I'm working in a space which has 8 dimensions (i.e. 8 features). I have plotted the data points in 2D by applying PCA as well as TSNE. Now I would like also to draw the borderlines of the classifiers I use as shown here. By the way, I'm using different classifiers (SVM, GNB, Logistic Regression).
This means that I have the different 8-dimensional points which I plot in 2D using PCA or TSNE. On top of this plot I would like to plot the different classification regions as shown in the link above.
Of course the classification boundaries/regions are also 8-dimensional. How can I turn the classification boundaries/regions into 2D matching my 2D data points?
Interesting question here, I once wondered it.
It can be answered several way, including more or less details depending whether you want to fully understand or to apply the method.
As you don't a lot of detail but you included a sklearn link, I will first answer on a technical point of view: "How can you do it with sklearn?"
You have a function for this: transform(X, y=None) which will apply the PCA projection (yes, PCA is a projection for high dimensional space to a lower one).
So you basically just need to give transform(your_boundaries) to apply it.
In term of pseudo code this would give:
pca = PCA(n_component=2).fit(data)
2dboundaries = pca.transform(boundaries)
Et voilà!
Do not hesitate to give more details or ask question. I could add some specific development if it is relevant.
Hope it helps
pltrdy
Related
I would like to know if in Python, and more precisely, in lmfit library, there is an option for fitting data by parts ? I would like to fit data defined in different ranges and then obtain a unique fit.
Thank you
Without a more concrete example, it is hard to give a concrete answer. But, if I understand your question correctly, you are looking to do a fit to one specific region of your data, then a fit (probably with a different functional form) to another region of your data, and then perhaps combine the multiple regions to get a final fit.
If that is correct, then yes, this can be done with lmfit (and probably with other libraries as well). Let's say you want to fit data that is sort of peak like with an exponential decaying background. First, isolate a region around that peak (it doesn't have to be perfect) and fit a peak (say, Gaussian to that). Then fit an exponential decay to all the data except the peak area. (Aside: numpy.where can be very useful in identifying the regions). Finally, combine the two and fit the whole curve to peak + background.
If that is too vague and doesn't point you in the right direction, please make the question more specific.
Let's say price of houses(target variable) can be easily plotted against area of houses(predictor variables) and we can see the data plotted and draw a best fit line through the data.
However, consider if we have predictor variables as ( size, no.of bedrooms,locality,no.of floors ) etc. How am I gonna plot all these against the
target variable and visualize them on a 2-D figure?
The computation shouldn't be an issue (the math works regardless of dimensionality), but the plotting definitely gets tricky. PCA can be hard to interpret and forcing orthogonality might not be appropriate here. I'd check out some of the advice provided here: https://stats.stackexchange.com/questions/73320/how-to-visualize-a-fitted-multiple-regression-model
Fundamentally, it depends on what you are trying to communicate. Goodness of fit? Maybe throw together multiple plots of residuals.
If you truly want a 2D figure, that's certainly not easy. One possible approach would be to reduce the dimensionality of your data to 2 using something like Principal Component Analysis. Then you can plot it in two dimensions again. Reducing to 3 dimensions instead of 2 might also still work, humans can understand 3D plots drawn on a 2D screen fairly well.
You don't normally need to do linear regression by hand though, so you don't need a 2D drawing of your data either. You can just let your computer compute the linear regression, and that works perfectly fine with way more than 2 or 3 dimensions.
I'm supposed to be doing a kmeans clustering implementation with some data. The example I looked at from http://glowingpython.blogspot.com/2012/04/k-means-clustering-with-scipy.html shows their test data in 2 columns... however, the data I'm given is 68 subjects with 78 features (so 68x78 matrix). How am I supposed to create an appropriate input for this?
I've basically just tried inputting the matrix anyway, but it doesn't seem to do what I want... and I don't know why it would. I'm pretty confused as to what to do.
data = np.rot90(data)
centroids,_ = kmeans(data,2)
# assign each sample to a cluster
idx,_ = vq(data,centroids)
# some plotting using numpy's logical indexing
plot(data[idx==0,0],data[idx==0,1],'ob',
data[idx==1,0],data[idx==1,1],'or')
plot(centroids[:,0],centroids[:,1],'sg',markersize=8)
show()
I honestly don't know what kind of code to show you.. the data format I told you was already described. Otherwise, it's the same as the tutorial I linked.
Your visualization only uses the first two dimensions.
That is why these points appear to be "incorrect" - they are closer in a different dimension.
Have a look at the next two dimensions:
plot(data[idx==0,2],data[idx==0,3],'ob',
data[idx==1,2],data[idx==1,3],'or')
plot(centroids[:,2],centroids[:,3],'sg',markersize=8)
show()
... repeat for all remaining of oyur 78 dimensions...
At this many features, (squared) Euclidean distance gets meaningless, and k-means results tend to become as good as random convex partitions.
To get a more representative view, consider using MDS to project the data into 2d for visualization. It should work reasonably fast with just 68 subjects.
Please include visualizations in your questions. We don't have your data.
I have some points that I need to classify. Given the collection of these points, I need to say which other (known) distribution they match best. For example, given the points in the top left distribution, my algorithm would have to say whether they are a better match to the 2nd, 3rd, or 4th distribution. (Here the bottom-left would be correct due to the similar orientations)
I have some background in Machine Learning, but I am no expert. I was thinking of using Gaussian Mixture Models, or perhaps Hidden Markov Models (as I have previously classified signatures with these- similar problem).
I would appreciate any help as to which approach to use for this problem. As background information, I am working with OpenCV and Python, so I would most likely not have to implement the chosen algorithm from scratch, I just want a pointer to know which algorithms would be applicable to this problem.
Disclaimer: I originally wanted to post this on the Mathematics section of StackExchange, but I lacked the necessary reputation to post images. I felt that my point could not be made clear without showing some images, so I posted it here instead. I believe that it is still relevant to Computer Vision and Machine Learning, as it will eventually be used for object identification.
EDIT:
I read and considered some of the answers given below, and would now like to add some new information. My main reason for not wanting to model these distributions as a single Gaussian is that eventually I will also have to be able to discriminate between distributions. That is, there might be two different and separate distributions representing two different objects, and then my algorithm should be aware that only one of the two distributions represents the object that we are interested in.
I think this depends on where exactly the data comes from and what sort of assumptions you would like to make as to its distribution. The points above can easily be drawn even from a single Gaussian distribution, in which case the estimation of parameters for each one and then the selection of the closest match are pretty simple.
Alternatively you could go for the discriminative option, i.e. calculate whatever statistics you think may be helpful in determining the class a set of points belongs to and perform classification using SVM or something similar. This can be viewed as embedding these samples (sets of 2d points) in a higher-dimensional space to get a single vector.
Also, if the data is actually as simple as in this example, you could just do the principle component analysis and match by the first eigenvector.
You should just fit the distributions to the data, determine the chi^2 deviation for each one, look at F-Test. See for instance these notes on model fitting etc
You might want to consider also non-parametric techniques (e.g. multivariate kernel density estimation on each of your new data set) in order to compare the statistics or distances of the estimated distributions. In Python stats.kde is an implementation in SciPy.Stats.
Can someone please tell me if there is a good (easy) way to visualize high dimensional data? My data is currently 21 dimensions but I would like to see how whether it is dense or sparse. Are there techniques to achieve this?
Parallel coordinates are a popular method for visualizing high-dimensional data.
What kind of visualization is best for your data in particular will depend on its characteristics-- how correlated are the different dimensions?
Principal component analysis could be helpful if the dimensions are correlated.
The buzzword I would search for is multidimensional scaling. It is a technique to develop a projection from the high dimensional space to a lower space (2 or 3 dimensional) in such a way that points which are close in the full space will be close in the projection.
It is often used for visualising the output of clustering algorithms (i.e. if your clusters are compact in the MDS projection there is a good chance they are also in the full space).
Edit: This wouldn't necessarily help with determining if the data is dense or sparse, because you lose the scale in the projection, but it would show whether it is uniform or clumpy (perhaps thats what you mean).
Not sure what kind of patterns you would like to see from the data. t-SNE and its faster variant Barnes-Hut-SNE do a very good job in visualizing groups of related concepts for high-dimensional data. It is available through R.
There is a short tutorial on using it against high-dimensional data with about 300 dimensions.
http://www.codeproject.com/Tips/788739/Visualizing-High-Dimensional-Vector-using-T-SNE-wi
I was looking for ways to visualize high dimensional data and found this t-SNE technique that has been used effectively. Might help others as well.
Take a look at http://www.ggobi.org (tours, parallel coordinates, scatterplot matrices) can be used for real-valued variables. Also http://cranvas.org for more recent. The tourr package in R.
Try using http://hypertools.readthedocs.io/en/latest/.
HyperTools is a library for visualizing and manipulating high-dimensional data in Python.
Star Schema.
http://en.wikipedia.org/wiki/Star_schema
Works well for high-dimensional data.
If the cardinality of your fact table is close to the product of your dimension sizes, you have dense data.
If the cardinality of your fact table is smaller than the product of your dimension sizes, you have sparse data.
In the middle you have a judgement call.
The curios.IT data exploration software is designed for the visualization of high dimensional data: data is shown as a collection of 3D objects (one for each data group) which can show up to 13 variables at the same time. The relationships between data variables and visual features are much easier to remember than with other techniques (like parallel coordinates).