# lm.circular: Circular-Circular and Circular-Linear Regression In circular: Circular Statistics

## Description

Fits a regression model for a circular dependent and circular independent variable or for a circular dependent and linear independent variables.

## Usage

 ```1 2 3 4 5 6 7``` ```lm.circular(..., type=c("c-c", "c-l")) lm.circular.cc(y, x, order = 1, level = 0.05, control.circular = list()) lm.circular.cl(y, x, init = NULL, verbose = FALSE, tol = 1e-10, control.circular = list()) ## S3 method for class 'lm.circular.cl' print(x, digits = max(3, getOption("digits") - 3), signif.stars= getOption("show.signif.stars"), ...) ```

## Arguments

 `...` arguments passed to `lm.circular.cc` or to `lm.circular.cl` depending on the value of `type`. `type` if `type=="c-c"` then `lm.circular.cc` is called otherwise `lm.circular.cl` is called. `y` vector of data for the dependent circular variable. `x` vector of data for the independent circular variable if `type="c-c"` or `lm.circular.cc` is used otherwise a matrix or a vector containing the independent linear variables. `order` order of trigonometric polynomial to be fit. Order must be an integer value. By default, order=1. Used if `type="c-c"`. `level` level of the test for the significance of higher order trigonometric terms. Used if `type="c-c"`. `control.circular` the attribute of the resulting objects (`fitted`, `residuals` components in the case of `type=="c-c"` and `mu` and `se.mu`) otherwise. `init` a vector with initial values of length equal to the columns of `x`. `verbose` logical: if `TRUE` messages are printed while the function is running. `tol` the absolute accuracy to be used to achieve convergence of the algorithm. `digits` the number of digits to be printed. `signif.stars` logical; if `TRUE`, P-values are additionally encoded visually as “significance stars” in order to help scanning of long coefficient tables. It defaults to the `show.signif.stars` slot of `options`.

## Details

If `type=="c-c"` or `lm.circular.cc` is called directly a trigonometric polynomial of x is fit against the cosine and sine of y. The order of trigonometric polynomial is specified by order. Fitted values of y are obtained by taking the inverse tangent of the predicted values of sin(y) devided by the predicted values of cos(y). Details of the regression model can be found in Sarma and Jammalamadaka (1993).

If `type=="c-l"` or `lm.circular.cl` is called directly, this function implements the homoscedastic version of the maximum likelihood regression model proposed by Fisher and Lee (1992). The model assumes that a circular response variable theta has a von Mises distribution with concentration parameter kappa, and mean direction related to a vector of linear predictor variables according to the relationship: mu + 2*atan(beta'*x), where mu and beta are unknown parameters, beta being a vector of regression coefficients. The function uses Green's (1984) iteratively reweighted least squares algorithm to perform the maximum likelihood estimation of kappa, mu, and beta. Standard errors of the estimates of kappa, mu, and beta are estimated via large-sample asymptotic variances using the information matrix. An estimated circular standard error of the estimate of mu is then obtained according to Fisher and Lewis (1983, Example 1).

## Value

If `type=="c-c"` or `lm.circular.cc` is called directly an object of class `lm.circular.cc` is returned with the following components:

 `call` the `match.call` result. `rho` square root of the average of the squares of the estimated conditional concentration parameters of y given x. `fitted` fitted values of the model of class `circular`. `data` matrix whose columns correspond to x and y. `residuals` circular residuals of the model of class `circular`. `coefficients` matrix whose entries are the estimated coefficients of the model. The first column corresponds to the coefficients of the model predicting the cosine of y, while the second column contains the estimates for the model predicting the sine of y. The rows of the matrix correspond to the coefficients according to increasing trigonometric order. `p.values` p-values testing whether the (order + 1) trigonometric terms are significantly different from zero. `A.k` is mean of the cosines of the circular residuals. `kappa` assuming the circular residuals come from a von Mises distribution, kappa is the MLE of the concentration parameter.

If `type=="c-l"` or `lm.circular.cl` is called directly an object of class `lm.circular.cc` is returned with the following components:

 `call` the `match.call` result. `x` the independent variables. `y` the dependent variable. `mu` the circular mean of the dependent variable of class `circular`. `se.mu` an estimated standard error of the circular mean with the same units of measure used for `mu`. `kappa` the concentration parameter for the dependent variable. `se.kappa` an estimated standard error of the concentration parameter. `coefficients` the estimated coefficients. `cov.coef` covariance matrix of the estimated coefficients. `se.coef` standard errord of the estimated coefficients. `log.lik` log-likehood. `t.values` values of the t statistics for the coefficients. `p.values` p-values of the t statistics. Approximated values using Normal distribution.

## Author(s)

Claudio Agostinelli and Ulric Lund

## References

Fisher, N. and Lee, A. (1992). Regression models for an angular response. Biometrics, 48, 665-677.

Fisher, N. and Lewis, T. (1983). Estimating the common mean direction of several circular or spherical distributions with different dispersions. Biometrika, 70, 333-341.

Green, P. (1984). Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives. Journal of the Royal Statistical Society, B, 46, 149-192.

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 8.3, World Scientific Press, Singapore.

Sarma, Y. and Jammalamadaka, S. (1993). Circular Regression. Statistical Science and Data Analysis, 109-128. Proceeding of the Thrid Pacific Area Statistical Conference. VSP: Utrecht, Netherlands.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```# Generate a data set of dependent circular variables. x <- circular(runif(50, 0, 2*pi)) y <- atan2(0.15*cos(x) + 0.25*sin(x), 0.35*sin(x)) + rvonmises(n=50, mu=circular(0), kappa=5) # Fit a circular-circular regression model. circ.lm <- lm.circular(y, x, order=1) # Obtain a crude plot of the data and fitted regression line. plot.default(x, y) circ.lm\$fitted[circ.lm\$fitted>pi] <- circ.lm\$fitted[circ.lm\$fitted>pi] - 2*pi points.default(x[order(x)], circ.lm\$fitted[order(x)], type='l') # Fit a circular-linear regression model. set.seed(1234) x <- cbind(rnorm(10), rep(1, 10)) y <- circular(2*atan(c(x%*%c(5,1))))+rvonmises(10, mu=circular(0), kappa=100) lm.circular(y=y, x=x, init=c(5,1), type='c-l', verbose=TRUE) ```

### Example output

```Attaching package: 'circular'

The following objects are masked from 'package:stats':

sd, var

Iteration  1 :    Log-Likelihood =  24.56042
Iteration  2 :    Log-Likelihood =  24.56847
Iteration  3 :    Log-Likelihood =  24.57257
Iteration  4 :    Log-Likelihood =  24.57465
Iteration  5 :    Log-Likelihood =  24.5757
Iteration  6 :    Log-Likelihood =  24.57622
Iteration  7 :    Log-Likelihood =  24.57649
Iteration  8 :    Log-Likelihood =  24.57662
Iteration  9 :    Log-Likelihood =  24.57669
Iteration  10 :    Log-Likelihood =  24.57672
Iteration  11 :    Log-Likelihood =  24.57674
Iteration  12 :    Log-Likelihood =  24.57675
Iteration  13 :    Log-Likelihood =  24.57675
Iteration  14 :    Log-Likelihood =  24.57675
Iteration  15 :    Log-Likelihood =  24.57675
Iteration  16 :    Log-Likelihood =  24.57675
Iteration  17 :    Log-Likelihood =  24.57676
Iteration  18 :    Log-Likelihood =  24.57676
Iteration  19 :    Log-Likelihood =  24.57676
Iteration  20 :    Log-Likelihood =  24.57676
Iteration  21 :    Log-Likelihood =  24.57676
Iteration  22 :    Log-Likelihood =  24.57676
Iteration  23 :    Log-Likelihood =  24.57676
Iteration  24 :    Log-Likelihood =  24.57676
Iteration  25 :    Log-Likelihood =  24.57676
Iteration  26 :    Log-Likelihood =  24.57676
Iteration  27 :    Log-Likelihood =  24.57676
Iteration  28 :    Log-Likelihood =  24.57676
Iteration  29 :    Log-Likelihood =  24.57676
Iteration  30 :    Log-Likelihood =  24.57676
Iteration  31 :    Log-Likelihood =  24.57676
Iteration  32 :    Log-Likelihood =  24.57676
Iteration  33 :    Log-Likelihood =  24.57676
Iteration  34 :    Log-Likelihood =  24.57676
Iteration  35 :    Log-Likelihood =  24.57676
Iteration  36 :    Log-Likelihood =  24.57676
Iteration  37 :    Log-Likelihood =  24.57676
Iteration  38 :    Log-Likelihood =  24.57676
Iteration  39 :    Log-Likelihood =  24.57676
Iteration  40 :    Log-Likelihood =  24.57676
Iteration  41 :    Log-Likelihood =  24.57676
Iteration  42 :    Log-Likelihood =  24.57676
Iteration  43 :    Log-Likelihood =  24.57676
Iteration  44 :    Log-Likelihood =  24.57676
Iteration  45 :    Log-Likelihood =  24.57676
Iteration  46 :    Log-Likelihood =  24.57676
Iteration  47 :    Log-Likelihood =  24.57676
Iteration  48 :    Log-Likelihood =  24.57676
Iteration  49 :    Log-Likelihood =  24.57676
Iteration  50 :    Log-Likelihood =  24.57676
Iteration  51 :    Log-Likelihood =  24.57676
Iteration  52 :    Log-Likelihood =  24.57676
Iteration  53 :    Log-Likelihood =  24.57676
Iteration  54 :    Log-Likelihood =  24.57676
Iteration  55 :    Log-Likelihood =  24.57676
Iteration  56 :    Log-Likelihood =  24.57676
Iteration  57 :    Log-Likelihood =  24.57676

Call:
lm.circular.cl(y = ..1, x = ..2, init = ..3, verbose = TRUE)

Circular-Linear Regression

Coefficients:
Estimate Std. Error t value Pr(>|t|)
[1,]   5.1566     0.4184  12.325  < 2e-16 ***
[2,]   1.1203     0.2342   4.783 8.64e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-Likelihood:  24.58