering ?
So how to approach the problem ? What should be the minimisation function here ? HOW to map euclidean space to lab space if for the k mean euclidean formula to work ? Any other approach here ?
The reason that CIE LAB is often used for clustering is because it reduces the color to 2 dimensions (as opposed to RGB with 3 color channels). You can easily think of the color for each pixel in a Cartesian coordinate system, instead of points (x,y) you have points (a,b) From here you simply perform a 2d kmeans.
Exactly how you implement kmeans is up to you. The nice thing about reducing colors to a 2d space is we can imagine the data on a grid, and now we can use any regular distance measure we want. Mahalonobis, euclidean, 1 norm, city block, etc. The possibilities are really endless here.
You don't have to use CIELAB, you can just as easily use YCbCr, YUV, or any other colorspace that represents color in 2 dimensions. IF you wanted to try a 3d kmeans you could use rgb, hsv, etc. One problem with higher dimensionality is sparsity of clusters (large variance) and most importantly, increased computation time.
Just for fun I've included two images clustered using kmeans, one in LAB and one in YCbCr, you can see the clustering is nearly identical (except that the labels are different), just proving that the exact color space is irrelevant, the main point is to match the dimensionality of your kmeans with that of your data
EDIT
You made some good points in your comments. I was merely demonstrating that by abstracting the problem you can imagine many variations for the same basic clustering algorithm. But you are right, there are advantages to using CIELAB
Back to the distance measure. Kmeans has two steps, assignment, and update (it is very similar to the Expectation Maximization algorithm). This distance is used in assignment step of k-means. Here is some psuedo code
for each pixel 1 to rows*cols
for each cluster 1 to k
dist[k] = calculate_distance(pixel, mu[k])
pixel_id = index k of minimum dist
you would create a function calculate_distance that uses the delta_e calculation from cielab94. This formula uses all 3 channels to calculate distance. Hopefully this answers your questions
NOTE
My examples only use the 2 color channels, ignoring the luminance channel. I used this technique since often the goal is group colors despite lighting disparities(such as shadows). The delta_E measure is not lighting invariant. This may or may not be a concern for your application, but it is something to keep in mind.
results using square euclidean distance
results using cityblock distance
There are k-means variations for other distance functions.
In particular k-medoids (PAM) works with arbitrary distance functions.
I have some geographical trajectories sampled to analyze, and I calculated the histogram of data in spatial and temporal dimension, which yielded a time domain based feature for each spatial element. I want to perform a discrete FFT to transform the time domain based feature into frequency domain based feature (which I think maybe more robust), and then do some classification or clustering algorithms.
But I'm not sure using what descriptor as frequency domain based feature, since there are amplitude spectrum, power spectrum and phase spectrum of a signal and I've read some references but still got confused about the significance. And what distance (similarity) function should be used as measurement when performing learning algorithms on frequency domain based feature vector(Euclidean distance? Cosine distance? Gaussian function? Chi-kernel or something else?)
Hope someone give me a clue or some material that I can refer to, thanks~
Edit
Thanks to #DrKoch, I chose a spatial element with the largest L-1 norm and plotted its log power spectrum in python and it did show some prominent peaks, below is my code and the figure
import numpy as np
import matplotlib.pyplot as plt
sp = np.fft.fft(signal)
freq = np.fft.fftfreq(signal.shape[-1], d = 1.) # time sloth of histogram is 1 hour
plt.plot(freq, np.log10(np.abs(sp) ** 2))
plt.show()
And I have several trivial questions to ask to make sure I totally understand your suggestion:
In your second suggestion, you said "ignore all these values."
Do you mean the horizontal line represent the threshold and all values below it should be assigned to value zero?
"you may search for the two, three largest peaks and use their location and probably widths as 'Features' for further classification."
I'm a little bit confused about the meaning of "location" and "width", does "location" refer to the log value of power spectrum (y-axis) and "width" refer to the frequency (x-axis)? If so, how to combine them together as a feature vector and compare two feature vector of "a similar frequency and a similar widths" ?
Edit
I replaced np.fft.fft with np.fft.rfft to calculate the positive part and plot both power spectrum and log power spectrum.
code:
f, axarr = plt.subplot(2, sharex = True)
axarr[0].plot(freq, np.abs(sp) ** 2)
axarr[1].plot(freq, np.log10(np.abs(sp) ** 2))
plt.show()
figure:
Please correct me if I'm wrong:
I think I should keep the last four peaks in first figure with power = np.abs(sp) ** 2 and power[power < threshold] = 0 because the log power spectrum reduces the difference among each component. And then use the log spectrum of new power as feature vector to feed classifiers.
I also see some reference suggest applying a window function (e.g. Hamming window) before doing fft to avoid spectral leakage. My raw data is sampled every 5 ~ 15 seconds and I've applied a histogram on sampling time, is that method equivalent to apply a window function or I still need apply it on the histogram data?
Generally you should extract just a small number of "Features" out of the complete FFT spectrum.
First: Use the log power spec.
Complex numbers and Phase are useless in these circumstances, because they depend on where you start/stop your data acquisiton (among many other things)
Second: you will see a "Noise Level" e.g. most values are below a certain threshold, ignore all these values.
Third: If you are lucky, e.g. your data has some harmonic content (cycles, repetitions) you will see a few prominent Peaks.
If there are clear peaks, it is even easier to detect the noise: Everything between the peaks should be considered noise.
Now you may search for the two, three largest peaks and use their location and probably widths as "Features" for further classification.
Location is the x-value of the peak i.e. the 'frequency'. It says something how "fast" your cycles are in the input data.
If your cycles don't have constant frequency during the measuring intervall (or you use a window before caclculating the FFT), the peak will be broader than one bin. So this widths of the peak says something about the 'stability' of your cycles.
Based on this: Two patterns are similar if the biggest peaks of both hava a similar frequency and a similar widths, and so on.
EDIT
Very intersiting to see a logarithmic power spectrum of one of your examples.
Now its clear that your input contains a single harmonic (periodic, oscillating) component with a frequency (repetition rate, cycle-duration) of about f0=0.04.
(This is relative frquency, proprtional to the your sampling frequency, the inverse of the time beetween individual measurment points)
Its is not a pute sine-wave, but some "interesting" waveform. Such waveforms produce peaks at 1*f0, 2*f0, 3*f0 and so on.
(So using an FFT for further analysis turns out to be very good idea)
At this point you should produce spectra of several measurements and see what makes a similar measurement and how differ different measurements. What are the "important" features to distinguish your mesurements? Thinks to look out for:
Absolute amplitude: Height of the prominent (leftmost, highest) peaks.
Pitch (Main cycle rate, speed of changes): this is position of first peak, distance between consecutive peaks.
Exact Waveform: Relative amplitude of the first few peaks.
If your most important feature is absoulute amplitude, you're better off with calculating the RMS (root mean square) level of our input signal.
If pitch is important, you're better off with calculationg the ACF (auto-correlation function) of your input signal.
Don't focus on the leftmost peaks, these come from the high frequency components in your input and tend to vary as much as the noise floor.
Windows
For a high quality analyis it is importnat to apply a window to the input data before applying the FFT. This reduces the infulens of the "jump" between the end of your input vector ant the beginning of your input vector, because the FFT considers the input as a single cycle.
There are several popular windows which mark different choices of an unavoidable trade-off: Precision of a single peak vs. level of sidelobes:
You chose a "rectangular window" (equivalent to no window at all, just start/stop your measurement). This gives excellent precission of your peaks which now have a width of just one sample. Your sidelobes (the small peaks left and right of your main peaks) are at -21dB, very tolerable given your input data. In your case this is an excellent choice.
A Hanning window is a single cosine wave. It makes your peaks slightly broader but reduces side-lobe levels.
The Hammimg-Window (cosine-wave, slightly raised above 0.0) produces even broader peaks, but supresses side-lobes by -42 dB. This is a good choice if you expect further weak (but important) components between your main peaks or generally if you have complicated signals like speech, music and so on.
Edit: Scaling
Correct scaling of a spectrum is a complicated thing, because the values of the FFT lines depend on may things like sampling rate, lenght of FFT, window, and even implementation details of the FFT algorithm (there exist several different accepted conventions).
After all, the FFT should show the underlying conservation of energy. The RMS of the input signal should be the same as the RMS (Energy) of the spectrum.
On the other hand: if used for classification it is enough to maintain relative amplitudes. As long as the paramaters mentioned above do not change, the result can be used for classification without further scaling.
I understand Knn has a problem knows a "curse of dimensionality" when dealing with high dimension data and it justification is that it includes all features while calculating distance i.e. Euclidean distance where non important feature act as a noise and bias the results however i don't understand a few things
1) How cosine distance metric will be effected by this curse of dimensionality problem i.e. we define cosine distance as cosDistance = 1- cosSimilarity where cosSimilarity is favourable for high dimension data so how cosine distance may be effected by curse of dimensionality problem ?
2) Can we assign any weights to features in weka or can i apply feature selection locally to KNN ? Local to knn means i write my own class of K-NN where in classification i first convert training instance to lower dimension and then calculate test instance neighbors ?
Cosine does not fundamentally differ from Euclidean distance.
In fact it is trivial to show that on normalized data with Euclidean length 1, Cosine and Euclidean distance are the same. In other words, Cosine is computing the Euclidean distance on L2 normalized vectors...
Thus, cosine is not more robust to the curse of dimensionality than Euclidean distance. However, cosine is popular with e.g. text data that has a high apparent dimensionality - often thousands of dimensions - but the intrinsic dimensionality must be much lower. Plus, it's mostly used for ranking; the actual distance value is ignored.
I've been exploring and learning about KD Trees for KNN (K Nearest Neighbors problem)
when would the search not work? or would be worth or not improve the naive search.
are there any drawbacks of this approach?
K-d trees don't work too well in high dimensions (where you have to visit lots and lots of tree branches). One rule of thumb is that if your data dimensionality is k, a k-d tree is only going to be any good if you have many more than 2^k data points.
In high dimensions, you'll generally want to switch to approximate nearest-neighbor searches instead. If you haven't run across it already, FLANN ( github ) is a very useful library for this (with C, C++, python, and matlab APIs); it has good implementations of k-d trees, brute-force search, and several approximate techniques, and it helps you automatically tune their parameters and switch between them easily.
It depends on your distance function.
You can't use k-d-trees with arbitrary distance functions. Minkowski norms should be fine though. But in a lot of applications, you will want to use more advanced distance functions.
Plus, with increasing dimensionality, k-d-trees work much less good.
The reason is simple: k-d-trees avoid looking at points where the one-dimensional distance to the boundary is already larger than the desired threshold, i.e. where for Euclidean distances (where z is the nearest border, y the closes known point):
(x_j - z_j) <=> sqrt(sum_i((x_i - y_i)^2))
equivalently, but cheaper:
(x_j - z_j)^2 <=> sum_i((x_i - y_i)^2)
You can imagine that the chance of this pruning rule holding decrease drastically with the number of dimensions. If you have 100 dimensions, there is next to no chance that a single dimensions squared difference will be larger than the sum of squared differences.
Time complexity for knn :O(k * lg(n))
where k is k-nearest neighbours and lg(n) is kd-tree height
kd-trees will not work well if the dimensions of the data set is high because of such huge space.
lets consider you have many points around the origin ,for simplicity consider in 2-D
If you want to find k-nearest neighbours for any point ,then you have to search along 4 axes because all points are closer to each other which results in backtracking to other axis in kd-tree,
So for a 3-dimensional space we have to search along 8 directions
To generalize for n -dimensional it is 2^k
So the time-complexity becomes O(2^k * lg(n))
I have implemented k-means clustering for determining the clusters in 300 objects. Each of my object
has about 30 dimensions. The distance is calculated using the Euclidean metric.
I need to know
How would I determine if my algorithms works correctly? I can't have a graph which will
give some idea about the correctness of my algorithm.
Is Euclidean distance the correct method for calculating distances? What if I have 100 dimensions
instead of 30 ?
The two questions in the OP are separate topics (i.e., no overlap in the answers), so I'll try to answer them one at a time staring with item 1 on the list.
How would I determine if my [clustering] algorithms works correctly?
k-means, like other unsupervised ML techniques, lacks a good selection of diagnostic tests to answer questions like "are the cluster assignments returned by k-means more meaningful for k=3 or k=5?"
Still, there is one widely accepted test that yields intuitive results and that is straightforward to apply. This diagnostic metric is just this ratio:
inter-centroidal separation / intra-cluster variance
As the value of this ratio increase, the quality of your clustering result increases.
This is intuitive. The first of these metrics is just how far apart is each cluster from the others (measured according to the cluster centers)?
But inter-centroidal separation alone doesn't tell the whole story, because two clustering algorithms could return results having the same inter-centroidal separation though one is clearly better, because the clusters are "tighter" (i.e., smaller radii); in other words, the cluster edges have more separation. The second metric--intra-cluster variance--accounts for this. This is just the mean variance, calculated per cluster.
In sum, the ratio of inter-centroidal separation to intra-cluster variance is a quick, consistent, and reliable technique for comparing results from different clustering algorithms, or to compare the results from the same algorithm run under different variable parameters--e.g., number of iterations, choice of distance metric, number of centroids (value of k).
The desired result is tight (small) clusters, each one far away from the others.
The calculation is simple:
For inter-centroidal separation:
calculate the pair-wise distance between cluster centers; then
calculate the median of those distances.
For intra-cluster variance:
for each cluster, calculate the distance of every data point in a given cluster from
its cluster center; next
(for each cluster) calculate the variance of the sequence of distances from the step above; then
average these variance values.
That's my answer to the first question. Here's the second question:
Is Euclidean distance the correct method for calculating distances? What if I have 100 dimensions instead of 30 ?
First, the easy question--is Euclidean distance a valid metric as dimensions/features increase?
Euclidean distance is perfectly scalable--works for two dimensions or two thousand. For any pair of data points:
subtract their feature vectors element-wise,
square each item in that result vector,
sum that result,
take the square root of that scalar.
Nowhere in this sequence of calculations is scale implicated.
But whether Euclidean distance is the appropriate similarity metric for your problem, depends on your data. For instance, is it purely numeric (continuous)? Or does it have discrete (categorical) variables as well (e.g., gender? M/F) If one of your dimensions is "current location" and of the 200 users, 100 have the value "San Francisco" and the other 100 have "Boston", you can't really say that, on average, your users are from somewhere in Kansas, but that's sort of what Euclidean distance would do.
In any event, since we don't know anything about it, i'll just give you a simple flow diagram so that you can apply it to your data and identify an appropriate similarity metric.
To identify an appropriate similarity metric given your data:
Euclidean distance is good when dimensions are comparable and on the same scale. If one dimension represents length and another - weight of item - euclidean should be replaced with weighted.
Make it in 2d and show the picture - this is good option to see visually if it works.
Or you may use some sanity check - like to find cluster centers and see that all items in the cluster aren't too away of it.
Can't you just try sum |xi - yi| instead if (xi - yi)^2
in your code, and see if it makes much difference ?
I can't have a graph which will give some idea about the correctness of my algorithm.
A couple of possibilities:
look at some points midway between 2 clusters in detail
vary k a bit, see what happens (what is your k ?)
use
PCA
to map 30d down to 2d; see the plots under
calculating-the-percentage-of-variance-measure-for-k-means,
also SO questions/tagged/pca
By the way, scipy.spatial.cKDTree
can easily give you say 3 nearest neighbors of each point,
in p=2 (Euclidean) or p=1 (Manhattan, L1), to look at.
It's fast up to ~ 20d, and with early cutoff works even in 128d.
Added: I like Cosine distance in high dimensions; see euclidean-distance-is-usually-not-good-for-sparse-data for why.
Euclidean distance is the intuitive and "normal" distance between continuous variable. It can be inappropriate if too noisy or if data has a non-gaussian distribution.
You might want to try the Manhattan distance (or cityblock) which is robust to that (bear in mind that robustness always comes at a cost : a bit of the information is lost, in this case).
There are many further distance metrics for specific problems (for example Bray-Curtis distance for count data). You might want to try some of the distances implemented in pdist from python module scipy.spatial.distance.