# mud package#

## mud.base module#

class mud.base.BayesProblem(X: , y: , domain: = None)[source]#

Bases: object

Sets up Bayesian Inverse Problem for parameter identification

Parameters
• X (ndarray) – 2D array containing parameter samples from an initial distribution. Rows represent each sample while columns represent parameter values.

• y (ndarray) – array containing push-forward values of parameters samples through the forward model. These samples will form the data-likelihood distribution.

• domain (array_like, optional) – 2D Array containing ranges of each parameter value in the parameter space. Note that the number of rows must equal the number of parameters, and the number of columns must always be two, for min/max range.

Examples

>>> from mud.base import BayesProblem
>>> import numpy as np
>>> from scipy.stats import distributions as ds
>>> X = np.random.rand(100,1)
>>> num_obs = 50
>>> Y = np.repeat(X, num_obs, 1)
>>> y = np.ones(num_obs)*0.5 + np.random.randn(num_obs)*0.05
>>> B = BayesProblem(X, Y, np.array([[0,1]]))
>>> B.set_likelihood(ds.norm(loc=y, scale=0.05))
>>> np.round(B.map_point()[0],1)
0.5

estimate()[source]#
fit()[source]#
map_point()[source]#
property n_features#
property n_params#
property n_samples#
plot_obs_space(obs_idx=0, ax=None, y_range=None, aff=1000, ll_opts={'color': 'r', 'label': 'Data-Likelihood', 'linestyle': '-', 'linewidth': 4}, pf_opts={'color': 'g', 'label': 'PF of Posterior', 'linestyle': ':', 'linewidth': 4})[source]#

Plot probability distributions defined over observable space.

plot_param_space(param_idx=0, ax=None, x_range=None, aff=1000, pr_opts={'color': 'b', 'label': 'Prior', 'linestyle': '--', 'linewidth': 4}, ps_opts={'color': 'g', 'label': 'Posterior', 'linestyle': ':', 'linewidth': 4}, map_opts={'color': 'g', 'label': '$\\lambda^\\mathrm{MAP}$'}, true_opts={'color': 'r', 'label': '$\\lambda^{\\dagger}$', 'linestyle': '-.'}, true_val=None)[source]#

Plot probability distributions over parameter space

set_likelihood(distribution, log=False)[source]#
set_prior(distribution=None)[source]#
class mud.base.DensityProblem(X: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], y: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], domain: Optional[Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]] = None, weights: Optional[Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]] = None, normalize: bool = False, pad_domain: float = 0.1)[source]#

Bases: object

Sets up Data-Consistent Inverse Problem for parameter identification

Data-Consistent inversion is a way to infer most likely model parameters using observed data and predicted data from the model.

X#

Array containing parameter samples from an initial distribution. Rows represent each sample while columns represent parameter values. If 1 dimensional input is passed, assumed that it represents repeated samples of a 1-dimensional parameter.

Type

ArrayLike

y#

Array containing push-forward values of parameters samples through the forward model. These samples will form the predicted distribution.

Type

ArrayLike

domain#

Array containing ranges of each parameter value in the parameter space. Note that the number of rows must equal the number of parameters, and the number of columns must always be two, for min/max range.

Type

ArrayLike

weights#

Weights to apply to each parameter sample. Either a 1D array of the same length as number of samples or a 2D array if more than one set of weights is to be incorporated. If so the weights will be multiplied and normalized row-wise, so the number of columns must match the number of samples.

Type

ArrayLike, optional

Examples

Generate test 1-D parameter estimation problem. Model to produce predicted data is the identity map and observed signal comes from true value plus some random gaussian nose.

See mud.examples.identity_uniform_1D_density_prob() for more details

>>> from mud.examples.simple import identity_1D_density_prob as I1D


First we set up a well-posed problem. Note the domain we are looking over contains our true value. We take 1000 samples, use 50 observations, assuming a true value of 0.5 populated with gaussian noise $$\mathcal{N}(0,0.5)$$. Or initial uniform distribution is taken from a $$[0,1]$$ range.

>>> D = I1D(1000, 50, 0.5, 0.05, domain=[0,1])


Estimate mud_point -> Note since WME map used, observed implied to be the standard normal distribution and does not have to be set explicitly from observed data set.

>>> np.round(D.mud_point()[0],1)
0.5


Expectation value of r, ratio of observed and predicted distribution, should be near 1 if predictability assumption is satisfied.

>>> np.round(D.expected_ratio(),0)
1.0


Set up ill-posed problem -> Searching out of range of true value

>>> D = I1D(1000, 50, 0.5, 0.05, domain=[0.6,1])


Mud point will be close as we can get within the range we are searching for

>>> np.round(D.mud_point()[0],1)
0.6


Expectation of r is close to zero since predictability assumption violated.

>>> np.round(D.expected_ratio(),1)
0.0

estimate()[source]#

Estimate

Returns the best estimate for most likely parameter values for the given model data using the data-consistent framework.

Returns

mud_point – Maximal Updated Density (MUD) point.

Return type

np.ndarray

expected_ratio()[source]#

Expectation Value of R

Returns the expectation value of the R, the ratio of the observed to the predicted density values.

(1)#$R = \frac{\pi_{ob}(\lambda)} {\pi_{pred}(\lambda)}$

If the predictability assumption for the data-consistent framework is satisfied, then $$E[R]\approx 1$$.

Returns

expected_ratio – Value of the E(r). Should be close to 1.0.

Return type

float

fit(**kwargs)[source]#

Update Initial Distribution

Constructs the updated distribution by fitting observed data to predicted data with:

(2)#$\pi_{up}(\lambda) = \pi_{in}(\lambda) \frac{\pi_{ob}(Q(\lambda))}{\pi_{pred}(Q(\lambda))}$

Note that if initial, predicted, and observed distributions have not been set before running this method, they will be run with default values. To set specific predicted, observed, or initial distributions use the set_ methods.

Parameters

**kwargs (dict, optional) – If specified, optional arguments are passed to the set_predicted() call in the case that the predicted distribution has not been set yet.

mud_point()[source]#

Maximal Updated Density (MUD) Point

Returns the Maximal Updated Density or MUD point as the parameter sample from the initial distribution with the highest update density value:

(3)#$\lambda^{MUD} := \mathrm{argmax} \pi_{up}(\lambda)$

Note if the updated distribution has not been computed yet, this function will call fit() to compute it.

Returns

mud_point – Maximal Updated Density (MUD) point.

Return type

np.ndarray

property n_features#
property n_params#
property n_samples#
plot_obs_space(obs_idx: int = 0, ax: = None, y_range: Optional[Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]] = None, aff=100, ob_opts={'color': 'r', 'label': '$\\pi_\\mathrm{obs}$', 'linestyle': '-'}, pr_opts={'color': 'b', 'label': '$\\pi_\\mathrm{pred}$', 'linestyle': '-'}, pf_opts={'color': 'k', 'label': '$\\pi_\\mathrm{pf-pr}$', 'linestyle': '-.'})[source]#

Plot probability distributions over parameter space

Observed distribution is plotted using the distribution function passed to set_observed() (or default). The predicted distribution is plotted using the stored predicted distribution function set in set_predicted(). The push-forward of the updated distribution is computed as a gkde on the predicted samples y as well, but using the product of the update ratio (1) and the initial weights as weights.

Parameters
• obs_idx (int, default=0) – Index of observable value to plot.

• ax (matplotlib.axes.Axes, optional) – Axes to plot distributions on. If non specified, a figure will be initialized to plot on.

• y_range (list or np.ndarray, optional) – Range over parameter value to plot over.

• aff (int, default=100) – Number of points to plot within x_range, evenly spaced.

• ob_opts (dict, optional) – Plotting option for observed distribution line. Defaults to {'color':'r', 'linestyle':'-','linewidth':4, 'label':'Observed'}. To suppress plotting, pass in None.

• pr_opts (dict, optional) – Plotting option for predicted distribution line. Defaults to {'color':'b', 'linestyle':'--','linewidth':4, 'label':'PF of Initial'}. To suppress plotting, pass in None.

• pf_opts (dict, optional) – Plotting option for push-forward of updated distribution line. Defaults to {'color':'k', 'linestyle':'-.','linewidth':4, 'label':'PF of Updated'}. To suppress plotting, pass in None.

plot_param_space(param_idx: int = 0, true_val: Optional[Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]] = None, ax: = None, x_range: = None, ylim: = None, pad_ratio: float = 0.05, aff: int = 100, in_opts={'color': 'b', 'label': '$\\pi_\\mathrm{init}$', 'linestyle': '-'}, up_opts={'color': 'k', 'label': '$\\pi_\\mathrm{update}$', 'linestyle': '-.'}, win_opts=None, mud_opts={'color': 'g', 'label': '$\\lambda^\\mathrm{MUD}$'}, true_opts={'color': 'r', 'label': '$\\lambda^{\\dagger}$'})[source]#

Plot probability distributions over parameter space

Initial distribution is plotted using the distribution function passed to set_initial(). The updated distribution is plotted using a weighted gaussian kernel density estimate (gkde) on the initial samples, using the product of the update ratio (1) value times the initial weights as weights for the gkde. The weighted initial is built using a weighted gkde on the initial samples, but only using the initial weights.

Parameters
• param_idx (int, default=0) – Index of parameter value to plot.

• ax (matplotlib.axes.Axes, optional) – Axes to plot distributions on. If non specified, a figure will be initialized to plot on.

• x_range (list or np.ndarray, optional) – Range over parameter value to plot over.

• aff (int, default=100) – Number of points to plot within x_range, evenly spaced.

• in_opts (dict, optional) – Plotting option for initial distribution line. Defaults to {'color':'b', 'linestyle':'--','linewidth':4, 'label':'Initial'}. To suppress plotting, pass in None explicitly.

• up_opts (dict, optional) – Plotting option for updated distribution line. Defaults to {'color':'k', 'linestyle':'-.','linewidth':4, 'label':'Updated'}. To suppress plotting, pass in None explicitly.

• win_opts (dict, optional) – Plotting option for weighted initial distribution line. Defaults to {'color':'g', 'linestyle':'--','linewidth':4, 'label':'Weighted Initial'}. To suppress plotting, pass in None explicitly.

plot_params_2d(x_1: int = 0, x_2: int = 1, y: int = 0, contours: bool = False, colorbar: bool = True, ax: = None, label=True, **kwargs)[source]#

2D plot over two indices of param space, optionally contoured by a y value.

Parameters
• x_1 (int, default=0) – Index of param value (column of X) to use as x value in plot.

• x_2 (int, default=1) – Index of param value (column of X) to use as y value in plot.

• y (int, optional) – Index of observable (column of y) to use as z value in contour plot.

• ax (matplotlib.axes.Axes, optional) – Axes to plot distributions on. If non specified, a figure will be initialized to plot on.

• kwargs (dict, optional) – Additional keyword arguments will be passed to matplotlib.pyplot.scatter.

plot_qoi(idx_x: int = 0, idx_y: int = 1, ax: = None, **kwargs)[source]#

Plot 2D plot over two indices of y space.

Parameters
• idx_x (int, default=0) – Index of observable value (column of y) to use as x value in plot.

• idx_y (int, default=1) – Index of observable value (column of y) to use as y value in plot.

• ax (matplotlib.axes.Axes, optional) – Axes to plot distributions on. If non specified, a figure will be initialized to plot on.

• kwargs (dict, optional) – Additional keyword arguments will be passed to matplotlib.pyplot.scatter.

set_initial(distribution: = None)[source]#

Set initial probability distribution of model parameter values $$\pi_{in}(\lambda)$$.

Parameters

distribution (scipy.stats.rv_continuous, optional) – scipy.stats continuous distribution object from where initial parameter samples were drawn from. If none provided, then a uniform distribution over domain of the density problem is assumed. If no domain is specified for density, then a standard normal distribution $$N(0,1)$$ is assumed.

Warning

Setting initial distribution resets the predicted and updated distributions, so make sure to set the initial first.

set_observed(distribution: ~scipy.stats._distn_infrastructure.rv_continuous = <scipy.stats._distn_infrastructure.rv_continuous_frozen object>)[source]#

Set distribution for the observed data.

The observed distribution is determined from assumptions on the collected data. In the case of using a weighted mean error map on sequential data from a single output, the distribution is stationary with respect to the number data points collected and will always be the standard normal d distribution $N(0,1)$.

Parameters

distribution (scipy.stats.rv_continuous, default=scipy.stats.norm()) – scipy.stats continuous distribution like object representing the likelihood of observed data. Defaults to a standard normal distribution N(0,1).

set_predicted(distribution: = None, bw_method: = None, weights: Optional[Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]] = None, **kwargs)[source]#

Set Predicted Distribution

The predicted distribution over the observable space is equal to the push-forward of the initial through the model $$\pi_{pr}(Q(\lambda)$$. If no distribution is passed, scipy.stats.gaussian_kde is used over the predicted values y to estimate the predicted distribution.

Parameters
• distribution (scipy.stats.rv_continuous, optional) – If specified, used as the predicted distribution instead of the default of using gaussian kernel density estimation on observed values y. This should be a frozen distribution if using scipy, and otherwise be a class containing a pdf() method return the probability density value for an array of values.

• bw_method (str, scalar, or Callable, optional) – Method to use to calculate estimator bandwidth. Only used if distribution is not specified, See documentation for scipy.stats.gaussian_kde for more information.

• weights (ArrayLike, optional) – Weights to use on predicted samples. Note that if specified, set_weights() will be run first to calculate new weights. Otherwise, whatever was previously set as the weights is used. Note this defaults to a weights vector of all 1s for every sample in the case that no weights were passed on upon initialization.

• **kwargs (dict, optional) – If specified, any extra keyword arguments will be passed along to the passed distribution.pdf() function for computing values of predicted samples.

• Note (distribution should be a frozen distribution if using scipy.) –

Warning

If passing a distribution argument, make sure that the initial distribution has been set first, either by having run set_initial() or fit() first.

set_weights(weights: Optional[Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]] = None, normalize: bool = False)[source]#

Set Sample Weights

Sets the weights to use for each sample. Note weights can be one or two dimensional. If weights are two dimensional the weights are combined by multiplying them row wise and normalizing, to give one weight per sample. This combining of weights allows incorporating multiple sets of weights from different sources of prior belief.

Parameters
• weights (np.ndarray, List) – Numpy array or list of same length as the n_samples or if two dimensional, number of columns should match n_samples

• normalize (bool, default=False) – Whether to normalize the weights vector.

Warning

Resetting weights will delete the predicted and updated distribution values in the class, requiring a re-run of adequate set_ methods and/or fit() to reproduce with new weights.

class mud.base.IterativeLinearProblem(A, b, y=None, mu_i=None, cov=None, data_cov=None, idx_order=None)[source]#
get_errors(ref_param)[source]#

Get errors with respect to a reference parameter

plot_chain(ref_param, ax=None, color='k', s=100, **kwargs)[source]#

Plot chain of solutions and contours

plot_chain_error(ref_param, ax=None, alpha=1.0, color='k', label=None, s=100, fontsize=12)[source]#

Plot error over iterations

solve(num_epochs=1, method='mud')[source]#

Iterative Solutions Performs num_epochs iterations of estimates

class mud.base.LinearGaussianProblem(A=array([[1], [1]]), b=None, y=None, mean_i=None, cov_i=None, cov_o=None, alpha=1.0)[source]#

Bases: object

Sets up inverse problems with Linear/Affine Maps

Class provides solutions using MAP, MUD, and least squares solutions to the linear (or affine) problem from p parameters to d observables.

(4)#$M(\mathbf{x}) = A\mathbf{x} + \mathbf{b}, A \in \mathbb{R}^{d\times p}, \mathbf{x}, \in \mathbb{R}^{p}, \mathbf{b}, \in \mathbb{R}^{d},$
A#

2D Array defining linear transformation from model parameter space to model output space.

Type

ArrayLike

y#

1D Array containing observed values of Q(lambda) Array containing push-forward values of parameters samples through the forward model. These samples will form the predicted distribution.

Type

ArrayLike

domain#

Array containing ranges of each parameter value in the parameter space. Note that the number of rows must equal the number of parameters, and the number of columns must always be two, for min/max range.

Type

ArrayLike

weights#

Weights to apply to each parameter sample. Either a 1D array of the same length as number of samples or a 2D array if more than one set of weights is to be incorporated. If so the weights will be multiplied and normalized row-wise, so the number of columns must match the number of samples.

Type

ArrayLike, optional

Examples

Problem set-up:

$\begin{split}A = \begin{bmatrix} 1 & 1 \end{bmatrix}, b = 0, y = 1 \lambda_0 = \begin{bmatrix} 0.25 & 0.25 \end{bmatrix}^T, \Sigma_{init} = \begin{bmatrix} 1 & -0.25 \\ -0.25 & 0.5 \end{bmatrix}, \Sigma_{obs} = \begin{bmatrix} 0.25 \end{bmatrix}\end{split}$
>>> from mud.base import LinearGaussianProblem as LGP
>>> lg1 = LGP(A=np.array([[1, 1]]),
...        b=np.array([[0]]),
...        y=np.array([[1]]),
...        mean_i=np.array([[0.25, 0.25]]).T,
...        cov_i=np.array([[1, -0.25], [-0.25, 0.5]]),
...        cov_o=np.array([[1]]))
>>> lg1.solve('mud')
array([[0.625],
[0.375]])

compute_functionals(X, terms='all')[source]#

For a given input and observed data, compute functionals or individual terms in functionals that are minimized to solve the linear gaussian problem.

property n_features#
property n_params#
property n_samples#
plot_contours(ref=None, subset=None, ax=None, annotate=False, note_loc=None, w=1, label='{i}', plot_opts={'color': 'k', 'fs': 20, 'ls': ':', 'lw': 1}, annotate_opts={'fontsize': 20})[source]#

Plot Linear Map Solution Contours

plot_fun_contours(mesh=None, terms='dc', ax=None, N=250, r=1, **kwargs)[source]#

Plot contour map of functionals being minimized over input space

plot_sol(point='mud', ax=None, label=None, note_loc=None, pt_opts={'color': 'k', 'marker': 'o', 's': 100}, ln_opts={'color': 'xkcd:blue', 'lw': 1, 'marker': 'd', 'zorder': 10}, annotate_opts={'backgroundcolor': 'w', 'fontsize': 14})[source]#

Plot solution points

solve(method='mud', output_dim=None)[source]#

Explicitly solve linear problem using given method.

updated_cov(A=None, init_cov=None, data_cov=None)[source]#

We start with the posterior covariance from ridge regression Our matrix R = init_cov^(-1) - X.T @ pred_cov^(-1) @ X replaces the init_cov from the posterior covariance equation. Simplifying, this is given as the following, which is not used due to issues of numerical stability (a lot of inverse operations).

up_cov = (X.T @ np.linalg.inv(data_cov) @ X + R )^(-1) up_cov = np.linalg.inv( X.T@(np.linalg.inv(data_cov) - inv_pred_cov)@X + np.linalg.inv(init_cov) )

We return the updated covariance using a form of it derived which applies Hua’s identity in order to use Woodbury’s identity.

Check using alternate for updated_covariance.

>>> from mud.base import LinearGaussianProblem as LGP
>>> lg2 = LGP(A=np.eye(2))
>>> lg2.updated_cov()
array([[1., 0.],
[0., 1.]])
>>> lg3 = LGP(A=np.eye(2)*2)
>>> lg3.updated_cov()
array([[0.25, 0.  ],
[0.  , 0.25]])
>>> lg3 = LGP(A=np.eye(3)[:, :2]*2)
>>> lg3.updated_cov()
array([[0.25, 0.  ],
[0.  , 0.25]])
>>> lg3 = LGP(A=np.eye(3)[:, :2]*2, cov_i=np.eye(2))
>>> lg3.updated_cov()
array([[0.25, 0.  ],
[0.  , 0.25]])

class mud.base.LinearWMEProblem(operators, data, sigma, y=None, mean_i=None, cov_i=None, cov_o=None, alpha=1.0)[source]#

Linear Inverse Problems using the Weighted Mean Error Map

class mud.base.SpatioTemporalProblem(df=None)[source]#

Bases: object

Class for parameter estimation problems related to spatio-temporal problems. equation models of real world systems. Uses a QoI map of weighted residuals between simulated data and measurements to do inversion

TODO#
Type

Finish

TODO: Finish
property data#
property domain#

Get closest simulated data point to measured data in $l^2$-norm.

get_closest_to_true_vals()[source]#

Get closest simulated data point to noiseless true values in $l^2$-norm.

Note for now no sub-sampling implemented here.

property lam#
property lam_ref#

Load data from a file on disk for a PDE parameter estimation problem.

Parameters

fname (str) – Name of file on disk. If ends in ‘.nc’ then assumed to be netcdf file and the xarray library is used to load it. Otherwise the data is assumed to be pickled data.

Returns

ds – Dictionary containing data from file for PDE problem class

Return type

dict,

property measurements#
measurements_from_reference(ref=None, std_dev=None, seed=None)[source]#

Add noise to a reference solution.

Build QoI Map Using Data and Measurements

property n_params: int#
property n_qoi: int#
property n_samples: int#
property n_sensors: int#
property n_ts: int#
plot_ts(ax=None, samples=None, times=None, sensor_idx=0, max_plot=100, meas_kwargs={}, samples_kwargs={}, alpha=0.1)[source]#

Plot time series data

property sample_dist#
sensor_contour_plot(idx=0, c_vals=None, ax=None, mask=None, fill=True, colorbar=True, **kwargs)[source]#

Plot locations of sensors in space

Plot locations of sensors in space

property true_vals#
validate(check_meas=True, check_true=False)[source]#

Validates if class has been set-up appropriately for inversion

## mud.funs module#

Python console script for mud, installed with pip install . or python setup.py install

mud.funs.data_prob(lam, qoi, qoi_true=None, measurements=None, std_dev=None, sample_dist='uniform', domain=None, lam_ref=None, times=None, sensors=None, idxs=None, method='wme', init_dist=<scipy.stats._distn_infrastructure.rv_continuous_frozen object>)[source]#

Data-Constructed Map Solve

Wrapper around SpatioTemporalProblem class to create and solve a MUD problem by first aggregating observed and siumalated data in a data-constructed qoi map.

mud.funs.iter_lin_solve(A, b, y=None, mean=None, cov=None, data_cov=None, method='mud', num_epochs=1, idx_order=None)[source]#

Iterative Linear Gaussian Problem Solver Entrypoint

mud.funs.iterate(A, b, y, initial_mean, initial_cov, data_cov=None, num_epochs=1, idx=None)[source]#
mud.funs.lin_prob(A, b, y=None, mean=None, cov=None, data_cov=None, alpha=None)[source]#

Linear Gaussian Problem Solver Entrypoint

mud.funs.map_problem(lam, qoi, qoi_true, domain, sd=0.05, num_obs=None, log=False)[source]#

Wrapper around map problem, takes in raw qoi + synthetic data and instantiates solver object

mud.funs.map_sol(A, b, y=None, mean=None, cov=None, data_cov=None, w=1)[source]#

MAP Linear Gaussian Problem Solve

mud.funs.map_sol_with_cov(A, b, y=None, mean=None, cov=None, data_cov=None, w=1)[source]#

MAP Linear Gaussian Problem Solve

mud.funs.mud_problem(lam, qoi, qoi_true, domain, sd=0.05, num_obs=None, split=None, weights=None)[source]#

Wrapper around mud problem, takes in raw qoi + synthetic data and performs WME transformation, instantiates solver object.

mud.funs.mud_sol(A, b, y=None, mean=None, cov=None, data_cov=None)[source]#

For SWE problem, we are inverting N(0,1). This is the default value for data_cov.

mud.funs.mud_sol_with_cov(A, b, y=None, mean=None, cov=None, data_cov=None)[source]#

Doesn’t use R directly, uses new equations. This presents the equation as a rank-k update to the error of the initial estimate.

mud.funs.performEpoch(A, b, y, initial_mean, initial_cov, data_cov=None, idx=None)[source]#
mud.funs.updated_cov(X, init_cov=None, data_cov=None)[source]#

We start with the posterior covariance from ridge regression Our matrix R = init_cov^(-1) - X.T @ pred_cov^(-1) @ X replaces the init_cov from the posterior covariance equation. Simplifying, this is given as the following, which is not used due to issues of numerical stability (a lot of inverse operations).

up_cov = (X.T @ np.linalg.inv(data_cov) @ X + R )^(-1) up_cov = np.linalg.inv( X.T@(np.linalg.inv(data_cov) - inv_pred_cov)@X + np.linalg.inv(init_cov) )

We return the updated covariance using a form of it derived which applies Hua’s identity in order to use Woodbury’s identity.

>>> updated_cov(np.eye(2))
array([[1., 0.],
[0., 1.]])
>>> updated_cov(np.eye(2)*2)
array([[0.25, 0.  ],
[0.  , 0.25]])
>>> updated_cov(np.eye(3)[:, :2]*2, data_cov=np.eye(3))
array([[0.25, 0.  ],
[0.  , 0.25]])
>>> updated_cov(np.eye(3)[:, :2]*2, init_cov=np.eye(2))
array([[0.25, 0.  ],
[0.  , 0.25]])

mud.funs.wme(predictions, data, sd=None)[source]#

Calculates Weighted Mean Error (WME) functional.

Parameters
• predictions (numpy.ndarray of shape (n_samples, n_features)) – Predicted values against which data is compared.

• data (list or numpy.ndarray of shape (n_features, 1)) – Collected (noisy) data

• sd (float, optional) – Standard deviation

Return type

numpy.ndarray of shape (n_samples, 1)

## mud.norm module#

mud.norm.full_functional(operator, inputs, data, initial_mean, initial_cov, observed_mean=0, observed_cov=1)[source]#
mud.norm.inner_product(X, mat)[source]#

Inner-product induced vector norm implementation.

Returns square of norm defined by the inner product (x,x)_C := x^T C^-1 x

Parameters
• X ((M, N) array_like) – Input array. N = number of samples, M = dimension

• mat ((M, M) array_like) – Positive-definite operator which induces the inner product

Returns

Z – inner-product of each column in X with respect to mat

Return type

(N, 1) ndarray

mud.norm.norm_data(operator, inputs, data, observed_mean, observed_cov)[source]#
mud.norm.norm_input(inputs, initial_mean, initial_cov)[source]#
mud.norm.norm_predicted(operator, inputs, initial_mean, initial_cov)[source]#

## mud.plot module#

MUD Plotting Module

Plotting utility functions for visualizing data-sets and distributions related to algorithm implemented within the MUD library.

### Functions#

mud.plot.build_nd_mesh_grid(domain, aff=100)[source]#

Build n-dimensional mesh grid

Parameters
• domain (List[List[int]]) – List of [min, max] ranges for each parameter to construct grid over.

• aff (int, default=100) – Number of points to use in each dimension.

Returns

(mesh, grid_points) – Tuple consistent of n-dimensional grid of points mesh and the list of coordinates $(x_1, x_2, …, x_n)$ for each grid point along.

Return type

Tuple[numpy.ndarray, numpy.ndarray]

Examples

Building 1d mesh”

>>> mesh_1d, pts_1d = build_nd_mesh_grid([[0,1]], aff=5)
>>> mesh_1d
[array([0.  , 0.25, 0.5 , 0.75])]
>>> pts_1d
array([[0.  , 0.25, 0.5 , 0.75]])

mud.plot.make_2d_normal_mesh(N=50, window=1)[source]#
mud.plot.make_2d_unit_mesh(N=50, window=1)[source]#
mud.plot.plotChain(mud_chain, ref_param, color='k', s=100)[source]#
mud.plot.plot_1D_vecs(vecs, markers=None, ax=None, label=True, **kwargs)[source]#

Plot components of 1D vectors

mud.plot.plot_contours(A, ref_param, subset=None, color='k', ls=':', lw=1, fs=20, w=1, s=100, **kwds)[source]#
mud.plot.plot_dist(dist, domain, ax=None, idx=0, source='kde', aff=100, **kwargs)[source]#

Plot a probability distribution over a given domain.

mud.plot.plot_vert_line(ax, x_loc, ylim=None, **kwargs)[source]#

Plot a vertical line on an existing axis at x_loc

mud.plot.save_figure(fname: str, save_path: = None, close_fig: bool = True, **kwargs)[source]#

Save Figure Utility

Utility to save figure to a given folder path if specified.

Parameters
• fname (str) – Name of image, with extension.

• save_path (str, optional) – Directory to save figure to. Assumed to exist. If not specified then the figure is saved to the current working directory.

• close_fig (bool, default=True) – Whether to close the figure after saving it.

• kwargs (dict, optional) – Arguments to pass to savefig()

## mud.preprocessing module#

MUD Pre-Processing Module

All functions for pre-processing QoI data-sets before applying inversion algorithms can be found in this module.

### Functions#

pca - Applly Principle Component Analysis transformation to QoI data.

mud.preprocessing.pca(data: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], n_components: int = 2, **kwargs) [source]#

Apply Principal Component Analysis

Uses sklearn.decomposition.PCA class to perform a truncated PCA transformation on input data using the first n_components principle components. Note sklearn.preprocessing.StandardScaler transformation is applied to the data first.

Parameters
Returns

pca_res – Tuple of (pca, X_train) where pca is the sklearn.decomposition.PCA class with principle component vectors accessible at pca.components_ and X_train being the transformed data-set, which should have same number of rows as original data, but now only n_components columns.

Return type

Examples

For a simple example lets apply the PCA transformation to the identity matrix in 2 dimensions, using first 1 principle component.

>>> data = np.eye(2)
>>> pca_1, X_train_1 = pca(data, n_components=1)
>>> np.around(X_train_1, decimals=1)
array([[-1.4],
[ 1.4]])
>>> np.around(pca_1.components_, decimals=1)
array([[-0.7,  0.7]])


Now lets try using two components

>>> pca_2, X_train_2 = pca(data, n_components=2)
>>> np.around(X_train_2, decimals=1)
array([[-1.4,  0. ],
[ 1.4,  0. ]])
>>> np.abs(np.around(pca_2.components_, decimals=1))
array([[0.7, 0.7],
[0.7, 0.7]])


Note that if we have three dimensional data we must flatten it before sending using pca()

>>> data = np.random.rand(2,2,2)
>>> pca, X_train = pca(data)
Traceback (most recent call last):
...
ValueError: Data is 3 dimensional. Must be 2D


Assuming the first dimension indicates each sample, and each sample contains 2D data within the 2nd and 3rd dimensions of the of the data set, then we can flatten this 2D data into a vector and then perform the PCA transformation.

>>> data = np.reshape(data, (2,-1))
>>> pca, X_train = pca(data)
>>> X_train.shape
(2, 2)

mud.preprocessing.svd(data: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], **kwargs) [source]#

Apply Singular Value Decomposition

Uses np.linalg.svd class to perform an SVD transformation on input data. Note sklearn.preprocessing.StandardScaler transformation is applied to the data first.

Parameters
• ds (numpy.typing.ArrayLike) – Data to apply SVD transformation to. Must be 2 dimensional.

• kwargs (dict, optional) – Additional keyword arguments will be passed to np.linalg.svd method.

Returns

svd_res – Tuple of (U, singular_values, singular_vectors) corresponding to the X = Sigma UV^T decomposition elements.

Return type

Tuple[numpy.ndarray,]

Examples

For a simple example lets apply the PCA transformation to the identity matrix in 2 dimensions, using first 1 principle component.

>>> data = np.eye(2)
>>> U, S, V = svd(data)
>>> np.around(U, decimals=1)
array([[-0.7,  0.7],
[ 0.7,  0.7]])
>>> np.around(S, decimals=1)
array([2., 0.])


Note that if we have three dimensional data we must flatten it before sending using pca()

>>> data = np.random.rand(2,2,2)
>>> U, S, V = svd(data)
Traceback (most recent call last):
...
ValueError: Data is 3 dimensional. Must be 2D


Assuming the first dimension indicates each sample, and each sample contains 2D data within the 2nd and 3rd dimensions of the of the data set, then we can flatten this 2D data into a vector and then perform svd().

>>> data = np.reshape(data, (2,-1))
>>> U, S, V = svd(data)
>>> U.shape
(2, 2)


## mud.util module#

mud.util.add_noise(signal: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], sd: float = 0.05, seed: = None)[source]#

Add noise to synthetic signal to model a real measurement device. Noise is assumed to be from a standard normal distribution std deviation sd:

$mathcal{N}(0,sigma)$

### Paramaters#

signalnumpy.typing.ArrayLike

sdfloat, default = 0.05

Standard deviation of error to add.

seedint, optional

Seed to use for numpy random number generator.

returns
• noisy_signal (numpy.typing.ArrayLike) – Signal with noise added to it.

• Example Usage

• ————-

• Generate test signal, add noise, check average distance

• >>> seed = 21

• >>> test_signal = np.ones(5)

• >>> noisy_signal = add_noise(test_signal, sd=0.05, seed=21)

• >>> np.round(1000*np.mean(noisy_signal-test_signal))

• 4.0

mud.util.fit_domain(x: = None, min_max_bounds: = None, pad_ratio: float = 0.1) [source]#

Fit domain bounding box to array x

Parameters
• x (ArrayLike) – 2D array to calculate min, max values along columns.

• pad_ratio (float, default=0.1) – What ratio of total range=max-min to subtract/add to min/max values to construct final domain range. Padding is done per x column dimension.

Returns

min_max_bounds – Domain fitted to values found in 2D array x, with padding added.

Return type

ArrayLike

Examples

Input must be 2D. Set pad_ratio = 0 to get explicit min/max bounds >>> fit_domain(np.array([[1, 10], [0, -10]]), pad_ratio=0.0) array([[ 0, 1],

[-10, 10]])

Can extend domain around the array values using the pad_ratio argument.

>>> fit_domain(np.array([[1, 10], [0, -10]]), pad_ratio=1)
array([[ -1,   2],
[-30,  30]])

mud.util.make_2d_unit_mesh(N: int = 50, window: int = 1)[source]#

Make 2D Unit Mesh

Constructs mesh based on uniform distribution to discretize each axis.

Parameters
• N (int, default=50) – Size of unit mesh. N points will be generated in each x,y direction.

• window (int, default=1) – Upper bound of mesh. Lower bound fixed at 0 always.

Returns

• grid (tuple of np.ndarray) – Tuple of (X, Y, XX), the grid X and Y and 2D mesh XX

• Example Usage

• ————-

• >>> x, y, XX = make_2d_unit_mesh(3)

• >>> print(XX)

• [[0. 0. ] – [0.5 0. ] [1. 0. ] [0. 0.5] [0.5 0.5] [1. 0.5] [0. 1. ] [0.5 1. ] [1. 1. ]]

mud.util.null_space(A, rcond=None)[source]#

Construct an orthonormal basis for the null space of A using SVD

Method is slight modification of scipy.linalg

Parameters
• A ((M, N) array_like) – Input array

• rcond (float, optional) – Relative condition number. Singular values s smaller than rcond * max(s) are considered zero. Default: floating point eps * max(M,N).

Returns

Z – Orthonormal basis for the null space of A. K = dimension of effective null space, as determined by rcond

Return type

(N, K) ndarray

Examples

One-dimensional null space:

>>> import numpy as np
>>> A = np.array([[1, 1], [1, 1]])
>>> ns = null_space(A)
>>> ns * np.sign(ns[0,0])  # Remove the sign ambiguity of the vector
array([[ 0.70710678],
[-0.70710678]])


Two-dimensional null space:

>>> B = np.random.rand(3, 5)
>>> Z = null_space(B)
>>> Z.shape
(5, 2)
>>> np.allclose(B.dot(Z), 0)
True


The basis vectors are orthonormal (up to rounding error):

>>> np.allclose(Z.T.dot(Z), np.eye(2))
True

mud.util.rank_decomposition(A: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]) [source]#

Build list of rank k updates of A

mud.util.set_shape(array: ndarray, shape: = (1, -1)) [source]#

Resizes inputs if they are one-dimensional.

mud.util.std_from_equipment(tolerance=0.1, probability=0.95)[source]#

Converts tolerance tolerance for precision of measurement equipment to a standard deviation, scaling so that (100probability) percent of measurements are within tolerance. A mean of zero is assumed. erfinv is imported from scipy.special

mud.util.transform_linear_map(operator, data, std)[source]#

Takes a linear map operator of size (len(data), dim_input) or (1, dim_input) for repeated observations, along with a vector data representing observations. It is assumed that data is formed with M@truth + sigma where sigma ~ N(0, std)

This then transforms it to the MWE form expected by the DCI framework. It returns a matrix A of shape (1, dim_input) and np.float b and transforms it to the MWE form expected by the DCI framework.

>>> X = np.ones((10, 2))
>>> x = np.array([0.5, 0.5]).reshape(-1, 1)
>>> std = 1
>>> d = X @ x
>>> A, b = transform_linear_map(X, d, std)
>>> np.linalg.norm(A @ x + b)
0.0
>>> A, b = transform_linear_map(X, d, [std]*10)
>>> np.linalg.norm(A @ x + b)
0.0
>>> A, b = transform_linear_map(np.array([[1, 1]]), d, std)
>>> np.linalg.norm(A @ x + b)
0.0
>>> A, b = transform_linear_map(np.array([[1, 1]]), d, [std]*10)
Traceback (most recent call last):
...
ValueError: For repeated measurements, pass a float for std

mud.util.transform_linear_setup(operator_list, data_list, std_list)[source]#