Add temporal error to data

Error is generated through a vector autoregressive model with its defining parameters parameters intercept, transition, and covariance, which should be specified by the user. Note that the temporal component of this model is defined on the residual level (i.e., residuals carry over over time), not on the data level (positions are not carried over over time). See the details for more information.

Usage

temporal(
  data,
  intercept = c(0, 0),
  transition = matrix(c(0.925, 0.085, 0.085, 0.87), nrow = 2, ncol = 2),
  covariance = matrix(c(0.015^2, 0, 0, 0.015^2), nrow = 2, ncol = 2),
  sampling_rate = 6.13
)

Arguments

data: A data.frame containing the data on which to base the parameters of the constant velocity model. This function assumes that this data.frame contains the columns "time", "x", and "y" containing the time at which the observed position (x, y) was measured respectively.
intercept: Numeric vector with 1 or 2 values containing the intercept of the vector autoregressive model.If only 1 number is provided, then this value will be assumed to represent the intercept for both the x- and y- dimensions. Defaults to c(0, 0), representing no bias in the system.
transition: Numeric matrix of dimensionality $2 \times 2$ having the autoregressive effects on the diagonal and crossregressive effects on the off-diagonal. Defaults to the autoregressive effects being equal to 0.925 and 0.87 for the x- and y-dimension respectively, and the cross-regressive effects being equal to 0.085 in a symmetric way. These values represent the means of these parameters as found across the four days of the calibration study that informed this project.
covariance: Numeric matrix of dimensionality $2 \times 2$, having the variances of the errors for x and y on the diagonal and the covariance of the errors on the off-diagonal. Defaults to having the variances of the x- and y-dimension being equal to 0.031^2 and 0.027^2 respectively, and the covariance being equal to 0.02. These values represent the means of the respective variances and covariances found across the four days of the calibration study that informed this project.
sampling_rate: Numeric denoting the sampling rate in Hz. Is used to scale the transition matrix according to the actual sampling rate. Defaults to 6.13, which is the mean sampling rate for the data obtained across the four days of the calibration study that informed this project. If you change the transition argument, it is recommended to ignore this argument.

Value

A data.frame in which the "x" and "y" columns have additional noise in them.

Details

According to this function, the observed positions $\mathbf{y}_i$ at time $t_i$ contain error that in itself is temporally related, that is carried over over time. Specifically, we assume that the error $\mathbf{\epsilon}_i$ at time $t_i$ depends on the error $\mathbf{\epsilon}_{i - 1}$ at previous time $t_{i - 1}$ and a white noise residual at the same timepoint, as specified in the vector autoregressive model. In symbols:

$$\mathbf{\epsilon}_i = \mathbf{\delta} + \Theta \mathbf{\epsilon}_{i - 1} + \mathbf{\omega}_i$$

with

$$\mathbf{\omega}_i \sim N(\mathbf{0}, \Sigma)$$ where $\mathbf{\delta}$ represents the intercept of the model, $\Theta$ is the transition matrix defining the temporal carry-over, and $\Sigma$ is the covariance matrix of the white noise residuals $\mathbf{\omega}$. Note that for the vector autoregressive model, the mean of the process is defined as:

$$\mathbf{\mu} = (I - \Theta)^{-1} \mathbf{\delta}$$ which is useful to keep in mind when specifying the intercept to add bias to the system. Furthermore note that the covariance $\Sigma_\epsilon$ of the process $\mathbf{\epsilon}$ is defined as:

$$\Sigma_\epsilon = \Sigma - \Theta \Sigma \Theta^T$$ which is useful to keep in mind when specifying the covariance matrix $\Sigma$, which is defined on the level of $\mathbf{\omega}$.

Once the errors $\mathbf{\epsilon}_i$ are defined, we compute the observed positions $\mathbf{y}_i$ as consisting of a systematic component (i.e., the latent positions, provided in data) and the unsystematic error component so that:

$$\mathbf{y}_i = \mathbf{x}_i + \mathbf{\epsilon}_i$$

Examples

# Generate data for illustration purposes. Movement in circular motion at a
# pace of 1.27m/s with some added noise of SD = 10cm.
angles <- seq(0, 4 * pi, length.out = 100)
coordinates <- 10 * cbind(cos(angles), sin(angles))

data <- data.frame(
  x = coordinates[, 1],
  y = coordinates[, 2],
  time = 1:100
)

# Add independent noise to these data with an error variance of 0.01 in each
# dimension and a transition matrix containing autoregressive effects of 0.15
temporal(
  data,
  intercept = c(0, 0),
  transition = diag(2) * 0.15,
  covariance = diag(2) * 0.01
) |>
  head()
#>          x           y time
#> 1 9.952147 -0.07084848    1
#> 2 9.745384  1.40280651    2
#> 3 9.657008  2.61651259    3
#> 4 9.324401  3.86177891    4
#> 5 8.671161  4.91355663    5
#> 6 8.100134  5.89624943    6

Usage

Arguments

Value

Details

See also

Examples