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"""
Optimal Transport Module - Wasserstein Distances
Provides geometric measures of how much "effort" is needed to move from
one geopolitical scenario to another using optimal transport theory.
Applications:
- Measure regime shifts
- Compare distributions of Monte Carlo futures
- Quantify shock impact
- Measure closeness of geopolitical scenarios
- Detect structural change
- Logistics modeling
"""
import numpy as np
from typing import Union, Tuple, Optional, List
from scipy.spatial.distance import cdist
from scipy.stats import wasserstein_distance
try:
import ot # Python Optimal Transport library
HAS_POT = True
except ImportError:
HAS_POT = False
print("Warning: POT library not available. Some features will be limited.")
class WassersteinDistance:
"""
Compute Wasserstein distances between probability distributions.
The Wasserstein distance (also known as Earth Mover's Distance) provides
a principled way to measure the distance between probability distributions,
accounting for the geometry of the underlying space.
"""
def __init__(self, metric: str = 'euclidean', p: int = 2):
"""
Initialize Wasserstein distance calculator.
Parameters
----------
metric : str
Distance metric to use for ground distance ('euclidean', 'cityblock', etc.)
p : int
Order of Wasserstein distance (1 or 2)
"""
self.metric = metric
self.p = p
def compute_1d(
self,
u_values: np.ndarray,
v_values: np.ndarray,
u_weights: Optional[np.ndarray] = None,
v_weights: Optional[np.ndarray] = None
) -> float:
"""
Compute 1D Wasserstein distance between two distributions.
Parameters
----------
u_values : np.ndarray
Values for first distribution
v_values : np.ndarray
Values for second distribution
u_weights : np.ndarray, optional
Weights for first distribution (defaults to uniform)
v_weights : np.ndarray, optional
Weights for second distribution (defaults to uniform)
Returns
-------
float
Wasserstein distance
"""
return wasserstein_distance(u_values, v_values, u_weights, v_weights)
def compute_nd(
self,
X_source: np.ndarray,
X_target: np.ndarray,
a: Optional[np.ndarray] = None,
b: Optional[np.ndarray] = None,
method: str = 'sinkhorn'
) -> float:
"""
Compute n-dimensional Wasserstein distance.
Parameters
----------
X_source : np.ndarray, shape (n_samples_source, n_features)
Source distribution samples
X_target : np.ndarray, shape (n_samples_target, n_features)
Target distribution samples
a : np.ndarray, optional
Weights for source distribution (defaults to uniform)
b : np.ndarray, optional
Weights for target distribution (defaults to uniform)
method : str
Method to use ('sinkhorn', 'emd', 'emd2')
Returns
-------
float
Wasserstein distance
"""
if not HAS_POT:
raise ImportError("POT library required for n-dimensional distances")
n_source = X_source.shape[0]
n_target = X_target.shape[0]
# Default to uniform distributions
if a is None:
a = np.ones(n_source) / n_source
if b is None:
b = np.ones(n_target) / n_target
# Compute cost matrix
M = cdist(X_source, X_target, metric=self.metric)
# Compute optimal transport
if method == 'sinkhorn':
# Sinkhorn algorithm (faster, approximate)
distance = ot.sinkhorn2(a, b, M, reg=0.1)
elif method == 'emd':
# Exact EMD
distance = ot.emd2(a, b, M)
elif method == 'emd2':
# Squared EMD
distance = ot.emd2(a, b, M**2)
else:
raise ValueError(f"Unknown method: {method}")
return float(distance)
def compute_barycenter(
self,
distributions: List[np.ndarray],
weights: Optional[np.ndarray] = None,
method: str = 'sinkhorn'
) -> np.ndarray:
"""
Compute Wasserstein barycenter of multiple distributions.
This finds the "average" distribution in Wasserstein space.
Parameters
----------
distributions : list of np.ndarray
List of distributions to average
weights : np.ndarray, optional
Weights for each distribution
method : str
Method to use ('sinkhorn')
Returns
-------
np.ndarray
Wasserstein barycenter
"""
if not HAS_POT:
raise ImportError("POT library required for barycenter computation")
n_distributions = len(distributions)
if weights is None:
weights = np.ones(n_distributions) / n_distributions
# Stack distributions
A = np.column_stack(distributions)
# Compute barycenter
if method == 'sinkhorn':
barycenter = ot.bregman.barycenter(A, M=None, reg=0.1, weights=weights)
else:
raise ValueError(f"Unknown method: {method}")
return barycenter
class ScenarioComparator:
"""
Compare geopolitical scenarios using optimal transport.
This class provides high-level methods for comparing scenarios,
detecting regime shifts, and quantifying shock impacts.
"""
def __init__(self, metric: str = 'euclidean'):
"""
Initialize scenario comparator.
Parameters
----------
metric : str
Distance metric for ground distance
"""
self.wasserstein = WassersteinDistance(metric=metric)
def compare_scenarios(
self,
scenario1: np.ndarray,
scenario2: np.ndarray,
weights1: Optional[np.ndarray] = None,
weights2: Optional[np.ndarray] = None
) -> float:
"""
Compare two geopolitical scenarios.
Parameters
----------
scenario1 : np.ndarray
First scenario (features x samples)
scenario2 : np.ndarray
Second scenario (features x samples)
weights1 : np.ndarray, optional
Weights for first scenario
weights2 : np.ndarray, optional
Weights for second scenario
Returns
-------
float
Distance between scenarios
"""
return self.wasserstein.compute_nd(scenario1, scenario2, weights1, weights2)
def detect_regime_shift(
self,
baseline: np.ndarray,
current: np.ndarray,
threshold: float = 0.1
) -> Tuple[bool, float]:
"""
Detect if a regime shift has occurred.
Parameters
----------
baseline : np.ndarray
Baseline scenario distribution
current : np.ndarray
Current scenario distribution
threshold : float
Threshold for detecting shift
Returns
-------
tuple
(shift_detected, distance)
"""
distance = self.compare_scenarios(baseline, current)
shift_detected = distance > threshold
return shift_detected, distance
def quantify_shock_impact(
self,
pre_shock: np.ndarray,
post_shock: np.ndarray
) -> dict:
"""
Quantify the impact of a shock event.
Parameters
----------
pre_shock : np.ndarray
Pre-shock scenario distribution
post_shock : np.ndarray
Post-shock scenario distribution
Returns
-------
dict
Dictionary with impact metrics
"""
distance = self.compare_scenarios(pre_shock, post_shock)
# Compute additional metrics
mean_shift = np.linalg.norm(np.mean(post_shock, axis=0) - np.mean(pre_shock, axis=0))
variance_change = np.abs(np.var(post_shock) - np.var(pre_shock))
return {
'wasserstein_distance': distance,
'mean_shift': mean_shift,
'variance_change': variance_change,
'impact_magnitude': distance * mean_shift
}
def compute_scenario_trajectory(
self,
scenarios: List[np.ndarray]
) -> np.ndarray:
"""
Compute trajectory of scenarios over time.
Parameters
----------
scenarios : list of np.ndarray
Time series of scenarios
Returns
-------
np.ndarray
Array of distances between consecutive scenarios
"""
n_scenarios = len(scenarios)
distances = np.zeros(n_scenarios - 1)
for i in range(n_scenarios - 1):
distances[i] = self.compare_scenarios(scenarios[i], scenarios[i + 1])
return distances
def logistics_optimal_transport(
self,
supply: np.ndarray,
demand: np.ndarray,
supply_locations: np.ndarray,
demand_locations: np.ndarray
) -> Tuple[np.ndarray, float]:
"""
Solve logistics problem using optimal transport.
Parameters
----------
supply : np.ndarray
Supply amounts at each location
demand : np.ndarray
Demand amounts at each location
supply_locations : np.ndarray
Coordinates of supply locations
demand_locations : np.ndarray
Coordinates of demand locations
Returns
-------
tuple
(transport_plan, total_cost)
"""
if not HAS_POT:
raise ImportError("POT library required for logistics optimization")
# Normalize supply and demand
supply_norm = supply / supply.sum()
demand_norm = demand / demand.sum()
# Compute cost matrix (distances)
M = cdist(supply_locations, demand_locations, metric=self.wasserstein.metric)
# Compute optimal transport plan
transport_plan = ot.emd(supply_norm, demand_norm, M)
total_cost = np.sum(transport_plan * M)
# Scale back to original quantities
transport_plan *= supply.sum()
return transport_plan, total_cost
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