Question 1178892: What is the correct way to write the following function:
K(x, xi) = Square of alpha multiplied with exp power minus of summation of k of (x - xi)² if we have following condition:
..
Vectors x1, x2, x3, ... , xn belongs to d dimensional Real space (R with superscript d) and d is from 1 to D dimensional.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down how to write this function in a clear and standard mathematical notation, along with explanations.
**Understanding the Components:**
* **K(x, xi):** This indicates that the function K takes two inputs:
* `x`: A vector in d-dimensional real space (ℝᵈ).
* `xi`: Another vector in d-dimensional real space (ℝᵈ).
* **α²:** "Square of alpha" means α raised to the power of 2 (α²).
* **exp(...):** This represents the exponential function (e raised to the power of...).
* **∑k (...):** This is a summation over an index k. We need to clarify what k is summing over.
* **(x - xi)²:** This represents the squared Euclidean distance between vectors x and xi.
**Clarifying the Summation:**
Since x and xi are d-dimensional vectors, the summation is likely over the components of these vectors. So, if:
* x = (x₁, x₂, ..., xᵈ)
* xi = (xi₁, xi₂, ..., xiᵈ)
Then:
* (x - xi)² = ∑(k=1 to d) (xk - xik)²
**Putting it All Together:**
The function K(x, xi) can be written as:
**K(x, xi) = α² * exp(-∑(k=1 to d) (xk - xik)²)**
**More Detailed Notation:**
If you want to be even more explicit, you can write:
**K(x, xi) = α² * exp(-∑(k=1)ᵈ (x<0xE2><0x82><0x96> - xi<0xE2><0x82><0x96>)²)**
Where:
* x<0xE2><0x82><0x96> represents the k-th component of vector x.
* xi<0xE2><0x82><0x96> represents the k-th component of vector xi.
**Python Example (for clarity):**
```python
import numpy as np
def K(x, xi, alpha):
"""
Calculates the function K(x, xi).
Args:
x: A numpy array representing vector x.
xi: A numpy array representing vector xi.
alpha: A scalar value.
Returns:
The value of K(x, xi).
"""
diff = x - xi
squared_distance = np.sum(diff**2)
return alpha**2 * np.exp(-squared_distance)
# Example usage:
x = np.array([1, 2, 3])
xi = np.array([4, 5, 6])
alpha = 2.0
result = K(x, xi, alpha)
print(result)
```
**Key Points:**
* The summation is crucial. It's summing the squared differences of the corresponding components of the vectors.
* The exponential function applies to the negative of the squared Euclidean distance.
* Alpha is a scaling factor.
* The function is a radial basis function (RBF) kernel, and is used in machine learning.
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