robclewley/Scenario-local-linear.yml

## readme.md

      
    Raw
  

              readme.md
            
          
    Literate modeling first demo

Start by reading the setup at https://gist.github.com/robclewley/0e54e7becf3bb017ea61#file-scenario-local-linear-yml

  
## index.html
<!doctype html>
<html>
  <head>
  </head>
  <body>
    <iframe src="http://nbviewer.ipython.org/gist/robclewley/0e54e7becf3bb017ea61/studycontext1.ipynb" width="600" height="800"></iframe>
  </body>
</html>

## model.py
"""
Van der Pol equations to generate a nonlinear 2D vector field
and save a black box object
"""
from PyDSTool import *
from PyDSTool.Toolbox import phaseplane as pp
import sys

# public exports
__all__ = ['F', 'target_dom', 'compute', 'L2_tol']

# ----

import yaml
with open('setup.yml') as f:
    setup = yaml.load(f)

L2_tol = setup['error_tol']

#pars = {'eps': 1e-1, 'a': 0.5}
pars = setup['pars']
# target_dom = {'x': [-2.5, 2.5], 'y': [-2, 2]}
target_dom = setup['target_dom']
# Convenience variables
xdom = target_dom['x']
ydom = target_dom['y']
xInterval = Interval('xdom', float, xdom, abseps=1e-3)
yInterval = Interval('ydom', float, ydom, abseps=1e-3)


def build():
    icdict = {'x': pars['a'],
              'y': pars['a'] - pars['a']**3/3}
    xstr = '(y - (x*x*x/3 - x))/eps'
    ystr = 'a - x'

    #DSargs.fnspecs = {'Jacobian': (['t','x','y'],
    #                               """[[(1-x*x)/eps, 1/eps ],
    #                                    [    -1,        0   ]]""")}

    algparams = {'max_pts': 3000, 'init_step': 0.01, 'stiff': True}

    compatGens = findGenSubClasses('ODEsystem')

    class ODEComponent(LeafComponent):
        compatibleGens=compatGens
        targetLangs=targetLangs
    #    compatibleSubcomponents=(sys,)

    vdpC = ODEComponent('vdp_gen')
    vdpC.add([Var(xstr, 'x', domain=target_dom['x'], specType='RHSfuncSpec'),
              Var(ystr, 'y', domain=target_dom['y'], specType='RHSfuncSpec'),
              Par(pars['a'], 'a'), Par(pars['eps'], 'eps')])
    vdpC.flattenSpec()

    MC = ModelConstructor('vdp',
                   generatorspecs={'vdp_gen':
                        {'modelspec': vdpC,
                         'target': 'Vode_ODEsystem',
                         'algparams': algparams}
                       },
                   indepvar=('t',[0,10]),
                   eventtol=1e-5, activateAllBounds=True,
                   withStdEvts={'vdp_gen': True},
                   stdEvtArgs={'term': True,
                               'precise': True,
                               'eventtol': 1e-5},
                   abseps=1e-7,
                   withJac={'vdp_gen': True})

    # withJac option for automatic calc of Jacobian
    return MC.getModel()

try:
    vdp = loadObjects('model.sav')[0]
except:
    vdp = build()
    saveObjects(vdp, 'model.sav')


def F(xarray):
    x, y = xarray
    assert x in xInterval
    assert y in yInterval
    return vdp.Rhs(0, {'x':x, 'y':y}, asarray=True)

def compute(xarray, trajname='test'):
    x, y = xarray
    assert x in xInterval
    assert y in yInterval
    icdict = {'x': x,
              'y': y}
    vdp.set(ics=icdict)
    vdp.compute(trajname, tdata=[0,5])
    pts = vdp.sample(trajname)
    return pts

def sanity_check():
    print target_dom
    x1 = sum(xdom)/2
    y1 = sum(ydom)/2
    print F((x1,y1))
    pts = compute((x1,y1), 'test')
    print len(pts)
    plt.plot(pts['x'], pts['y'], 'k-o')

if __name__ == '__main__':
    vdp_gen = vdp.registry.values()[0]
    pp.plot_PP_vf(vdp_gen, 'x', 'y', subdomain=target_dom,
                  N=20, scale_exp=-1)
    sanity_check()
    plt.show()

## Scenario-local-linear.yml
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


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

## studycontext1.ipynb

      
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## studycontext1.py
"""
studycontext-metadata:
  ID: 1
  tag: solve scenario
"""
import PyDSTool as dst
from PyDSTool.Toolbox import phaseplane as pp
import numpy as np
from matplotlib import pyplot as plt

"""
studycontext-givens:
  tag: import public interface to the target model
"""
from model import F, target_dom, compute, L2_tol

# Convenience variables
xdom = target_dom['x']
ydom = target_dom['y']
xdom_half = sum(xdom)/2
ydom_half = sum(ydom)/2
xdom_width = xdom[1]-xdom[0]
ydom_width = ydom[1]-ydom[0]
xInterval = dst.Interval('xdom', float, xdom, abseps=1e-3)
yInterval = dst.Interval('ydom', float, ydom, abseps=1e-3)

# Convenience function for orientation of new users
# (not explicit part of study context)
def demo_sim():
    print target_dom
    print F((xdom_half,ydom_half))
    pts = compute((xdom_half,ydom_half), 'test')
    print len(pts)
    plt.plot(pts['x'], pts['y'])
    plt.show()


"""
studycontext-step:
  tag: build linear model
  notes:
    - define build function
"""
def build_lin():
    # make local linear system spec
    DSargs = dst.args(name='lin')
    xfn_str = '(x0+yfx*y - x)/taux'
    yfn_str = '(y0+xfy*x - y)/tauy'
    DSargs.varspecs = {'x': xfn_str, 'y': yfn_str}
    DSargs.xdomain = {'x': xdom, 'y': ydom}
    DSargs.pars = {'x0': xdom_half, 'y0': ydom_half,
                   'xfy': 1, 'yfx': 1,
                   'taux': 1, 'tauy': 1}
    DSargs.algparams = {'init_step':0.001,
                        'max_step': 0.001,
                        'max_pts': 10000}
    DSargs.checklevel = 0
    DSargs.tdata = [0, 10]
    DSargs.ics = {'x': xdom_half*1.1, 'y': ydom_half*1.1}
    DSargs.fnspecs = {'Jacobian': (['t', 'x', 'y'],
                                   """[[-1/taux, -yfx/taux],
                                       [-xfy/tauy, -1/tauy]]""")}
    return dst.Generator.Vode_ODEsystem(DSargs)

"""
    - instantiate local model
"""
lin = build_lin()

"""
studycontext-uac:
  tag: define uac
  notes:
    - test sample 30 x 30 grid of ICs in domain
"""
def make_mesh_pts(N=30):
    xsamples = xInterval.uniformSample(xdom_width/N, avoidendpoints=True)
    ysamples = yInterval.uniformSample(ydom_width/N, avoidendpoints=True)
    xmesh, ymesh = np.meshgrid(xsamples,ysamples)
    return xmesh, ymesh, np.dstack((xmesh,ymesh)).reshape(N*N,2)

xmesh5, ymesh5, mesh_pts_test5 = make_mesh_pts(5)
xmesh30, ymesh30, mesh_pts_test30 = make_mesh_pts(30)

"""
    - mirror signature of F for linear model
"""
def LF(pt):
    x, y = pt
    return lin.Rhs(0, {'x':x, 'y':y})

"""
    - L2 metric between vector fields
"""
def metric(pt):
    # could usefully vectorize this
    return pp.dist(LF(pt), F(pt))

def condition(m, tol):
    return m < tol

"""
studycontext-goal:
  tag: define goal
  notes: functions with richer diagnostic feedback about spatial errors
"""
def error_pts(mesh_pts):
    return np.array([metric(pt) for pt in mesh_pts])

def test_goal(mesh_pts, goal_tol=L2_tol):
    errors_array = error_pts(mesh_pts)
    result = condition(np.max(errors_array), goal_tol)
    return dst.args(result=result,
                    errors=errors_array)

"""
studycontext-test:
  tag: initial test that goal fails with a small sample
"""
test = test_goal(mesh_pts_test5)
assert not test.result


"""
studycontext-step:
  tag: visualize mesh of errors
  notes:
    - figure 1
"""
def viz_errors(mesh_pts, errors, fignum, scaling=2):
    fig = plt.figure(fignum)
    fig.clf()
    for pt, err in zip(mesh_pts, errors):
        plt.plot(pt[0], pt[1], 'ko', markersize=scaling*err)
    plt.show()

test30 = test_goal(mesh_pts_test30)
viz_errors(mesh_pts_test30, test30.errors, 1)
fig1 = plt.figure(1)
fig1.savefig('viz_errors1.png')

"""
studycontext-step:
  tag: visualize mesh of linear vectors overlaid with actual vectors
  notes:
    - figure 2
"""
def get_all_Fs(F, mesh_pts):
    return np.array([F(pt) for pt in mesh_pts])

# store globally for later convenience
all_Fs = get_all_Fs(F, mesh_pts_test30)
all_LFs = get_all_Fs(LF, mesh_pts_test30)

def Fmesh(xmesh, ymesh, F):
    dxs, dys = np.zeros(xmesh.shape, float), np.zeros(ymesh.shape, float)
    for xi, x in enumerate(xmesh[0]):
        for yi, y in enumerate(ymesh.T[0]):
            dx, dy = F((x,y))
            # note order of indices
            dxs[yi,xi] = dx
            dys[yi,xi] = dy
    return dxs, dys

Fmeshes5 = Fmesh(xmesh5, ymesh5, F)
LFmeshes5 = Fmesh(xmesh5, ymesh5, LF)


Fmeshes30 = Fmesh(xmesh30, ymesh30, F)
LFmeshes30 = Fmesh(xmesh30, ymesh30, LF)

def viz_VF(Fmeshes, meshes, fignum, col):
    plt.figure(fignum)
    Fxmesh, Fymesh = Fmeshes
    xmesh, ymesh = meshes
    plt.quiver(xmesh, ymesh, Fxmesh, Fymesh, angles='xy', pivot='middle',
               scale=10, lw=0.5, width=0.005*max(xdom_width,ydom_width),
               minshaft=2, minlength=0.1,
               units='inches', color=col)

# test case
viz_VF(Fmeshes5, (xmesh5, ymesh5), 2, 'r')
viz_VF(LFmeshes5, (xmesh5, ymesh5), 2, 'k')

#viz_VF(Fmeshes30, (xmesh30, ymesh30), 2, 'r')

fig2 = plt.figure(2)
fig2.savefig('viz_VF_overlay1.png')

## viz_errors1.png

      
    Raw
  

              viz_errors1.png
            
          
## viz_VF_overlay1.png

      
    Raw
  

              viz_VF_overlay1.png
	<!doctype html>
	<html>
	<head>
	</head>
	<body>
	<iframe src="http://nbviewer.ipython.org/gist/robclewley/0e54e7becf3bb017ea61/studycontext1.ipynb" width="600" height="800"></iframe>
	</body>
	</html>
	"""
	Van der Pol equations to generate a nonlinear 2D vector field
	and save a black box object
	"""
	from PyDSTool import *
	from PyDSTool.Toolbox import phaseplane as pp
	import sys

	# public exports
	__all__ = ['F', 'target_dom', 'compute', 'L2_tol']

	# ----

	import yaml
	with open('setup.yml') as f:
	setup = yaml.load(f)

	L2_tol = setup['error_tol']

	#pars = {'eps': 1e-1, 'a': 0.5}
	pars = setup['pars']
	# target_dom = {'x': [-2.5, 2.5], 'y': [-2, 2]}
	target_dom = setup['target_dom']
	# Convenience variables
	xdom = target_dom['x']
	ydom = target_dom['y']
	xInterval = Interval('xdom', float, xdom, abseps=1e-3)
	yInterval = Interval('ydom', float, ydom, abseps=1e-3)


	def build():
	icdict = {'x': pars['a'],
	'y': pars['a'] - pars['a']**3/3}
	xstr = '(y - (xxx/3 - x))/eps'
	ystr = 'a - x'

	#DSargs.fnspecs = {'Jacobian': (['t','x','y'],
	# """[[(1-x*x)/eps, 1/eps ],
	# [ -1, 0 ]]""")}

	algparams = {'max_pts': 3000, 'init_step': 0.01, 'stiff': True}

	compatGens = findGenSubClasses('ODEsystem')

	class ODEComponent(LeafComponent):
	compatibleGens=compatGens
	targetLangs=targetLangs
	# compatibleSubcomponents=(sys,)

	vdpC = ODEComponent('vdp_gen')
	vdpC.add([Var(xstr, 'x', domain=target_dom['x'], specType='RHSfuncSpec'),
	Var(ystr, 'y', domain=target_dom['y'], specType='RHSfuncSpec'),
	Par(pars['a'], 'a'), Par(pars['eps'], 'eps')])
	vdpC.flattenSpec()

	MC = ModelConstructor('vdp',
	generatorspecs={'vdp_gen':
	{'modelspec': vdpC,
	'target': 'Vode_ODEsystem',
	'algparams': algparams}
	},
	indepvar=('t',[0,10]),
	eventtol=1e-5, activateAllBounds=True,
	withStdEvts={'vdp_gen': True},
	stdEvtArgs={'term': True,
	'precise': True,
	'eventtol': 1e-5},
	abseps=1e-7,
	withJac={'vdp_gen': True})

	# withJac option for automatic calc of Jacobian
	return MC.getModel()

	try:
	vdp = loadObjects('model.sav')[0]
	except:
	vdp = build()
	saveObjects(vdp, 'model.sav')


	def F(xarray):
	x, y = xarray
	assert x in xInterval
	assert y in yInterval
	return vdp.Rhs(0, {'x':x, 'y':y}, asarray=True)

	def compute(xarray, trajname='test'):
	x, y = xarray
	assert x in xInterval
	assert y in yInterval
	icdict = {'x': x,
	'y': y}
	vdp.set(ics=icdict)
	vdp.compute(trajname, tdata=[0,5])
	pts = vdp.sample(trajname)
	return pts

	def sanity_check():
	print target_dom
	x1 = sum(xdom)/2
	y1 = sum(ydom)/2
	print F((x1,y1))
	pts = compute((x1,y1), 'test')
	print len(pts)
	plt.plot(pts['x'], pts['y'], 'k-o')

	if __name__ == '__main__':
	vdp_gen = vdp.registry.values()[0]
	pp.plot_PP_vf(vdp_gen, 'x', 'y', subdomain=target_dom,
	N=20, scale_exp=-1)
	sanity_check()
	plt.show()
	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
	pars:
	eps: 0.1
	a: 0.5
	target_dom:
	x: [1, 1.3]
	y: [-0.9, -0.75]
	error_tol: 0.002
	"""
	studycontext-metadata:
	ID: 1
	tag: solve scenario
	"""
	import PyDSTool as dst
	from PyDSTool.Toolbox import phaseplane as pp
	import numpy as np
	from matplotlib import pyplot as plt

	"""
	studycontext-givens:
	tag: import public interface to the target model
	"""
	from model import F, target_dom, compute, L2_tol

	# Convenience variables
	xdom = target_dom['x']
	ydom = target_dom['y']
	xdom_half = sum(xdom)/2
	ydom_half = sum(ydom)/2
	xdom_width = xdom[1]-xdom[0]
	ydom_width = ydom[1]-ydom[0]
	xInterval = dst.Interval('xdom', float, xdom, abseps=1e-3)
	yInterval = dst.Interval('ydom', float, ydom, abseps=1e-3)

	# Convenience function for orientation of new users
	# (not explicit part of study context)
	def demo_sim():
	print target_dom
	print F((xdom_half,ydom_half))
	pts = compute((xdom_half,ydom_half), 'test')
	print len(pts)
	plt.plot(pts['x'], pts['y'])
	plt.show()


	"""
	studycontext-step:
	tag: build linear model
	notes:
	- define build function
	"""
	def build_lin():
	# make local linear system spec
	DSargs = dst.args(name='lin')
	xfn_str = '(x0+yfx*y - x)/taux'
	yfn_str = '(y0+xfy*x - y)/tauy'
	DSargs.varspecs = {'x': xfn_str, 'y': yfn_str}
	DSargs.xdomain = {'x': xdom, 'y': ydom}
	DSargs.pars = {'x0': xdom_half, 'y0': ydom_half,
	'xfy': 1, 'yfx': 1,
	'taux': 1, 'tauy': 1}
	DSargs.algparams = {'init_step':0.001,
	'max_step': 0.001,
	'max_pts': 10000}
	DSargs.checklevel = 0
	DSargs.tdata = [0, 10]
	DSargs.ics = {'x': xdom_half1.1, 'y': ydom_half1.1}
	DSargs.fnspecs = {'Jacobian': (['t', 'x', 'y'],
	"""[[-1/taux, -yfx/taux],
	[-xfy/tauy, -1/tauy]]""")}
	return dst.Generator.Vode_ODEsystem(DSargs)

	"""
	- instantiate local model
	"""
	lin = build_lin()

	"""
	studycontext-uac:
	tag: define uac
	notes:
	- test sample 30 x 30 grid of ICs in domain
	"""
	def make_mesh_pts(N=30):
	xsamples = xInterval.uniformSample(xdom_width/N, avoidendpoints=True)
	ysamples = yInterval.uniformSample(ydom_width/N, avoidendpoints=True)
	xmesh, ymesh = np.meshgrid(xsamples,ysamples)
	return xmesh, ymesh, np.dstack((xmesh,ymesh)).reshape(N*N,2)

	xmesh5, ymesh5, mesh_pts_test5 = make_mesh_pts(5)
	xmesh30, ymesh30, mesh_pts_test30 = make_mesh_pts(30)

	"""
	- mirror signature of F for linear model
	"""
	def LF(pt):
	x, y = pt
	return lin.Rhs(0, {'x':x, 'y':y})

	"""
	- L2 metric between vector fields
	"""
	def metric(pt):
	# could usefully vectorize this
	return pp.dist(LF(pt), F(pt))

	def condition(m, tol):
	return m < tol

	"""
	studycontext-goal:
	tag: define goal
	notes: functions with richer diagnostic feedback about spatial errors
	"""
	def error_pts(mesh_pts):
	return np.array([metric(pt) for pt in mesh_pts])

	def test_goal(mesh_pts, goal_tol=L2_tol):
	errors_array = error_pts(mesh_pts)
	result = condition(np.max(errors_array), goal_tol)
	return dst.args(result=result,
	errors=errors_array)

	"""
	studycontext-test:
	tag: initial test that goal fails with a small sample
	"""
	test = test_goal(mesh_pts_test5)
	assert not test.result


	"""
	studycontext-step:
	tag: visualize mesh of errors
	notes:
	- figure 1
	"""
	def viz_errors(mesh_pts, errors, fignum, scaling=2):
	fig = plt.figure(fignum)
	fig.clf()
	for pt, err in zip(mesh_pts, errors):
	plt.plot(pt[0], pt[1], 'ko', markersize=scaling*err)
	plt.show()

	test30 = test_goal(mesh_pts_test30)
	viz_errors(mesh_pts_test30, test30.errors, 1)
	fig1 = plt.figure(1)
	fig1.savefig('viz_errors1.png')

	"""
	studycontext-step:
	tag: visualize mesh of linear vectors overlaid with actual vectors
	notes:
	- figure 2
	"""
	def get_all_Fs(F, mesh_pts):
	return np.array([F(pt) for pt in mesh_pts])

	# store globally for later convenience
	all_Fs = get_all_Fs(F, mesh_pts_test30)
	all_LFs = get_all_Fs(LF, mesh_pts_test30)

	def Fmesh(xmesh, ymesh, F):
	dxs, dys = np.zeros(xmesh.shape, float), np.zeros(ymesh.shape, float)
	for xi, x in enumerate(xmesh[0]):
	for yi, y in enumerate(ymesh.T[0]):
	dx, dy = F((x,y))
	# note order of indices
	dxs[yi,xi] = dx
	dys[yi,xi] = dy
	return dxs, dys

	Fmeshes5 = Fmesh(xmesh5, ymesh5, F)
	LFmeshes5 = Fmesh(xmesh5, ymesh5, LF)


	Fmeshes30 = Fmesh(xmesh30, ymesh30, F)
	LFmeshes30 = Fmesh(xmesh30, ymesh30, LF)

	def viz_VF(Fmeshes, meshes, fignum, col):
	plt.figure(fignum)
	Fxmesh, Fymesh = Fmeshes
	xmesh, ymesh = meshes
	plt.quiver(xmesh, ymesh, Fxmesh, Fymesh, angles='xy', pivot='middle',
	scale=10, lw=0.5, width=0.005*max(xdom_width,ydom_width),
	minshaft=2, minlength=0.1,
	units='inches', color=col)

	# test case
	viz_VF(Fmeshes5, (xmesh5, ymesh5), 2, 'r')
	viz_VF(LFmeshes5, (xmesh5, ymesh5), 2, 'k')

	#viz_VF(Fmeshes30, (xmesh30, ymesh30), 2, 'r')

	fig2 = plt.figure(2)
	fig2.savefig('viz_VF_overlay1.png')