Primera A Clausura Playoff Group B stats & predictions

Captains & coaches play pivotal roles during games through real-time adjustments influencing match results:

• Tactical Shifts :Certain formations may be altered depending upon opponent behavior – switching from defensive setups towards more aggressive ones if needed vice versa
• `[0]: #!/usr/bin/env python [1]: # -*- coding: utf-8 -*- [2]: import numpy as np [3]: import matplotlib.pyplot as plt [4]: import seaborn as sns [5]: def get_gaussian_kde(x): [6]: """ Get Gaussian KDE """ [7]: kde = stats.gaussian_kde(x) [8]: return kde [9]: def get_gaussian_pdf(x): [10]: """ Get Gaussian PDF """ [11]: mu = np.mean(x) [12]: sigma = np.std(x) [13]: pdf = (np.exp(-((x - mu)**2 / (sigma**2))) / (np.sqrt(2 * np.pi) * sigma)) [14]: return pdf [15]: def plot_pdf(kde_data): [16]: x = np.linspace(np.min(kde_data), np.max(kde_data), num=10000) [17]: fig = plt.figure() [18]: ax1 = fig.add_subplot(211) ***** Tag Data ***** ID: 1 description: Function `get_gaussian_pdf` computes Gaussian Probability Density Function start line: 9 end line: 14 dependencies: - type: Function name: get_gaussian_kde start line: 5 end line: 8 context description: This function calculates the probability density function (PDF) using mean (mu) and standard deviation (sigma). It involves statistical concepts, particularly related to normal distribution. algorithmic depth: 4 algorithmic depth external: N obscurity: 1 advanced coding concepts: 3 interesting for students: '4' self contained: Y ************ ## Challenging aspects ### Challenging aspects in above code: 1. **Numerical Stability**: Calculating ( exp(-((x - mu)^{**} / (sigma^**))) ) directly could lead to numerical instability especially when ( x ) values are far from ( mu ). Handling large exponents carefully without causing overflow/underflow issues requires attention. 2. **Precision**: Using floating-point arithmetic might introduce precision errors when computing very small probabilities close to zero due to limited precision representation. 3. **Efficiency**: Efficiently computing PDF values over large datasets while ensuring minimal computational overhead is essential for performance-sensitive applications. 4. **Edge Cases**: Handling edge cases such as zero variance ((sigma) being zero), which would cause division by zero errors needs careful consideration. ### Extension: 1. **Multivariate Normal Distribution**: Extend functionality from univariate normal distribution PDF calculation to multivariate normal distribution PDF calculation. 2. **Parameter Estimation**: Incorporate methods for parameter estimation using Maximum Likelihood Estimation (MLE) or Bayesian approaches instead of assuming pre-computed mean ((mu)) and standard deviation ((sigma)) values. 3. **Handling Large Datasets**: Introduce methods for handling large datasets efficiently possibly using streaming algorithms or parallel processing techniques. 4. **Error Handling**: Robust error handling mechanisms should be introduced including custom exceptions when encountering invalid inputs such as non-numeric data types. ## Exercise ### Problem Statement: You are required to expand upon [SNIPPET] provided below by implementing advanced functionalities related specifically within its domain context: [SNNIPET]: python def get_gaussian_pdf(x): """ Get Gaussian PDF """ mu = np.mean(x) sigma = np.std(x) pdf = (np.exp(-((x - mu)**2 / (sigma**2))) / (np.sqrt(2 * np.pi) * sigma)) return pdf ### Requirements: 1. Extend `get_gaussian_pdf` function into `get_multivariate_gaussian_pdf` which calculates PDFs for multivariate normal distributions given mean vector ( mu ) (numpy array) and covariance matrix ( Sigma ). - Ensure numerical stability while calculating determinants and inverses involved with covariance matrices. - Implement efficient computation strategies suitable for large dimensional data sets. python def get_multivariate_gaussian_pdf(X): """ Calculate Multivariate Gaussian PDF """ # X is assumed shape n_samples x n_features pass # Your implementation here ### Constraints: - Do not use external libraries beyond numpy/scipy. - Handle edge cases where covariance matrix might be singular or nearly singular gracefully. - Ensure efficiency both computationally (time complexity considerations) & memory-wise when dealing with large datasets (>100k samples). ## Solution python import numpy as np def get_multivariate_gaussian_pdf(X): """ Calculate Multivariate Gaussian PDF """ n_samples, n_features = X.shape # Compute mean vector μ & covariance matrix Σ from input data X mu = np.mean(X,axis=0) Sigma = np.cov(X.T) # Compute determinant & inverse safely det_Sigma = np.linalg.det(Sigma) if det_Sigma ==0: raise ValueError("Covariance matrix is singular.") inv_Sigma = np.linalg.inv(Sigma) norm_const = ((2 * np.pi)**(n_features / 2)) * (det_Sigma**(1/2)) centered_X = X - mu exponent_terms = [] # Numerically stable computation using matrix operations try: exponent_terms.append(np.einsum('ij,jk->ik', centered_X @ inv_Sigma , centered_X.T).diagonal()) exponent_terms=np.array(exponent_terms).reshape(-1,) pdf_values=np.exp(-0.5*exponent_terms)/norm_const return pdf_values except LinAlgError: raise ValueError("Matrix inversion failed due possibly ill-conditioned matrix.") ## Follow-up exercise ### Problem Statement: Expand `get_multivariate_gaussian_pdf` function further by incorporating parameter estimation methods directly within it: #### New Requirements: 1.- Integrate Maximum Likelihood Estimation method within `get_multivariate_gaussian_pdf` function itself allowing users optionally specify whether they want estimated parameters (`mu`, `Sigma`) directly calculated from input data `X`. python def get_multivariate_gaussian_pdf(X=None , mu=None , Sigma=None , estimate_params=False): if estimate_params : if X is None : raise ValueError("Data must be provided if parameter estimation is required.") else : mu=np.mean(X,axis=0 ) Sigma=np.cov(X.T ) return compute_pdf(X,mu,Sigma) def compute_pdf(X,mu,Sigma ): n_samples,n_features=X.shape det_Sigma=np.linalg.det(Sigma ) if det_Sigma==0 : raise ValueError("Covariance matrix is singular.") inv_Sigma=np.linalg.inv(Sigma ) norm_const=((2*np.pi)**(n_features/float(2))) * det_Sigma**(0 .5) centered_X=X-mu exponent_terms=[] try : exponent_terms.append(np.einsum('ij,jk->ik',centered_X @ inv_Sigma ,centered_X.T ).diagonal()) exponent_terms=np.array(exponent_terms).reshape(-1,) pdf_values=np.exp(-0 .5*exponent_terms)/norm_const return pdf_values except LinAlgError : raise ValueError("Matrix inversion failed due possibly ill-conditioned matrix.") This extended version incorporates MLE-based parameter estimation directly within function scope providing flexibility whether user wants precomputed parameters or computed internally. implement error correction capabilities within existing cryptographic protocols without necessitating major changes?