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Add independent error to data

Source: R/measurement_models.R
independent.Rd

Error is generated through a multivariate normal distribution with a mean and covariance that are specified by the user. Primary assumption of this model is that the errors are independent over time, which stands in contrast to the temporal() model.

Usage

independent(
  data,
  mean = c(0, 0),
  covariance = matrix(c(0.031^2, 0, 0, 0.027^2), nrow = 2, ncol = 2)
)

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.

mean

Numeric vector with 1 or 2 values containing the mean of the error process. If only 1 number is provided, then this value will be assumed to represent the mean for both the x- and y-dimensions. Defaults to c(0, 0), representing no bias in the system.

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.

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 some white noise error. In symbols, this comes down to saying that:

$$\mathbf{\epsilon}_{i} \sim N(\mathbf{\mu}, \Sigma)$$ where \(\mathbf{\mu}\) represents the mean of the noise and \(\Sigma\) the covariance matrix of the noise. The observed positions \(\mathbf{y}_i\) then consist of a systematic component (i.e., the latent positions, provided in data) and an unsystematic error component so that:

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

See also

noiser() temporal()

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
independent(
  data,
  mean = c(0, 0),
  covariance = diag(2) * 0.01
) |>
  head()
#>          x         y time
#> 1 9.861939 0.1534916    1
#> 2 9.752047 1.2243047    2
#> 3 9.561796 2.4594255    3
#> 4 9.298569 3.8016829    4
#> 5 8.756276 4.8954170    5
#> 6 7.972390 5.8461441    6