# decomposition of time series into many time series

I have two time series

• $x_t$ represents expected default rate from one data provider
• $y_t$ represent expected default rate from another data provider.

Correlation between $y_t$ and $x_t$ is 0.9 However, the level between $x_t$ and $y_t$ is significantly different due to the data provider. One pattern of $x_t$ is

• the data provider provides some components of $x_t$ as well (expected default rates for 5 different countries): $x_{t,k}$ for $k=1$ to 5

I would like to decompose $y_t$ into the 5 different countries given the information in $x_t$ (with the inclusion of noise since the coleration coefficient is not perfect). Do you know a statistical method?

• Could you elaborate on what you want to do and what data you have? Currently, your question is not clear enough to be answerable. Feb 11 '17 at 1:09
• @MatthewGunn I updated the question, is it better now? thanks for your precious help Feb 11 '17 at 17:39
• Does $x_t = \sum_{i=1}^5 x_{t,i}$? How do the components $x_{t,k}$ aggregate to form $x_t$? Feb 11 '17 at 17:43
• @MatthewGunn correct, they aggregate using your formula, do you have any idea? Feb 11 '17 at 21:48

### Quick reactions:

It sounds like you're asking about how to separate tomatoes from olive oil from peppers etc... after it's all been mixed together in tomato sauce $y_t$. It's actually even worse than that because at least you know what peppers look like, there's some way to identify individual components. If you tell me $a + b + c + d + e = 10$, how can I possibly determine what $a$, $b$, $c$, $d$, and $e$ are? It's an undetermined system.

There are quite a few questions on here of the form, "I don't have some data Z, but I'd like to have it. Is there something magic I can do?" And the answer generally is no.

You do have the $x_{t,1}$, $x_{t,2}$, $\ldots$, $x_{t,5}$ series though. If you assume that $\frac{\hat{y}_{i,t}}{y_t} \approx \frac{x_{i,t}}{x_t}$ then you perhaps could try the following:

For $i=1,\ldots,k$, let $a_{t,i} = \frac{x_{t,i}}{x_t}$ where $x_t = \sum_{i=1}^5 x_{t,i}$. You could try something ad-hoc like $\hat{y}_{t,i} = a_{t,i} y_t = x_{t,i} \left(\frac{y_t}{x_t}\right)$, the idea being the shares are the same $\frac{\hat{y}_{i,t}}{y_t} = \frac{x_{i,t}}{x_t}$.
Realize though that you'd just be rescaling the five $x_{t,i}$ series so that they total to $y_t$ instead. This isn't magic. And I don't know if it would be at all useful.
Let $\mathbf{y}_t = \begin{bmatrix} y_{t,1} \\ y_{t,2} \\ \ldots \\ y_{t,5} \end{bmatrix}$. You don't observe $\mathbf{y}_t$ but if you have other data $\mathbf{z}_t = f(\mathbf{y}_t)$, and process $\mathbf{y}_t$ satisfies the Markov property, then you could treat this is as a Hidden Markov Model and try to infer the hidden state through some Bayesian filter.
For example, $\mathbf{y}_t$ could be a state variable describing the location, speed of a rocket, and you try to infer this hidden state based upon instrument readings. For this to work, you need some strong theory about how $\mathbf{y}_t$ evolves and how other data you do observe is related to $\mathbf{y}_t$.