Introduction

There is a number of R functions available in the basic, pre-loaded packages, such as base, utils or stats and there is a huge number of functions available via third-party packages. Today, we will work a bit with some of the functions that are commonly used by R users coming from different disciplines. In particular, we will:

Exercise 1. Create and work with sets

  1. Open R-studio and create three sets of genes:
    G1 <- {‘ANK-1’, ‘ANK-c’, ‘GALNTL-1’}
    G2 <- {‘ANK-1’, ‘FMA’, ‘RHO’, ‘GRP’}
    G3 <- {‘GALNTL-1’, ‘ANK-c’, ‘HQX’}.
    Visualise membership relations between all elements of the sets using Venn diagram. Use package venn.
    :key: Click to see how 
     library('venn')
 G1 <- c('ANK-1', 'ANK-c', 'GALNTL-1')
 G2 <- c('ANK-1', 'FMA', 'RHO', 'GRP')
 G3 <- c('GALNTL-1', 'ANK-c', 'HQX')
 venn(list(G1 = G1, G2 = G2, G3 = G3))


  2. What is the:
    • Union of G1 and G3?
    • Union of intersections: G1 with G2 and G1 with G3?
    • Difference between G2 and G3?
    • Is union of G1 and G2 equal to the intersection of G2 with G3?
    • Are genes ANK-c and GALNTL-1 members of the intersection of G1 with G3?
    :key: Click to see how 
     union(G1, G3)
 union(intersect(G1, G2), intersect(G2, G3))
 setdiff(G2, G3)
 setequal(union(G1, G2), intersect(G2, G3))
 is.element(c('ANK-c', 'GALNTL-1'), intersect(G1, G3))


Exercise 2. Define and work with polynomials

  1. Define the following polynomials p1 and p2: 5x^3 + 4x^2 + 7 and 2x^2 + 3x - 11.
    :key: Click to see how 
     library(polynom)
 p1 <- polynomial(c(7, 0, 4, 5))
 p2 <- polynomial(c(-11, 3, 2))


  2. Define a polynomial (p3) with the following zeros: -3, 4, 7.
    :key: Click to see how 
     library(polynom)
 p3 <- poly.calc(c(-3, 4, 7))


  3. Define a polynomial (p4) passing through the following points: A(-3,7), B(24,-9), C(7,4).
    :key: Click to see how 
     p4 <- poly.calc(c(-3, 24, 7), c(7, -9, 4))


  4. Find approximate maximum (value) of the p4 polynomial using its visualisation.
    :key: Click to see how 
     plot(p4, ylim=c(-1, 8))
 #The maximum is between 6 and 8.


  5. Perform the following operations:
    • Find the sum of p1 and p3.
    • Divide p4 by p2.
    • Find the area under p4 on the [-10, 0] interval.
    • Find the second order derivative of p1.
    • Find the Greatest Common Divisor of p1 and p2.
    • Find the Least Common Multiple of p1 and the sum of p2 and p3.
    :key: Click to see how 
     p1 + p3
 p4 / p2
 integral(p4, c(-10, 0))
 deriv(deriv(p1))
 GCD(p1, p2)
 LCM(p1, p2 + p3)


Exercise 3. Formulas

Observe that there is a convenient way of validating your formulas:

# First, define the formula  
frm <- formula(y ~ (v1 + v2 + v3) %in% v)  
# Check the validity of the above  
terms <- attr(terms.formula(frm), "term.labels")  
# See all terms that will be added together  
terms  
# Here: y ~ v1:v + v2:v + v3:v

Write formulas describing the following relations of y:

:key: Click to see how 
	y ~ var1 + var2
	y ~ +0 + height
	frm <- formula(y ~ (b + c + d) %in% a)
	#Check the validity of the above
	terms <- attr(terms.formula(frm), "term.labels")
	y ~ a * (b + c + d)
	y ~ a * (b + c + d) - c


Exercise 4. Define and work with functions

  1. Define function: f(x) = 2x^3 + 3x^2 + sin(x/2) + 7 and finds its value for x = 42.
    :key: Click to see how 
     f <- function(x) {
     y = 2*x^3 + 3*x^2 + sin(x/2) + 7
     return(y)
 }
 f(42)

 f(42) = 153476


  2. Find the f(x) zeros on the [-10, 10] interval, values at zeros and plot the function for the same interval.
    :key: Click to see how 
     uniroot(f, lower=-10, upper=10)
 curve(f, from=-10, to=10)
	
 One zero: $f(-2.16) = 5.24\times10^{-5}$


  3. Find the first order derivative of f(x) with respect to x. Use both D() and deriv(). Are the resulting derivatives the same? What is the difference between the functions?
    :key: Click to see how 
     D(expression(2*x^3 + 3*x^2 + sin(x/2) + 7), name='x')
 deriv(~2*x^3 + 3*x^2 + sin(x/2) + 7, 'x')
 
 The resulting derivatives are the same, just written in different
 ways. *D()* takes an expression as argument and it returns an
 expression while *deriv()* works on formulas.


  4. Evaluate the derivative from FNS4.3 at x = {1, 3, 7}. What values does it take at these points?
    :key: Click to see how 
     my.call <- D(expression(2*x^3 + 3*x^2 + sin(x/2) + 7), name='x')
 x <- c(1, 3, 7)
 eval(my.call)
 
 [1]  12.4  72.0 335.5


  5. Perform numerical integration: \int_1^7 f(x).
    • What is the value?
    • What is the absolute estimation error?
    :key: Click to see how 
     integrate(f, lower = 1, upper = 7)

 1588 with absolute error < 1.8e-11


Exercise 5. Statistical tests

Draw N=30 random observations from N(0,1) (normal distribution with mean=0 and std. dev.=1) and N=38 random observations from N(0.02,1.1).

Exercise 6. Linear model and diagnostics

Package UsingR contains a data with fat measurements in 252 males.

:key: Click to see how 
library('UsingR')
data('fat')
?fat
model <- lm(body.fat ~ age + weight + height + neck + hip,  data=fat)
summary(model)
plot(model)
#Remove outliers
fat2 <- fat[-c(54, 39, 42), ]
model2 <- lm(body.fat ~ age + weight + height + neck + hip,  data=fat2)
summary(model2)
plot(model2)