This is Part 1 of a series of posts.

As I discussed recently in a blog post and comments, I want to see the early development process for mathematical models deconstructed into somewhat formalized workflows that can be meaningfully captured by version control systems (VCS). To be precise, I want to structure my workflow and filesystem and add structured metadata using a project management system. Unlike existing focus on the simulation of existing models, I want to capture provenance of model development ‘steps,’ their diagnostics, and their results, not just of the primary data (model code, final simulation results, parameter sets), and not just of unstructuted text descriptions of notes that might accompany that primary data (like a simple journal).

In this, my first technical and tutorial-like post, I’ll share a prototype workflow for achieving this. I will use a contrived and minimal model project example and a mostly aspirational set of management tools (you’ll see what I mean).

My literate modeling “solution” embodies some compromises and has many shortcomings that we can discuss and hopefully build from. But, I also think it effectively demonstrates the principles. It is also executable and already version controlled for posterity as a GitHub ‘gist’.

## Dependencies

I have to assume you know something about applied math, differential equations, optimization, and python programming, otherwise we’ll be here all month.

You can execute the gist files provided you have installed my PyDSTool package for simulating dynamical systems models. The Stage 3 revisions of the GitHub repo are immiediately Python 3 compatible. I forgot in the earlier stages to ensure that all print statements were written as function calls. If you correct those then all the stages are Python 3 compatible.

The simple and cool pyfscache file system object cache manager is an optional dependency. This package minimizes code re-execution to build objects that can be reused across scripts in a common folder. Such a facility would be important in the kind of workflow that I’m proposing, but in this simple example you can get away without it. My code runs either way. (Edit: to use this package on my system and with numpy objects, I needed to alter some things in this package and add some convenience features. This is my forked version for use with this project.)

All of the development was done in a python-friendly IDE (Wing IDE) alongside an interactive iPython session. Interactive sessions are key, here. Running pre-written scripts from the OS prompt is only useful when you have everything already figured out and ready to go. With iPython, I can mock up and test functions or arrays quickly, and introspect objects to figure out what to do next. Then I can copy snippets back to the IDE and compose a clean version for future reference and reuse. Or I edit such snippets in the IDE and %paste them into the session to update its state. iPython has color syntax highlighting, special meta-commands, and extensive command history tools. Wing IDE lets me interactively debug problem elements in my code as I develop it. When I make more fundamental changes to my code or pollute my session’s memory, I just restart the session and re-run whichever script I’m currently working on. The combination of these two tools make me very productive.

If you were actually performing the modeling steps yourself, you’d obviously want to have Git locally installed, and a GitHub account as an online repository.

## Viewing and understanding the example

Ideally, the gist would be presented online in a convenient fashion that makes it self-documenting and amenable to easy sharing, discussion, and collaboration, in the same way that the notebook viewer changed how we can share iPython notebooks online. However, this will require some additional technology that doesn’t exist yet. So, for now, you’ll have to dig into the gist files individually and reconstruct some of the view of the project by yourselves. This post will help to guide you through it, in an admittedly inadequate way.

For lack of the appropriate tool chain to present the major stages of model development within the revision history of a single gist, I moved over to a regular GitHub project and tagged major stages:

The primary files to take a look at the beginning are in the gist of Stage 1:

These already reflect a few commits, partly to assist in creating backups in case I messed something up. (See later.)

## The math modeling problem

Suppose we are given experimental details of an unknown Black Box containing a nonlinear dynamical system. We are asked to approximate its dynamics in a region where its behavior is only weakly nonlinear (phew). Therefore, we do not have access to the equations of motion or other background information, only raw sample data. In the real world, the sample data could be non-stationary and noisy, but let’s start with stationary (repeated samples over time yield identical results) and deterministic data.

For simplicity, let’s assume that we know we are working in two dimensions only, and that a suitable sub-domain D of two-dimensional space is given to us. We are given a black box function from which we can request spatial samples of the vector field F (measuring flow at a location), i.e. using a call F(x,y) that returns a 2D flow vector.

We will find a 2D linear system of ordinary differential equations (ODEs) that matches this flow as best as possible. Its vector field will be named LF. How well should the match be? We must be given User Acceptance Criteria (UAC): let’s assume we’re given a maximum error tolerance L2_tol that our model must achieve in the standard Euclidean (L2) metric d(x,y) applied to any sampled vectors in D compared to the linear model.

 $\forall (x,y) \in D, \quad d(x,y) = F(x,y) - LF(x,y) < L2tol$

Without getting too technical, if we can make guarantees about the continuity of F and the amount of nonlinearity that it may possess in D (i.e. about its local variance, or bounds on its derivatives), then it is reasonable to only require that this condition is met on a finite sample of points in order to guarantee it will work for all points in D. Let’s keep this simple for the sake of exposition.

These details constitute a problem scenario. It is easy to write this up in YAML imagining what a hypothetical project management program might automatically process for me. YAML is a simple and great markup language for easily creating structure out of unstructured text.

scenario-metadata:
name: local linear fit
difficulty: easy
time: short
tags: graphical, ODE

Given:
- 2D, weakly nonlinear, determininstic vector field given as a black box function of (x,y) -> (f_x(x,y), f_y(x,y))
- finite, rectangular domain [x0,x1] X [y0,y1]
- Euclidean error tolerance of forward trajectories (chosen to be feasible for the given function f, unless task scenario is open to being proven inconsistent)

Goal:
- Derive a UAC-satisfying local linear ODE approximation in the domain (where nonlinearity is fairly weak)

UAC:
- Predict all forward trajectories up to given max tolerance of Euclidean error


These seem to be the minimal, necessary elements to make a modeling problem well posed.

### Implementation

For the sake of an easy example, I reused code for the classic van der Pol nonlinear oscillator to create the model in model.py. Since the internal details are meant to be hidden from the user, I simulate this by being clear about what is publically exported from the script according to the list of givens in the scenario YAML file.

In particular, the exports are:

__all__ = ['F', 'target_dom', 'compute', 'L2_tol']


For easy reconfiguration of the test problem, the model setup loads values for the ODEs’ free parameters and domain D from setup.yml:

pars:
eps: 0.1
a: 0.5
target_dom:
x: [1, 1.3]
y: [-0.9, -0.75]
error_tol: 0.002


I wrote a couple of sanity check tests and demonstrations in that module to check that I had everything set up right before attempting the model development scenario. Those were used to generate the above diagram of the local, nonlinear vector field in the chosen sub-domain. The diagram doesn’t show arrow heads right now, but they are pointing to the left. The thick line shows a numerically integrated trajectory that starts from the center of the domain, with the individual sample points shown as black dots.

## Other project preparations

At this point, it was time to initialize a new Wing IDE python project in a new folder, and clone the gist I began online into it. I then configured the Git integration tool within Wing and synchronized all the files between the local folder and GitHub.

## Steps towards a goal: a workflow of test-driven development

### The ‘StudyContext’

For want of a better name (I’ve considered several), a StudyContext (SC) is what I call the primary unit block of activity in my workflow. It is meant to represent a frame of reference, a situation, scene, or setting for studying a given problem or question. Every SC wraps up a goal-directed project activity, and I organize my file system around the SC tasks towards achieving whatever goal. The SC goal could be as high level as “learning” or “discovery” providing the endpoint criteria can be specified in some way as a UAC. Thus, in some cases, it may be qualitative rather than quantitative. This follows the same principle as a “sprint” in Agile software development. Equally, an SC should be chosen to be as concrete as possible, and achievable in a reasonably short time. If the problem will take much longer to solve, then multiple SCs are recommended. A set of unresolved SCs become the project’s ‘backlog’ of work to be allocated to investigators!

### Flexible workflow

Since, by the nature of early model development, a lot is unknown, it makes sense not to enforce a rigid workflow. The only rules that I need are 1) to set up the goals of an SC before starting, and 2) to immediately code representations of the goals and the objects needed to create an executable test of the goals. As with test-driven software development, the test is expected to fail before the actual model development work is done. But the existence of the test is an objective goal to define the UAC of the SC.

### Open ontology

By the same token, it also makes sense not to limit the user to a small set of workflow object types. Activities, inputs, and outputs will vary greatly depending on the problem and the user. Thus, the ontology for demarking these is kept general and simple.

To get into the right frame of mind, let’s quickly recap the stereotypical structure of an academic mathematics article. There is an introduction that sets up a problem that will be addressed and its background context. This provides the ‘given’ data for the problem and allows assumptions and definitions to be made. From there on, it’s very context dependent, but we typically see a logical progression of hypotheses, questions, remarks, calculations, validations, lemmas, theorems, and their proofs. Hopefully, at the end of the article, conclusions can be drawn from all of this work that answers the original question. Unlike most articles in science, a math article unashamedly uses structured markup throughout the document to allow the reader to keep track of the beginning and end of each logical step, and what its role is.

With this presentation model in mind, a SC script should have the following structure:

• Definitions / declarations
• Given data or functions (probably a necessity)
• (optional) Assumptions
• (optional) Hypothesis, Question, or Need – if not obvious from Goal statement
• Goal or Target
• (optional) Constraints
• UAC statement
• (one or more) work Steps of various types ending in a Terminal Step for the script

This list is a work in progress. Discussion is welcome.

Further definitions that are marked ‘optional’ can be made, where appropriate, later in the document, but the others must be made once and at the top of the file.

Thus, the primary, ongoing project activity is to iteratively convert as much of the plain-text definition of the scenario to executable code blocks associated with each part of the definition. The success of the code itself (since it is a formal system) can be a validation of the logic behind the scenario’s original definition.

At the beginning, all we have is the the scenario metadata YAML, which is pre-pseudo code and hardly a formal specification. As the project develops, the mapping to the code blocks becomes more concrete as we lay down declarations of Steps that we will take to achieve the goal, and write more code to fill in the blocks we need.

### Example Step types

Types of ‘Step’ are entirely user-defined, depending on need. Here are some useful ideas of common types:

• Validation Test (expect a result): graphic visualization or text output
• Diagnostic Test (no expectations): graphic visualization or text output
• Calculation
• Header-like preamble and boilerplate: package imports, etc.
• Footer-like clean-up or admin Step at very end of file, e.g. matplotlib’s show() command

In the true CS tradition of “divide and conquer”, Steps can be defined hierarchically, if needed. Steps that turn into research problems of their own should spawn a new StudyContext to address it (see more about this below). These should be broken out into sub-folders in the file system, and linked to from the metadata in the parent document, and from the header metadata in the child documents back to the parent. Again, management software should assist in tracking and validating such links. For now, we’ll enforce it as a convention.

### StudyContext object declaration syntax

As there is no actual project management program visiting and parsing the SC, there is no need for a formal syntax yet. Using YAML, I have suggested one way to make the declarations using global-level triple-quoted comment strings to begin a Step’s code block. This is just a convention that I’m exploring. In the earlier versions, I referred separately to tests, etc. in the YAML declarations. Later, they are merged as step-test, and so on. In the files presented here, I didn’t attempt to be consistent with how to specify Step types.

"""
studycontext-goal:
tag: define goal
notes: functions with richer diagnostic feedback about spatial errors
"""
# python code block here

"""
studycontext-step:
tag: visualize mesh of errors
notes:
- figure 1
"""
# python code block here


### Commit the project when you’d commit code

As in a software development project, I committed the project code whenever I had reached a local “stopping point” in my day’s work. For instance, when I had cleaned up a script that now runs successfullly, and I’m ready to move on to other tasks; or, when I had finished refactoring some of the library code and everything is back to normal in the status quo.

Neglecting some initial commits where I was fiddling about and syncing with GitHub, the terminal command git log --oneline -10 produces:

4e21abf Step 3: fix files created
423b48e Step 3: visualizations part 1
b854606 Step 2: setup linear model spec, goal, and test conditions; verify initial failure
2215d75 Step 1: scenario implementation setup


Alas, at the time I was doing this, I hadn’t yet chosen the term ‘Step’ to generically refer to all the different blocks of the SC script. The numbered ‘Steps’ in this revision history refer to time checkpoints in the progression of the project overall. My apologies! This is very complicated to present as a rapid communication with full disclosure of my thought process for implementing a first working example.

You can see what changed between commits using commands like:

git diff b854606 2215d75


### Bifurcate the StudyContext when it gets large or complex

During rapid prototyping, I want to use meaningful script file names and have all my figure output data and model input data in the same working directory. This avoids the overhead in coding up systematic access to sub-folders for each of the relevant data types (logs, parameters, figures, raw data, etc.), and avoids needing to track file names through formal file system naming and organization rules. While the argument for a priori organization of the local file system is about avoiding difficulty in search or intuitive comprehension of the filespace, my alternative solution is that we have 1) great algorithmic search capabilities locally available on our computers and 2) good visual search of folders if we KISS: “keep it simple, stupid”.

The SC file in the gist is a single script. But, by the time I finished setting it up, I realized that it really was achieving several different sub-goals, such as diagnostic visualization and initial testing.

That might not matter a lot, but during debugging or tweaking of reusable function definitions, I wanted to re-run parts (Steps) of the script again. I didn’t want to re-run other Steps, some of which are more computationally costly or generate unnecessary output. I have always found myself commenting out parts of scripts while I work on other parts. Instead of doing that, I can recognize that some of the Steps are really Terminal Steps in their own right, and ‘parallel’ to each other as elements of the same StudyContext. It’s easier to break the file into many.

One reason I hadn’t split up files in the past is the proliferation of boilerplate code that must be run to set up any Step. Copying this into each file is not a good solution. Code should be re-used not duplicated. Thus, the next step was to use an object cache and a common library convention in order to move reusable objects and functions to a module from which I can import *. A drawback is that I lose the record of the order in which I created those Steps in the past, but the revision history (and branch tags, for instance) at least retains their provenance.

## Summary so far

We are nearly half-way done, but we haven’t even tackled the mathematical problem yet. Summarizing so far, you’ve seen a mock up of some project metadata attached to code blocks that uses an open ontology for tracking the major elements of a task. Tasks are executed by Steps, and organized into common StudyContexts, which are scripts or folders of scripts, like a python package’s organization. We used Git behind the scenes to commit intuitively meaningful units of work and to help create the “origin story” of the Steps taken. We have mentioned a hypothetical project management software package that would take care of the administration of some of these tasks and enforcement of the logical operations that modify and track the project elements through the project’s lifetime.

Come back soon!