Steady State Estimate — ssEst • BayesMassBal

Allows for the estimation of process steady state of a single stream for a process using flow rate data.

Usage

ssEst(y, BTE = c(100, 1000, 1), stationary = FALSE)

Arguments

y: Vector of mass flow rate observations. Must be specified sequentially with y[1] as the initial observation.
BTE: Numeric vector giving c(Burn-in, Total-iterations, and Every) for MCMC approximation of target distributions. The function BMB produces a total number of samples of $(T - B)/E$. $E$ specifies that only one of every $E$ draws are saved. $E > 1$ reduces autocorrelation between obtained samples at the expense of computation time.
stationary: Logical indicating if stationarity will be imposed when generating posterior draws. See Details.

Value

Returns a list of outputs

samples: List of vectors containing posterior draws of model parameters
stationary: Logical indicating the setting of the stationary argument provided to the ssEst function
y: Vector of observations initially passed to the ssEst function.
type: Character string giving details of the model fit. Primarily included for use with plot.BayesMassBal

Details

The model of the following form is fit to the data:

$$y_t = \mu + \alpha y_{t-1} + \epsilon$$

Where $\epsilon \sim \mathcal{N}(0,\sigma^2)$ and $t$ indexes the time step.

A time series is stationary, and predictable, when $|\alpha|< 1$. Stationarity can be enforced, using the argument setting stationary = TRUE. This setting utilizes the priors $p(\alpha) \sim \mathcal{N}$(0, 1000) truncated at (-1,1), and $p(\mu) \sim \mathcal{N}$(0, var(y)*100) for inference, producing a posterior distribution for $\alpha$ constrained to be within (-1,1).

When fitting a model where stationarity is not enforced, the Jeffreys prior of $p(\mu,\alpha)\propto 1$ is used.

The Jeffreys prior of $p(\sigma^2)\propto 1/\sigma^2$ is used for all inference of $\sigma^2$

A stationary time series will have an expected value of:

$$\frac{\mu}{1-\alpha}$$

Samples of this expectation are included in the output if stationary = TRUE or if none of the samples of $\alpha$ lie outside of (-1,1).

The output list is a BMB object, passing the output to plot.BayesMassBal allows for observation of the results.

Examples


## Generating Data
y <- rep(NA, times = 21)

y[1] <- 0
mu <- 3
alpha <- 0.3
sig <- 2
for(i in 2:21){
 y[i] <- mu + alpha*y[i-1] + rnorm(1)*sig
}

## Generating draws of model parameters

fit <- ssEst(y, BTE = c(100,500,1))