![]() # Calcualte Mahalanobis distances for the grid space and points D2gd <- mahalanobis(grid, dat.center, dat.cov)ĭ2gd <- matrix(D2gd, nrow = dim) # reform grid xy locations into a matrix D2xy <- mahalanobis(dat, dat.center, cov = dat.cov) # Create a grid of values for the underlaying space dim <- 200 xmin <-40 xmax <- 160 xseq <- seq(xmin, xmax, ((xmax - xmin) / (dim - 1))) The four points outside the ellipse are identified as outliers at a significance level of 0.95 based on Mahalanobis distance. The black points are the measured observations for ozone and temperature. The large blue point on the plot is the center of the data. Subtitle = 'Outlier detection using Mahalanobis distances', # Create scatterplot with Mahalanobis outliers ggplot(dat, aes(x = Ozone, y = Temp, color = Outlier)) geom_point(size = 3) geom_point( aes(dat.center, dat.center), size = 5, color = 'blue') geom_polygon(data = ellipse, fill = 'grey80', color = 'black', alpha = 0.3) scale_x_continuous(limits = c( -40, 180),īreaks = seq( 50, 100, 10)) scale_color_manual(values = c( 'black', 'orange3')) labs(title = 'New York Air Quality Measurements', The function returns D-squared distances. The variable x is the multivariate data (matrix or data frame), center is the vector of center points of the variables, and cov is the covariance matrix of the data. This function takes 3 arguments: x, center, and cov. The mahalanobis function, mahalanobis(), that comes in the R stats package returns distances between each point and the given center point. Plot.caption = element_text(size = 8, color = 'black'), Y = 'Temperature (F)') theme_bw() theme(plot.title = element_text(margin = margin(b = 3), size = 13, Subtitle = 'Daily air quality measurements May to September 1973', # Create scatter plot ggplot(dat, aes(x = Ozone, y = Temp)) geom_point(size = 3) scale_x_continuous(limits = c( -40, 180),īreaks = seq( 0, 160, 40)) scale_y_continuous(limits = c( 50, 105),īreaks = seq( 50, 100, 10)) labs(title = 'New York Air Quality Measurements', 'Daily readings of air quality values for to September 30, 1973' ) Mean ozone in parts per billion (ppm) was measured at Roosevelt Island and maximum daily temperature in degrees Fahrenheit (☏) was measured at La Guardia Airport. This dataset is composed of daily readings of air quality values collected between and September 30, 1973. Plot.subtitle = element_text(margin = margin(b = 3), size = 11,Ī = element_text(size = 11, color = 'black'),Ī = element_text(size = 11, color = 'black'),Ī = element_text(size = 9, color = 'black'),Ī = element_text(size = 9, color = 'black'))įor this example, let’s use the temperature and ozone measurements from the airquality dataset contained within base R. Hjust = 0, color = 'black', face = quote(bold)), Y = 'Chi-Squared Quantile') theme_bw() theme(plot.title = element_text(margin = margin(b = 3), size = 13, Subtitle = 'Mahalanobis distance of samples follows Chi-Square distribution', # Create plot ggplot(data = dat) geom_point( aes(x = qEmp, y = qChiSq)) geom_abline(slope = 1) labs(title = 'Quantile-Quantile Plot', Qq2 <- sapply(X = pps, FUN = qchisq, df = ncol(Sigma))ĭat <- ame(qEmp = qq1, qChiSq = qq2) The Euclidean distance between the two points \((p_1, q_1)\) and \((p_2, q_2)\) is given by: Mahalanobis DistanceĮuclidean distance is commonly used to find distance between two points. Larger values indicate that an observation is farther from where most of the points cluster. In multivariate space, Mahalanobis distance is the distance of each observation from the the center of the data cloud, taking into account the shape (covariance) of the cloud. A popular way to identify and deal with multivariate outliers is to use Mahalanobis distance. In other cases, outliers might be mistakes in the data that may adversely affect statistical analysis of the data. ![]() In some cases, outliers can be beneficial to understanding special characteristics of the data. Often they are extreme values that fall outside the “normal” range of the data. Outliers are data points that do not match the general character of the data. About Blog Project Outlier Identification Using Mahalanobis Distance
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