def make_plate_buckle():
r"""Initialize a buckling plate model
Variables (deterministic):
w (in): Plate width
h (in): Plate height
t (in): Plate thickness
m (-): Wavenumber
L (kips): Applied (compressive) load;
uniformly applied along top and bottom edges
Variables (random):
E (kips/in^2): Elasticity
mu (-): Poisson's ratio
Outputs:
k_cr (-): Prefactor for buckling stress
g_buckle (kips/in^2): Buckling limit state:
critical stress - applied stress
"""
md = (
Model("Plate Buckling")
>> cp_vec_function(
fun=lambda df: df_make(
k_cr=(df.m*df.h/df.w + df.w/df.m/df.h)**2
),
var=["w", "h", "m"],
out=["k_cr"],
)
>> cp_vec_function(
fun=lambda df: df_make(
g_buckle=df.k_cr * pi**2/12 * df.E / (1 - df.mu**2) * (df.t/df.h)**2
- df.L / df.t / df.w
),
var=["k_cr", "t", "h", "w", "E", "mu", "L"],
out=["g_buckle"],
name="limit state",
)
>> cp_bounds(
t=(0.5 * THICKNESS, 2 * THICKNESS),
h=(6, 18),
w=(6, 18),
m=(1, 5),
L=(LOAD / 2, LOAD * 2),
)
>> cp_marginals(
E=marg_fit("norm", df_stang.E),
mu=marg_fit("beta", df_stang.mu),
)
>> cp_copula_gaussian(df_data=df_stang)
)
return md
def make_plate_buckle():
md = (gr.Model("Plate Buckling") >> gr.cp_function(
fun=function_buckle_state,
var=["t", "h", "w", "E", "mu", "L"],
out=["g_buckle"],
name="limit state",
) >> gr.cp_bounds(
t=(0.5 * THICKNESS, 2 * THICKNESS),
h=(6, 18),
w=(6, 18),
L=(LOAD / 2, LOAD * 2),
) >> gr.cp_marginals(E=gr.marg_named(df_stang.E, "norm"),
mu=gr.marg_named(df_stang.mu, "beta")) >>
gr.cp_copula_gaussian(df_data=df_stang))
return md
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
var_applied = ["L", "w", "t"]
out_applied = ["sig_app"]
def fun_limit(x):
sig_cr, sig_app = x
return sig_cr - sig_app
var_limit = ["sig_cr", "sig_app"]
out_limit = ["safety"]
## Build model
md_plate = (
gr.Model("Plate under buckling load") >> gr.cp_function(
fun=fun_critical, var=var_critical, out=out_critical, name="Critical")
>> gr.cp_function(
fun=fun_applied, var=var_applied, out=out_applied, name="Applied") >>
gr.cp_function(fun=fun_limit, var=var_limit, out=out_limit,
name="Safety") >> gr.cp_bounds( # Deterministic variables
t=(0.03, 0.12), # Thickness
w=(6, 18), # Width
h=(6, 18), # Height
L=(2.5e-1, 4.0e-1), # Load
) >> gr.cp_marginals( # Random variables
E=gr.marg_gkde(df_stang.E),
mu=gr.marg_gkde(df_stang.mu)) >>
gr.cp_copula_gaussian(df_data=df_stang)) # Dependence