A population model for PCA

We know that in econometrics it is common to work with population models and relationships. Thus, when we are faced with the data, we appeal to the analogy technique to emulate the population condition. This is the principle of analogy (see, for example this).

Well, I'm reading a little about Principal Component Analysis (PCA) and one motivation is when we have more regressors ($$n$$) than the number or sample size ($$T$$). But first, the population model is: $$y= x'\beta + u_t, \quad x= (x_1,..., x_n)$$ Denote the covariance matrix of $$x$$ as $$\Sigma= [cov(x_i,x_j)]_{n\times n}$$.

If we have a sample $$(x_{t}= (x_{1t},.., x_{nt}))_{t = 1}^T$$ with $$n>>T$$, tipically the problem of PCA begins first to find $$\gamma$$ such that $$$$\max_{\gamma \in \mathbb R^n} \gamma' \hat{\Sigma} \gamma, \quad \hbox{ s.t. } |\gamma|^2 = \gamma' \gamma = 1$$$$ where $$\gamma' \hat{\Sigma} \gamma$$ is nothing more that the covariance matrix of $$X\gamma$$ with $$X$$ being the matrix sample of the $$x_t$$. Moreover, $$\hat \Sigma = \frac{1}{T} X'X$$.

As you can see, this is a problem that uses the available samples given by $$X$$. But I would like to know if there is behind this a maximization problem with population objects only.

• I'm afraid I don't understand what you might mean by "with population objects only." Part of my problem is that PCA neither requires nor uses a statistical model. (That's just an extra complication introduced in some applications.) So I am left wondering what you mean by the "population" and by "population objects."
– whuber
May 2, 2023 at 21:42
• Nothing else. Just something analogous to a relationship between population moments that are crucial for identification. My question was just out of curiosity that arose when I was studying the subject, but I think I understand that it might be an unnecessary complication. I just wanted to know if it was something important to note. May 2, 2023 at 22:18