Develop Reference

Approximate maximum of an unknown curve

I have a data set that looks like this:
I used the scipy.signal.find_peaks function to determine the peaks of the data set, and it works out fine enough, but since this function determines the local maxima of the data, it is not neglecting the noise in the data which causes overshoot. Therefore what I'm determining isn't actually the location of the most likely maxima, but rather the location of an 'outlier'.
Is there another, more exact way to approximate the local maxima?

I'm not sure that you can consider those points to be outliers so easily, as they look to be close to the place I would expect them to be. But if you don't think they are a valid approximation let me tell you three other ways you can use.
First option
I would construct a physical model of these peaks (a mathematical formula) and do a fitting analysis around the peaks. You can for instance, suppose that the shape of the plot is the sum of some background model (maybe constant or maybe more complicated) plus some Gaussian peaks (or Lorentzian).
This is what we usually do in physics. Of course it will be more accurate taking knowledge from the underlying processes, which I don't have.
Having a good model, this way is definitely better as taking the maximum values, as even if they are not outliers, they still have errors which you want to reduce.
Second option
But if you want a easier way, just a rough estimation and you already found the location of the three peaks programmatically, you can make the average of a few points around the maximum. If you do it so, the function np.where or np.argwhere tend to be useful for this kind of things.
Third option
The easiest option is taking the value by hand. It could sound unacceptable for academic proposes and it probably is. Even worst, it is not a programmatic way, and this is SO. But at the end, it depends on why and for what you need those values and on the confidence interval you need for your measurement.

Is it possible to increase the number of centroids in KMeans during fitting?

I am attempting to use MiniBatchKMeans to stream NLP data in and cluster it, but have no way of determining how many clusters I need. What I would like to do is periodically take the silhouette score and if it drops below a certain threshold, increase the number of centroids. But as far as I can tell, n_clusters is set when you initialize the clusterer and can't be changed without restarting. Am I wrong here? Is there another way to approach this problem that would avoid this issue?

It is not a good idea to do this during optimization, because it changes the optimization procedure substantially. It will essentially reset the whole optimization. There are strategies such as bisecting k-means that try to learn the value of k during clustering, but they are a bit more tricky than increasing k by one - they decide upon one particular cluster to split, and try to choose good initial centroids for this cluster to keep things somewhat stable.
Furthermore, increasing k will not necessarily improve Silhouette. It will trivially improve SSQ, so you cannot use SSQ as a heuristic for choosing k, either.
Last but not least, computing the Silhouette is O(n^2). It is too expensive to run often. If you have large enough amount of data to require MiniBatchKMeans (which really is only for massive data), then you clearly cannot afford to compute Silhouette at all.

Machine learning : find the closest results to a queried vector

I have thousands of vectors of about 20 features each.
Given one query vector, and a set of potential matches, I would like to be able to select the best N matches.
I have spent a couple of days trying out regression (using SVM), training my model with a data set I have created myself : each vector is the concatenation of the query vector and a result vector, and I give a score (subjectively evaluated) between 0 and 1, 0 for perfect match, 1 for worst match.
I haven't had great results, and I believe one reason could be that it is very hard to subjectively assign these scores. What would be easier on the other hand is to subjectively rank results (score being an unknown function):
score(query, resultA) > score(query, resultB) > score(query, resultC)
So I believe this is more a problem of Learning to rank and I have found various links for Python:
http://fa.bianp.net/blog/2012/learning-to-rank-with-scikit-learn-the-pairwise-transform/
https://gist.github.com/agramfort/2071994
...
but I haven't been able to understand how it works really. I am really confused with all the terminology, pairwise ranking, etc ... (note that I know nothing about machine learning hence my feeling of being a bit lost), etc ... so I don't understand how to apply this to my problem.
Could someone please help me clarify things, point me to the exact category of problem I am trying to solve, and even better how I could implement this in Python (scikit-learn) ?

It seems to me that what you are trying to do is to simply compute the distances between the query and the rest of your data, then return the closest N vectors to your query. This is a search problem.
There is no ordering, you simply measure the distance between your query and "thousands of vectors". Finally, you sort the distances and take the smallest N values. These correspond to the most similar N vectors to your query.
For increased efficiency at making comparisons, you can use KD-Trees or other efficient search structures: http://scikit-learn.org/stable/modules/neighbors.html#kd-tree
Then, take a look at the Wikipedia page on Lp space. Before picking an appropriate metric, you need to think about the data and its representation:
What kind of data are you working with? Where does it come from and what does it represent? Is the feature space comprised of only real numbers or does it contain binary values, categorical values or all of them? Wiki for homogeneous vs heterogeneous data.
For a real valued feature space, the Euclidean distance (L2) is usually the choice metric used, with 20 features you should be fine. Start with this one. Otherwise you might have to think about cityblock distance (L1) or other metrics such as Pearson's correlation, cosine distance, etc.
You might have to do some engineering on the data before you can do anything else.
Are the features on the same scale? e.g. x1 = [0,1], x2 = [0, 100]
If not, then try scaling your features. This is usually a matter of trial and error since some features might be noisy in which case scaling might not help.
To explain this, think about a data set with two features: height and weight. If height is in centimeters (10^3) and weight is in kilograms (10^1), then you should aim to convert the cm to meters so both features weigh equally. This is generally a good idea for feature spaces with a wide range of values, meaning you have a large sample of values for both features. You'd ideally like to have all your features normally distributed, with only a bit of noise - see central limit theorem.
Are all of the features relevant?
If you are working with real valued data, you can use Principal Component Analysis (PCA) to rank the features and keep only the relevant ones.
Otherwise, you can try feature selection http://scikit-learn.org/stable/modules/classes.html#module-sklearn.feature_selection
Reducing the dimension of the space increases performance, although it is not critical in your case.
If your data consists of continuous, categorical and binary values, then aim to scale or standardize the data. Use your knowledge about the data to come up with an appropriate representation. This is the bulk of the work and is more or less a black art. Trial and error.
As a side note, metric based methods such as knn and kmeans simply store data. Learning begins where memory ends.

Utilising Genetic algorithm to overcome different size datasets in model

SO I realise the question I am asking here is large and complex.
A potential solution to variences in sizes of
In all of my searching through statistical forums and posts I haven't come across a scientifically sound method of taking into account the type of data that I am encountering,
but I have thought up a (novel?) potential solutions to account perfectly (in my mind) for large and small datasets within the same model.
The proposed method involves using a genetic algorithm to alter two numbers defining a relationship between the size of the dataset making up an implied strike rate and the
percentage of the implied strike to be used, with the target of the model to maximise the homology of the number 1 in two columns of the following csv. (ultra simplified
but hopefully demonstrates the principle)
Example data
Date,PupilName,Unique class,Achieved rank,x,y,x/y,Average xy
12/12/2012,PupilName1,UniqueClass1,1,3000,9610,0.312174818,0.08527
12/12/2012,PupilName2,UniqueClass1,2,300,961,0.312174818,0.08527
12/12/2012,PupilName3,UniqueClass1,3,1,3,0.333333333,0.08527
13/12/2012,PupilName1,UniqueClass2,1,2,3,0.666666667,0.08527
13/12/2012,PupilName2,UniqueClass2,2,0,1,0,0.08527
13/12/2012,PupilName3,UniqueClass2,3,0,5,0,0.08527
13/12/2012,PupilName4,UniqueClass2,4,0,2,0,0.08527
13/12/2012,PupilName5,UniqueClass2,5,0,17,0,0.08527
14/12/2012,PupilName1,UniqueClass3,1,1,2,0.5,0.08527
14/12/2012,PupilName2,UniqueClass3,2,0,1,0,0.08527
14/12/2012,PupilName3,UniqueClass3,3,0,5,0,0.08527
14/12/2012,PupilName4,UniqueClass3,4,0,6,0,0.08527
14/12/2012,PupilName5,UniqueClass3,5,0,12,0,0.08527
15/12/2012,PupilName1,UniqueClass4,1,0,0,0,0.08527
15/12/2012,PupilName2,UniqueClass4,2,1,25,0.04,0.08527
15/12/2012,PupilName3,UniqueClass4,3,1,29,0.034482759,0.08527
15/12/2012,PupilName4,UniqueClass4,4,1,38,0.026315789,0.08527
16/12/2012,PupilName1,UniqueClass5,1,12,24,0.5,0.08527
16/12/2012,PupilName2,UniqueClass5,2,1,2,0.5,0.08527
16/12/2012,PupilName3,UniqueClass5,3,13,59,0.220338983,0.08527
16/12/2012,PupilName4,UniqueClass5,4,28,359,0.077994429,0.08527
16/12/2012,PupilName5,UniqueClass5,5,0,0,0,0.08527
17/12/2012,PupilName1,UniqueClass6,1,0,0,0,0.08527
17/12/2012,PupilName2,UniqueClass6,2,2,200,0.01,0.08527
17/12/2012,PupilName3,UniqueClass6,3,2,254,0.007874016,0.08527
17/12/2012,PupilName4,UniqueClass6,4,2,278,0.007194245,0.08527
17/12/2012,PupilName5,UniqueClass6,5,1,279,0.003584229,0.08527
So I have created a tiny model dataset, which contains some good examples of where my current methods fall short and how I feel a genetic algorithm can be used to fix this. If we look in the dataset above it contains 6 unique classes the ultimate objective of the algorithm is to create as high as possible correspondence between a rank of an adjusted x/y and the achieved rank in column 3 (zero based referencing.) In uniqueclass1 we have two identical x/y values, now these are comparatively large x/y values if you compare with the average (note the average isn't calculated from this dataset) but it would be common sense to expect that the 3000/9610 is more significant and therefore more likely to have an achieved rank of 1 than the 300/961. So what I want to do is make an adjusted x/y to overcome these differences in dataset sizes using a logarithmic growth relationship defined by the equation:
adjusted xy = ((1-exp(-y*α)) * x/y)) + ((1-(1-exp(-y*α)))*Average xy)
Where α is the only dynamic number
If I can explain my logic a little and open myself up to (hopefully) constructive criticsm. This graph below shows is an exponential growth relationship between size of the data set and the % of x/y contributing to the adjusted x/y. Essentially what the above equation says is as the dataset gets larger the percentage of the original x/y used in the adjusted x/y gets larger. Whatever percentage is left is made up by the average xy. Could hypothetically be 75% x/y and 25% average xy for 300/961 and 95%/5% for 3000/9610 creating an adjusted x/y which clearly demonstrates
For help with understanding the lowering of α would produce the following relationship where by a larger dataset would be requred to achieve the same "% of xy contributed"
Conversly increasing α would produce the following relationship where by a smaller dataset would be requred to achieve the same "% of xy contributed"
So I have explained my logic. I am also open to code snippets to help me overcome the problem. I have plans to make a multitude of genetic/evolutionary algorithms in the future and could really use a working example to pick apart and play with in order to help my understanding of how to utilise such abilities of python. If additional detail is required or further clarification about the problem or methods please do ask, I really want to be able to solve this problem and future problems of this nature.
So after much discussion about the methods available to overcome the problem presented here I have come to the conclusion that he best method would be a genetic algorithm to iterate α in order to maximise the homology/correspondance between a rank of an adjusted x/y and the achieved rank in column 3. It would be greatly greatly appreciated if anyone be able to help in that department?
So to clarify, this post is no longer a discussion about methodology
I am hoping someone can help me produce a genetic algorithm to maximise the homology between the results of the equation
adjusted xy = ((1-exp(-y*α)) * x/y)) + ((1-(1-exp(-y*α)))*Average xy)
Where adjusted xy applies to each row of the csv. Maximising homology could be achieved by minimising the difference between the rank of the adjusted xy (where the rank is by each Unique class only) and Achieved rank.
Minimising this value would maximise the homology and essentially solve the problem presented to me of different size datasets. If any more information is required please ask, I check this post about 20 times a day at the moment so should reply rather promptly. Many thanks SMNALLY.

The problem you are facing sounds to me like "Bias Variance Dilemna" from a general point of view. In a nutshell, a more precise model favours variance (sensitivity to change in a single training set), a more general model favours bias (model works for many training sets)
May I suggest not to focus on GA but look at Instance Base Learning and advanced regression techniques. The Andrew moore page at CMU is a good entry point.
And particularly those slides.
[EDIT]
After a second reading, here is my second understanding:
You have a set of example data with two related attributes X and Y.
You do not want X/Y to dominate when Y is small, (considered as less representative).
As a consequence you want to "weigth" the examples with a adapted value adjusted_xy .
You want adjusted_xy to be related to a third attribute R (rank). Related such as,per class, adjusted_xy is sorted like R.
To do so you suggest to put it as an optimization problem, searching for PARAMS of a given function F(X,Y,PARAMS)= adjusted_xy .
With the constraint that D=Distance( achieved rank for this class, rank of adjusted_xy for this class ) is minimal.
Your question, at least for me, is in the field of attribute selection/attribute adaptation. (I guess the data set will later be used for supervised learning ).
One problem that I see in your approach (if well understood) is that, at the end, rank will be highly related to adjusted_xy which will bring therefore no interesting supplementary information.
Once this said, I think you surely know how GA works . You have to
define the content of the chromosome : this appears to be your alpha parameter.
define an appropriate fitness function
The fitness function for one individual can be a sum of distances over all examples of the dataset.
As you are dealing with real values , other metaheuristics such as Evolution Strategies (ES) or Simulated Anealing may be more adapted than GA.
As solving optimization problems is cpu intensive, you might eventually consider C or Java instead of Python. (as fitness at least will be interpreted and thus cost a lot).
Alternatively I would look at using Y as a weight to some supervised learning algorithm (if supervised learning is the target).

Let's start by the problem: You consider the fact that some features lead to some of your classes a 'strike'. You are taking a subset of your data and try to establish a rule for the strikes. You do establish one but then you notice that the accuracy of your rule depends on the volume of the dataset that was used to establish the 'strike' rate anyway. You are also commenting on the effect of some samples in biasing your 'strike' estimate.
The immediate answer is that it looks like you have a lot of variation in your data, therefore you will in one way or another need to collect more to account for that variation. (That is, variation that is inherent to the problem).
The fact that in some cases the numbers end up in 'unusable cases' could also be down to outliers. That is, measurements that are 'out of bounds' for a number of reasons and which you would have to find a way to either exclude them or re-adjust them. But this depends a lot on the context of the problem.
'Strike rates' on their own will not help but they are perhaps a step towards the right direction. In any case, you can not compare strike rates if they are coming from samples of different sizes as you have found out too. If your problem is purely to determine the size of your sample so that your results conform to some specific accuracy then i would recommend that you have a look at Statistical Power and how does the sample size affects it. But still, to determine the sample size you need to know a bit more about your data, which brings us back to point #1 about the inherent variation.
Therefore, my attempt to an answer is this: If i have understood your question correctly, you are dealing with a classification problem in which you seek to assign a number of items (patients) to a number of classes (types of cancer) on the evidence of some features (existence of genetic markers, or frequency of their appearance or any other quantity anyway) about these items. But, some features might not exist for all items or, there is a core group of features but there might be some more that do not appear all the time. The question now is, which classifier do you use to achieve this? Logistic regression was mentioned previously and has not helped. Therefore, what i would suggest is going for a Naive Bayesian Classifier. The classifier can be trained with the datasets you have used to derive the 'strike rates' which will provide the a-priori probabilities. When the classifier is 'running' it will be using the features of new data to construct a likelihood that the patient who provided this data should be assigned to each class.
Perhaps the more common example for such a classifier is the spam-email detectors where the likelihood that an email is spam is judged on the existence of specific words in the email (and a suitable training dataset that provides a good starting point of course).
Now, in terms of trying this out practically (and since your post is tagged with python related tags :) ), i would like to recommend Weka. Weka contains a lot of related functionality including bootstrapping that could potentially help you with those differences in the size of the datasets. Although Weka is Java, bindings exist for it in Python too. I would definitely give it a go, the Weka package, book and community are very helpful.

No. Don't use a genetic algorithm.
The bigger the search space of models and parameters, the better your chances of finding a good fit for your data points. But the less this fit will mean. Especially since for some groups your sample sizes are small and therefore the measurements have a high random component to them. This is why, somewhat counterintuitively, it is often actually harder to find a good model for your data after collecting it than before.
You have taken the question to the programmer's lair. This is not the place for it. We solve puzzles.
This is not a puzzle to find the best line through the dots. You are searching for a model that makes sense and brings understanding on the subject matter. A genetic algorithm is very creative at line-through-dot drawing but will bring you little understanding.
Take the problem back where it belongs and ask the statisticians instead.
For a good model should be based on theory behind the data. It'll have to match the points on the right side of the graph, where (if I understand you right) most of the samples are. It'll be able to explain in hard probabilities how likely the deviations on the left are and tell you if they are significant or not.
If you do want to do some programming, I'd suggest you take the simplest linear model, add some random noise, and do a couple simulation runs for a population like your subjects. See if the data looks like the data you're looking at or if it generally 'looks' different, in which case there really is something nonlinear (and possibly interesting) going on on the left.

I once tackled a similar problem (as similar as problems like this ever are), in which there were many classes and high variance in features per data point. I personally used a Random Forest classifier (which I wrote in Java). Since your data is highly variant, and therefore hard to model, you could create multiple forests from different random samples of your large dataset and put a control layer on top to classify data against all the forests, then take the best score. I don't write python, but i found this link
http://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html
which may give you something to play with.

Following Occam's razor, you must select a simpler model for small dataset and may want to switch to a more complex model as your dataset grows.
There are no [good] statistical tests that show you if a given model, in isolation, is a good predictor of your data. Or rather, a test may tell you that given model fitness is N, but you can never tell what the acceptable value of N is.
Thus, build several models and pick one with better tradeoff of predictive power and simplicity using Akaike information criterion. It has useful properties and not too hard to understand. :)
There are other tests of course, but AIC should get you started.
For a simple test, check out p-value

Curve_fit not converging means...?

I need to crossmatch a list of astronomical coordinates with different catalogues, and I want to decide a maximum radius for the crossmatch. This will avoid mismatches between my list and the catalogues.
To do this, I compute the separation between the best match with the catalogue for each object in my list. My initial list is supossed to be the position of a known object, but it could happend that it is not detected in the catalog, and my coordinates may suffer from small offsets.
They way I am computing the maximum radius is by fitting the gaussian kernel density of the separation with a gaussian, and use the center + 3sigmas value. The method works nicely for most of the cases, but when a small subsample of my list has an offset, I have two gaussians instead. In these cases, I will specify the max radius in a different way.
My problem is that when this happens, curve_fit can't normally do the fit with one gaussian. For a scientific publication, I will need to justify the "no fit" in curve_fit, and in which cases the "different way" is used. Could someone give me a hand on what this means in mathematical terms?

There are varying lengths to which you can go justifying this or that fitting ansatz --- which strongly depends on the details of your specific case (eg: why do you expect a gaussian to work in a first place? to what depth you need/want to delve into why exactly a certain fitting procedure fails and what exactly is a fail etc).
If the question is really about the curve_fit and its failure to converge, then show us some code and some input data which demonstrate the problem.
If the question is about how to evaluate the goodness-of-fit, you're best off going back to the library and picking a good book on statistics.
If all you look for is way of justifying why in a certain case a gaussian is not a good fitting ansatz, one way would be to calculate the moments: for a gaussian distribution 1st, 2nd, 3rd and higher moments are related to each other in a very precise way. If you can demonstrate that for your underlying data the relation between moments is very different, it sounds reasonable that these data can't be fit by a gaussian.