# Methods for time-series prediction depending on multiple parameters

We have hourly time-series data of the status of a system: number of people present at different train stations. We collected it for a year, and we want to use it to train a model to predict the status of this system in future.

We know that several parameters influence the system status, such as: temperature, type of day (holiday/working day/weekend), period of the year, etc. Thus, we collected data for these parameters in parallel to the data of our system status.

We are interested in implementing a tool that predicts the status of the system in the next hours based on the previous observations of the system status and on the external parameters that affect it.

What methods are most suitable for such a problem?

A Vector Autoregressive (VAR) might be suitable for this problem. Consider a vector of $\mathbf{y}_t=(y_{1,t},y_{2,t},\dots,y_{n,t})'$ where the different $y$s are the variables you have collected, and for example $y_1$ is the variable you are interested in modelling, status of a system, and the $n$ other variables you mentioned. You can fit the following model, $\mathbf{y_t}=\beta_0+\beta_1 \mathbf{y}_{t-1}+\beta_2 \mathbf{y}_{t-2}+\cdots+\beta_p \mathbf{y}_{t-p}+\epsilon_t$, by OLS/GLS, where $\epsilon_t$ is the residuals which is $IIDN~N(0,\Sigma_\epsilon)$. You can use this model to predict the future values of you dependent variable, by regressing on the past values of the dependent variable and the other variables you have collected. Is this what you are looking for? I'd recommend testing the individual series for unit roots first, and difference the series if there are unit roots present.
• Variables such as holiday or working day can be included by transforming them into dummy variables, for example in one week there is $(1,1,1,1,1,0,0)$ where $1$ is a work day and $0$ is holiday, and you can see the difference on the coefficient of the dummy. If there are variables with several ordinal categories, you should considered using a ordinal logistic regression. – fredrikhs Jul 8 '13 at 12:34
• Thank you, this is very interesting. Thus, I am planning to transform the qualitative variables into 0/1 dummy variables. I would then apply a regression method to predict the status at time $t$ based on: - the quantitative variables (e.g. temperature) at $t$ - the dummified variables at $t$ (e.g. holiday) - the status (number of people at station) at the previous times $t-1$, $t-2$,... Therefore, I will embed the status of the previous times in the vector $y_t$ Is this method correct? – Marco Jul 9 '13 at 11:43