סדרות עתיות ושיטות חיזוי – הרצאה 2

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(0) # For reproducibility

N = 500
u_noise = np.random.uniform(-0.5, 0.5, N)
g_noise = np.random.normal(0, 1, N)

def plot_lags(series, title):
    fig, axes = plt.subplots(1, 3, figsize=(15, 4))
    
    # Time series plot
    axes[0].plot(series)
    axes[0].set_title(f"{title} - Time Series")
    axes[0].set_xlabel("Time (t)")
    axes[0].set_ylabel("Value")
    
    # Lag 1 plot
    axes[1].scatter(series[:-1], series[1:], alpha=0.5)
    axes[1].set_title("Lag 1 Plot")
    axes[1].set_xlabel("x[t]")
    axes[1].set_ylabel("x[t+1]")
    
    # Lag 5 plot
    axes[2].scatter(series[:-5], series[5:], alpha=0.5)
    axes[2].set_title("Lag 5 Plot")
    axes[2].set_xlabel("x[t]")
    axes[2].set_ylabel("x[t+5]")
    
    plt.tight_layout()
    plt.show()

plot_lags(u_noise, "Uniform IID")
plot_lags(g_noise, "Gaussian IID")

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(0) # For reproducibility

N = 500
noise = np.random.normal(0, 1, N)
random_walk = np.cumsum(noise)
first_difference = np.diff(random_walk)

fig, axes = plt.subplots(2, 1, figsize=(10, 6))

axes[0].plot(random_walk, color='blue')
axes[0].set_title("Random Walk (Non-Stationary)")
axes[0].set_ylabel("Value")

axes[1].plot(first_difference, color='green')
axes[1].set_title("First Difference (Stationary / White Noise)")
axes[1].set_ylabel("Difference")
axes[1].set_xlabel("Time (t)")

plt.tight_layout()
plt.show()

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(0) # For reproducibility

def acf(x, H):
    N = len(x)
    x_mean = np.mean(x)
    x_centered = x - x_mean
    
    # Calculate gamma_0 (variance)
    gamma_0 = np.sum(x_centered**2) / N
    
    rhos = []
    for h in range(H + 1):
        # Calculate gamma_h
        gamma_h = np.sum(x_centered[h:] * x_centered[:N-h]) / N
        rho_h = gamma_h / gamma_0
        rhos.append(rho_h)
        
    return np.array(rhos)

# Data from previous tasks
N = 500
iid_noise = np.random.normal(0, 1, N)
rw = np.cumsum(iid_noise)

H = 20
acf_iid = acf(iid_noise, H)
acf_rw = acf(rw, H)

# Plotting
lags = np.arange(H + 1)
fig, axes = plt.subplots(1, 2, figsize=(12, 4))

axes[0].bar(lags, acf_iid, color='purple')
axes[0].set_title("ACF of IID Noise")
axes[0].set_xlabel("Lag (h)")
axes[0].set_ylabel("ACF")

axes[1].bar(lags, acf_rw, color='orange')
axes[1].set_title("ACF of Random Walk")
axes[1].set_xlabel("Lag (h)")

plt.tight_layout()
plt.show()

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(0) # For reproducibility

t = np.arange(100)
trend = 0.5 * t
noise = np.random.normal(0, 3, 100)
jumps = np.zeros(100)
jumps[50:] = 20  # Sudden marketing success!

data = trend + noise + jumps

# Fit linear trend
coeffs = np.polyfit(t, data, 1)
trend_est = np.polyval(coeffs, t)

# Detrending
detrended = data - trend_est

fig, axes = plt.subplots(2, 1, figsize=(10, 8))

axes[0].plot(t, data, label="Original Data (Sales)")
axes[0].plot(t, trend_est, color='red', linestyle='--', label="Fitted Trend")
axes[0].set_title("Time Series with Linear Trend Fit")
axes[0].legend()

axes[1].plot(t, detrended, color='green')
axes[1].axhline(0, color='black', linestyle='--')
axes[1].set_title("Detrended Series (Residuals)")

plt.tight_layout()
plt.show()

import numpy as np

np.random.seed(0) # For reproducibility

def simulate_casino(p, num_sims=1000):
    success_count = 0
    for _ in range(num_sims):
        balance = 50
        while 0 < balance < 100:
            if np.random.rand() < p:
                balance += 1
            else:
                balance -= 1
        if balance == 100:
            success_count += 1
    return success_count / num_sims * 100

print(f"Win probability (p=0.49): {simulate_casino(0.49)}%")
print(f"Win probability (p=0.51): {simulate_casino(0.51)}%")