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import numpy as np
import pandas as pd
from sklearn.decomposition import PCA
from scipy.stats import norm
import pymc3 as pm

def local_pca(data, n_components=2):
    """
    Apply PCA on local areas of the spatial field data.
    :param data: DataFrame with spatial field data.
    :param n_components: Number of principal components to keep.
    :return: DataFrame with PCA components for each local area.
    """
    pca_results = pd.DataFrame()
    for area in data['Area'].unique():
        local_data = data[data['Area'] == area].drop(['Area'], axis=1)
        pca = PCA(n_components=n_components)
        pca.fit(local_data)
        pca_data = pca.transform(local_data)
        pca_df = pd.DataFrame(pca_data, columns=[f'PC{i+1}' for i in range(n_components)])
        pca_df['Area'] = area
        pca_results = pd.concat([pca_results, pca_df], ignore_index=True)
    return pca_results

def hierarchical_bayesian_model(pca_data):
    """
    Define and run a hierarchical Bayesian model using Local PCA data.
    :param pca_data: DataFrame with PCA components for each local area.
    """
    n_areas = pca_data['Area'].nunique()
    with pm.Model() as model:
        # Hyperparameters
        mu = pm.Normal('mu', mu=0, sd=10, shape=n_areas)
        sigma = pm.HalfNormal('sigma', sd=10, shape=n_areas)
        
        # Model for each area
        for area in range(n_areas):
            area_data = pca_data[pca_data['Area'] == area]
            area_index = area_data.index
            Y_obs = pm.Normal('Y_obs' + str(area), mu=mu[area], sd=sigma[area], observed=area_data['PC1'].values)
        
        # Sampling
        trace = pm.sample(1000)
    return trace

# Example usage
data = pd.read_csv('your_spatial_data.csv')  # Load your spatial field data
pca_data = local_pca(data)
trace = hierarchical_bayesian_model(pca_data)