Suppose that I have a data set S that contains the service time for different jobs, like S={t1,t2,t3,...,tn}, where ti is the service time for the ith job; and n the total number in my data set. This S is only a sample from a population. n here is 300k. I would like to study the impact of long service time as some jobs takes very long and some not. My intuition is to study this impact based on data gathered from real system. The system in study has thousands of millions of jobs, and this number is increasing by 100 new jobs each several seconds. Also, service time is measured via benchmarking the jobs on a local machine. So practically it is expensive to keep expanding your data set. Thus, i decided to randomly pick up 300k.
I am conducting simulation experiments where I have to generate a large number of jobs with their service times (say millions) and then do some other calculations.
How to use S as a population in my simulation, I came across the following:
1- use S itself. I could use bootstrapping 'sample with replacement' or ' sample without replacement'.
2- fit a theoretical distribution model to S and then draw from it.
Am I correct? which approach is best (pros and cons)? the first approach seems easy as just picking a random service time from S each time? is it reliable? Any suggestion is appreciated as I am not got at stats.
Quoting from this tutorial in the 2007 Winter Simulation Conference:
At first glance, trace-driven simulation seems appealing. That is
where historical data are used directly as inputs. It’s hard to argue
about the validity of the distributions when real data from the
real-world system is used in your model. In practice, though, this
tends to be a poor solution for several reasons. Historical data may
be expensive or impossible to extract. It certainly won’t be available
in unlimited quantities, which significantly curtails the statistical
analysis possible. Storage requirements are high. And last, but not
least, it is impossible to assess “what-if?” strategies or try to
simulate a prospective system, i.e., one which doesn’t yet exist.
One of the major uses of simulation is to study alternative configurations or policies, and trace data is not suitable for that—it can only show you how you're currently operating. Trace data cannot be used for studying systems which are under consideration but don't yet exist.
Bootstrapping resamples your existing data. This removes the data quantity limitations, but at a potential cost. Bootstrapping is premised on the assumption that your data are representative and independent. The former may not be an issue with 300k observations, but often comes up when your sample size is smaller due to cost or availability issues. The latter is a big deal if your data come from a time series where the observations are serially correlated or non-homogeneous. In that case, independent random sampling (rather than sequential playback) can lose significant information about the behaviors being studied.
If sequential playback is required you're back to being limited to 300k observations, and that may not be nearly as much data as you think for statistical measures. Variance estimation is essential to calculating margins of error for confidence intervals, and serial correlation has a huge impact on the variance of a sample mean. Getting valid confidence interval estimates can take several orders of magnitude more data than is required for independent data.
In summary, distribution fitting takes more work up front but is usually more useful in the long run.
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