def fit_nls(
df_data,
md=None,
out=None,
var_fix=None,
df_init=None,
verbose=True,
uq_method=None,
**kwargs,
):
r"""Fit a model with Nonlinear Least Squares (NLS)
Estimate best-fit variable levels with nonlinear least squares (NLS), and
return an executable model with those frozen best-fit levels. Optionally,
fit a distribution on the parameters to quantify parametric uncertainty.
Note: This is a *synonym* for eval_nls(); see the documentation for
eval_nls() for keyword argument options available beyond those listed here.
Args:
df_data (DataFrame): Data for estimating best-fit variable levels.
Variables not found in df_data optimized for fitting.
md (gr.Model): Model to analyze. All model variables
selected for fitting must be bounded or random. Deterministic
variables may have semi-infinite bounds.
var_fix (list or None): Variables to fix to nominal levels. Note that
variables with domain width zero will automatically be fixed.
df_init (DataFrame): Initial guesses for parameters; overrides n_restart
n_restart (int): Number of restarts to try; the first try is at
the nominal conditions of the model. Returned model will use
the least-error parameter set among restarts tested.
n_maxiter (int): Optimizer maximum iterations
verbose (bool): Print best-fit parameters to console?
uq_method (str OR None): If string, select method to quantify parameter
uncertainties. If None, provide best-fit values only. Methods:
uq_method = "linpool": assume normal errors; linearly approximate
parameter effects; equally pool variance matrices for each output
Returns:
gr.Model: Model for evaluation with best-fit variables frozen to
optimized levels.
Examples:
>>> import grama as gr
>>> from grama.data import df_trajectory_windowed
>>> from grama.models import make_trajectory_linear
>>> X = gr.Intention()
>>>
>>> md_trajectory = make_trajectory_linear()
>>> md_fitted = (
>>> df_trajectory_windowed
>>> >> gr.ft_nls(
>>> md=md_trajectory,
>>> uq_method="linpool",
>>> )
>>> )
"""
## Check `out` invariants
if out is None:
out = md.out
print("... fit_nls setting out = {}".format(out))
## Check invariants
if md is None:
raise ValueError("Must provide model md")
## Determine variables to be fixed
if var_fix is None:
var_fix = set()
else:
var_fix = set(var_fix)
for var in md.var_det:
wid = md.domain.get_width(var)
if wid == 0:
var_fix.add(var)
## Run eval_nls to fit model parameter values
df_fit = eval_nls(
md,
df_data=df_data,
var_fix=var_fix,
df_init=df_init,
append=True,
verbose=verbose,
**kwargs,
)
## Select best-fit values
df_best = df_fit.sort_values(by="mse",
axis=0).iloc[[0]].reset_index(drop=True)
if verbose:
print(df_fit.sort_values(by="mse", axis=0))
## Determine variables that were fitted
var_fitted = list(set(md.var).intersection(set(df_best.columns)))
var_remain = list(set(md.var).difference(set(var_fitted)))
if len(var_remain) == 0:
raise ValueError("Resulting model is constant!")
## Assemble and return fitted model
if md.name is None:
name = "(Fitted Model)"
else:
name = md.name + " (Fitted)"
## Calibrate parametric uncertainty, if requested
if uq_method == "linpool":
## Precompute data
df_nom = eval_nominal(md, df_det="nom")
df_base = tran_outer(
df_data, concat((df_best[var_fitted], df_nom[var_fix]), axis=1))
df_pred = eval_df(md, df=df_base)
df_grad = eval_grad_fd(md, df_base=df_base, var=var_fitted)
## Pool variance matrices
n_obs = df_data.shape[0]
n_fitted = len(var_fitted)
Sigma_pooled = zeros((n_fitted, n_fitted))
for output in out:
## Approximate sigma_sq
sigma_sq = npsum(
nppow(df_data[output].values - df_pred[output].values,
2)) / (n_obs - n_fitted)
## Approximate (pseudo)-inverse hessian
var_grad = list(map(lambda v: "D" + output + "_D" + v, var_fitted))
Z = df_grad[var_grad].values
Hinv = pinv(Z.T.dot(Z), hermitian=True)
## Add variance matrix to pooled Sigma
Sigma_pooled = Sigma_pooled + sigma_sq * Hinv / n_fitted
## Check model for identifiability
kappa_out = cond(Sigma_pooled)
if kappa_out > 1e10:
warn(
"Model is locally unidentifiable as measured by the " +
"condition number of the pooled covariance matrix; " +
"kappa = {}".format(kappa_out),
RuntimeWarning,
)
## Convert to std deviations and correlation
sigma_comp = npsqrt(diag(Sigma_pooled))
corr_mat = Sigma_pooled / (atleast_2d(sigma_comp).T.dot(
atleast_2d(sigma_comp)))
corr_data = []
I, J = triu_indices(n_fitted, k=1)
for ind in range(len(I)):
i = I[ind]
j = J[ind]
corr_data.append([var_fitted[i], var_fitted[j], corr_mat[i, j]])
df_corr = DataFrame(data=corr_data, columns=["var1", "var2", "corr"])
## Assemble marginals
marginals = {}
for ind, var_ in enumerate(var_fitted):
marginals[var_] = {
"dist": "norm",
"loc": df_best[var_].values[0],
"scale": sigma_comp[ind],
}
## Construct model with Gaussian copula
if len(var_fix) > 0:
md_res = (Model(name) >> cp_function(
lambda x: df_nom[var_fix].values,
var=set(var_remain).difference(var_fix),
out=var_fix,
name="Fix variable levels",
) >> cp_md_det(md=md) >> cp_marginals(**marginals) >>
cp_copula_gaussian(df_corr=df_corr))
else:
md_res = (Model(name) >> cp_md_det(md=md) >> cp_marginals(
**marginals) >> cp_copula_gaussian(df_corr=df_corr))
## Return deterministic model
elif uq_method is None:
md_res = (Model(name) >> cp_function(
lambda x: df_best[var_fitted].values,
var=var_remain,
out=var_fitted,
name="Fix variable levels",
) >> cp_md_det(md=md))
else:
raise ValueError(
"uq_method option {} not recognized".format(uq_method))
return md_res
def eval_nls(
model,
df_data=None,
out=None,
var_fix=None,
df_init=None,
append=False,
tol=1e-6,
ftol=1e-9,
gtol=1e-5,
n_maxiter=100,
n_restart=1,
n_process=1,
method="L-BFGS-B",
seed=None,
verbose=True,
):
r"""Estimate with Nonlinear Least Squares (NLS)
Estimate best-fit variable levels with nonlinear least squares (NLS).
Args:
model (gr.Model): Model to analyze. All model variables
selected for fitting must be bounded or random. Deterministic
variables may have semi-infinite bounds.
df_data (DataFrame): Data for estimating parameters. Variables not
found in df_data optimized in fitting.
out (list or None): Output contributions to consider in computing MSE.
Assumed to be model.out if left as None.
var_fix (list or None): Variables to fix to nominal levels. Note that
variables with domain width zero will automatically be fixed.
df_init (DataFrame): Initial guesses for parameters; overrides n_restart
append (bool): Append metadata? (Initial guess, MSE, optimizer status)
tol (float): Optimizer convergence tolerance
n_maxiter (int): Optimizer maximum iterations
n_restart (int): Number of restarts; beyond n_restart=1 random
restarts are used.
seed (int OR None): Random seed for restarts
verbose (bool): Print messages to console?
Returns:
DataFrame: Results of estimation
Examples:
>>> import grama as gr
>>> from grama.data import df_trajectory_full
>>> from grama.models import make_trajectory_linear
>>>
>>> md_trajectory = make_trajectory_linear()
>>>
>>> df_fit = (
>>> md_trajectory
>>> >> gr.ev_nls(df_data=df_trajectory_full)
>>> )
>>>
>>> print(df_fit)
"""
## Check `out` invariants
if out is None:
out = model.out
if verbose:
print("... eval_nls setting out = {}".format(out))
set_diff = set(out).difference(set(df_data.columns))
if len(set_diff) > 0:
raise ValueError("out must be subset of df_data.columns\n" +
"difference = {}".format(set_diff))
## Determine variables to be fixed
if var_fix is None:
var_fix = set()
else:
var_fix = set(var_fix)
for var in model.var_det:
wid = model.domain.get_width(var)
if wid == 0:
var_fix.add(var)
if verbose:
print("... eval_nls setting var_fix = {}".format(list(var_fix)))
var_fix = list(var_fix)
## Determine variables for evaluation
var_feat = set(model.var).intersection(set(df_data.columns))
if verbose:
print("... eval_nls setting var_feat = {}".format(var_feat))
var_feat = list(var_feat)
## Determine variables for fitting
var_fit = set(model.var).difference(set(var_fix).union(set(var_feat)))
if len(var_fit) == 0:
raise ValueError("No var selected for fitting!\n" +
"Try checking model bounds and df_data.columns.")
var_fit = list(var_fit)
## Separate var_fit into det and rand
var_fit_det = list(set(model.var_det).intersection(var_fit))
var_fit_rand = list(set(model.var_rand).intersection(var_fit))
## Construct bounds, fix var_fit order
var_fit = var_fit_det + var_fit_rand
bounds = []
var_prob = []
for var in var_fit_det:
if not isfinite(model.domain.get_nominal(var)):
var_prob.append(var)
bounds.append(model.domain.get_bound(var))
if len(var_prob) > 0:
raise ValueError(
"all variables to be fitted must finite nominal value\n" +
"offending var = {}".format(var_prob))
for var in var_fit_rand:
bounds.append((
model.density.marginals[var].q(0),
model.density.marginals[var].q(1),
))
## Determine initial guess points
df_nom = eval_nominal(model, df_det="nom", skip=True)
## Use specified initial guess(es)
if not (df_init is None):
# Check invariants
set_diff = list(set(var_fit).difference(set(df_init.columns)))
if len(set_diff) > 0:
raise ValueError("var_fit must be subset of df_init.columns\n" +
"difference = {}".format(set_diff))
# Pull n_restart
n_restart = df_init.shape[0]
## Generate initial guess(es)
else:
df_init = df_nom[var_fit]
if n_restart > 1:
if not (seed is None):
setseed(seed)
## Collect sweep-able deterministic variables
var_sweep = list(
filter(
lambda v: isfinite(model.domain.get_width(v))
& (model.domain.get_width(v) > 0),
model.var_det,
))
## Generate pseudo-marginals
dicts_var = {}
for v in var_sweep:
dicts_var[v] = {
"dist": "uniform",
"loc": model.domain.get_bound(v)[0],
"scale": model.domain.get_width(v),
}
## Overwrite model
md_sweep = comp_marginals(model, **dicts_var)
md_sweep = comp_copula_independence(md_sweep)
## Generate random start points
df_rand = eval_sample(
md_sweep,
n=n_restart - 1,
df_det="nom",
skip=True,
)
df_init = concat((df_init, df_rand[var_fit]),
axis=0).reset_index(drop=True)
## Iterate over initial guesses
df_res = DataFrame()
def fun_mp(i):
x0 = df_init[var_fit].iloc[i].values
## Build evaluator
def objective(x):
"""x = [var_fit]"""
## Evaluate model
df_var = tran_outer(
df_data[var_feat],
concat(
(df_nom[var_fix].iloc[[0]],
df_make(**dict(zip(var_fit, x)))),
axis=1,
),
)
df_tmp = eval_df(model, df=df_var)
## Compute joint MSE
return ((df_tmp[out].values - df_data[out].values)**2).mean()
## Run optimization
res = minimize(
objective,
x0,
args=(),
method=method,
jac=False,
tol=tol,
options={
"maxiter": n_maxiter,
"disp": False,
"ftol": ftol,
"gtol": gtol,
},
bounds=bounds,
)
df_tmp = df_make(
**dict(zip(var_fit, res.x)),
**dict(zip(map(lambda s: s + "_0", var_fit), x0)),
)
df_tmp["success"] = [res.success]
df_tmp["message"] = [res.message]
df_tmp["n_iter"] = [res.nit]
df_tmp["mse"] = [res.fun]
return df_tmp
df_res = DataFrame()
for i in range(n_restart):
df_tmp = fun_mp(i)
df_res = concat((df_res, df_tmp), axis=0).reset_index(drop=True)
## Post-process
if append:
return df_res
return df_res[var_fit]
def eval_min(
model,
out_min=None,
out_geq=None,
out_leq=None,
out_eq=None,
method="SLSQP",
tol=1e-6,
n_restart=1,
n_maxiter=50,
seed=None,
df_start=None,
):
r"""Constrained minimization using functions from a model
Perform constrained minimization using functions from a model. Model must
have deterministic variables only.
Wrapper for scipy.optimize.minimize
Args:
model (gr.Model): Model to analyze. All model variables must be
deterministic.
out_min (str): Output to use as minimization objective.
out_geq (None OR list of str): Outputs to use as geq constraints; out >= 0
out_leq (None OR list of str): Outputs to use as leq constraints; out <= 0
out_eq (None OR list of str): Outputs to use as equality constraints; out == 0
method (str): Optimization method; see the documentation for
scipy.optimize.minimize for options.
tol (float): Optimization objective convergence tolerance
n_restart (int): Number of restarts; beyond n_restart=1 random
restarts are used.
df_start (None or DataFrame): Specific starting values to use; overrides
n_restart if non None provided.
Returns:
DataFrame: Results of optimization
Examples:
>>> import grama as gr
>>> md = (
>>> gr.Model("Constrained Rosenbrock")
>>> >> gr.cp_function(
>>> fun=lambda x: (1 - x[0])**2 + 100*(x[1] - x[0]**2)**2,
>>> var=["x", "y"],
>>> out=["c"],
>>> )
>>> >> gr.cp_function(
>>> fun=lambda x: (x[0] - 1)**3 - x[1] + 1,
>>> var=["x", "y"],
>>> out=["g1"],
>>> )
>>> >> gr.cp_function(
>>> fun=lambda x: x[0] + x[1] - 2,
>>> var=["x", "y"],
>>> out=["g2"],
>>> )
>>> >> gr.cp_bounds(
>>> x=(-1.5, +1.5),
>>> y=(-0.5, +2.5),
>>> )
>>> )
>>> md >> gr.ev_min(
>>> out_min="c",
>>> out_leq=["g1", "g2"]
>>> )
"""
## Check that model has only deterministic variables
if model.n_var_rand > 0:
raise ValueError("model must have no random variables")
## Check that objective is in model
if not (out_min in model.out):
raise ValueError("model must contain out_min")
## Check that constraints are in model
if not (out_geq is None):
out_diff = set(out_geq).difference(set(model.out))
if len(out_diff) > 0:
raise ValueError(
"model must contain each out_geq; missing {}".format(out_diff))
if not (out_leq is None):
out_diff = set(out_leq).difference(set(model.out))
if len(out_diff) > 0:
raise ValueError(
"model must contain each out_leq; missing {}".format(out_diff))
if not (out_eq is None):
out_diff = set(out_eq).difference(set(model.out))
if len(out_diff) > 0:
raise ValueError(
"model must contain each out_eq; missing {}".format(out_diff))
## Formulate initial guess
df_nom = eval_nominal(model, df_det="nom", skip=True)
if df_start is None:
df_start = df_nom[model.var]
if n_restart > 1:
if not (seed is None):
setseed(seed)
## Collect sweep-able deterministic variables
var_sweep = list(
filter(
lambda v: isfinite(model.domain.get_width(v))
& (model.domain.get_width(v) > 0),
model.var_det,
))
## Generate pseudo-marginals
dicts_var = {}
for v in var_sweep:
dicts_var[v] = {
"dist": "uniform",
"loc": model.domain.get_bound(v)[0],
"scale": model.domain.get_width(v),
}
## Overwrite model
md_sweep = comp_marginals(model, **dicts_var)
md_sweep = comp_copula_independence(md_sweep)
## Generate random start points
df_rand = eval_sample(
md_sweep,
n=n_restart - 1,
df_det="nom",
skip=True,
)
df_start = concat((df_start, df_rand[model.var]),
axis=0).reset_index(drop=True)
else:
n_restart = df_start.shape[0]
## Factory for wrapping model's output
def make_fun(out, sign=+1):
def fun(x):
df = DataFrame([x], columns=model.var)
df_res = eval_df(model, df)
return sign * df_res[out]
return fun
## Create helper functions for constraints
constraints = []
if not (out_geq is None):
for out in out_geq:
constraints.append({
"type": "ineq",
"fun": make_fun(out),
})
if not (out_leq is None):
for out in out_leq:
constraints.append({
"type": "ineq",
"fun": make_fun(out, sign=-1),
})
if not (out_eq is None):
for out in out_eq:
constraints.append({
"type": "eq",
"fun": make_fun(out),
})
## Parse the bounds for minimize
bounds = list(map(lambda k: model.domain.bounds[k], model.var))
## Run optimization
df_res = DataFrame()
for i in range(n_restart):
x0 = df_start[model.var].iloc[i].values
res = minimize(
make_fun(out_min),
x0,
args=(),
method=method,
jac=False,
tol=tol,
options={
"maxiter": n_maxiter,
"disp": False
},
constraints=constraints,
bounds=bounds,
)
df_opt = df_make(
**dict(zip(model.var, res.x)),
**dict(zip(map(lambda s: s + "_0", model.var), x0)),
)
df_tmp = eval_df(model, df=df_opt)
df_tmp["success"] = [res.success]
df_tmp["message"] = [res.message]
df_tmp["n_iter"] = [res.nit]
df_res = concat((df_res, df_tmp), axis=0).reset_index(drop=True)
return df_res
def eval_nls(
model,
df_data=None,
out=None,
var_fix=None,
append=False,
tol=1e-3,
maxiter=25,
nrestart=1,
):
r"""Estimate with Nonlinear Least Squares (NLS)
Estimate best-fit variable levels with nonlinear least squares (NLS).
Args:
model (gr.Model): Model to analyze. All model variables
selected for fitting must be bounded or random. Deterministic
variables may have semi-infinite bounds.
df_data (DataFrame): Data for estimating parameters. Variables not
found in df_data optimized in fitting.
out (list or None): Output contributions to consider in computing MSE.
Assumed to be model.out if left as None.
var_fix (list or None): Variables to fix to nominal levels. Note that
variables with domain width zero will automatically be fixed.
append (bool): Append metadata? (Initial guess, MSE, optimizer status)
tol (float): Optimizer convergence tolerance
maxiter (int): Optimizer maximum iterations
nrestart (int): Number of restarts; beyond nrestart=1 random
restarts are used.
Returns:
DataFrame: Results of estimation
Examples:
>>> import grama as gr
>>> from grama.data import df_trajectory_full
>>> from grama.models import make_trajectory_linear
>>>
>>> md_trajectory = make_trajectory_linear()
>>>
>>> df_fit = (
>>> md_trajectory
>>> >> gr.ev_nls(df_data=df_trajectory_full)
>>> )
>>>
>>> print(df_fit)
"""
## Check `out` invariants
if out is None:
out = model.out
print("... eval_nls setting out = {}".format(out))
set_diff = set(out).difference(set(df_data.columns))
if len(set_diff) > 0:
raise ValueError("out must be subset of df_data.columns\n" +
"difference = {}".format(set_diff))
## Determine variables to be fixed
if var_fix is None:
var_fix = set()
else:
var_fix = set(var_fix)
for var in model.var_det:
wid = model.domain.get_width(var)
if wid == 0:
var_fix.add(var)
print("... eval_nls setting var_fix = {}".format(list(var_fix)))
## Determine variables for evaluation
var_feat = set(model.var).intersection(set(df_data.columns))
print("... eval_nls setting var_feat = {}".format(list(var_feat)))
## Determine variables for fitting
var_fit = set(model.var).difference(var_fix.union(var_feat))
if len(var_fit) == 0:
raise ValueError("No var selected for fitting!\n" +
"Try checking model bounds and df_data.columns.")
## Separate var_fit into det and rand
var_fit_det = list(set(model.var_det).intersection(var_fit))
var_fit_rand = list(set(model.var_rand).intersection(var_fit))
## Construct bounds, fix var_fit order
var_fit = var_fit_det + var_fit_rand
bounds = []
var_prob = []
for var in var_fit_det:
if not isfinite(model.domain.get_nominal(var)):
var_prob.append(var)
bounds.append(model.domain.get_bound(var))
if len(var_prob) > 0:
raise ValueError(
"all variables to be fitted must finite nominal value\n" +
"offending var = {}".format(var_prob))
for var in var_fit_rand:
bounds.append((
model.density.marginals[var].q(0),
model.density.marginals[var].q(1),
))
## Determine initial guess points
df_nom = eval_nominal(model, df_det="nom", skip=True)
df_init = df_nom[var_fit]
if nrestart > 1:
raise NotImplementedError()
## Iterate over initial guesses
df_res = DataFrame()
for i in range(df_init.shape[0]):
x0 = df_init[var_fit].iloc[i].values
## Build evaluator
def objective(x):
"""x = [var_fit]"""
## Evaluate model
df_var = tran_outer(
df_data[var_feat],
concat(
(df_nom[var_fix].iloc[[0]],
df_make(**dict(zip(var_fit, x)))),
axis=1,
),
)
df_res = eval_df(model, df=df_var)
## Compute joint MSE
return ((df_res[out].values - df_data[out].values)**2).mean()
## Run optimization
res = minimize(
objective,
x0,
args=(),
method="SLSQP",
jac=False,
tol=tol,
options={
"maxiter": maxiter,
"disp": False
},
bounds=bounds,
)
df_res = concat(
(
df_res,
df_make(
**dict(zip(var_fit, res.x)),
**dict(zip(map(lambda s: s + "_0", var_fit), x0)),
status=res.status,
mse=res.fun,
),
),
axis=0,
)
## Post-process
if append:
return df_res
else:
return df_res[var_fit]