What is the best statistical test for an effect across groups when the data are nested but may not be of normal distribution? I get a highly significant effect using Kruskall Wallis test, but using it there is no account to that the data points are from several locations, each contributed for several years, and in every year the data were pulled into age groups.
I think you can categorize the data by year, and change the data structure so that the data will be non-nested, making it easier to process. I agree that Kruskal-Wallis' test is a good choice of the cross-group effect test.
In below post of Analytics Vidya, ANOVA test has been performed on COVID data, to check whether the difference in posotive cases of denser region is statistically significant.
I believe ANOVA test can’t be performed on this COVID time series data, atleast not in way as it has been done in this post.
Sample data has been consider randomly from different groups(denser1, denser2…denser4). The data is time series so it is more likely that number of positive cases in random sample of groups will be from different point of time.
There might be the case denser1 has random data from early covid time and another region has random data from another point of time. If this is the case, then F-Statistics will high certainly.
Can anyone explain if you have other opinions?
https://www.analyticsvidhya.com/blog/2020/06/introduction-anova-statistics-data-science-covid-python/
ANOVA should not be applied to time-series data, as the independence assumption is violated. The issue with independence is that days tend to correlate very highly. For example, if you know that today you have 1400 positive cases, you would expect tomorrow to have a similar number of positive cases, regardless of any underlying trends.
It sounds like you're trying to determine causality of different treatments (ie mask mandates or other restrictions etc) and their effects on positive cases. The best way to infer causality is usually to perform A-B testing, but obviously in this case it would not be reasonable to give different populations different treatments. One method that is good for going back and retro-actively inferring causality is called "synthetic control".
https://economics.mit.edu/files/17847
Above is linked a basic paper on the methodology. The hard part of this analysis will be in constructing synthetic counterfactuals or "controls" to test your actual population against.
If this is not what you're looking for, please reply with a clarifying question, but I think this should be an appropriate method that is well-suited to studying time-series data.
Preface: I'm a self-taught coder, so a lot of my knowledge is limited to my research. I'm hoping to have other opinions as I want to build things right the first time. I need help with determining an appropriate solution and how to implement the solution.
I'm looking to build a least cost alternative model (essentially a shortest path) for delivering between locations (nodes), based on different modes of transportation (vehicles) and the different roads taken (paths). Another consideration is the product price (value) to determine the least cost path.
Here are my important data items:
nodes: cities where the product will travel to and from.
paths: roads have different costs, depending on the road.
vehicles: varying vehicles have differing rental costs when transporting (motorbike, car, truck). Note that the cost of a vehicle is not constant, it is highly dependent on the to/from nodes. For example, using a car to go from city A to city B will have a different cost than using a car to go from city B to A or city A to city C.
value: Product value. Again, a product's value is highly dependent on its destination node. The same product can have a different value at City A, B or C.
Problem Statement
How to setup data structure to best determine where the least cost path would be to get a product from one location to every other location.
Possible Solutions
From my research, I believe a weighted graph data structure would be most suitable for my situation in combination with dijkstra's algorithm. I believe breaking the problem down simpler would be essential, to first create a simple weighted graph of only nodes and paths.
From there, adding the vehicle cost and the product value considerations afterwards. Perhaps just adding the two values as a cost to "visit" a node? (aka incorporate it into the path cost?)
Thoughts on my current solution? Other considerations I overlooked? Perhaps a better solution?
Implementation
I'd love to be able to build this within Excel VBA (as that is how I learned how to code) and Excel is what I use for my tools. Would VBA be too limited in this task? How else can I incorporate my analysis with Excel with another language?
Try the book Practical Management Science by Winston & Albright and check out the chapter on Operations Management - lots of models explained in there from the simple onwards. Available online as a pdf : http://ingenieria-industrial.net/downloads/practicalmanagementscience.pdf
VBA is more a scripting language than a full-fledged one, though one may contend that the underlying framework is .NET. Why don't you give a shot at C++ or Java? If you intuitively understand the data structure and the algorithm, then it'll be a breeze coding in these. Chapter 4 of Algorithms by Sedgewick and Wayne has a beautiful explanation of Shortest Paths. You may also consider studying Bellman-Ford algorithm if you foresee any negative weight cycles on a vertex.
I am seeking a method to allow me to analyse/search for patterns in asset price movements using 5 variables that move and change with price (from historical data).
I'd like to be able to assign a probability to a forecasted price move when for example, var1 and var2 do this and var3..5 do this, then price should do this with x amount of certainty.
Q1: Could someone point me in the right direction as to what framework / technique can help me achieve this?
Q2: Would this be a multivariate continuous random series analysis?
Q3: A Hidden Markov modelling?
Q4: Or perhaps is it a data-mining problem?
I'm looking for what rather then how.
One may opt to use Machine-Learning tools to build a learner to either
both classify of what kind the said "asset price movement" will beand serve also statistical probability measures for such a Classifier prediction
both regress a real target value, to which the asset price will moveandserve also statistical probability measures for such a Regressor prediction
A1: ( while StackOverflow strongly discourages users to ask about an opinion about a tool or a particular framework ) there would be not much damages or extra time to be spent, if one performs academia papers research and there would be quite a remarkable list of repeatedly used tools, used for ML in the context of academic R&D. For a reason, there would not be a surprise to meet scikit-learn ML-classes a lot, some other papers may work with R-based quantitative finance / statistical libraries. The tools, however, with all due respect, are not the core to answer all the doubts and inital confusion present in a mix of your questions. The subject confusion is.
A2: No, it would not. Well, unless you beat all the advanced quantitative research and happen to prove that the Market exhibits a random behaviour ( which it is not and for which it would be waste of time to re-cite remarkable research published about why it is not indeed a random process ).
A3: Do not try to jump on any wagon just because of it's attractive Tag or "contemporary popularity" in marketing minded texts. With all due respect, understanding HMM is outside of your sight while you now appear to move just to the nearest horizons to first understand what to look for.
A4: This is a nice proof of a missed target. Your question shows in this particular point better than in others, how small amount of own research efforts were put into covering the problem-domain and acquiring at least some elementary knowledge before typing the last two questions.
StackOverflow encourages users to ask high quality questions, so do not hesitate to re-edit your post to add some polishing efforts to this subject.
If in a need for an inspiration, try to review a nice and a powerful approach for a fast Machine Learning process, where both Classification and Regression tasks obtain also probability estimates for each predicted target value.
To have some idea about highly performant ML-predictors, these typically operate on much more than a set of 5 variables ( called in the ML-domain "features" ) . ( Think rather about some large hundreds to small thousands features, typically heavily non-linear transformations from the original TimeSeries' data ).
There you go, if indeed willing to master ML for algorithmic trading.
May like to read about a state-of-art research in this direction:
[1] Mondrian Forests: Efficient Online Random Forests
>>> arXiv:1406.2673v2 [stat.ML] 16 Feb 2015
[2] Mondrian Forests for Large-Scale Regression when Uncertainty Matters
>>> arXiv:1506.03805v4 [stat.ML] 27 May 2016 >>>
May also enjoy other posts on subject: >>> StackOverflow Algorithmic-Trading >>>
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Basically, I have a reasonably large list (a year's worth of data) of times that a single discrete event occurred (for my current project, a list of times that someone printed something). Based on this list, I would like to construct a statistical model of some sort that will predict the most likely time for the next event (the next print job) given all of the previous event times.
I've already read this, but the responses don't exactly help out with what I have in mind for my project. I did some additional research and found that a Hidden Markov Model would likely allow me to do so accurately, but I can't find a link on how to generate a Hidden Markov Model using just a list of times. I also found that using a Kalman filter on the list may be useful but basically, I'd like to get some more information about it from someone who's actually used them and knows their limitations and requirements before just trying something and hoping it works.
Thanks a bunch!
EDIT: So by Amit's suggestion in the comments, I also posted this to the Statistics StackExchange, CrossValidated. If you do know what I should do, please post either here or there
I'll admit it, I'm not a statistics kind of guy. But I've run into these kind of problems before. Really what we're talking about here is that you have some observed, discrete events and you want to figure out how likely it is you'll see them occur at any given point in time. The issue you've got is that you want to take discrete data and make continuous data out of it.
The term that comes to mind is density estimation. Specifically kernel density estimation. You can get some of the effects of kernel density estimation by simple binning (e.g. count the number events in a time interval such as every quarter hour or hour.) Kernel density estimation just has some nicer statistical properties than simple binning. (The produced data is often 'smoother'.)
That only takes care of one of your problems, though. The next problem is still the far more interesting one -- how do you take a time line of data (in this case, only printer data) and produced a prediction from it? First thing's first -- the way you've set up the problem may not be what you're looking for. While the miracle idea of having a limited source of data and predicting the next step of that source sounds attractive, it's far more practical to integrate more data sources to create an actual prediction. (e.g. maybe the printers get hit hard just after there's a lot of phone activity -- something that can be very hard to predict in some companies) The Netflix Challenge is a rather potent example of this point.
Of course, the problem with more data sources is that there's extra legwork to set up the systems that collect the data then.
Honestly, I'd consider this a domain-specific problem and take two approaches: Find time-independent patterns, and find time-dependent patterns.
An example time-dependent pattern would be that every week day at 4:30 Suzy prints out her end of the day report. This happens at specific times every day of the week. This kind of thing is easy to detect with fixed intervals. (Every day, every week day, every weekend day, every Tuesday, every 1st of the month, etc...) This is extremely simple to detect with predetermined intervals -- just create a curve of the estimated probability density function that's one week long and go back in time and average the curves (possibly a weighted average via a windowing function for better predictions).
If you want to get more sophisticated, find a way to automate the detection of such intervals. (Likely the data wouldn't be so overwhelming that you could just brute force this.)
An example time-independent pattern is that every time Mike in accounting prints out an invoice list sheet, he goes over to Johnathan who prints out a rather large batch of complete invoice reports a few hours later. This kind of thing is harder to detect because it's more free form. I recommend looking at various intervals of time (e.g. 30 seconds, 40 seconds, 50 seconds, 1 minute, 1.2 minutes, 1.5 minutes, 1.7 minutes, 2 minutes, 3 minutes, .... 1 hour, 2 hours, 3 hours, ....) and subsampling them via in a nice way (e.g. Lanczos resampling) to create a vector. Then use a vector-quantization style algorithm to categorize the "interesting" patterns. You'll need to think carefully about how you'll deal with certainty of the categories, though -- if your a resulting category has very little data in it, it probably isn't reliable. (Some vector quantization algorithms are better at this than others.)
Then, to create a prediction as to the likelihood of printing something in the future, look up the most recent activity intervals (30 seconds, 40 seconds, 50 seconds, 1 minute, and all the other intervals) via vector quantization and weight the outcomes based on their certainty to create a weighted average of predictions.
You'll want to find a good way to measure certainty of the time-dependent and time-independent outputs to create a final estimate.
This sort of thing is typical of predictive data compression schemes. I recommend you take a look at PAQ since it's got a lot of the concepts I've gone over here and can provide some very interesting insight. The source code is even available along with excellent documentation on the algorithms used.
You may want to take an entirely different approach from vector quantization and discretize the data and use something more like a PPM scheme. It can be very much simpler to implement and still effective.
I don't know what the time frame or scope of this project is, but this sort of thing can always be taken to the N-th degree. If it's got a deadline, I'd like to emphasize that you worry about getting something working first, and then make it work well. Something not optimal is better than nothing.
This kind of project is cool. This kind of project can get you a job if you wrap it up right. I'd recommend you do take your time, do it right, and post it up as function, open source, useful software. I highly recommend open source since you'll want to make a community that can contribute data source providers in more environments that you have access to, will to support, or time to support.
Best of luck!
I really don't see how a Markov model would be useful here. Markov models are typically employed when the event you're predicting is dependent on previous events. The canonical example, of course, is text, where a good Markov model can do a surprisingly good job of guessing what the next character or word will be.
But is there a pattern to when a user might print the next thing? That is, do you see a regular pattern of time between jobs? If so, then a Markov model will work. If not, then the Markov model will be a random guess.
In how to model it, think of the different time periods between jobs as letters in an alphabet. In fact, you could assign each time period a letter, something like:
A - 1 to 2 minutes
B - 2 to 5 minutes
C - 5 to 10 minutes
etc.
Then, go through the data and assign a letter to each time period between print jobs. When you're done, you have a text representation of your data, and that you can run through any of the Markov examples that do text prediction.
If you have an actual model that you think might be relevant for the problem domain, you should apply it. For example, it is likely that there are patterns related to day of week, time of day, and possibly date (holidays would presumably show lower usage).
Most raw statistical modelling techniques based on examining (say) time between adjacent events would have difficulty capturing these underlying influences.
I would build a statistical model for each of those known events (day of week, etc), and use that to predict future occurrences.
I think the predictive neural network would be a good approach for this task.
http://en.wikipedia.org/wiki/Predictive_analytics#Neural_networks
This method is also used for predicting f.x. weather forecasting, stock marked, sun spots.
There's a tutorial here if you want to know more about how it works.
http://www.obitko.com/tutorials/neural-network-prediction/
Think of a markov chain like a graph with vertex connect to each other with a weight or distance. Moving around this graph would eat up the sum of the weights or distance you travel. Here is an example with text generation: http://phpir.com/text-generation.
A Kalman filter is used to track a state vector, generally with continuous (or at least discretized continuous) dynamics. This is sort of the polar opposite of sporadic, discrete events, so unless you have an underlying model that includes this kind of state vector (and is either linear or almost linear), you probably don't want a Kalman filter.
It sounds like you don't have an underlying model, and are fishing around for one: you've got a nail, and are going through the toolbox trying out files, screwdrivers, and tape measures 8^)
My best advice: first, use what you know about the problem to build the model; then figure out how to solve the problem, based on the model.