Mersenne twister: limitations used in agent based models - seed
I am using the Mersnenne Twister as the engine to generate random numbers in an agent based model: it is fast and has an extremely long period before repeating.
Recently I did a literature review on this, while Colt library Java API recommends the Mersenne twister, I came across two limitations:
the seed should not be 0. Is this something suggested in the Apache Commons Math library ?
based on a cryptography paper, it was mentioned that "if the initial state has too many zeros then the generated sequence may also contain many zeros for more than 10000 generations and if the seeds are chosen systematically such as 0, 20, 30….. the output sequences will be correlated".
Has anyone come across such issues, or is it something fixed and not the case anymore ?
Is there any literature showing the spectral analysis of the Mersenne Twister vs the others like the Linear Congruential Generator?
SFMT has a better characteristic of zero-excess initial state.
A usual tip to get rid of zero-excess initialization of the seed is to use another PRNG (which might have near-equal probability of zeros and ones in the output) to generate the seed itself.
See also a comment on "How to properly seed a mersenne twister RNG?"
Related
Scikit learn models gives weight to random variable? Should I remove features with less importance?
I do some feature selection by removing correlated variables and backwards elimination. However, after all that is done as a test I threw in a random variable, and then trained logistic regression, random forest and XGBoost. All 3 models have the feature importance of the random feature as greater than 0. First, how can that be? Second, all models have it ranked toward the bottom, but it's not the lowest feature. Is this a valid step for another round of feature selection -i.e. remove all those who score below the random feature? The random feature is created with model_data['rand_feat'] = random.randint(100, size=(model_data.shape[0]))
This can happen, What random is the number you sample, but this random sampling can still generate a pattern by chance. I dont know whether you are doing classification or regression but lets consider the simple example of binary classification. We have class 1 and 0 and 1000 data point from each. When you sample a random number for each data point, it can happen that for example a majority of class 1 gets some value higher than 50, whereas majority of class 0 gets a random number smaller than 50. So in the end effect, this might result into some pattern. So I would guess everytime you run your code the random feature importance changes. It is always ranked low because it is very unlikely that a good pattern is generated(e.g all 1s get higher than 50 and all 0s get lower than 50). Finally, yes you should consider to drop the features with low value
I agree with berkay's answer that a random variable can have patterns that are by chance associated to your outcome variable. Secondly, I will neither include random variable in model building nor as my filtering threshold because if random variable has by chance significant or nearly significant association to the outcome it will suppress the expression of important features of original data and you probably end up losing those important features.
In early phase of model development I always include two random variables. For me it is like a 'sanity check' since these are in effect junk variables or junk features. If any of my features are worse in importance than the junk features then that is a warning sign that I need to look more carefully at the worth* of those features or to do some better feature engineering. For example what does theory suggest about the inclusion of those features?
Audio signal source separation with neural network
What I am trying to do is separating the audio sources and extract its pitch from the raw signal. I modeled this process myself, as represented below: Each sources oscillate in normal modes, often makes its component peaks' frequency integer multiplication. It's known as Harmonic. And then resonanced, finally combined linearly. As seen in above, I've got many hints in frequency response pattern of audio signals, but almost no idea how to 'separate' it. I've tried countless of my own models. This is one of them: FFT the PCM Get peak frequency bins and amplitudes. Calculate pitch candidate frequency bins. For each pitch candidates, using recurrent neural network analyze all the peaks and find appropriate combination of peaks. Separate analyzed pitch candidates. Unfortunately, I've got non of them successfully separates the signal until now. I want any of advices to solve these kind of problem. Especially in modeling of source separation like my one above.
Because no one has really attempted to answer this, and because you've marked it with the neural-network tag, I'm going to address the suitability of a neural network to this kind of problem. As the question was somewhat non-technical, this answer will also be "high level". Neural networks require some sort of sample set from which to learn. In order to "teach" a neural net to solve this problem you would essentially need to have a working set of known solutions to work from. Do you have this? If so, read on. If not, a neural is probably not what you are seeking. You stated that you have "many hints" but no real solution. This leads me to believe you probably don't have sample sets. If you can get them, great, otherwise you might be out of luck. Supposing now that you have a sample set of Raw Signal samples and corresponding Source 1 and Source 2 outputs... Well, now you're going to need a method for deciding on a topology. Assuming you don't know a lot about how neural nets work (and don't want to), and assuming you also don't know the exact degree of complexity of the problem, I would probably recommend the open source NEAT package to get you started. I am not affiliated in any way with this project, but I have used it, and it allows you to (relatively) intelligently evolve neural network topologies to fit the problem. Now, in terms of how a neural net would solve this specific problem. The first thing that comes to mind is that all audio signals are essentially time-series. That is to say, the information they convey is somehow dependent and related to the data at previous timesteps (e.g. the detection of some waveform cannot be done from a single time-point; it requires information about previous timesteps as well). Again, there's a million ways of solving this problem, but since I'm already recommending NEAT I'd probably suggest you take a look at the C++ NEAT Time Series mod. If you're going down this route, you'll probably be wanting to use some sort of sliding window to provide information about the recent past at each time step. For a quick and dirty intro to sliding windows, check out this SO question: Time Series Prediction via Neural Networks The size of the sliding window can be important, especially if you're not using recurrent neural nets. Recurrent networks allow neural nets to remember previous time steps (at the cost of performance - NEAT is already recurrent so that choice is made for you here). You will probably want the sliding window length (ie. the number of timesteps in the past provided at every time step) to be roughly equal to your conservative guess of the largest number of previous timesteps required to gain enough information to split your waveform. I'd say this is probably enough information to get you started. When it comes to deciding how to provide the neural net with the data, you'll first want to normalise the input signals (consider a sigmoid function) and experiment with different transfer functions (sigmoid would probably be a good starting point). I would imagine you'll want to have 2 output neurons, providing normalised amplitude (which you would denormalise via the inverse of the sigmoid function) as the output representing Source 1 and Source 2 respectively. For the fitness value (the way you judge the ability of each tested network to solve the problem) would be something along the lines of the negative of the RMS error of the output signal against the actual known signal (ie. tested against the samples I was referring to earlier that you will need to procure). Suffice to say, this will not be a trivial operation, but it could work if you have enough samples to train the network against. What is a good number of samples? Well as a rule of thumb it's roughly a number that is large enough such that a simple polynomial function of order N (where N is the number of neurons in the netural network you require to solve the problem) cannot fit all of the samples accurately. This is basically to ensure you are not simply overfitting the problem, which is a serious issue with neural networks. I hope this has been helpful! Best of luck. Additional note: your work to date wouldn't have been in vain if you go down this route. A neural network is likely to benefit from additional "help" in the form of FFTs and other signal modelling "inputs", so you might want to consider taking the signal processing you have already done, organising into an analog, continuous representation and feeding it as an input alongside the input signal.
I need a function that describes a set of sequences of zeros and ones?
I have multiple sets with a variable number of sequences. Each sequence is made of 64 numbers that are either 0 or 1 like so: Set A sequence 1: 0,0,0,0,0,0,1,1,0,0,0,0,1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0 sequence 2: 0,0,0,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0 sequence 3: 0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0 ... Set B sequence1: 0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1 sequence2: 0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,0 ... I would like to find a mathematical function that describes all possible sequences in the set, maybe even predict more and that does not contain the sequences in the other sets. I need this because I am trying to recognize different gestures in a mobile app based on the cells in a grid that have been touched (1 touch/ 0 no touch). The sets represent each gesture and the sequences a limited sample of variations in each gesture. Ideally the function describing the sequences in a set would allow me to test user touches against it to determine which set/gesture is part of. I searched for a solution, either using Excel or Mathematica, but being very ignorant about both and mathematics in general I am looking for the direction of an expert. Suggestions for basic documentation on the subject is also welcome.
It looks as if you are trying to treat what is essentially 2D data in 1D. For example, let s1 represent the first sequence in set A in your question. Then the command ArrayPlot[Partition[s1, 8]] produces this picture: The other sequences in the same set produce similar plots. One of the sequences from the second set produces, in response to the same operations, the picture: I don't know what sort of mathematical function you would like to define to describe these pictures, but I'm not sure that you need to if your objective is to recognise user gestures. You could do something much simpler, such as calculate the 'average' picture for each of your gestures. One way to do this would be to calculate the average value for each of the 64 pixels in each of the pictures. Perhaps there are 6 sequences in your set A describing gesture A. Sum the sequences element-by-element. You will now have a sequence with values ranging from 0 to 6. Divide each element by 6. Now each element represents a sort of probability that a new gesture, one you are trying to recognise, will touch that pixel. Repeat this for all the sets of sequences representing your set of gestures. To recognise a user gesture, simply compute the difference between the sequence representing the gesture and each of the sequences representing the 'average' gestures. The smallest (absolute) difference will direct you to the gesture the user made. I don't expect that this will be entirely foolproof, it may well result in some user gestures being ambiguous or not recognisable, and you may want to try something more sophisticated. But I think this approach is simple and probably adequate to get you started.
In Mathematica the following expression will enumerate all the possible combinations of {0,1} of length 64.
Tuples[{1, 0}, {64}]
But there are 2^62 or 18446744073709551616 of them, so I'm not sure what use that will be to you.
Maybe you just wanted the unique sequences contained in each set, in that case all you need is the Mathematica Union[] function applied to the set. If you have a the sets grouped together in a list in Mathematica, say mySets, then you can apply the Union operator to every set in the list my using the map operator.
Union/#mySets
If you want to do some type of prediction a little more information might be useful.
Thanks you for the clarifications.
Machine Learning
The task you want to solve falls under the disciplines known by a variety of names, but probably most commonly as Machine Learning or Pattern Recognition and if you know which examples represent the same gestures, your case would be known as supervised learning.
Question: In your case do you know which gesture each example represents ?
You have a series of examples for which you know a label ( the form of gesture it is ) from which you want to train a model and use that model to label an unseen example to one of a finite set of classes. In your case, one of a number of gestures. This is typically known as classification.
Learning Resources
There is a very extensive background of research on this topic, but a popular introduction to the subject is machine learning by Christopher Bishop.
Stanford have a series of machine learning video lectures Standford ML available on the web.
Accuracy
You might want to consider how you will determine the accuracy of your system at predicting the type of gesture for an unseen example. Typically you train the model using some of your examples and then test its performance using examples the model has not seen. The two of the most common methods used to do this are 10 fold Cross Validation or repeated 50/50 holdout. Having a measure of accuracy enables you to compare one method against another to see which is superior.
Have you thought about what level of accuracy you require in your task, is 70% accuracy enough, 85%, 99% or better?
Machine learning methods are typically quite sensitive to the specific type of data you have and the amount of examples you have to train the system with, the more examples, generally the better the performance.
You could try the method suggested above and compare it against a variety of well proven methods, amongst which would be Random Forests, support vector machines and Neural Networks. All of which and many more are available to download in a variety of free toolboxes.
Toolboxes
Mathematica is a wonderful system, is infinitely flexible and my favourite environment, but out of the box it doesn't have a great deal of support for machine learning.
I suspect you will make a great deal of progress more quickly by using a custom toolbox designed for machine learning. Two of the most popular free toolboxes are WEKA and R both support more than 50 different methods for solving your task along with methods for measuring the accuracy of the solutions.
With just a little data reformatting, you can convert your gestures to a simple file format called ARFF, load them into WEKA or R and experiment with dozens of different algorithms to see how each performs on your data. The explorer tool in WEKA is definitely the easiest to use, requiring little more than a few mouse clicks and typing some parameters to get started.
Once you have an idea of how well the established methods perform on your data you have a good starting point to compare a customised approach against should they fail to meet your criteria.
Handwritten Digit Recognition
Your problem is similar to a very well researched machine learning problem known as hand written digit recognition. The methods that work well on this public data set of handwritten digits are likely to work well on your gestures.
Compare and Contrast Monte-Carlo Method and Evolutionary Algorithms
What's the relationship between the Monte-Carlo Method and Evolutionary Algorithms? On the face of it they seem to be unrelated simulation methods used to solve complex problems. Which kinds of problems is each best suited for? Can they solve the same set of problems? What is the relationship between the two (if there is one)?
"Monte Carlo" is, in my experience, a heavily overloaded term. People seem to use it for any technique that uses a random number generator (global optimization, scenario analysis (Google "Excel Monte Carlo simulation"), stochastic integration (the Pi calculation that everybody uses to demonstrate MC). I believe, because you mentioned evolutionary algorithms in your question, that you are talking about Monte Carlo techniques for mathematical optimization: You have a some sort of fitness function with several input parameters and you want to minimize (or maximize) that function. If your function is well behaved (there is a single, global minimum that you will arrive at no matter which inputs you start with) then you are best off using a determinate minimization technique such as the conjugate gradient method. Many machine learning classification techniques involve finding parameters that minimize the least squares error for a hyperplane with respect to a training set. The function that is being minimized in this case is a smooth, well behaved, parabaloid in n-dimensional space. Calculate the gradient and roll downhill. Easy peasy. If, however, your input parameters are discrete (or if your fitness function has discontinuties) then it is no longer possible to calculate gradients accurately. This can happen if your fitness function is calculated using tabular data for one or more variables (if variable X is less than 0.5 use this table else use that table). Alternatively, you may have a program that you got from NASA that is made up of 20 modules written by different teams that you run as a batch job. You supply it with input and it spits out a number (think black box). Depending on the input parameters that you start with you may end up in a false minimum. Global optimization techniques attempt to address these types of problems. Evolutionary Algorithms form one class of global optimization techniques. Global optimization techniques typically involve some sort of "hill climbing" (accepting a configuration with a higher (worse) fitness function). This hill climbing typically involves some randomness/stochastic-ness/monte-carlo-ness. In general, these techniques are more likely to accept less optimal configurations early on and, as the optimization progresses, they are less likely to accept inferior configurations. Evolutionary algorithms are loosely based on evolutionary analogies. Simulated annealing is based upon analogies to annealing in metals. Particle swarm techniques are also inspired by biological systems. In all cases you should compare results to a simple random (a.k.a. "monte carlo") sampling of configurations...this will often yield equivalent results. My advice is to start off using a deterministic gradient-based technique since they generally require far fewer function evaluations than stochastic/monte-carlo techniques. When you hear hoof steps think horses not zebras. Run the optimization from several different starting points and, unless you are dealing with a particularly nasty problem, you should end up with roughly the same minimum. If not, then you might have zebras and should consider using a global optimization method.
well I think Monte Carlo methods is the general name for these methods which use random numbers in order to solve optimization problems. In this ways, even the evolutionary algorithms are a type of Monte Carlo methods if they use random numbers (and in fact they do). Other Monte Carlo methods are: metropolis, wang-landau, parallel tempering,etc OTOH, Evolutionary methods use 'techniques' borrowed from nature such as mutation, cross-over, etc.
Supervised Learning for User Behavior over Time
I want to use machine learning to identify the signature of a user who converts to a subscriber of a website given their behavior over time. Let's say my website has 6 different features which can be used before subscribing and users can convert to a subscriber at any time. For a given user I have stats which represent the intensity on a continuous range of that user's interaction with features 1-6 on a daily basis so: D1: f1,f2,f3,f4,f5,f6 D2: f1,f2,f3,f4,f5,f6 D3: f1,f2,f3,f4,f5,f6 D4: f1,f2,f3,f4,f5,f6 Let's say on day 5, the user converts. What machine using algorithms would help me identify which are the most common patterns in feature usage which lead to a conversion? (I know this is a super basic classification question, but I couldn't find a good example using longitudinal data, where input vectors are ordered by time like I have) To develop the problem further, let's assume that each feature has 3 intensities at which the user can interact (H, M, L). We can then represent each user as a string of states of interaction intensity. So, for a user: LLLLMM LLMMHH LLHHHH Would mean on day one they only interacted significantly with features 5 and 6, but by the third day they were interacting highly with features 3 through 6. N-gram Style I could make these states words and the lifetime of a user a sentence. (Would probably need to add a "conversion" word to the vocabulary as well) If I ran these "sentences" through an n-gram model, I could get the likely future state of a user given his/her past few state which is somewhat interesting. But, what I really want to know the most common sets of n-grams that lead to the conversion word. Rather than feeding in an n-gram and getting the next predicted word, I want to give the predicted word and get back the 10 most common n-grams (from my data) which would be likely to lead to the word. Amaç Herdağdelen suggests identifying n-grams to practical n and then counting how many n-gram states each user has. Then correlating with conversion data (I guess no conversion word in this example). My concern is that there would be too many n-grams to make this method practical. (if each state has 729 possibilities, and we're using trigrams, thats a lot of possible trigrams!) Alternatively, could I just go thru the data logging the n-grams which led to the conversion word and then run some type of clustering on them to see what the common paths are to a conversion? Survival Style Suggested by Iterator, I understand the analogy to a survival problem, but the literature here seems to focus on predicting time to death as opposed to the common sequence of events which leads to death. Further, when looking up the Cox Proportional Hazard model, I found that it does not event accommodate variables which change over time (its good for differentiating between static attributes like gender and ethnicity)- so it seems very much geared toward a different question than mine. Decision Tree Style This seems promising though I can't completely wrap my mind around how to structure the data. Since the data is not flat, is the tree modeling the chance of moving from one state to another down the line and when it leads to conversion or not? This is very different than the decision tree data literature I've been able to find. Also, need clarity on how to identify patterns which lead to conversion instead a models predicts likely hood of conversion after a given sequence.
Theoretically, hidden markov models may be a suitable solution to your problem. The features on your site would constitute the alphabet, and you can use the sequence of interactions as positive or negative instances depending on whether a user finally subscribed or not. I don't have a guess about what the number of hidden states should be, but finding a suitable value for that parameter is part of the problem, after all. As a side note, positive instances are trivial to identify, but the fact that a user has not subscribed so far doesn't necessarily mean s/he won't. You might consider to limit your data to sufficiently old users. I would also consider converting the data to fixed-length vectors and apply conceptually simpler models that could give you some intuition about what's going on. You could use n-grams (consecutive interaction sequences of length n). As an example, assuming that the interaction sequence of a given user ise "f1,f3,f5", "f1,f3,f5" would constitute a 3-gram (trigram). Similarly, for the same user and the same interaction sequence you would have "f1,f3" and "f3,f5" as the 2-grams (bigrams). In order to represent each user as a vector, you would identify all n-grams up to a practical n, and count how many times the user employed a given n-gram. Each column in the vector would represent the number of times a given n-gram is observed for a given user. Then -- probably with the help of some suitable normalization techniques such as pointwise mutual information or tf-idf -- you could look at the correlation between the n-grams and the final outcome to get a sense of what's going on, carry out feature selection to find the most prominent sequences that users are involved in, or apply classification methods such as nearest neighbor, support machine or naive Bayes to build a predictive model.
This is rather like a survival analysis problem: over time the user will convert or will may drop out of the population, or will continue to appear in the data and not (yet) fall into neither camp. For that, you may find the Cox proportional hazards model useful. If you wish to pursue things from a different angle, namely one more from the graphical models perspective, then a Kalman Filter may be more appealing. It is a generalization of HMMs, suggested by #AmaçHerdağdelen, which work for continuous spaces. For ease of implementation, I'd recommend the survival approach. It is the easiest to analyze, describe, and improve. After you have a firm handle on the data, feel free to drop in other methods.
Other than Markov chains, I would suggest decision trees or Bayesian networks. Both of these would give you a likely hood of a user converting after a sequence. I forgot to mention this earlier. You may also want to take a look at the Google PageRank algorithm. It would help you account for the user completely disappearing [not subscribing]. The results of that would help you to encourage certain features to be used. [Because they're more likely to give you a sale]
I think Ngramm is most promising approach, because all sequnce in data mining are treated as elements depndent on few basic steps(HMM, CRF, ACRF, Markov Fields) So I will try to use classifier based on 1-grams and 2 -grams.