Kalman Filter

🔥 Vibe Prompt

"Track a vehicle's position and velocity from noisy GPS (σ=30m) and IMU (σ=20m). Compare raw vs filtered."

import numpy as np

class KalmanFilter:
    def __init__(self):
        self.x = np.array([[0], [0], [0], [0]])  # pos_x, pos_y, vel_x, vel_y
        self.P = np.eye(4) * 100
        self.F = np.array([[1,0,1,0],[0,1,0,1],[0,0,1,0],[0,0,0,1]])
        self.H = np.array([[1,0,0,0],[0,1,0,0]])
        self.R = np.eye(2) * 30  # GPS noise
        self.Q = np.eye(4) * 0.1
    
    def predict(self):
        self.x = self.F @ self.x
        self.P = self.F @ self.P @ self.F.T + self.Q
    
    def update(self, z):
        y = z - self.H @ self.x
        S = self.H @ self.P @ self.H.T + self.R
        K = self.P @ self.H.T @ np.linalg.inv(S)
        self.x = self.x + K @ y
        self.P = (np.eye(4) - K @ self.H) @ self.P

kf = KalmanFilter()
true_pos = np.array([100, 100])
true_vel = np.array([5, 3])

for t in range(20):
    true_pos = true_pos + true_vel  # True movement
    gps = true_pos + np.random.normal(0, 30, 2)  # Noisy GPS
    
    kf.predict()
    kf.update(gps)
    
    error_raw = np.linalg.norm(gps - true_pos)
    error_filt = np.linalg.norm(kf.x[:2].flatten() - true_pos)
    
    print(f"t={t}: raw_error={error_raw:.1f}, filt_error={error_filt:.1f}")

Summary

| Aspect | Detail | |--------|--------| | State | What we're tracking (position, velocity, etc.) | | Predict | Motion model (F) + process noise (Q) | | Update | Measurement (z) + measurement noise (R) | | Kalman Gain (K) | Balances trust between prediction and measurement | | Extensions | EKF (non-linear), UKF (unscented), particle filter (non-Gaussian) |

Applications

| Domain | Application | |--------|-------------| | 🛰️ GPS/IMU | Drone, self-driving car, smartphone position | | 🚀 Aerospace | Rocket guidance, satellite orbit determination | | 🤖 Robotics | SLAM (Simultaneous Localization And Mapping) | | 📈 Finance | Hidden Markov models for regime detection | | 🫀 Medical | ECG denoising, blood glucose monitoring |

Gradient Descent Course Complete! 🎉

✅ Gradient Descent Basics
✅ Momentum & Adam
✅ FFT / DFT
✅ PID Control
✅ Kalmanum & Adam\n- ✅ FFT / DFT\n- ✅ PID Control\n- ✅ Kalman Filter"}]}

Key Points

Understand the core concepts thoroughly
Practice with hands-on code examples
Apply knowledge to real-world problems
Review and reinforce through exercises

Further Learning

Official documentation
Open source projects on GitHub
Community forums and discussions
Related courses and tutorials

梯度下降實戰要點

梯度下降是機器學習中最核心的最佳化演算法。理解它的變體（SGD、Momentum、Adam）對於訓練深度學習模型至關重要。

核心概念

梯度方向是函數增加最快的方向
學習率決定每一步的大小
SGD 用隨機樣本近似梯度
Momentum 加速收斂
Adam 結合 Momentum + RMSProp

卡爾曼濾波：從雜訊中過濾出真相

GPS 訊號有誤差（5-15 公尺）、速度感測器有雜訊。卡爾曼濾波結合不完美的資訊，估計出比任何單一感測器都準確的狀態。

預測-更新循環

預測：根據物理模型預測下一時刻狀態
       計算預測的不確定性
更新：取得感測器讀數
       計算 Kalman Gain（相信預測 vs 相信感測器）
       用感測器讀數修正預測

與梯度下降的關聯

Kalman Filter 可以視為「線上版本的梯度下降」——每次有新資料就更新估計。用在自駕車、機器人定位、金融時間序列。

課程總結

這堂課從基本梯度下降到 Kalman Filter——涵蓋了最佳化和控制領域的核心演算法。