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1361 lines (997 loc) · 36.2 KB
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"""
Code from Modeling and Simulation in Python.
Copyright 2017 Allen Downey
License: https://creativecommons.org/licenses/by/4.0)
"""
import logging
logger = logging.getLogger(name='modsim.py')
#TODO: Make this Python 3.7 when conda is ready
# make sure we have Python 3.6 or better
import sys
if sys.version_info < (3, 6):
logger.warn('modsim.py depends on Python 3.6 features.')
import inspect
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import scipy
import sympy
import seaborn as sns
sns.set(style='white', font_scale=1.2)
import pint
UNITS = pint.UnitRegistry()
Quantity = UNITS.Quantity
# expose some names so we can use them without dot notation
from copy import copy
from numpy import sqrt, log, exp, pi
from pandas import DataFrame, Series
from time import sleep
from scipy.interpolate import interp1d
from scipy.interpolate import InterpolatedUnivariateSpline
from scipy.integrate import odeint
from scipy.integrate import solve_ivp
from scipy.optimize import leastsq
from scipy.optimize import minimize_scalar
import scipy.optimize
def flip(p=0.5):
"""Flips a coin with the given probability.
p: float 0-1
returns: boolean (True or False)
"""
return np.random.random() < p
# For all the built-in Python functions that do math,
# let's use the NumPy version instead.
abs = np.abs
min = np.min
max = np.max
pow = np.power
sum = np.sum
round = np.round
def cart2pol(x, y, z=None):
"""Convert Cartesian coordinates to polar.
x: number or sequence
y: number or sequence
z: number or sequence (optional)
returns: theta, rho OR theta, rho, z
"""
x = np.asarray(x)
y = np.asarray(y)
# TODO: use hypot?
rho = np.sqrt(x**2 + y**2)
theta = np.arctan2(y, x)
if z is None:
return theta, rho
else:
return theta, rho, z
def pol2cart(theta, rho, z=None):
"""Convert polar coordinates to Cartesian.
theta: number or sequence
rho: number or sequence
z: number or sequence (optional)
returns: x, y OR x, y, z
"""
if hasattr(theta, 'units'):
if theta.units == UNITS.degree:
theta = theta.to(UNITS.radian)
if theta.units != UNITS.radian:
msg = """In pol2cart, theta must be either a number or
a Quantity in degrees or radians."""
raise ValueError(msg)
x = rho * np.cos(theta)
y = rho * np.sin(theta)
if z is None:
return x, y
else:
return x, y, z
def linspace(start, stop, num=50, **options):
"""Returns an array of evenly-spaced values in the interval [start, stop].
start: first value
stop: last value
num: number of values
Also accepts the same keyword arguments as np.linspace. See
https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html
returns: array or Quantity
"""
underride(options, dtype=np.float64)
# see if either of the arguments has units
units = getattr(start, 'units', None)
units = getattr(stop, 'units', units)
array = np.linspace(start, stop, num, **options)
if units:
array = array * units
return array
def linrange(start=0, stop=None, step=1, **options):
"""Returns an array of evenly-spaced values in the interval [start, stop].
This function works best if the space between start and stop
is divisible by step; otherwise the results might be surprising.
By default, the last value in the array is `stop-step`
(at least approximately).
If you provide the keyword argument `endpoint=True`,
the last value in the array is `stop`.
start: first value
stop: last value
step: space between values
returns: array or Quantity
"""
if stop is None:
stop = start
start = 0
# TODO: what breaks if we don't make the dtype float?
#underride(options, endpoint=True, dtype=np.float64)
underride(options, endpoint=False)
# see if any of the arguments has units
units = getattr(start, 'units', None)
units = getattr(stop, 'units', units)
units = getattr(step, 'units', units)
n = np.round((stop - start) / step)
if options['endpoint']:
n += 1
array = np.full(int(n), magnitude(step))
array[0] = magnitude(start)
array = np.cumsum(array)
if units:
array = array * units
return array
def magnitude(x):
"""Returns the magnitude of a Quantity or number.
x: Quantity or number
returns: number
"""
return x.magnitude if isinstance(x, Quantity) else x
def magnitudes(x):
"""Returns the magnitude of a Quantity or number, or sequence.
x: Quantity or number, or sequence
returns: number
"""
try:
return [magnitude(elt) for elt in x]
except TypeError: # not iterable
return magnitude(x)
def units(x):
"""Returns the units of a Quantity or number.
x: Quantity or number
returns: Unit object or 1
"""
return x.units if isinstance(x, Quantity) else 1
def remove_units(series):
"""Removes units from the values in a Series.
Only removes units from top-level values;
does not traverse nested values.
returns: new Series object
"""
res = copy(series)
print(type(res))
for label, value in res.iteritems():
res[label] = magnitude(value)
return res
def require_units(x, units):
"""Apply units to `x`, if necessary.
x: Quantity or number
units: Pint Units object
returns: Quantity
"""
if isinstance(x, Quantity):
return x.to(units)
else:
return Quantity(x, units)
def fit_leastsq(error_func, params, *args, **options):
"""Find the parameters that yield the best fit for the data.
`params` can be a sequence, array, or Series
Whatever arguments are provided are passed along to `error_func`
error_func: function that computes a sequence of errors
params: initial guess for the best parameters
data: the data to be fit; will be passed to min_fun
options: any other arguments are passed to leastsq
"""
# if any of the params are quantities, strip the units
x0 = [magnitude(x) for x in params]
# override `full_output` so we get a message if something goes wrong
options['full_output'] = True
# run leastsq
with units_off():
best_params, cov_x, infodict, mesg, ier = leastsq(error_func,
x0=x0, args=args, **options)
details = ModSimSeries(infodict)
details.set(cov_x=cov_x, mesg=mesg, ier=ier)
# if we got a Params object, we should return a Params object
if isinstance(params, Params):
best_params = Params(Series(best_params, params.index))
# return the best parameters and details
return best_params, details
def min_bounded(min_func, bounds, *args, **options):
"""Finds the input value that minimizes `min_func`.
Wrapper for https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize_scalar.html
min_func: computes the function to be minimized
bounds: sequence of two values, lower and upper bounds of the
range to be searched
args: any additional positional arguments are passed to min_func
options: any keyword arguments are passed as options to minimize_scalar
returns: ModSimSeries object
"""
# try:
# print(bounds[0])
# min_func(bounds[0], *args)
# except Exception as e:
# msg = """Before running scipy.integrate.min_bounded, I tried
# running the slope function you provided with the
# initial conditions in system and t=0, and I got
# the following error:"""
# logger.error(msg)
# raise(e)
underride(options, xatol=1e-3)
# TODO: Do we need to remove units from bounds?
with units_off():
res = minimize_scalar(min_func,
bracket=bounds,
bounds=bounds,
args=args,
method='bounded',
options=options)
if not res.success:
msg = """scipy.optimize.minimize_scalar did not succeed.
The message it returned is %s""" % res.message
raise Exception(msg)
return ModSimSeries(res)
def max_bounded(max_func, bounds, *args, **options):
"""Finds the input value that maximizes `max_func`.
Wrapper for https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize_scalar.html
min_func: computes the function to be maximized
bounds: sequence of two values, lower and upper bounds of the
range to be searched
args: any additional positional arguments are passed to max_func
options: any keyword arguments are passed as options to minimize_scalar
returns: ModSimSeries object
"""
def min_func(*args):
return -max_func(*args)
res = min_bounded(min_func, bounds, *args, **options)
# we have to negate the function value before returning res
res.fun = -res.fun
return res
def minimize(min_func, x0, *args, **options):
"""Finds the input value that minimizes `min_func`.
Wrapper for https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html
min_func: computes the function to be minimized
x0: initial guess
args: any additional positional arguments are passed to min_func
options: any keyword arguments are passed as options to minimize_scalar
returns: ModSimSeries object
"""
underride(options, tol=1e-3)
with units_off():
res = scipy.optimize.minimize(min_func, x0, *args, **options)
return ModSimSeries(res)
def run_odeint(system, slope_func, **options):
"""Integrates an ordinary differential equation.
`system` should contain system parameters and `ts`, which
is an array or Series that specifies the time when the
solution will be computed.
system: System object
slope_func: function that computes slopes
returns: TimeFrame
"""
# make sure `system` contains `ts`
if not hasattr(system, 'ts'):
msg = """It looks like `system` does not contain `ts`
as a system variable. `ts` should be an array
or Series that specifies the times when the
solution will be computed:"""
raise ValueError(msg)
# make sure `system` contains `ts`
if not hasattr(system, 'init'):
msg = """It looks like `system` does not contain `init`
as a system variable. `init` should be a State
object that specifies the initial condition:"""
raise ValueError(msg)
# make the system parameters available as globals
unpack(system)
# try running the slope function with the initial conditions
try:
slope_func(init, ts[0], system)
except Exception as e:
msg = """Before running scipy.integrate.odeint, I tried
running the slope function you provided with the
initial conditions in system and t=0, and I got
the following error:"""
logger.error(msg)
raise(e)
# when odeint calls slope_func, it should pass `system` as
# the third argument. To make that work, we have to make a
# tuple with a single element and pass the tuple to odeint as `args`
args = (system,)
# now we're ready to run `odeint` with `init` and `ts` from `system`
with units_off():
array = odeint(slope_func, list(init), ts, args, **options)
# the return value from odeint is an array, so let's pack it into
# a TimeFrame with appropriate columns and index
frame = TimeFrame(array, columns=init.index, index=ts, dtype=np.float64)
return frame
def run_ode_solver(system, slope_func, **options):
"""Computes a numerical solution to a differential equation.
`system` must contain `init` with initial conditions,
`t_0` with the start time, and `t_end` with the end time.
It can contain any other parameters required by the slope function.
`options` can be any legal options of `scipy.integrate.solve_ivp`
system: System object
slope_func: function that computes slopes
returns: TimeFrame
"""
# make sure `system` contains `init`
if not hasattr(system, 'init'):
msg = """It looks like `system` does not contain `init`
as a system variable. `init` should be a State
object that specifies the initial condition:"""
raise ValueError(msg)
# make sure `system` contains `t_end`
if not hasattr(system, 't_end'):
msg = """It looks like `system` does not contain `t_end`
as a system variable. `t_end` should be the
final time:"""
raise ValueError(msg)
# make the system parameters available as globals
unpack(system)
# the default value for t_0 is 0
t_0 = getattr(system, 't_0', 0)
# try running the slope function with the initial conditions
# try:
# slope_func(init, t_0, system)
# except Exception as e:
# msg = """Before running scipy.integrate.solve_ivp, I tried
# running the slope function you provided with the
# initial conditions in `system` and `t=t_0` and I got
# the following error:"""
# logger.error(msg)
# raise(e)
# wrap the slope function to reverse the arguments and add `system`
f = lambda t, y: slope_func(y, t, system)
def wrap_event(event):
"""Wrap the event functions.
Make events terminal by default.
"""
wrapped = lambda t, y: event(y, t, system)
wrapped.terminal = getattr(event, 'terminal', True)
wrapped.direction = getattr(event, 'direction', 0)
return wrapped
# wrap the event functions so they take the right arguments
events = options.pop('events', [])
try:
events = [wrap_event(event) for event in events]
except TypeError:
events = wrap_event(events)
# remove dimensions from the initial conditions.
# we need this because otherwise `init` gets copied into the
# results array along with its units
y_0 = [magnitude(x) for x in init]
# run the solver
with units_off():
bunch = solve_ivp(f, [t_0, t_end], y_0, events=events, **options)
# separate the results from the details
y = bunch.pop('y')
t = bunch.pop('t')
details = ModSimSeries(bunch)
# pack the results into a TimeFrame
results = TimeFrame(np.transpose(y), index=t, columns=init.index)
return results, details
def fsolve(func, x0, *args, **options):
"""Return the roots of the (non-linear) equations
defined by func(x) = 0 given a starting estimate.
Uses scipy.optimize.fsolve, with extra error-checking.
func: function to find the roots of
x0: scalar or array, initial guess
args: additional positional arguments are passed along to fsolve,
which passes them along to func
returns: solution as an array
"""
# make sure we can run the given function with x0
try:
func(x0, *args)
except Exception as e:
msg = """Before running scipy.optimize.fsolve, I tried
running the error function you provided with the x0
you provided, and I got the following error:"""
logger.error(msg)
raise(e)
# make the tolerance more forgiving than the default
underride(options, xtol=1e-6)
x0 = magnitude(x0)
# run fsolve
with units_off():
result = scipy.optimize.fsolve(func, x0, args=args, **options)
return result
def crossings(series, value):
"""Find the labels where the series passes through value.
The labels in series must be increasing numerical values.
series: Series
value: number
returns: sequence of labels
"""
interp = InterpolatedUnivariateSpline(series.index, series-value)
return interp.roots()
def interpolate(series, **options):
"""Creates an interpolation function.
series: Series object
options: any legal options to scipy.interpolate.interp1d
returns: function that maps from the index of the series to values
"""
# TODO: add error checking for nonmonotonicity
if sum(series.index.isnull()):
msg = """The Series you passed to interpolate contains
NaN values in the index, which would result in
undefined behavior. So I'm putting a stop to that."""
raise ValueError(msg)
# make the interpolate function extrapolate past the ends of
# the range, unless `options` already specifies a value for `fill_value`
underride(options, fill_value='extrapolate')
# call interp1d, which returns a new function object
interp_func = interp1d(series.index, series.values, **options)
units = getattr(series, 'units', None)
if units:
return lambda x: Quantity(interp_func(x), units)
else:
return interp_func
def interp_inverse(series, **options):
"""Interpolate the inverse function of a Series.
series: Series object, represents a mapping from `a` to `b`
kind: string, which kind of iterpolation
options: keyword arguments passed to interpolate
returns: interpolation object, can be used as a function
from `b` to `a`
"""
inverse = Series(series.index, index=series.values)
T = interpolate(inverse, **options)
return T
def unpack(series):
"""Make the names in `series` available as globals.
series: Series with variables names in the index
"""
# TODO: Make this a context manager, so the syntax is
# with series:
# and maybe even add an __exit__ that copies changes back
frame = inspect.currentframe()
caller = frame.f_back
caller.f_globals.update(series)
def source_code(obj):
"""Prints the source code for a given object.
obj: function or method object
"""
print(inspect.getsource(obj))
def underride(d, **options):
"""Add key-value pairs to d only if key is not in d.
If d is None, create a new dictionary.
d: dictionary
options: keyword args to add to d
"""
if d is None:
d = {}
for key, val in options.items():
d.setdefault(key, val)
return d
def plot(*args, **options):
"""Makes line plots.
args can be:
plot(y)
plot(y, style_string)
plot(x, y)
plot(x, y, style_string)
options are the same as for pyplot.plot
"""
# TODO: add lines to REPLOT_CACHE
x, y, style = parse_plot_args(*args, **options)
if isinstance(x, pd.DataFrame) or isinstance(y, pd.DataFrame):
raise ValueError("modsimpy.plot can't handle DataFrames.")
if x is None:
if isinstance(y, np.ndarray):
x = np.arange(len(y))
if isinstance(y, pd.Series):
x = y.index
y = y.values
x = magnitudes(x)
y = magnitudes(y)
underride(options, linewidth=3, alpha=0.6)
if style is not None:
lines = plt.plot(x, y, style, **options)
else:
lines = plt.plot(x, y, **options)
return lines
REPLOT_CACHE = {}
def replot(*args, **options):
"""
"""
try:
label = options['label']
except KeyError:
raise ValueError('To use replot, you must provide a label argument.')
axes = plt.gca()
key = (axes, label)
if key not in REPLOT_CACHE:
lines = plot(*args, **options)
if len(lines) != 1:
raise ValueError('Replot only works with a single plotted element.')
REPLOT_CACHE[key] = lines[0]
return lines
line = REPLOT_CACHE[key]
x, y, style = parse_plot_args(*args, **options)
line.set_xdata(x)
line.set_ydata(y)
def parse_plot_args(*args, **options):
"""Parse the args the same way plt.plot does."""
x = None
y = None
style = None
if len(args) == 1:
y = args[0]
elif len(args) == 2:
if isinstance(args[1], str):
y, style = args
else:
x, y = args
elif len(args) == 3:
x, y, style = args
return x, y, style
def contour(df, **options):
"""Makes a contour plot from a DataFrame.
Note: columns and index must be numerical
df: DataFrame
"""
x = results.columns
y = results.index
X, Y = np.meshgrid(x, y)
cs = plt.contour(X, Y, results, **options)
plt.clabel(cs, inline=1, fontsize=10)
def savefig(filename, **options):
"""Save the current figure.
Keyword arguments are passed along to plt.savefig
https://matplotlib.org/api/_as_gen/matplotlib.pyplot.savefig.html
filename: string
"""
print('Saving figure to file', filename)
plt.savefig(filename, **options)
def decorate(**options):
"""Decorate the current axes.
Call decorate with keyword arguments like
decorate(title='Title',
xlabel='x',
ylabel='y')
The keyword arguments can be any of the axis properties
https://matplotlib.org/api/axes_api.html
In addition, you can use `legend=False` to suppress the legend.
And you can use `loc` to indicate the location of the legend
(the default value is 'best')
"""
loc = options.pop('loc', 'best')
if options.pop('legend', True):
legend(loc=loc)
plt.gca().set(**options)
plt.tight_layout()
def legend(**options):
"""Draws a legend only if there is at least one labeled item.
options are passed to plt.legend()
https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html
"""
underride(options, loc='best')
ax = plt.gca()
handles, labels = ax.get_legend_handles_labels()
ax.legend(handles, labels, **options)
def remove_from_legend(bad_labels):
"""Removes some labels from the legend.
bad_labels: sequence of strings
"""
ax = plt.gca()
handles, labels = ax.get_legend_handles_labels()
handle_list, label_list = [], []
for handle, label in zip(handles, labels):
if label not in bad_labels:
handle_list.append(handle)
label_list.append(label)
ax.legend(handle_list, label_list)
# TODO: Either finish SubPlots or remove it
class SubPlots:
def __init__(self, fig, axes_seq):
self.fig = fig
self.axes_seq = axes_seq
self.current_axes_index = 0
def current_axes():
return self.axes_seq(self.current_axes_index)
# TODO: consider making SubPlots iterable
def next_axes(self):
self.current_axes_index += 1
return current_axes()
def subplots(*args, **options):
fig, axes_seq = plt.subplots(*args, **options)
return SubPlots(fig, axes_seq)
def subplot(nrows, ncols, plot_number, **options):
figsize = {(2, 1): (8, 8),
(3, 1): (8, 10)}
key = nrows, ncols
default = (8, 5.5)
width, height = figsize.get(key, default)
plt.subplot(nrows, ncols, plot_number, **options)
fig = plt.gcf()
fig.set_figwidth(width)
fig.set_figheight(height)
class ModSimSeries(pd.Series):
"""Modified version of a Pandas Series,
with a few changes to make it more suited to our purpose.
In particular:
1. I provide a more consistent __init__ method.
2. Series provides two special variables called
`dt` and `T` that cause problems if we try to use those names
as variables. I override them so they can be used variable names.
3. Series doesn't provide a good _repr_html, so it doesn't look
good in Jupyter notebooks.
4. ModSimSeries provides a set() method that takes keyword arguments.
"""
def __init__(self, *args, **kwargs):
"""Initialize a Series.
Note: this cleans up a weird Series behavior, which is
that Series() and Series([]) yield different results.
See: https://github.com/pandas-dev/pandas/issues/16737
"""
if args or kwargs:
underride(kwargs, copy=True)
super().__init__(*args, **kwargs)
else:
super().__init__([], dtype=np.float64)
def _repr_html_(self):
"""Returns an HTML representation of the series.
Mostly used for Jupyter notebooks.
"""
df = pd.DataFrame(self.values, index=self.index, columns=['values'])
return df._repr_html_()
def __copy__(self, deep=True):
series = super().copy(deep=deep)
return self.__class__(series)
copy = __copy__
def set(self, **kwargs):
"""Uses keyword arguments to update the Series in place.
Example: series.set(a=1, b=2)
"""
for name, value in kwargs.items():
self[name] = value
@property
def dt(self):
"""Intercept the Series accessor object so we can use `dt`
as a row label and access it using dot notation.
https://pandas.pydata.org/pandas-docs/stable/generated/pandas.Series.dt.html
"""
return self.loc['dt']
@property
def T(self):
"""Intercept the Series accessor object so we can use `T`
as a row label and access it using dot notation.
https://pandas.pydata.org/pandas-docs/stable/generated/pandas.Series.T.html
"""
return self.loc['T']
def get_first_label(series):
"""Returns the label of the first element."""
return series.index[0]
def get_last_label(series):
"""Returns the label of the first element."""
return series.index[-1]
def get_index_label(series, i):
"""Returns the ith label in the index."""
return series.index[i]
def get_first_value(series):
"""Returns the value of the first element."""
return series.values[0]
def get_last_value(series):
"""Returns the value of the first element."""
return series.values[-1]
def gradient(series):
"""Computes the numerical derivative of a series."""
a = np.gradient(series, series.index)
return TimeSeries(a, series.index)
class TimeSeries(ModSimSeries):
"""Represents a mapping from times to values."""
pass
class SweepSeries(ModSimSeries):
"""Represents a mapping from parameter values to metrics."""
pass
class System(ModSimSeries):
"""Contains system variables and their values.
Takes keyword arguments and stores them as rows.
"""
def __init__(self, *args, **kwargs):
"""Initialize the series.
If there are no positional arguments, use kwargs.
If there is one positional argument, copy it and add
in the kwargs.
More than one positional argument is an error.
"""
if len(args) == 0:
super().__init__(list(kwargs.values()), index=kwargs)
elif len(args) == 1:
super().__init__(*args, copy=True)
self.set(**kwargs)
else:
msg = '__init__() takes at most one positional argument'
raise TypeError(msg)
class State(System):
"""Contains state variables and their values.
Takes keyword arguments and stores them as rows.
"""
pass
class Condition(System):
"""Represents the condition of a system.
Condition objects are often used to construct a System object.
"""
pass
class Params(System):
"""Represents a set of parameters.
"""
pass
def compute_abs_diff(seq):
xs = np.asarray(seq)
diff = np.ediff1d(xs, np.nan)
if isinstance(seq, Series):
return Series(diff, seq.index)
else:
return diff