Olympique Lyonnais U19 vs AS Monaco U19
Date: [Insert Date]
Prediction Overview
Olympique Lyonnais U19
- Recent Form: W-W-D-L-W (Winning momentum)
- Head-to-Head Record: Drawn last three encounters
- Key Player: [Player Name] - Leading goal scorer
- Tactical Strength: Solid defense with quick counter-attacks
- Injuries/Suspensions: None critical
- Home Advantage: Strong home record this season
- Recent Tactical Changes: Switched to a more attacking formation
- Motivation Level: High stakes game for promotion spot
- Weather Conditions: Clear skies expected; no impact anticipated [Manager Name] has been pivotal in shaping Lyon's youth setup.
AS Monaco U19
&nb &sp; &nb sp;- &nb sp; &nb sp;< li >< span clas s = "st at-label" > Recent Form :< /s pan > W-D-L-W-L (Struggling with consistency) &nb sp; &nb sp;< li >< span clas s = "st at-label" > Head-to-Head Record :< /s pan > Drawn last three encounters< /l i > &nb sp; &nb sp;< li >< span clas s = "st at-label" > Key Player :< /s pan > [Player Name] - Dynamic midfielder< /l i > &nb sp; &nb sp;< li >< span clas s = "st at-label" > Tactical Strength :< /s pan > Strong midfield control< /l i > &nb sp; &nb sp;< li >< span clas s = "st at-label" > Injuries/Suspensions :< /s pan > One key midfielder suspended< /l i > [0]: import os
[1]: import time
[2]: import numpy as np
[3]: import pandas as pd
[4]: from matplotlib import pyplot as plt
[5]: from PIL import Image
[6]: from IPython.display import display
[7]: from scipy.optimize import minimize [8]: # plot_image(X) plots an image X
[9]: def plot_image(X):
[10]: plt.imshow(X.reshape(28,28), cmap='gray')
[11]: plt.show() [12]: # plot_images(X) plots all images in X side by side
[13]: def plot_images(X):
[14]: plt.figure(figsize=(10,10))
[15]: for i in range(X.shape[0]):
[16]: plt.subplot(10,int(np.ceil(X.shape[0]/10)),i+1)
[17]: plt.xticks([])
[18]: plt.yticks([])
[19]: plt.grid(False)
[20]: plt.imshow(X[i].reshape(28,28), cmap=plt.cm.binary)
[21]: plt.show() [22]: # compute_cost(W,b,X,Y) computes cost function value for given W,b,X,Y
[23]: def compute_cost(W,b,X,Y):
[24]: cost = (1/len(X)) * (-1) * np.sum((Y * np.log(sigmoid(np.dot(W.T,X.T)+b))) + ((1-Y) * np.log(1 - sigmoid(np.dot(W.T,X.T)+b))))
[25]: return cost [26]: # gradient_descent(alpha,W,b,X,Y) performs gradient descent on W,b using learning rate alpha
[27]: def gradient_descent(alpha,W,b,X,Y): # sigmoid(z) returns sigmoid of z
def sigmoid(z): # predict(W,b,X) returns vector of predictions given W,b,X
def predict(W,b,X): # plot_decision_boundary(plot_data,W,b) plots decision boundary using weights W,b along with data points plot_data
def plot_decision_boundary(plot_data,W,b): # main() loads MNIST dataset into train_X (training data features), train_Y (training data labels), test_X (test data features), test_Y (test data labels)
def main(): ## DO NOT EDIT CODE BELOW THIS LINE ##
if __name__ == '__main__':
main() ***** Tag Data *****
ID: 3
description: Gradient descent function performing parameter updates for logistic regression.
start line: 26
end line: 209
dependencies:
- type: Function
name: compute_cost
start line: 22
end line: 25
- type: Function
name: sigmoid
start line: 23
end line: 23
context description: The `gradient_descent` function performs iterative updates on
weights `W` and bias `b` using gradient descent algorithm. This involves computing
gradients of the cost function w.r.t parameters `W` and `b`, followed by updating
these parameters using learning rate `alpha`. This code is crucial for training
logistic regression models.
algorithmic depth: 4
algorithmic depth external: N
obscurity: 3
advanced coding concepts: 4
interesting for students: 5
self contained: N ************
## Challenging aspects: ### Challenging aspects in above code:
The provided snippet contains several intricate components that make it complex: 1. **Gradient Calculation**:
- Properly calculating gradients of the cost function concerning both weights (W) and bias (b) requires careful handling of matrix operations.
- Ensuring that dimensions align correctly during matrix multiplication is crucial. 2. **Iterative Update**:
- Implementing iterative updates correctly involves managing convergence criteria (e.g., number of iterations or tolerance level).
- Ensuring numerical stability during updates can be challenging. 3. **Cost Function Computation**:
- Computing cost using log-loss requires understanding of logarithmic operations on probabilities which can lead to numerical issues if not handled properly. 4. **Sigmoid Function**:
- Implementing a numerically stable version of the sigmoid function to avoid overflow/underflow issues. 5. **Prediction Functionality**:
- Correctly implementing prediction logic based on learned parameters (W) and (b). 6. **Decision Boundary Plotting**:
- Visualizing decision boundaries requires understanding how logistic regression maps inputs to outputs. ### Extension:
Here are some ways this problem can be extended: 1. **Regularization**:
- Introduce L1 or L2 regularization terms into both cost computation and gradient calculations. 2. **Mini-batch Gradient Descent**:
- Implement mini-batch gradient descent instead of full-batch gradient descent to improve convergence speed. 3. **Adaptive Learning Rates**:
- Implement adaptive learning
