Table of Contents

This is Part 2 of my posts about capturing the provenance of model prototyping and development. Part 1 is here.

Steps towards the goal, continued

Breaking into Terminal Steps

At this point, we switch to the GitHub repo, where I separated the StudyContext (SC) into multiple files describing Terminal Steps, which now populate a folder named studycontext1/ instead of a single file1. This ensures that each file has one primary purpose. I also added a common library and made the whole file space into a python package to allow local imports to work. These are jobs that the hypothetical project manager would do for me.

If you cloned the whole repo locally, you’ll have to reset the project’s state to the initial tag for this point in the development:

git reset --hard "Stage2"

We now have this file system tree:

 - project/
   -    # unused
   - setup.yml
   - studycontext1/
	  - viz_errors1.png
	  - viz_VF_overlay.png
   - Target_VF_and_test_traj.png

We have two study context files with the following Terminal Steps:

  tag: initial test that goal fails with a small sample
  tag: visualize mesh of errors
    - figure 1
  tag: visualize mesh of linear vectors overlaid with actual vectors
    - figure 2

As mentioned before, the initial_test script just validates that the UAC fails, and that the goal test code is set up correctly, before we begin solving the actual problem.

The visualizations produce the following figures:

  • Figure 1: Initial linear VF overlay (compare Figure 3)

Initial linear VF overlay

This figure overlays a 5 x 5 sample of vectors from F (pointing left, as we saw before) with those from LF (pointing up, arrowheads not shown!).

  • Figure 2: Initial error measure (compare Figure 4)

Initial error measure

This figure shows a 30 x 30 mesh of absolute errors for the metric d(x,y), which is implemented as,y). The blob size is proportional to the error magnitude.

When Steps need to be decomposed further

The tasks in scope of a StudyContext and its respective input and output data should be well chosen to involve only a modest number of files that are easy to manage in one folder. If a task of the SC is later understood to require much more setup, work steps, or additional files, then my convention is to make a sub-StudyContext in a new sub-folder and work in there. If files and data need to be inherited, then make it a python sub-package and import from the parent’s common library.

The markup of the SC files should indicate where a Step in the parent SC leads to work in a named sub-SC that probably corresponds to the sub-folder’s name. In lieu of a formal project management system, this user-enforced rule can work for small sized projects. This idea is not demonstrated here: I will exemplify it in a future post.

Completing the model development

Recap of the linear model

A 2D linear model requires 6 parameters to specify it. In simplest terms, it would look like:

dx/dt = a*x + b*y + c
dy/dt = d*x + e*y + f

However, with a tiny bit of algebra, this can be rewritten in a convenient form that accentuates the timescales associated with each variable:

dx/dt = (x0+yfx*y - x)/taux
dy/dt = (y0+xfy*x - y)/tauy

This ODE system is implemented in the project as the PyDSTool.Model object lin.

Major repository changes

We can now switch to Stage 3, which is the HEAD of the GitHub repo, and where the development has ended.

git reset HEAD

We now have this file system tree 2:

 - project/
   -    # unused
   - setup.yml
   - studycontext1/
	  - viz_errors1.png
	  - viz_VF_overlay1.png
	  - viz_VF_overlay2.png
   - Target_VF_and_test_traj.png is the additional studycontext1 script that sets up an optimization process.

Optimization step

In order that each script does just one task, some data must be accrued in the project that can be accessed between scripts. This involves ‘serializing’ or ‘marshalling’ data and code objects to the file system for future use. The pyfscache package achieves this in a very convenient way, while also avoiding inefficient recalculation of identical objects that have already been computed once before.

The prior dependencies of these objects are noted in the metadata for a SC comment string.

We use the error_pts function, and the meshes we set up in Stage 1, to form the residual error function for a minimization routine:

def residual_fn(p):
    # fmin will take the norm for us
    return error_pts(mesh_pts_test5)

This function takes a sequence of parameter values and returns the vector of the errors at all the mesh sample points.

We create two constraints to see if we can ensure that the two timescale parameters (taux and tauy) are both positive numbers. 3

Then, we run the optimizer (fmin_cobyla, which can accept simple inequality constraints). Fortunately, we quickly converge to a useful solution. If not, we could have created a sub-SC to diagnose and explore the best parameter choices to make the algorithm perform. Sometimes, no progress is made at all in a complex optimization problem without learning some ‘magic’ parameter values. Documenting this is not fun, as it is already a tiresome task to have to tweak an algorithm’s parameters in a nonlinear problem.

But, it’s also not fun to explain to someone (e.g., your PI/colleague) how you arrived at a set of choices, and which ones you tried out. You could write them out in a notebook but you could also commit each trial (both input and output) in the VCS with one quick keystroke as you work, and keep all that provenance inherently digital and alongside the project code.

Furthermore, if you used any cunning logic to figure out the parameter choices, you might like to markup a discussion of what you did and share your provenance record with others to help educate them.

Visualizations of the results

In Stage 2, we observed that the default state of the linear system’s vector field is almost perpendicular to that of the target vector field. In comparison, the visualizations of the “optimized” state, shown below, indicate a great subjective improvement in the alignment of the vectors and a reduction in the error metric:

  • Figure 3: Final linear VF overlay (compare Figure 1)

Final linear VF overlay

  • Figure 4: Final error measure (compare Figure 2)

Final error measure

The visualizations are grouped into one script, and of course they depend on previous scripts having been called. Again, these are noted in the SC comment metadata.

We have not addressed the original UAC, which is to ensure that trajectories computed from LF do not diverge more than L2_tol from those from F, but we will see that it’s not even necessary to assess that to make a conclusion.

Conclusions and discussion

Based on the original error tolerance requested (L2_tol = 0.002), the scenario appears to be unfeasible just from looking at the vector field itself! We achieved a visually similar-fitting vector field with a linear system, but the maximum error in the vectors across the mesh sample was still 1.0. There is no way such an error will satisfy the UAC for trajectories across this domain, which will only be compounded given the local vector error.

Without turning this demonstration into a protracted interlude on numerical analysis, you can verify that the optimization algorithm had converged to a solution and was not going to make any large enough improvements to get the error down another order of magnitude. It is, of course, possible that we are getting trapped in a local minimum, and ideally we would create some further tasks to explore this: e.g., repeat over randomized initial conditions for the optimization in a simple Monte-Carlo sampling, or perform simulated annealing. We could also have tried to determine appropriate rescalings of each parameter based on the sensitivity of the residual error vectors, and provided this rescaling to the optimizer to help it choose step sizes more effectively.

We may also wish to close this SC and setup a new one that changes the permitted steps in order to really achieve the requested tolerance. This could involve sub-dividing the domain further and using piecewise linear models to fit in each sub-domain.

These ideas are beyond the scope of this demonstration, but it might be worth revisiting them if the extension of this demo will highlight more sophisticated and realistic StudyContext usage.

Admittedly, there was nothing so interesting about the linear model we derived that deserved such detailed attention. It was a pedagogically convenient example with no surprises along the way and no difficult choices to explore and resolve. If there were, and there would be in more realistic problems, I argue that this methodology would better facilitate the planning, organization, execution, and documentation of work to be done to address those choices and problems.

Later, I will introduce more realistic worked examples that display such properties. In the meantime, I hope this minimal demonstration initiates some discussion and maybe even a contribution to creating a python platform for supporting literate modeling of this kind.


These two posts have been an almost stream-of-consciousness look at a way to capture the provenance of mathematical model prototyping. We set out a need, a problem space, and a task, and set up an organized set of files and workflow conventions to document how we arrived, piece by piece, to our conclusion. We did derive a visually reasonable linear model for the nonlinear system in the given domain.

I really don’t think it’s as difficult to imagine doing literate modeling at this early model development stage as I’ve heard people lament. Although, I haven’t demonstrated how to capture aspects of the discovery and learning process that involve more graphical interactions. I will return to this.

The essence of my approach to literate modeling is to treat model development steps as the kinds of task work done during software development, in particular to apply concepts such as test-driven development, version control, and highly iterative workflows such as that espoused in the Agile approach. The version control aspect is inspired, in part, by previous work on the Sumatra simulation provenance tool, and simulation execution workflow design tools such as Taverna and Kepler.

By establishing a simple convention about markup for model development steps and file system structure, I hope to have made a convincing argument that such prototyping and early development can realistically be captured and organized in a useful way.

Note that we could apply this provenance methodology to other forms of mathematical or computational work, including data analysis.

Limitations of the implementation

Several limitations and problems that I encountered are recorded as comments on the gist github page, but expanded upon here.

Obviously, I need scripts or a proper project management tool to automate git add/commit/push with automated message headers in the commit message. Right now, it’s all being done by hand, and it’s difficult to keep track of my self-imposed conventions.

How do I add math markup in the documentation of the goals, etc., in a way that will be transformed to readable math in the browser (online, I know how to achieve it on localhost)? I think this can be done with github pages and jekyll, but I’m not sure which third party extension to invoke yet.

What to do when functions defined during development of need to be pushed to a common library to enable reuse? How does the provenance markup using the triple-quoted strings (or any other solution), which shows the order of sub-tasks (Steps) in that file get affected? Well, for now, a solution is to leave it to the historical record in the git revision history. But it’s not an entirely satisfactory solution.

Another fundamental problem is, when working interactively, how do we interpret the SC Step declarations that may be repeated (even out of order) during pasted-code execution? A simple way to resolve this is to only retain the changes to the (hypothetical) project manager’s database state at commit time, whenever we are ready to run a SC script once through and move on.

Another issue is that there is no obvious way to mark when a StudyContext is complete, except to leave a note in the document or use a commit message. Presumably, the project management software would keep metadata in special files to track progress just like project management tools for software development.

Naturally, though, it makes sense that there should be an executable script that runs a test on whatever solution to check the UAC have passed. This is a long-winded way to tell if the SC’s work is complete. The code may still need to be tidied up before committing and publishing.

Weaving a tangled web of unreadability

Finally, as Tony Fast pointed out in our discussion, python code files are meant for, er, code, and also for direct documentation for code. They really are not the place for storing metadata for other systems. I agree, but separating the metadata from the code makes it more difficult to track where code blocks begin an end as part of Steps.

But, it’s true. Triple-quote docstrings belong at the top of the file and inside function definitions, and I end up repeating some of that documentation outside for the studycontext metadata. And I don’t want to keep track of indenting exactly the same amount for the next note as part of the same Step, several code lines below where it started in the previous comment (e.g. see line 86 of the gist’s It looks weird.

The literate programming community have provided some tools, but I am not excited about working with a setup like WEB – via noweb, for instance, in which you write markup for the metadata and put code into special markup tag environments. On one hand, it’s good that code and metadata are all stored together, but that is not a productive way to write exploratory code. On most editors, even with noweb markup modes, you’ll lose the code’s native syntax highlighting and other editing conveniences (e.g., auto-indenting blocks). Then, the file is not executable (i.e., even testable) until it has been ‘tangled’. For my personal view on rapid development, this situation is inconvenient and backwards. I want to get code snippets written down and tested ASAP to build up the pieces I’ll need to work on the problem. Then, I’ll generate the documented version from the embedded notes and present the results. I want to keep notes in comments to the side of the immediately-executable code. Maybe a custom editor or IDE would solve this problem.

A project management system’s GUI editor could use templates to insert the bulk of the strings for each Step declaration, and precede them with some kind of coded directive to indicate that the string is not a regular docstring or comment.

A possible alternative is that, instead of using comment strings for each Step declaration, there are method calls on a global “Scenario” object that provides access to the project manager’s API. E.g.:

# code block here

Scenario.Step(type='test', tag='check initial model state',
    notes=['use 5x5 grid for now', 'needs optimizing for speed'])
# code block here

Then, the metadata is made part of the code from the beginning.

What the calls would do: Write a YAML file entry in sequence, essentially building the docstrings I wrote in. These can be diff’ed to check for structural changes. The presence of the call in the source code allows a pre- or post-processor to select the computational python code to put in a JSON block (like ipynb).

Pasting the SC calls into an interactive script should be innocuous because the calls may repeat and/or be out of order. We only care about the results from running the whole script. This is actually easy to imagine taking care of, because the post-processor would only act on the ordered declarations when invoked at the end of a script. The YAML file being formed can safely get messed up during the interaction session but won’t be “committed” until after the whole, completed script is run to the endpoint. Each run rewrites the whole YAML record for that script.

An interestingly similar tool concept is Watson, a software development issue manager that is based on inline code markup. If a python version matures (this one is not there yet), I’d consider basing my project manager on this code.

iPython notebooks to the rescue?

However, iPython notebooks, as I currently appreciate them, are also not the solution to this problem of mixing StudyContext metadata and code. Notebooks do well at separating text and other data formats from code, and presenting their combination in a clean and pretty way. This makes a notebook a great presentation tool for a finalized, static version of the code and output (including intermediate states), but not for active, rapid development. A notebook is like the result of a weave process.

On the other hand, Tony mentioned that I might ‘magic parse’ the SC’s YAML metadata on execute in a notebook using this fancy snippet, although I have yet to understand if that will help or even how to do it. (There is no documentation provided with that gist.)

Regardless, Ipython notebooks seem inappropriate for my use if I want to explore interactively in the command line, or especially in a graphical UI (I don’t just mean resizing or zooming plots). I often do not want my graphs to show up inline, and there may be too many to conveniently manage in a single window as sub-plots. I may also want to structure my file system around a local library of common elements and different scripts for different tasks. These are not so easy to present as a single notebook document. When I discuss interactive tools more in future posts, and I learn more about the notebooks, my arguments will hopefully become clearer one way or another.

Version control issues

In terms of version control, the gist system is not sophisticated enough to support all the present needs. The git history is not as easily viewable as a regular git project on GitHub (e.g., no network diagram) as there is no GUI for gist revision history in the same way. This means that git tags and branches are not as easy to discover or navigate with online GUI tools. It’s not clear to me how they could be usefully applied to tracking this mini project. A proper model project manager would have to wrap all the git operations and UI into a customized interface.

Further work

Merging and combining results and artifacts of sub-SCs that resolve back into the parent SCs to allow them to continue. This poses a data import problem. For now, we must use a convention. Having committed the state of the project at the resolution of the sub-SCs, the new artifacts can be tagged by their original SC file and then merged into the parent’s common library for continued use in the parent SC.


  1. We won’t bother to record the structure of the study contexts in a metadata file in this example, but a real project management system would be keeping track of the change from a single to the folder studycontext1/ and a database of all project SCs. 

  2. Note that the script is a meta-level check for me to verify that the linearization made sense in the context of the original model. This isn’t part of the main workflow. 

  3. They needn’t be in the general case, and this is an artificial addition here, although it doesn’t affect the result.