# Random Walk Class Cover Catch Digraph Classifier `rwcccd_classifier` and `rwcccd_classifier_2` fits a Random Walk Class Cover Catch Digraph (RWCCCD) classification model. `rwcccd_classifier` uses C++ for speed and `rwcccd_classifier_2` uses R language to determine balls. ```r rwcccd_classifier(x, y, method = "default", m = 1, proportion = 0.99) rwcccd_classifier_2( x, y, method = "default", m = 1, proportion = 0.99, partial_ordering = FALSE ) ``` ## Arguments - `x`: feature matrix or dataframe. - `y`: class factor variable. - `method`: "default" or "balanced". - `m`: penalization parameter. Takes value in $[0,\infty)$. - `proportion`: proportion of covered samples. A real number between $(0,1]$. - `partial_ordering`: `TRUE` or `FALSE` Default is `FALSE` `TRUE` uses partial ordering in determining dominant points. It orders incompletely but faster. Only for `rwcccd_classifier_2`. ## Returns a rwcccd_classifier object - **i_dominant_list**: dominant sample indexes. - **x_dominant_list**: dominant samples from feature matrix, x - **radii_dominant_list**: Radiuses of the circle for dominant samples - **class_names**: class names - **k_class**: number of classes - **proportions**: proportions each class covered ## Details Random Walk Class Cover Catch Digraphs (RWCCD) are determined by calculating $T_{\text{target}}$ score for each class as target class as $$ T_{\text{target}}=R_{\text{target}}(r_{\text{target}})-\frac{r_{\text{target}}n_u}{2d_m(x)}. $$ Here, $r_{\text{target}}$ is radius and determined by maximum $R_{\text{target}}(r) - P_{\text{target}}(r)$ calculated for each target sample. $R_{\text{target}}(r)$ is $$ R_{\text{target}}(r):=w_{target}|{z\in X^{\text{target}}_{n_{\text{target}}}:d(x^{\text{target}},z)\leq r}| -w_{non-target}|{z\in X^{\text{non-target}}_{n_{\text{non-target}}}:d(x^{\text{target}},z)\leq r}| $$ and $P_{\text{target}}(r)$ is $$ P_{\text{target}}(r) = m\times d(x^{\text{target}},z)^p. $$ $m=0$ removes penalty. $w_{target}=1$ for default and $w_{target}=n_{\text{target}/n_{\text{non-target}}}$ for balanced method. $n_u$ is the number of uncovered samples in the current iteration and $d_m(x)$ is $\max{d(x^{\text{target}},x^{\text{uncovered}})}$. This method is more robust to noise compared to PCCCD However, balls covers classes improperly and $r = 0$ can be selected. For detail, please refer to Priebe et al. (2001), Priebe et al. (2003), and Manukyan and Ceyhan (2016). ## Examples ```r n <- 500 x1 <- runif(n, 1, 10) x2 <- runif(n, 1, 10) x <- cbind(x1, x2) y <- as.factor(ifelse(3 < x1 & x1 < 7 & 3 < x2 & x2 < 7, "A", "B")) # dataset m_rwcccd_1 <- rwcccd_classifier(x = x, y = y, method = "default", m = 1) plot(x, col = y, asp = 1, main = "default") # dominant samples of second class x_center <- m_rwcccd_1$x_dominant_list[[2]] # radii of balls for second class radii <- m_rwcccd_1$radii_dominant_list[[2]] # balls for (i in 1:nrow(x_center)) { xx <- x_center[i, 1] yy <- x_center[i, 2] r <- radii[i] theta <- seq(0, 2*pi, length.out = 100) xx <- xx + r*cos(theta) yy <- yy + r*sin(theta) lines(xx, yy, type = "l", col = "green") } # dataset m_rwcccd_2 <- rwcccd_classifier_2(x = x, y = y, method = "default", m = 1, partial_ordering = TRUE) plot(x, col = y, asp = 1, main = "default, prartial_ordering = TRUE") # dominant samples of second class x_center <- m_rwcccd_2$x_dominant_list[[2]] # radii of balls for second class radii <- m_rwcccd_2$radii_dominant_list[[2]] # balls for (i in 1:nrow(x_center)) { xx <- x_center[i, 1] yy <- x_center[i, 2] r <- radii[i] theta <- seq(0, 2*pi, length.out = 100) xx <- xx + r*cos(theta) yy <- yy + r*sin(theta) lines(xx, yy, type = "l", col = "green") } # dataset m_rwcccd_3 <- rwcccd_classifier(x = x, y = y, method = "balanced", m = 1, proportion = 0.5) plot(x, col = y, asp = 1, main = "balanced, proportion = 0.5") # dominant samples of second class x_center <- m_rwcccd_3$x_dominant_list[[2]] # radii of balls for second class radii <- m_rwcccd_3$radii_dominant_list[[2]] # balls for (i in 1:nrow(x_center)) { xx <- x_center[i, 1] yy <- x_center[i, 2] r <- radii[i] theta <- seq(0, 2*pi, length.out = 100) xx <- xx + r*cos(theta) yy <- yy + r*sin(theta) lines(xx, yy, type = "l", col = "green") } # testing the performance i_train <- sample(1:n, round(n*0.8)) x_train <- x[i_train,] y_train <- y[i_train] x_test <- x[-i_train,] y_test <- y[-i_train] m_rwcccd <- rwcccd_classifier(x = x_train, y = y_train, method = "balanced") pred <- predict(object = m_rwcccd, newdata = x_test) # confusion matrix table(y_test, pred) # accuracy sum(y_test == pred)/nrow(x_test) ``` ## References Priebe, C. E., DeVinney, J., & Marchette, D. J. (2001). On the distribution of the domination number for random class cover catch digraphs. Statistics & Probability Letters, 55(3), 239–246. https://doi.org/10.1016/s0167-7152(01)00129-8 Priebe, C. E., Marchette, D. J., DeVinney, J., & Socolinsky, D. A. (2003). Classification Using Class Cover Catch Digraphs. Journal of Classification, 20(1), 3–23. https://doi.org/10.1007/s00357-003-0003-7 Manukyan, A., & Ceyhan, E. (2016). Classification of imbalanced data with a geometric digraph family. Journal of Machine Learning Research, 17(1), 6504–6543. https://jmlr.org/papers/volume17/15-604/15-604.pdf ## Author(s) Fatih Saglam, saglamf89@gmail.com

rwcccd_classifier function