# Example of resampling

This process is commonly used in **particle resampling** during particle filtering. Here's an example to illustrate the steps:

### Scenario:
We have 5 particles with the following weights:

$$
\text{weights} = [0.1, 0.2, 0.4, 0.2, 0.1]$$

We want to resample these particles based on their weights to produce 5 new particles.

---

### Step-by-Step Explanation:

#### **1. Normalize the weights:**
Ensure the weights sum to 1 (if not already). In this case, they are already normalized:

$$
\text{normalized weights} = [0.1, 0.2, 0.4, 0.2, 0.1]
$$

---

#### **2. Calculate the cumulative sum of the weights:**
The cumulative sum creates intervals that represent the range of selection for each particle:

$$
\text{cumulative weights} = [0.1, 0.3, 0.7, 0.9, 1.0]
$$

This means:
- Particle 1 corresponds to the range \( [0, 0.1) \)
- Particle 2 corresponds to the range \( [0.1, 0.3) \)
- Particle 3 corresponds to the range \( [0.3, 0.7) \)
- Particle 4 corresponds to the range \( [0.7, 0.9) \)
- Particle 5 corresponds to the range \( [0.9, 1.0] \)

---

#### **3. Randomly generate numbers:**
Generate 5 random numbers between 0 and 1 to select particles. For example:

$$
\text{random numbers} = [0.15, 0.85, 0.05, 0.45, 0.75]
$$

---

#### **4. Determine which range each number belongs to:**
Check each random number against the cumulative weights to determine the corresponding particle:

1. **0.15** falls in the range \( [0.1, 0.3) \) → Select Particle 2.
2. **0.85** falls in the range \( [0.7, 0.9) \) → Select Particle 4.
3. **0.05** falls in the range \( [0, 0.1) \) → Select Particle 1.
4. **0.45** falls in the range \( [0.3, 0.7) \) → Select Particle 3.
5. **0.75** falls in the range \( [0.7, 0.9) \) → Select Particle 4.

---

#### **5. Repeat until the desired number of samples is achieved:**
The resampled particles based on this example would be:

$$
\text{resampled particles} = [2, 4, 1, 3, 4]
$$

---

### Summary of Steps in Code:

import numpy as np

# Initial weights
weights = np.array([0.1, 0.2, 0.4, 0.2, 0.1])

# Cumulative sum of weights
cumulative_weights = np.cumsum(weights)

# Generate random numbers
random_numbers = np.random.uniform(0, 1, size=5)

# Resample particles
resampled_indices = np.searchsorted(cumulative_weights, random_numbers)

print("Random Numbers:", random_numbers)
print("Resampled Indices:", resampled_indices)

This will output a similar result, resampling particles based on their weights.