Applied Example with Bison Movement Data

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This vignette illustrates a compact workflow using the bison data included in CircularRegression. The response y.dir is the movement direction at the current time. The covariate y.prec is the previous movement direction, and x.meadow:z.meadow represents the direction toward the nearest meadow weighted by the associated non-negative distance variable.

library(CircularRegression)

data(bison)

d <- bison[seq_len(200), c("y.dir", "y.prec", "x.meadow", "z.meadow")]
d <- na.omit(d)
str(d)
#> Classes 'tbl_df', 'tbl' and 'data.frame':    182 obs. of  4 variables:
#>  $ y.dir   : num  -2.27 -2.27 -2.27 -2.27 -2.26 ...
#>  $ y.prec  : num  -2.27 -2.27 -2.27 -2.27 -2.27 ...
#>  $ x.meadow: num  -2.22 -2.22 -2.22 -2.22 -2.22 ...
#>  $ z.meadow: num  600 590 580 570 560 ...
#>  - attr(*, "na.action")= 'omit' Named int [1:18] 1 2 66 67 74 75 99 100 121 122 ...
#>   ..- attr(*, "names")= chr [1:18] "1" "2" "66" "67" ...

Fit the default two-step model

fit <- circular_regression(
  y.dir ~ y.prec + x.meadow:z.meadow,
  data = d
)

fit
#> Call:
#> circular_regression(formula = y.dir ~ y.prec + x.meadow:z.meadow, 
#>     data = d)
#> 
#> Method: two_step 
#> Number of observations: 182 
#> 
#> Call:
#> angular(formula = formula, data = data, reference = c("name", 
#>     ref_name), initbeta = initbeta, control = control_angular, 
#>     na.action = na.action)
#> 
#> Reference angle: y.prec 
#> Maximum cosine: 0.9813 
#> Concentration parameter: 27.05 
#> 
#> Parameters (non-reference terms):
#>                     Estimate Std. Error z value P(|z|>.)   
#> x.meadow:z.meadow 0.00051640 0.00017127  3.0151 0.002569 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The default fit uses the consensus model to select a reference direction and initialize the homogeneous angular fit.

summary(fit)
#> Circular regression summary
#> Method: two_step 
#> Number of observations: 182 
#> 
#> Call:
#> angular(formula = formula, data = data, reference = c("name", 
#>     ref_name), initbeta = initbeta, control = control_angular, 
#>     na.action = na.action)
#> 
#> Reference angle: y.prec 
#> Maximum cosine: 0.9813 
#> Concentration parameter: 27.05 
#> 
#> Residuals:
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -8.859e-01 -6.634e-02 -1.000e-08 -6.543e-03  7.021e-02  8.253e-01 
#> 
#> Parameters:
#>                     estimate     stderr    z value P(|z|>.)
#> x.meadow:z.meadow 0.00051640 0.00017127 3.01506345   0.0026
#> 
#> Number of observations: 182
coef(fit)
#> [1] 0.0005163958

Predictions and residuals

Predicted values are angles in radians. Residuals are wrapped circular differences between the observed and fitted directions.

pred <- predict(fit)
res <- residuals(fit)

head(data.frame(observed = d$y.dir, fitted = pred, residual = res))
#>    observed    fitted     residual
#> 1 -2.268730 -2.257728 -0.011002591
#> 2 -2.268787 -2.257641 -0.011146246
#> 3 -2.268723 -2.257640 -0.011082586
#> 4 -2.265709 -2.257558 -0.008150492
#> 5 -2.264139 -2.255194 -0.008945436
#> 6 -2.264147 -2.253935 -0.010211883
summary(res)
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -8.859e-01 -6.634e-02 -1.000e-08 -6.543e-03  7.021e-02  8.253e-01

A simple base-R diagnostic plot can be useful before calling the package plot method.

plot(
  pred,
  res,
  xlab = "Fitted direction",
  ylab = "Circular residual",
  pch = 19,
  col = "gray30"
)
abline(h = 0, lty = 2)

Direct consensus fit

The consensus model can be fitted directly when the interest is in the consensus coefficients.

fit_cons <- consensus(
  y.dir ~ y.prec + x.meadow:z.meadow,
  data = d
)

coef(fit_cons)
#>            y.prec x.meadow:z.meadow 
#>       2.115901258       0.004961799
head(predict(fit_cons))
#> [1] -2.241449 -2.241157 -2.240906 -2.240630 -2.239088 -2.238146

This applied example is intentionally small so that it can be built quickly as part of package checks. Larger analyses should keep the same structure: inspect the angular variables, fit the model, examine circular residuals, and verify that conclusions are stable under reasonable changes to the formula.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.