Notation

$$N$$ $$t \in \left\lbrace 1, 2, \ldots, N \right\rbrace $$ $$t' \in \left\lbrace 31/1/1970 , 28/2/1970, \ldots, 31/12/2024 \right\rbrace $$ $$r$$ $$R_i$$ $$r_i(t)$$ $$S_i$$ $$s_i(t)$$ $$S'$$ $$S'_i$$ $$s'_i(t)$$ $$(t-1,t)$$ $$i \in \left\lbrace 1, 2, \ldots, m \right\rbrace $$ $$j \in \left\lbrace 1, 1.5, 2, \ldots, 19.5, 20 \right\rbrace $$ $$\bar{{s}}(t)$$ $$\Sigma$$ $$\lambda$$

\bar{{s}}(t)=\frac{{\sum_{{t=1}}^{{N}} s(t)}}{{N}})" i \in {1, 2, \ldots, m} \mathbf{r_1},\mathbf{r_2},...,\mathbf{r_m} \right)$" s_i(t)=\log{{ r_i(t)}}-\log {{ r_{{i-1}}(t)}} )" s'_i(t)=s_i(t)-\bar{{s}}(t))"

\Sigma = \text{Cov}(S') = \begin{bmatrix} \text{Var}(s'_1) & \text{Cov}(s'_1, s'_2) & \cdots & \text{Cov}(s'_1, s'_n) \ \text{Cov}(s'_2, s'_1) & \text{Var}(s'_2) & \cdots & \text{Cov}(s'_2, s'_n) \ \vdots & \vdots & \ddots & \vdots \ \text{Cov}(s'_n, s'_1) & \text{Cov}(s'_n, s'_2) & \cdots & \text{Var}(s'_n) \end{bmatrix}

\text{Cov}(S'{i_1}, S'{i_2}) = \frac{1}{N-1} \sum_{t=1}^{N} (s'{i_1}(t) - \bar{s'{i_1}}) (s'{i_2}(t) - \bar{s'{i_2}}) )" \Sigma_{{i_1},{i_2}} = \mathrm{Cov}(S'{i_1}, S'{i_2}))" \Sigma_{1,1} = \text{Var}(s'1) = \frac{1}{N-1} \sum{t=1}^{N} (s_1'(t) - \bar{s_1'}) (s_1'(t) - \bar{s_1'}) )" \Sigma v = \lambda v)" \mathbf{v} = \lambda \mathbf{v})" \det(A - \lambda I) = 0)" ((A - \lambda I)\mathbf{v} = \mathbf{0})"