Philippe J. S. De Brouwer - The Big R-Book

# my_mode # Finds the mode(s) of a vector v # Arguments: # v -- numeric vector or factor # return.all -- boolean -- set to true to return all modes # Returns: # the modal elementsmy_mode <- function(v, return.all = FALSE) { uniqv <- unique(v) tabv <- tabulate( match(v, uniqv)) if(return.all) { uniqv[tabv == max(tabv)] } else{ uniqv[ which.max(tabv)] } } # example:x <- c(1,2,2,3,3,4,5) my_mode(x) ## [1] 2 my_mode(x, return.all = TRUE) ## [1] 2 3

We were confident that it was fine to over-ride the definition of the function my_mode. Indeed, if the function was already used in some older code, then one would expect to see only one mode appear. That behaviour is still the same, because we chose the default value for the optional parameter return.allto be FALSE. If the default choice would be TRUE, then older code would produce wrong results and if we would not use a default value, then older code would fail to run.

measures of spread

Variation or spread measures how different observations are compared to the mean or other central measure. If variation is small, one can expect observations to be closer to each other.

variance

spread – standard deviation

standard deviation

The estimator for standard deviation is:

t <- rnorm(100, mean=, sd=20) var(t) ## [1] 248.2647 sd(t) ## [1] 15.75642 sqrt( var(t)) ## [1] 15.75642 sqrt( sum((t - mean(t)) ∧2) /( length(t) -1)) ## [1] 15.75642

sd()

mad

median absolute deviation

mad(t) ## [1] 14.54922 mad(t,constant=1) ## [1] 9.813314

mad()

The default “constant=1.4826” (approximately ensures consistency, i.e.,

for X idistributed as N( μ , σ 2) and large n.

covariation

When there is more than one variable, it is useful to understand what the interdependencies of variables are. For example when measuring the size of peoples hands and their length, one can expect that people with larger hands on average are taller than people with smaller hands. The hand size and length are positively correlated.

The basic measure for linear interdependence is covariance, defined as

An important metric for linear relationship is the Pearson correlation coefficient ρ .

correlation – Pearson

Definition: Pearson Correlation Coefficient

cor(mtcars $hp,mtcars $wt) ## [1] 0.6587479

cor()

Of course, we also have functions that provide the covariance matrix and functions that convert the one in the other.

d <- data.frame(mpg = mtcars $mpg, wt = mtcars $wt, hp = mtcars $hp) # Note that we can feed a whole data-frame in the functions. var(d) ## mpg wt hp ## mpg 36.324103 -5.116685 -320.73206 ## wt -5.116685 0. 957379 44.19266 ## hp -320.732056 44.192661 4700.86694 cov(d) ## mpg wt hp ## mpg 36.324103 -5.116685 -320.73206 ## wt -5.116685 0.957379 44.19266 ## hp -320.732056 44.192661 4700.86694 cor(d) ## mpg wt hp ## mpg 1.0000000 -0.8676594 -0.7761684 ## wt -0.8676594 1.0000000 0.6587479 ## hp -0.7761684 0.6587479 1.0000000

var()

cov()

cor()

cov2cor( cov(d)) ## mpg wt hp ## mpg 1.0000000 -0.8676594 -0.7761684 ## wt -0.8676594 1.0000000 0.6587479 ## hp -0.7761684 0.6587479 1.0000000

cov2cor()

The measure for correlation, as defined in previous section, actually tests for a linear relation. This means that even the presence of a strong non-linear relationship can go undetected.

x <- c( -10 :10) df <- data.frame(x=x, x_sq=x ∧2, x_abs= abs(x), x_exp= exp(x)) cor(df) ## x x_sq x_abs x_exp ## x 1.000000 0.0000000 0.0000000 0.5271730 ## x_sq 0.000000 1.0000000 0.9671773 0.5491490 ## x_abs 0.000000 0.9671773 1.0000000 0.4663645 ## x_exp 0.527173 0.5491490 0.4663645 1.0000000

The correlation between x and x 2is zero, and the correlation between x and exp ( x ) is a meagre 0.527173.

correlation – Spearman

The Spearman correlation is the correlation applied to the ranks of the data. It is one if an increase in the variable X is always accompanied with an increase in variable Y .

cor(rank(df$x), rank(df$x_exp)) ## [1] 1

The Spearman correlation checks for a relationship that can bemore general than only linear. It will be one if X increases when Y increases.

Consider the vectors