# Core numerical computing
import numpy as np
import pandas as pd
from scipy.stats import multivariate_normal, multivariate_t

# Financial data handling
import yfinance as yf

# Book data (pip install "git+https://github.com/dppalomar/pob.git#subdirectory=python")
from pob_python import SP500_stocks_2015to2020, SP500_index_2015to2020

# Visualization
import matplotlib.pyplot as plt
import seaborn as sns
sns.set_theme(style="darkgrid")
from tqdm import tqdm  # For progress tracking

# Use S&P 500 index from the book data
SP500_index_2015to2020.head()

sp500_prices = SP500_index_2015to2020.values.squeeze()

# Compute log returns and drop the first difference
sp500_returns = np.diff(np.log(sp500_prices))[1:]

# Calculate sample mean and sample variance
mu = np.mean(sp500_returns)
var = np.var(sp500_returns)

# Generate synthetic data
np.random.seed(42)
T = 500
x = np.random.normal(mu, np.sqrt(var), T)

# Create a DataFrame with a time index and the synthetic data
df = pd.DataFrame({'t': np.arange(1, T + 1), 'x': x})

# Plot the time series
fig, ax = plt.subplots(figsize=(10, 6))
plt.plot(df['t'], df['x'], linewidth=0.8, color='blue')
plt.title('i.i.d. time series')
plt.xlabel('')
plt.ylabel('')
plt.show()

# Generate Gaussian synthetic return data
np.random.seed(42)
N = 2
T = 10
mu_true = np.zeros(N)
Sigma_true = np.array([[0.0012, 0.0011],
                       [0.0011, 0.0014]])
X = np.random.multivariate_normal(mean=mu_true, cov=Sigma_true, size=200)
#X = multivariate_normal.rvs(mean=mu_true, cov=Sigma_true, size=200)
X_ = X[:T, :]


# Estimators
mu_sm = np.mean(X_, axis=0)  # classical mean estimator (sample mean)
X_centered = X_ - mu_sm
Sigma_scm = np.cov(X_centered.T) # classical covariance estimator (sample covariance matrix)

# Print errors
print(np.linalg.norm(mu_sm - mu_true))
print(np.linalg.norm(Sigma_scm - Sigma_true, 'fro') / np.linalg.norm(Sigma_true, 'fro'))

from matplotlib.patches import Ellipse

def confidence_ellipse(cov, mean, ax, n_std=2, **kwargs):
    """Create covariance ellipse with proper scaling"""
    vals, vecs = np.linalg.eigh(cov)
    theta = np.degrees(np.arctan2(*vecs[:,0][::-1]))
    width, height = 2 * n_std * np.sqrt(vals)
    return Ellipse(xy=mean, width=width, height=height, angle=theta, fill=False, **kwargs)

# Scatter plot with corrected ellipses
fig, ax = plt.subplots(figsize=(10, 8))

# Plot data points
ax.scatter(X[:, 0], X[:, 1], color='black', s=2)

# Add ellipses with 95% confidence (n_std=2)
ax.add_patch(confidence_ellipse(Sigma_scm, mu_sm, ax, n_std=2, 
                                edgecolor='red', linewidth=2, label='sample estimator'))
ax.add_patch(confidence_ellipse(Sigma_true, mu_true, ax, n_std=2,
                                edgecolor='blue', linewidth=2, label='true'))

# Plot mean estimates with big colored dots
ax.scatter(mu_sm[0], mu_sm[1], color='red', s=100)
ax.scatter(mu_true[0], mu_true[1], color='blue', s=100)

# Configure plot
ax.set_xlabel('X-axis')
ax.set_ylabel('Y-axis')
ax.legend()
plt.tight_layout()
plt.show()

# Configuration
np.random.seed(42)
N = 100
T_max = 2000

# Get realistic mu and Sigma
X_market = np.log(SP500_stocks_2015to2020).diff().dropna().to_numpy()[:, :N]
mu_true = np.mean(X_market, axis=0)
Sigma_true = np.cov(X_market.T)

# # Alternatively, create synthetic true parameters
# mu_true = np.random.randn(N) * 0.01
# Sigma_true = np.diag(np.abs(np.random.randn(N))) + 0.1*np.eye(N)

# Generate synthetic Gaussian data
X_gaussian = multivariate_normal.rvs(mean=mu_true, cov=Sigma_true, size=10*T_max)

# Benchmarking framework
T_sweep = np.linspace(50, T_max, 50, dtype=int)
error_mu_vs_T = []
error_Sigma_vs_T = []

for T in T_sweep:
    X_ = X_gaussian[:T,]
    
    # Sample estimators
    mu_sm = np.mean(X_, axis=0)
    Sigma_scm = np.cov(X_.T)
    
    # Compute error
    error_mu_vs_T.append(np.linalg.norm(mu_sm - mu_true) / np.linalg.norm(mu_true))
    error_Sigma_vs_T.append(np.linalg.norm(Sigma_scm - Sigma_true, 'fro') / np.linalg.norm(Sigma_true, 'fro'))
    
error_mu_vs_T = np.asarray(error_mu_vs_T)
error_Sigma_vs_T = np.asarray(error_Sigma_vs_T)

# Plots
fig, ax = plt.subplots(figsize=(10, 6))
plt.plot(T_sweep, 100*error_mu_vs_T)
plt.title("Estimation error of sample mean")
plt.xlabel("T")
plt.ylabel("normalized error (%)")
plt.show()

fig, ax = plt.subplots(figsize=(10, 6))
plt.plot(T_sweep, 100*error_Sigma_vs_T)
plt.title("Estimation error of sample covariance matrix")
plt.xlabel("T")
plt.ylabel("normalized error (%)")
plt.show()

# Configuration
np.random.seed(42)
N = 2
T = 10
mu_true = np.zeros(N)
Sigma_true = np.array([[0.0012, 0.0011],
                       [0.0011, 0.0014]])
nu = 4
scatter_true = (nu - 2)/nu * Sigma_true

# Generate synthetic data
X = multivariate_t.rvs(
    loc=mu_true,
    shape=scatter_true,
    df=nu,
    size=800
)

def spatial_median(X, max_iter=100, tol=1e-5):
    """Weiszfeld algorithm for spatial median"""
    med = np.median(X, axis=0)
    for _ in range(max_iter):
        dist = np.linalg.norm(X - med, axis=1)
        np.clip(dist, 1e-12, None, out=dist)  # Avoid division by zero
        weights = 1 / dist
        new_med = X.T @ weights / weights.sum()
        if np.linalg.norm(new_med - med) < tol:
            break
        med = new_med
    return med

# Calculate estimators using first T samples
X_est = X[:T]
estimators = {
    "true": mu_true,
    "sample mean": np.mean(X_est, axis=0),
    "median": np.median(X_est, axis=0),
    "spatial median": spatial_median(X_est)  # Assume this function exists
}

# Create comparison dataframe
df = pd.DataFrame([
    {"location": key, "x": val[0], "y": val[1]} 
    for key, val in estimators.items()
])
df["location"] = pd.Categorical(
    df["location"], 
    categories=["true", "sample mean", "median", "spatial median"]
)

# Visualization
fig, ax = plt.subplots(figsize=(10, 8))

# Plot data points
ax.scatter(X[:, 0], X[:, 1], alpha=0.7, s=10, label='Data points')

# Plot estimators with custom markers
marker_map = {
    "true": "X",         # Big X for ground truth
    "sample mean": "o",  # Circle for sample mean
    "median": "s",       # Square for median
    "spatial median": "D" # Diamond for spatial median
}

for location in df.location.cat.categories:
    subset = df[df.location == location]
    ax.scatter(subset.x, subset.y, 
               s=150, 
               edgecolor='w',
               linewidth=1.5,
               marker=marker_map[location],
               label=location)

# Formatting
ax.set_xlim(-0.08, 0.08)
ax.set_ylim(-0.08, 0.08)
ax.set_title("Scatter plot of data points with different location estimators", pad=20)
ax.set_xlabel("")
ax.set_ylabel("")
ax.grid(True, alpha=0.3)
ax.legend(loc='upper right', frameon=True)

plt.tight_layout()
plt.show()

# Configuration
np.random.seed(42)
N = 100
T_max = 2000
n_trials = 100

# Get realistic mu and Sigma
X_market = np.log(SP500_stocks_2015to2020).diff().dropna().to_numpy()[:, :N]
mu_true = np.mean(X_market, axis=0)
Sigma_true = np.cov(X_market.T)

# Generate synthetic Gaussian data
X_gaussian = multivariate_normal.rvs(mean=mu_true, cov=Sigma_true, size=10*T_max)

# Generate synthetic heavy-tailed distribution data
nu = 4
scatter_true = ((nu-2)/nu) * Sigma_true
X_heavy = multivariate_t.rvs(loc=mu_true, shape=scatter_true, df=nu, size=10*T_max)

# Benchmarking framework
T_sweep = np.linspace(50, T_max, 50, dtype=int)
results = []

for T in T_sweep:
    for dist_name, data in [('Gaussian', X_gaussian), 
                            ('heavy-tailed', X_heavy)]:
        for _ in range(n_trials):
            # Random subsample
            idx = np.random.choice(len(data), size=T, replace=False)
            X = data[idx]
            
            # Calculate estimators
            mu_sm = np.mean(X, axis=0)
            mu_med = np.median(X, axis=0)
            mu_spmed = spatial_median(X)
            
            # Calculate relative errors
            error = lambda x: 100*np.linalg.norm(x - mu_true)/np.linalg.norm(mu_true)
            
            results.append({
                'T': T,
                'distribution': dist_name,
                'method': 'sample mean',
                'error': error(mu_sm)
            })
            results.append({
                'T': T,
                'distribution': dist_name,
                'method': 'median',
                'error': error(mu_med)
            })
            results.append({
                'T': T,
                'distribution': dist_name,
                'method': 'spatial median',
                'error': error(mu_spmed)
            })

# Create DataFrame and aggregate results
df = pd.DataFrame(results)
df_agg = df.groupby(['method', 'T', 'distribution']).mean().reset_index()

# Visualization
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 10), sharex=True)

for method in df_agg['method'].unique():
    subset = df_agg[(df_agg['distribution'] == 'Gaussian') & 
                   (df_agg['method'] == method)]
    ax1.plot(subset['T'], subset['error'], label=method)
    
    subset = df_agg[(df_agg['distribution'] == 'heavy-tailed') & 
                   (df_agg['method'] == method)]
    ax2.plot(subset['T'], subset['error'], label=method)

ax1.set_title('Error of location estimators under Gaussian data')
ax2.set_title('Error of location estimators under heavy-tailed data')
ax2.set_xlabel('Sample Size (T)')
for ax in (ax1, ax2):
    ax.set_ylabel('Relative Error (%)')
    ax.legend()
    ax.grid(True)

plt.tight_layout()
plt.show()

# Configuration
np.random.seed(42)
N = 100
T = 200
n_trials = 200
nu_values = np.arange(3, 31, 2)

# Get realistic mu and Sigma
X_market = np.log(SP500_stocks_2015to2020).diff().dropna().to_numpy()[:, :N]
mu_true = np.mean(X_market, axis=0)
Sigma_true = np.cov(X_market.T)


# Benchmarking framework
results = []
for nu in tqdm(nu_values, desc="Processing ν values"):
    scatter_matrix = ((nu - 2)/nu) * Sigma_true
    
    for _ in range(n_trials):
        # Generate multivariate t-distributed data
        X = multivariate_t.rvs(loc=mu_true, shape=scatter_matrix, df=nu, size=T)
        
        # Calculate estimators
        methods = {
            'sample mean': np.mean(X, axis=0),
            'median': np.median(X, axis=0),
            'spatial median': spatial_median(X)
        }
        
        # Calculate errors
        for method, estimate in methods.items():
            error = 100 * np.linalg.norm(estimate - mu_true) / np.linalg.norm(mu_true)
            results.append({
                'ν': nu,
                'method': method,
                'error': error
            })

# Create and process DataFrame
df = pd.DataFrame(results)
df_agg = df.groupby(['method', 'ν'], as_index=False).agg(
    mean_error=('error', 'mean'),
    std_error=('error', 'std')
)

# Visualization
fig, ax = plt.subplots(1, 1, figsize=(12, 7))
colors = {'sample mean': '#1f77b4', 'median': '#ff7f0e', 'spatial median': '#2ca02c'}

for method in df_agg['method'].unique():
    subset = df_agg[df_agg['method'] == method]
    ax.plot(subset['ν'], subset['mean_error'], label=method, color=colors[method], lw=2)
ax.set_title('Error of location estimators under heavy-tailed data')
ax.set_xlabel('ν')
ax.set_ylabel('Relative Error (%)')
plt.legend()
plt.tight_layout()
plt.show()

# Configuration
np.random.seed(42)
N = 100
T_max = 1000
n_trials = 100

# Get realistic mu and Sigma
X_market = np.diff(np.log(SP500_stocks_2015to2020.values), axis=0)[1:, :N]
mu_true = X_market.mean(axis=0)
Sigma_true = np.cov(X_market, rowvar=False)
nu = 4
scatter_true = (nu-2)/nu * Sigma_true

# Generate synthetic datasets
X_gaussian = multivariate_normal.rvs(
    mean=mu_true, cov=Sigma_true, size=10*T_max
)
X_heavy = multivariate_t.rvs(
    loc=mu_true, shape=scatter_true, df=nu, size=10*T_max
)

# Benchmarking framework
T_sweep = np.linspace(50, T_max, num=20, dtype=int)
results = []

for T in T_sweep:
    for dist_name, data in [('Gaussian', X_gaussian),
                            ('heavy-tailed', X_heavy)]:
        for _ in range(n_trials):
            # Subsample data
            idx = np.random.choice(len(data), size=T, replace=False)
            X = data[idx]
            
            # Calculate estimators
            mu_hat = X.mean(axis=0)
            Sigma_hat = (T-1)/T * np.cov(X, rowvar=False)
            
            # Calculate relative errors
            rel_error_mu = (np.linalg.norm(mu_hat - mu_true, 2) / np.linalg.norm(mu_true, 2))
            rel_error_Sigma = (np.linalg.norm(Sigma_hat - Sigma_true, 'fro') / np.linalg.norm(Sigma_true, 'fro'))
            
            results.append({
                'T': T,
                'distribution': dist_name,
                'error_mu': 100 * rel_error_mu,
                'error_Sigma': 100 * rel_error_Sigma
            })

# Create DataFrame and aggregate
df = pd.DataFrame(results)
df_agg = df.groupby(['distribution', 'T']).mean().reset_index()

# Visualization
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 8), sharex=True)

for dist in ['Gaussian', 'heavy-tailed']:
    subset = df_agg[df_agg['distribution'] == dist]
    ax1.plot(subset['T'], subset['error_mu'], label=dist)
    ax2.plot(subset['T'], subset['error_Sigma'], label=dist)

ax1.set_title('Error of Gaussian ML mean estimator')
ax2.set_title('Error of Gaussian ML covariance estimator')
ax2.set_xlabel('Sample Size (T)')

for ax in (ax1, ax2):
    ax.set_ylabel('Normalized Error (%)')
    ax.legend(title='Data distribution')
    ax.grid(True)
    ax.label_outer()

plt.tight_layout()
plt.show()

# Configuration
np.random.seed(42)
N = 100
T = 200
n_trials = 200
nu_values = np.arange(3, 31, 2)

# Get realistic mu and Sigma (assuming SP500_stocks_2015to2020 is a DataFrame)
X_market = np.log(SP500_stocks_2015to2020).diff().dropna().to_numpy()[:, :N]
mu_true = np.mean(X_market, axis=0)
Sigma_true = np.cov(X_market, rowvar=False)

# Benchmarking framework
results = []
for nu in tqdm(nu_values, desc="Processing ν values"):
    scatter_matrix = ((nu - 2)/nu) * Sigma_true
    
    for _ in range(n_trials):
        # Generate multivariate t-distributed data
        X = multivariate_t.rvs(
            loc=mu_true, 
            shape=scatter_matrix, 
            df=nu, 
            size=T,
        )
        
        # Calculate estimators
        mu_est = np.mean(X, axis=0)
        Sigma_est = np.cov(X, rowvar=False)  # MLE covariance
        
        # Calculate normalized errors
        mu_error = np.linalg.norm(mu_est - mu_true) / np.linalg.norm(mu_true)
        Sigma_error = np.linalg.norm(Sigma_est - Sigma_true, 'fro') / np.linalg.norm(Sigma_true, 'fro')
        
        results.append({
            'ν': nu,
            'method': 'Gaussian ML',
            'error mu': 100 * mu_error,
            'error Sigma': 100 * Sigma_error
        })

# Create and process DataFrame
df = pd.DataFrame(results)
df_agg = df.groupby(['method', 'ν'], as_index=False).agg({
    'error mu': 'mean',
    'error Sigma': 'mean'
})

# Visualization
plt.figure(figsize=(12, 10))

# Plot for mean estimation error
plt.subplot(2, 1, 1)
plt.plot(df_agg['ν'], df_agg['error mu'], color='#1f77b4', lw=2)
plt.title('Estimation error of Gaussian ML mean estimator')
plt.xlabel('ν')
plt.ylabel('Normalized Error (%)')
plt.ylim(200, 250)

# Plot for covariance estimation error
plt.subplot(2, 1, 2)
plt.plot(df_agg['ν'], df_agg['error Sigma'], 
         color='#2ca02c', lw=2)
plt.title('Estimation error of Gaussian ML covariance estimator')
plt.xlabel('ν')
plt.ylabel('Normalized Error (%)')

plt.tight_layout()
plt.show()

def create_2covariance_plot(X, Sigma_true, Sigma_scm, title):
    """Helper function to create consistent plots"""
    fig, ax = plt.subplots(figsize=(10, 8))
    
    # Scatter plot
    ax.scatter(X[:,0], X[:,1], s=8, color='black', alpha=0.8)
    
    # Add ellipses
    for cov, color, label in zip([Sigma_true, Sigma_scm], 
                               ['blue', 'red'], 
                               ['True', 'Gaussian ML']):
        ellipse = confidence_ellipse(cov, [0,0], ax=ax,
                                     edgecolor=color, 
                                     linewidth=1.5)
        ax.add_patch(ellipse)
    
    # Plot configuration
    ax.set_xlim(-0.2, 0.2)
    ax.set_ylim(-0.2, 0.2)
    ax.set_title(title, pad=20)
    ax.legend(handles=[plt.Line2D([0], [0], color=c, lw=2) for c in ['blue', 'red']],
             labels=['True covariance', 'Gaussian ML estimator'])
    plt.tight_layout()
    return fig

# Parameters
mu_true = np.array([0.0, 0.0])
Sigma_true = np.array([[0.0012, 0.0011], 
                       [0.0011, 0.0014]])
T = 10
np.random.seed(42)

X_gaussian = multivariate_normal.rvs(
    mean=mu_true, cov=Sigma_true, size=200
)
X_subset = X_gaussian[:T]
Sigma_scm = (X_subset.T @ X_subset) / T
create_2covariance_plot(X_gaussian, Sigma_true, Sigma_scm, "Gaussian Data")
plt.show()

nu = 4
scatter_matrix = (nu-2)/nu * Sigma_true  # For multivariate_t shape param

X_heavy = multivariate_t.rvs(
    loc=mu_true,
    shape=scatter_matrix,
    df=nu,
    size=200
)
X_subset = X_heavy[:T]
Sigma_scm = (X_subset.T @ X_subset) / T
create_2covariance_plot(X_heavy, Sigma_true, Sigma_scm, "Heavy-tailed Data (ν=4)")
plt.show()

X_gaussian = multivariate_normal.rvs(
    mean=mu_true, cov=Sigma_true, size=200
)
# Add outlier at (-0.15, 0.05)
X_outlier = np.vstack([[-0.15, 0.05], X_gaussian])
X_subset = X_outlier[:T]
Sigma_scm = (X_subset.T @ X_subset) / T
create_2covariance_plot(X_outlier, Sigma_true, Sigma_scm, "Gaussian Data with Outlier")
plt.show()

def nu_POP_estimator(r2, nu, N):
    """
    Estimate the degrees of freedom of a heavy-tailed t distribution based on the POP estimator.

    Based on:
      Frédéric Pascal, Esa Ollila, and Daniel P. Palomar, "Improved estimation of the degree of freedom parameter of
      multivariate t-distribution," in Proc. European Signal Processing Conference (EUSIPCO), Dublin, Ireland, Aug. 23-27, 2021.
      <https://doi.org/10.23919/EUSIPCO54536.2021.9616162>
    """
    T = len(r2)
    u = (N + nu) / (nu + r2 * T / (T - 1))
    r2i = r2 / (1 - r2 * u / T)
    theta = (1 - N / T) * np.sum(r2i) / T / N
    nu = 2 * theta / (theta - 1)
    nu = min(100, max(2.5, nu))
    return nu
    
def fit_mvt(X, verbose=False, max_iter=100, ptol=1e-3):
    """
    Fit of a multivariate t distribution to the data X.

    References:
    Chuanhai Liu and Donald B. Rubin, "ML estimation of the t-distribution using EM and its extensions, ECM and ECME,"
    Statistica Sinica (5), pp. 19-39, 1995.

    Chuanhai Liu, Donald B. Rubin, and Ying Nian Wu, "Parameter Expansion to Accelerate EM: The PX-EM Algorithm,"
    Biometrika, Vol. 85, No. 4, pp. 755-770, Dec., 1998

    Rui Zhou, Junyan Liu, Sandeep Kumar, and Daniel P. Palomar, "Robust factor analysis parameter estimation,"
    Lecture Notes in Computer Science (LNCS), 2019. <https://arxiv.org/abs/1909.12530>

    Esa Ollila, Daniel P. Palomar, and Frédéric Pascal, "Shrinking the Eigenvalues of M-estimators of Covariance Matrix,"
    IEEE Trans. on Signal Processing, vol. 69, pp. 256-269, Jan. 2021. <https://doi.org/10.1109/TSP.2020.3043952>

    Frédéric Pascal, Esa Ollila, and Daniel P. Palomar, "Improved estimation of the degree of freedom parameter of
    multivariate t-distribution," in Proc. European Signal Processing Conference (EUSIPCO), Dublin, Ireland, Aug. 23-27, 2021.
    <https://doi.org/10.23919/EUSIPCO54536.2021.9616162>
    """
    X = np.asarray(X)
    T, N = np.shape(X)

    # initial points
    nu = 4
    mu = np.mean(X, axis=0)
    cov = np.cov(X, rowvar=False)
    Sigma = (nu - 2)/nu * cov
    def fnu(nu):
      return nu/(nu - 2)

    #
    #  loop
    #
    Sigma_diff_record = np.empty(max_iter)
    for k in range(max_iter):
        nu_prev    = np.copy(nu)
        mu_prev    = np.copy(mu)
        Sigma_prev = np.copy(Sigma)

        # intermediate variables
        Xc = X - mu
        #r2 = np.sum(Xc * np.dot(Xc, np.linalg.inv(Sigma)), axis=1)  # np.diag( Xc @ np.linalg.inv(Sigma) @ t(Xc) )
        r2 = np.sum(Xc * np.linalg.solve(Sigma + 1e-6 * np.eye(N), Xc.T).T, axis=1)

        # update nu, mu, Sigma
        nu = nu_POP_estimator(r2=r2, nu=nu, N=N)
        E_tau = (N + nu) / (nu + r2)
        ave_E_tau = np.mean(E_tau)
        ave_E_tau_X = np.mean(X * E_tau[:, None], axis=0)
        mu = ave_E_tau_X / ave_E_tau
        Xc = X - mu
        ave_E_tau_XX = (1 / T) * np.dot((Xc * np.sqrt(E_tau)[:, None]).T, Xc * np.sqrt(E_tau)[:, None])
        #ave_E_tau_XX_ = (1 / T) * Xc.T @ np.diag(E_tau) @ Xc
        Sigma = ave_E_tau_XX
        Sigma = Sigma / ave_E_tau  # PX_EM_acceleration

        # stopping criterion
        Sigma_diff_record[k] = np.linalg.norm(Sigma - Sigma_prev) / np.linalg.norm(Sigma_prev)
        if (np.all(np.abs(mu - mu_prev) <= .5 * ptol * (np.abs(mu_prev) + np.abs(mu))) and
                np.abs(fnu(nu) - fnu(nu_prev)) <= .5 * ptol * (np.abs(fnu(nu_prev)) + np.abs(fnu(nu))) and
                np.all(np.abs(Sigma - Sigma_prev) <= .5 * ptol * (np.abs(Sigma_prev) + np.abs(Sigma)))):
            break
    Sigma_diff_record = Sigma_diff_record[:k]

    Sigma_cov = nu / (nu - 2) * Sigma

    if verbose:
        fig, ax = plt.subplots(figsize=(16, 6))
        ax.plot(np.arange(k), Sigma_diff_record)
        plt.xlabel("Iterations")
        plt.ylabel("Sigma differences")
        plt.show()

    return mu, Sigma_cov

# Parameters
mu_true = np.array([0.0, 0.0])
Sigma_true = np.array([[0.0012, 0.0011], 
                       [0.0011, 0.0014]])
T = 10
np.random.seed(42)

def create_3covariance_plot(X, Sigma_true, Sigma_scm, Sigma_heavytail, title):
    """Helper function to create consistent plots"""
    fig, ax = plt.subplots(figsize=(10, 8))
    
    # Scatter plot
    ax.scatter(X[:,0], X[:,1], s=8, color='black', alpha=0.8)
    
    # Add ellipses
    for cov, color, label in zip([Sigma_true, Sigma_scm, Sigma_heavytail], 
                               ['blue', 'red', 'green'], 
                               ['True', 'Gaussian ML', 'Heavy-tailed ML']):
        ellipse = confidence_ellipse(cov, [0,0], ax=ax,
                                    edgecolor=color, 
                                    linewidth=1.5)
        ax.add_patch(ellipse)
    
    # Plot configuration
    ax.set_xlim(-0.2, 0.2)
    ax.set_ylim(-0.2, 0.2)
    ax.set_title(title, pad=20)
    ax.legend(handles=[plt.Line2D([0], [0], color=c, lw=2) 
                      for c in ['blue', 'red', 'green']],
             labels=['True covariance', 'Gaussian ML estimator', 'Heavy-tailed ML estimator'])
    plt.tight_layout()
    return fig

# Data generation
X_gaussian = multivariate_normal.rvs(
    mean=mu_true, cov=Sigma_true, size=200
)
X_subset = X_gaussian[:T]

# Estimators
Sigma_scm = (X_subset.T @ X_subset) / T
mu_heavytail, Sigma_heavytail = fit_mvt(X_subset)

# Plot
create_3covariance_plot(X_gaussian, Sigma_true, Sigma_scm, Sigma_heavytail, "Gaussian Data")
plt.show()

# Data generation
nu = 4
scatter_matrix = (nu-2)/nu * Sigma_true  # For multivariate_t shape param
X_heavy = multivariate_t.rvs(
    loc=mu_true,
    shape=scatter_matrix,
    df=nu,
    size=200
)
X_subset = X_heavy[:T]

# Estimators
Sigma_scm = (X_subset.T @ X_subset) / T
mu_heavytail, Sigma_heavytail = fit_mvt(X_subset)

# Plot
create_3covariance_plot(X_gaussian, Sigma_true, Sigma_scm, Sigma_heavytail, "Heavy-tailed Data (ν=4)")
plt.show()

# Data generation
X_gaussian = multivariate_normal.rvs(
    mean=mu_true, cov=Sigma_true, size=200
)
# Add outlier at (-0.15, 0.05)
X_outlier = np.vstack([[-0.15, 0.05], X_gaussian])
X_subset = X_outlier[:T]

# Estimators
Sigma_scm = (X_subset.T @ X_subset) / T
mu_heavytail, Sigma_heavytail = fit_mvt(X_subset)

# Plot
create_3covariance_plot(X_gaussian, Sigma_true, Sigma_scm, Sigma_heavytail, "Gaussian Data with Outlier")
plt.show()

# Configuration
np.random.seed(42)
N = 100
T_max = 1000
n_trials = 100
nu = 4

# Get realistic mu and Sigma
X_market = np.log(SP500_stocks_2015to2020).diff().dropna().to_numpy()[:, :N]
mu_true = np.mean(X_market, axis=0)
Sigma_true = np.cov(X_market.T)

# Generate synthetic heavy-tailed data
scatter_true = ((nu-2)/nu) * Sigma_true
X_heavtytailed = multivariate_t.rvs(
    loc=mu_true,
    shape=scatter_true,
    df=nu,
    size=10*T_max
)

# Benchmarking framework
T_sweep = np.arange(150, T_max+1, 50)
results = []

for T in T_sweep:
    for _ in range(n_trials):
        # Random subsample
        idx = np.random.choice(len(X_heavtytailed), size=T, replace=False)
        X_sub = X_heavtytailed[idx]
        
        # Gaussian MLE
        mu_gauss = np.mean(X_sub, axis=0)
        Sigma_gauss = (T-1)/T * np.cov(X_sub.T)
        
        # Heavy-tailed MLE (assuming fit_mvt exists)
        mu_heavy, Sigma_heavy = fit_mvt(X_sub)
        
        # Calculate errors
        errors = {
            'Gaussian MLE': {
                'mu': np.linalg.norm(mu_gauss - mu_true)/np.linalg.norm(mu_true),
                'Sigma': np.linalg.norm(Sigma_gauss - Sigma_true, 'fro')/np.linalg.norm(Sigma_true, 'fro')
            },
            'heavy-tailed MLE': {
                'mu': np.linalg.norm(mu_heavy - mu_true)/np.linalg.norm(mu_true),
                'Sigma': np.linalg.norm(Sigma_heavy - Sigma_true, 'fro')/np.linalg.norm(Sigma_true, 'fro')
            }
        }
        
        for method in errors:
            results.append({
                'T': T,
                'method': method,
                'error mu': errors[method]['mu'],
                'error Sigma': errors[method]['Sigma']
            })

# Process results
df = pd.DataFrame(results)
df_agg = df.groupby(['method', 'T'], as_index=False).agg({
    'error mu': lambda x: 100*np.mean(x),
    'error Sigma': lambda x: 100*np.mean(x)
})

# Visualization
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 10), sharex=True)

for method in df_agg['method'].unique():
    subset = df_agg[df_agg['method'] == method]
    ax1.plot(subset['T'], subset['error mu'], label=method)
    ax2.plot(subset['T'], subset['error Sigma'], label=method)

ax1.set_title('Error of Mean Estimators')
ax2.set_title('Error of Covariance Estimators')
ax2.set_xlabel('Sample Size (T)')
ax1.set_ylabel('Normalized Error (%)')
ax2.set_ylabel('Normalized Error (%)')
ax1.legend()
ax1.grid(True)
ax2.grid(True)

plt.tight_layout()
plt.show()

# Configuration
np.random.seed(42)
N = 100
T = 200
n_trials = 200
nu_values = np.arange(3, 31, 2)


# Get realistic mu and Sigma
X_market = np.log(SP500_stocks_2015to2020).diff().dropna().to_numpy()[:, :N]
mu_true = np.mean(X_market, axis=0)
Sigma_true = np.cov(X_market.T)

# Benchmarking framework
results = []
for nu in tqdm(nu_values, desc="Processing ν values"):
    scatter_matrix = ((nu - 2)/nu) * Sigma_true
    
    for _ in range(n_trials):
        # Generate multivariate t-distributed data
        X = multivariate_t.rvs(
            loc=mu_true, 
            shape=scatter_matrix, 
            df=nu, 
            size=T,
        )
        
        # Gaussian MLE
        mu_gauss = np.mean(X, axis=0)
        Sigma_gauss = (T-1)/T * np.cov(X, rowvar=False)
        
        mu_error_gauss = np.linalg.norm(mu_gauss - mu_true) / np.linalg.norm(mu_true)
        Sigma_error_gauss = np.linalg.norm(Sigma_gauss - Sigma_true, 'fro') / np.linalg.norm(Sigma_true, 'fro')
        
        results.append({
            'ν': nu,
            'method': 'Gaussian MLE',
            'error mu': 100 * mu_error_gauss,
            'error Sigma': 100 * Sigma_error_gauss
        })
        
        # Heavy-tailed MLE (assuming fit_mvt exists)
        mu_heavy, Sigma_heavy = fit_mvt(X)  # Replace with actual implementation of fit_mvt
        
        mu_error_heavy = np.linalg.norm(mu_heavy - mu_true) / np.linalg.norm(mu_true)
        Sigma_error_heavy = np.linalg.norm(Sigma_heavy - Sigma_true, 'fro') / np.linalg.norm(Sigma_true, 'fro')
        
        results.append({
            'ν': nu,
            'method': 'Heavy-tailed MLE',
            'error mu': 100 * mu_error_heavy,
            'error Sigma': 100 * Sigma_error_heavy
        })

# Create and process DataFrame
df = pd.DataFrame(results)
df_agg = df.groupby(['method', 'ν'], as_index=False).agg({
    'error mu': 'mean',
    'error Sigma': 'mean'
})

# Visualization
plt.figure(figsize=(12, 10))

# Plot for mean estimation error
plt.subplot(2, 1, 1)
for method in df_agg['method'].unique():
    subset = df_agg[df_agg['method'] == method]
    plt.plot(subset['ν'], subset['error mu'], label=method)

plt.title('Estimation error of mean estimators')
plt.xlabel('ν')
plt.ylabel('Normalized Error (%)')
plt.legend()
plt.grid(True)

# Plot for covariance estimation error
plt.subplot(2, 1, 2)
for method in df_agg['method'].unique():
    subset = df_agg[df_agg['method'] == method]
    plt.plot(subset['ν'], subset['error Sigma'], label=method)

plt.title('Estimation error of covariance estimators')
plt.xlabel('ν')
plt.ylabel('Normalized Error (%)')
plt.legend()
plt.grid(True)

plt.tight_layout()
plt.show()

# Configuration
np.random.seed(42)
N = 100
T = 200
n_trials = 200
nu_values = np.arange(3, 31, 2)

# Get realistic mu and Sigma
X_market = np.log(SP500_stocks_2015to2020).diff().dropna().to_numpy()[:, :N]
mu_true = np.mean(X_market, axis=0)
Sigma_true = np.cov(X_market.T)
nu <- 4
scatter_matrix = (nu-2)/nu * Sigma_true

# Generate multivariate t-distributed data
X = multivariate_t.rvs(
    loc=mu_true, 
    shape=scatter_matrix, 
    df=nu, 
    size=T,
)

# heavy-tailed MLE
mu_heavy, Sigma_heavy = fit_mvt(X, verbose=True)