Approximate new acorrelation given previous acorrelation and a new set of data?

[member 428835]

Hi PF!

The autocorrelation coefficient $\rho$ is defined as $$\rho_k \equiv \frac{\sum_{t=k+1}^T (x_t - \bar x)(x_{t-k} - \bar x)}{\sum_{t=1}^T(x_t-\bar x)^2}$$

Now suppose we calculate $\rho$ through $T$, but are then given a new data at time $T + \Delta t$. Is there a way to approximate the new autocorrelation without recalculating from scratch? Obviously if we use the definition we'd have to recalculate the mean $\bar x$ and therefore the entire computation.

 

[pbuk]

Is there a way to approximate the new autocorrelation without recalculating from scratch?

Yes, can you multiply out the terms in $(x_t - \bar x)(x_{t-k} - \bar x)$ and $(x_t-\bar x)^2$?

 

[member 428835]

Yes, can you multiply out the terms in $(x_t - \bar x)(x_{t-k} - \bar x)$ and $(x_t-\bar x)^2$?

$\bar x^2 - \bar x x_{t-k}- \bar x x_{t} + x_t x_{t-k}$ and $\bar x^2 -2 \bar x x_{t} + x_t ^2$

 

[pbuk]

$\bar x^2 - \bar x x_{t-k}- \bar x x_{t} + x_t x_{t-k}$ and $\bar x^2 -2 \bar x x_{t} + x_t ^2$

Excellent, now you can sum over the terms separately:
$$ \rho_k \approx \frac{
\sum_{t=k+1}^T \bar x^2
- \sum_{t=k+1}^T \bar x x_{t-k}
- \sum_{t=k+1}^T \bar x x_{t}
+ \sum_{t=k+1}^T + x_t x_{t-k}
}{
\sum_{t=1}^T \bar x^2
- 2 \sum_{t=1}^T \bar x x_{t}
+ \sum_{t=1}^T x_t^2
} $$
and finally rewrite all the $\bar x$ terms e.g.
$$ \sum_{t=k+1}^T \bar x x_{t-k} = \bar x \sum_{t=k+1}^T x_{t-k} = \frac 1 T \sum_{t=1}^T x_t \sum_{t=k+1}^T x_{t-k} $$
Then you "just" need to

1. translate that all into code which keeps track of the running sums
2. test it thoroughly
3. rewrite it to pick up mistakes in my or your algebra
4. test it again

Good luck!