# Intuition Behind Correlation Function in Kriging Models

I'm thinking and researching extensively to interpret the parameter $$\theta$$ (activeness parameter) in Gaussian correlation function in a Kriging model, namely as:
$$K(h;\theta)=exp(-h^2/(2\theta^2))$$ or in some literatures: $$K(h;\theta)=exp(-\theta h^2)$$

The random variables are correlated with each other using the basis function expression:

$$cor[Y(x^{(j)}),Y(x^{(l)})]=exp(-\sum_{i=1}^k\theta_i|x_i^{(j)}-x_i^{(l)}|^2)$$

I know this parameter $$\theta_i$$ represents the activeness of each feature $$x_i, i=1,...k$$ where $$k$$ is the number of features.
My question here is, after we estimate this $$\theta$$ for each feature, how do we interpret this parameter $$\theta$$ for every feature in the first and second equation? For example, if we find $$\theta_{x_1} = 10$$ for $$x_1$$, what does 10 mean here?

• What is the meaning of the subscript $i$ and how do you suppose the $\theta_i$ are related to $\theta$? – whuber May 9 at 17:06
• $i$ refers to $i$th feature in data. – user273192 May 9 at 17:41
• @whuber I updated my question to clarify how $\theta$ relates to $\theta_i$. – user273192 May 9 at 17:54
• This blog post gives some helpful intuition. yugeten.github.io/posts/2019/09/GP – Sycorax May 10 at 18:44
• This post has some relevant information about the question. – user273192 May 16 at 20:00

$$K(h;\theta)=exp(-\theta h^2)$$

The overall intuition is that $$h$$ is represents the distance between two points. It is always non-negative.

As $$\theta$$ multiplies $$h$$, the larger $$\theta$$ is the more negative $$-\theta h^2$$ becomes.

The more negative the exponent becomes, the smaller $$K$$ becomes.

Therefore, increasing $$\theta$$ makes your 'correlation' drop to zero more quickly the further apart the points are.

You'd have to do a similar walkthrough for each formulation, depending on signs, whether $$\theta$$ divides or multiplies etc.

But note for the third formulation that, holding $$x_i^{(k)}, i>1, k \in \{j,l\}$$ constant, the equation could be written:

$$K = exp(-\sum_{i=1}^k\theta_i|x_i^{(j)}-x_i^{(l)}|^2) = exp(-\theta_1 (x_1^{(j)}-x_1^{(l)})^2) c = exp(-\theta_1 h_1^2) c$$

Which again has the above interpretation.

• Thank you "@conjectures" for your response. I can follow how $\theta$ would affect the correlation between samples. What I cannot infer is to relate $\theta$ value to each feature. Since in Kriging, $\theta$ is the activeness parameter which I cannot interpret it intuitively. For example, if we find $\theta_{x_1} = 10$ for $x_1$, what does 10 mean here? – user273192 May 9 at 18:36
• Exactly the same interpretation applies as above, but holding the other elements constant. That's why I picked your second formulation rather than your first ;) Anyway, added a comment. – conjectures May 10 at 18:24
• So can we say, as $\theta_i$ becomes larger, the corresponding variable $x_i$ encourages the points to be more uncorrelated, given other variables constant? – user273192 May 11 at 4:56
• > 'the corresponding variable xi'. Actually the distance between the $i^{th}$ elements of a pair of points $x^{(j)}, x^{(l)}$ – conjectures May 14 at 17:51
• Can you please elaborate more your last comment? I did not get what you mean. Thank you! – user273192 May 15 at 20:04