- Suppose I have $n$ items.
- Each item has some baseline properties that are constant. For example, the $i^{th}$ item has properties (i.e. covariates) $w_i$ and $z_i$ that are constant. Each item has a property $X_t$ which follows a stochastic process.
- For the $i^{th}$ item, $X_t$ can be defined as :
$$ X_{t+1} = \begin{cases} X_t + \epsilon_t & \text{if } X_t > 0 \\ 0 & \text{if } X_t \leq 0 \end{cases} $$
$$ \epsilon_t \sim N(\mu_t, \sigma^2) $$
$$ \mu_t = -kX_t $$
I showed some simulations as to how this stochastic process will look - I purposefully wanted this stochastic process to decay towards 0 as time goes on:
Without taking the baseline properties into consideration, to estimate the parameters of this stochastic process, I can write the likelihood and optimize it:
For a single transition when $X_t > 0$, the probability density is:
$$ f(X_{t+1}|X_t; k, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} > \exp\left(-\frac{(X_{t+1} - X_t - k)^2}{2\sigma^2}\right) $$
When $X_t \leq 0$ (by definition):
$$ f(X_{t+1}|X_t; k, \sigma^2) = \mathbb{1}(X_{t+1} = 0) $$
For multiple trajectories $i = 1,...,N$, each of length $T$, the complete likelihood function is (to be numerically solved):
$$ \mathcal{L}(k, \sigma^2) = \prod_{i=1}^N \prod_{t=1}^{T-1} \left[ > \left(\frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(X^i_{t+1} - X^i_t > - k)^2}{2\sigma^2}\right)\right)^{\mathbb{1}(X^i_t > 0)} \cdot (1)^{\mathbb{1}(X^i_t \leq 0)} \right] $$
$$ \log \mathcal{L}(k, \sigma^2) = \sum_{i=1}^N \sum_{t=1}^{T-1} > \mathbb{1}(X^i_t > 0) \left[-\frac{1}{2}\log(2\pi\sigma^2) - > \frac{(X^i_{t+1} - X^i_t - k)^2}{2\sigma^2}\right] $$
$$ \frac{\partial \log \mathcal{L}}{\partial k} = ...$$
$$ \frac{\partial \log \mathcal{L}}{\partial \sigma^2} = ...$$
I have the following question: Suppose I believe that the rate of decay depends on the baseline properties of each item. Can I model this using a mixed effects regression model and then link this to the stochastic process?
Naively, I would write the decay as a mixed effects model:
$$ k_i = \beta_0 + \beta_1w_i + \beta_2z_i + u_i $$ $$ u_i \sim N(0, \tau^2) $$
This would then result in a modified stochastic process:
$$ X_{t+1} = \begin{cases} X_t + \epsilon_t & \text{if } X_t > 0 \\ 0 & \text{if } X_t \leq 0 \end{cases} $$ $$ \epsilon_t \sim N(\mu_t, \sigma^2) $$ $$ \mu_t = -(k_i)X_t = -(\beta_0 + \beta_1w_i + \beta_2z_i + u_i)X_t $$
Finally, as is done in mixed effects regression, I would write the likelihood with an integral to take into consideration the probabilistic element of the random effects (numerically optimized using more involved methods, e.g. Hermite quadrature, Laplace):
$$ f(X_{t+1}|X_t, w_i, z_i; \beta_0, \beta_1, \beta_2, \tau^2, \sigma^2) = \int_{u_i} \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(X_{t+1} - X_t + (\beta_0 + \beta_1w_i + \beta_2z_i + u_i)X_t)^2}{2\sigma^2}\right) \cdot \frac{1}{\tau\sqrt{2\pi}} \exp\left(-\frac{u_i^2}{2\tau^2}\right) du_i $$
$$ \log \mathcal{L}(\beta_0, \beta_1, \beta_2, \tau^2, \sigma^2) = \sum_{i=1}^N \log \int_{u_i} \prod_{t=1}^{T-1} \left[ \left(\frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(X^i_{t+1} - X^i_t + (\beta_0 + \beta_1w_i + \beta_2z_i + u_i)X^i_t)^2}{2\sigma^2}\right)\right)^{\mathbb{1}(X^i_t > 0)} \cdot (1)^{\mathbb{1}(X^i_t \leq 0)} \right] \cdot \frac{1}{\tau\sqrt{2\pi}} \exp\left(-\frac{u_i^2}{2\tau^2}\right) du_i $$
Are these kinds of approaches ever taken in statistics? Can stochastic processes have parameters that are estimated using mixed effects regression?
