R Tutorial: Mean vector and variance-covariance matrix

Key Takeaways

now that we have read multivariate data we will explore the data set by looking at various summary statistics like means and variances the univariate mean identifies the location of a distribution on the number line for multivariate data the mean identifies the location of the distribution in a multi-dimensional space in the figure on the right the red dot represents the bivariate mean the univariate variance specifies how spread out the observations are from the mean given by the red line segment the variance covariance measures spread of multivariate data in several directions the angle and width of the two red line segments on the right specify the spread of the data along major sources of variation which is also shown by the gray ellipse to calculate the mean of all four measurements disregarding the species we use call means function with the first four columns of the iris underscore law data set since these observations come from different species we should also calculate the mean vector for each species there are two functions by an aggregate which subset the data and calculate the mean with the subset the first argument of the by command is the variables whose mean we want to calculate the second argument indeces is the variable we want to group by and the third argument is the function that we want to use to calculate the means for example call means here we calculate the means of the first four columns grouping by species in contrast the aggregate function uses a formula interface the period on the left-hand side of formula indicates that we want the means of all variables in the data frame except the variables specified as the one to group by which is listed immediately after the tilde the variance covariance matrix of the data frame can be calculated using the VAR function it produces a variance covariance matrix where the rows and columns have the same names as the data set columns the diagonal elements of this matrix are the individual variances for example 0.6 8 5 7 the first entry is the variance of sepal length the off diagonal elements give the covariance between the corresponding variables for example minus zero point zero 3 9 3 is the covariance between sepal width and sepal length the correlation matrix is a generalization of the concept of correlation between two variables the off diagonal elements give the correlation between the ight and the jade variables and the diagonal entries are all this one since it is a variables correlation with itself the Col plot function from the core plot library enables us to visualize correlations here we use the argument method equals ellipse the sign of the correlation is specified by the seed and the tilt of the ellipse and the magnitude is specified by how much it differs in Sep from a circle for example a very high positive correlation between petal width and petal length is represented by the team right tilted dark blue ellipse in contrast the circle like shape representing the correlation between sepal width and length signifies a small correlation between the corresponding variables next we provide some interpretation for mean as the location parameter plotting the mean of the petal length and petal width using colored triangles we see that the species means differ between the three groups visualizing the variance-covariance matrix using the colored ellipsis provides information regarding how compact or dispersed the values are for each group the small red ellipse correspond to C Tosa with the variances 0.03 and 0.01 1 whereas the comparatively thinner ellipse correspond to the stronger correlation in versicolor now