## Autoregressive

Given a time series ${T^i}$, a simple predictive model can be constructed using an autoregressive model.

$$\begin{equation} T^t = \sum_{i=1}^p \beta_i T^{t - i} + \beta^t + \beta^0. \end{equation}$$

Such a model is usually called an AR(p) model due to the fact that we are using data back in $p$ steps.

Differential Equation

For simplicity we will look at a AR(1) model. Assume the time series has a step size of $dt$, our model can be rewritten as

$$T^t = \beta_1 T^{t - 1} + \beta^t + \beta^0$$

which can be rewritten in the following way

$$(1 - \beta_1) T^t = \beta_1 T^{t - 1} - \beta_1 T^t + \beta^t + \beta^0.$$

We can cast it into a differential equation form

$$T(t) = - dt \frac{\beta_1}{1 - \beta_1} T’(t) + \frac{\beta^t + \beta^0}{1 - \beta_1}.$$

For AR(2), we have

$$T^t = \beta_1 T^{t - 1} + \beta_2 T^{t - 2} + \beta^t + \beta^0$$

casted as

\begin{align*} &(1-\beta_1 - \beta_2) T^t = -\beta_1 (T^t - T^{t - 1}) - \beta_2 (T^t - T^{t-1} + T^{t-1} - T^{t - 2}) + \beta^t + \beta^0 \\
\Rightarrow &(1-\beta_1 - \beta_2) T^t = -dt \beta_1 (T^t - T^{t - 1})/dt - 2dt\beta_2 (T^t - T^{t-1} + T^{t-1} - T^{t - 2})/(2dt) + \beta^t + \beta^0 \\
\Rightarrow &T^t = - dt \frac{\beta_1 + 2\beta_2}{1-\beta_1 - \beta_2} T’(t) + \frac{\beta^t + \beta^0}{1-\beta_1 - \beta_2} \end{align*}

We could also write this into a combination of first order derivative and second order derivative form but I think it is better to be only first order derivative.

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