# A test of segregation/association based on arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data An object of class `"htest"` (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of `Xp` points in the range (i.e., range) of `Yp` points against the alternatives of segregation (where `Xp` points cluster away from `Yp` points) and association (where `Xp` points cluster around `Yp` points) based on the normal approximation of the arc density of the CS-PCD for uniform 1D data. The function yields the test statistic, $p$-value for the corresponding `alternative`, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used. Under the null hypothesis of uniformity of `Xp` points in the range of `Yp` points, arc density of CS-PCD whose vertices are `Xp` points equals to its expected value under the uniform distribution and `alternative` could be two-sided, or left-sided (i.e., data is accumulated around the `Yp` points, or association) or right-sided (i.e., data is accumulated around the centers of the intervals, or segregation). CS proximity region is constructed with the expansion parameter $t > 0$ and centrality parameter `c` which yields $M$-vertex regions. More precisely, for a middle interval $(y_{(i)},y_{(i+1)})$, the center is $M=y_{(i)}+c(y_{(i+1)}-y_{(i)})$ for the centrality parameter $c \in (0,1)$. If there are duplicates of `Yp` points, only one point is retained for each duplicate value, and a warning message is printed. **Caveat:** This test is currently a conditional test, where `Xp` points are assumed to be random, while `Yp` points are assumed to be fixed (i.e., the test is conditional on `Yp` points). Furthermore, the test is a large sample test when `Xp` points are substantially larger than `Yp` points, say at least 5 times more. This test is more appropriate when supports of `Xp` and `Yp` have a substantial overlap. Currently, the `Xp` points outside the range of `Yp` points are handled with a range correction (or end-interval correction) factor (see the description below and the function code.) However, in the special case of no `Xp` points in the range of `Yp` points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing line of research of the author of the package. `end.int.cor` is for end-interval correction, recommended when both `Xp` and `Yp` have the same interval support (default is "no end-interval correction", i.e., `end.int.cor=FALSE`). ```r CSarc.dens.test1D( Xp, Yp, t, c = 0.5, support.int = NULL, end.int.cor = FALSE, alternative = c("two.sided", "less", "greater"), conf.level = 0.95 ) ``` ## Arguments - `Xp`: A set of 1D points which constitute the vertices of the CS-PCD. - `Yp`: A set of 1D points which constitute the end points of the partition intervals. - `t`: A positive real number which serves as the expansion parameter in CS proximity region. - `c`: A positive real number which serves as the centrality parameter in CS proximity region; must be in $(0,1)$ (default `c=.5`). - `support.int`: Support interval $(a,b)$ with $a

CSarc.dens.test1D function