Mean -- a measure of center

A numeric data set may have a central tendency — where some of the most typical data points reside.^[19] The arithmetic mean (average) is the most commonly used measure of central tendency.^[19] The mean of a numeric data set is the sum of the data points divided by the number of data points.^[19] It is represented by the symbol ${\displaystyle {\bar {x}}}$.

Let ${\displaystyle x}$ = a list of data points.

Let ${\displaystyle n}$ = the number of data points.

${\displaystyle {\bar {x}}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}}$

Suppose a sample of four observations of Celsius temperature measurements were taken 12 hours apart.

Let ${\displaystyle x}$ = a list of degrees Celsius data points of 30, 27, 31, 28.

This R computer program will output the mean of ${\displaystyle x}$:

# The c() function "combines" a list into a single object.
x <- c( 30, 27, 31, 28 )

sum <- sum( x )
length <- length( x )
mean <- sum / length

message( "Mean:" )
print( mean )

Note: R can have the same identifier represent both a function name and its result. For more information, visit scope.

Output:

Mean:
[1] 29

This R program will execute the native mean() function to output the mean of ${\displaystyle }$x:

x <- c( 30, 27, 31, 28 )

message( "Mean:" )
print( mean( x ) )

Output:

Mean:
[1] 29

Standard Deviation -- a measure of dispersion

A standard deviation of a numeric data set is an indication of the average distance all the data points are from the mean.^[20] For a data set with a small amount of variation, then each data point will be close to the mean, so the standard deviation will be small.^[20] Standard deviation is represented by the symbol ${\displaystyle s}$.

Let ${\displaystyle x}$ = a list of data points.

Let ${\displaystyle n}$ = the number of data points.

${\displaystyle s={\sqrt {\frac {\sum \left(x_{i}-{\bar {x}}\right)^{2}}{n-1}}}}$

Suppose a sample of four observations of Celsius temperature measurements were taken 12 hours apart.

Let ${\displaystyle x}$ = a list of degrees Celsius data points of 30, 27, 31, 28.

This R program will output the standard deviation of ${\displaystyle x}$:

x <- c( 30, 27, 31, 28 )
distanceFromMean <- x - mean( x )
distanceFromMeanSquared <- distanceFromMean ** 2
distanceFromMeanSquaredSum <- sum( distanceFromMeanSquared )
variance <- distanceFromMeanSquaredSum / ( length( x ) - 1 )
standardDeviation <- sqrt( variance )

message( "Standard Deviation:" )
print( standardDeviation )

Output:

Standard Deviation:
[1] 1.825742

This R program will execute the native sd() function to output the standard deviation of ${\displaystyle x}$:

x <- c( 30, 27, 31, 28 )

message( "Standard Deviation:" )
print( sd( x ) )

Output:

Standard Deviation:
[1] 1.825742

Linear regression -- a measure of relation

A scatter plot resembling a linear relationship has infinitely many^[21] straight lines that will pass close to all the data points (depicted in red). The blue regression line (generally called curve fit) is the one straight line that has the minimum average distance (depicted in green) from all the points to the line.

A phenomenon may be the result of one or more observable events. For example, the phenomenon of skiing accidents may be the result of having snow in the mountains. A method to measure whether or not a numeric data set is related to another data set is linear regression.^[22]

Let ${\displaystyle x}$ = a data set of independent data points, in which each point occurred at a specific time.

Let ${\displaystyle y}$ = a data set of dependent data points, in which each point occurred at the same time of an independent data point.

If a linear relationship exists, then a scatter plot of the two data sets will show a pattern that resembles a straight line.^[23] If a straight line is embedded into the scatter plot such that the average distance from all the points to the line is minimal, then the line is called a regression line. The equation of the regression line is called the regression equation.^[24]

The regression equation is a linear equation; therefore, it has a slope and y-intercept. The format of the regression equation is ${\displaystyle {\hat {y}}=b_{0}+b_{1}x}$.^{[25][lower-alpha 1]}

Let ${\displaystyle b_{1}}$ = the slope of the regression equation.

${\displaystyle b_{1}={\frac {\sum \left(x-{\bar {x}}\right)\left(y-{\bar {y}}\right)}{\sum \left(x-{\bar {x}}\right)^{2}}}}$

Let ${\displaystyle b_{0}}$ = the y-intercept of the regression equation.

${\displaystyle b_{0}={\bar {y}}-b_{1}{\bar {x}}}$

Suppose a sample of four observations of Celsius temperature measurements were taken 12 hours apart. At the same time, the thermometer was switched to Fahrenheit temperature and another measurement was taken.

Let ${\displaystyle x}$ = a list of degrees Celsius data points of 30, 27, 31, 28.

Let ${\displaystyle y}$ = a list of degrees Fahrenheit data points of 86.0, 80.6, 87.8, 82.4.

This R program will output the slope and y-intercept of the linear relationship between ${\displaystyle x}$ and ${\displaystyle y}$:

x <- c( 30, 27, 31, 28 )
y <- c( 86.0, 80.6, 87.8, 82.4 )

# Build the numerator
independentDistanceFromMean <- x - mean( x )
sampledDependentDistanceFromMean <- y - mean( y )

independentDistanceTimesSampledDistance <-
    independentDistanceFromMean *
    sampledDependentDistanceFromMean

independentDistanceTimesSampledDistanceSum <-
    sum( independentDistanceTimesSampledDistance )

# Build the denominator
independentDistanceFromMeanSquared <-
    independentDistanceFromMean ** 2

independentDistanceFromMeanSquaredSum <-
    sum( independentDistanceFromMeanSquared )

# Slope is rise over run
slope <-
    independentDistanceTimesSampledDistanceSum /
    independentDistanceFromMeanSquaredSum

yIntercept <- mean( y ) - slope * ( mean( x ) )

message( "Slope:" )
print( slope )

message( "Y-intercept" )
print( yIntercept )

Output:

Slope:
[1] 1.8
Y-intercept:
[1] 32

This R program will execute the native functions to output the slope and y-intercept:

x <- c( 30, 27, 31, 28 )
y <- c( 86.0, 80.6, 87.8, 82.4 )

# Execute lm() with Fahrenheit depends upon Celsius
linearModel <- lm( y ~ x )

# coefficients() returns a structure containing the slope and y intercept
coefficients <- coefficients( linearModel )

# Extract the slope from the structure
slope <- coefficients[["x"]]

# Extract the y intercept from the structure
yIntercept <- coefficients[["(Intercept)"]]

message( "Slope:" )
print( slope )

message( "Y-intercept" )
print( yIntercept )

Output:

Slope:
[1] 1.8
Y-intercept:
[1] 32

The coefficient of determination determines the percentage of variation explained by the independent variable.^[26] It always lies between 0 and 1.^[27] A value of 0 indicates no relationship between the two data sets, and a value near 1 indicates the regression equation is extremely useful for making predictions.^[28]

Let ${\displaystyle {\hat {y}}}$ = the data set of predicted response data points when the independent data points are passed through the regression equation.

Let ${\displaystyle r^{2}}$ = the coefficient of determination in a relationship between an independent variable and a dependent variable.

${\displaystyle r^{2}={\frac {\sum \left({\hat {y}}-{\bar {y}}\right)^{2}}{\sum \left(y-{\bar {y}}\right)^{2}}}}$

This R program will output the coefficient of determination of the linear relationship between ${\displaystyle x}$ and ${\displaystyle y}$:

x <- c( 30, 27, 31, 28 )
y <- c( 86.0, 80.6, 87.8, 82.4 )

# Build the numerator
linearModel <- lm( y ~ x )
coefficients <- coefficients( linearModel )
slope <- coefficients[["x"]]
yIntercept <- coefficients[["(Intercept)"]]
predictedResponse <- yIntercept + ( slope * x )

predictedResponseDistanceFromMean <-
    predictedResponse - mean( y )

predictedResponseDistanceFromMeanSquared <-
    predictedResponseDistanceFromMean ** 2

predictedResponseDistanceFromMeanSquaredSum <-
    sum( predictedResponseDistanceFromMeanSquared )

# Build the denominator
sampledResponseDistanceFromMean <- y - mean( y )

sampledResponseDistanceFromMeanSquared <-
    sampledResponseDistanceFromMean ** 2

sampledResponseDistanceFromMeanSquaredSum <-
    sum( sampledResponseDistanceFromMeanSquared )

coefficientOfDetermination <-
    predictedResponseDistanceFromMeanSquaredSum /
    sampledResponseDistanceFromMeanSquaredSum

message( "Coefficient of Determination:" )
print( coefficientOfDetermination )

Output:

Coefficient of Determination:
[1] 1

This R program will execute the native functions to output the coefficient of determination:

x <- c( 30, 27, 31, 28 )
y <- c( 86.0, 80.6, 87.8, 82.4 )

linearModel <- lm( y ~ x )
summary <- summary( linearModel )
coefficientOfDetermination <- summary[["r.squared"]]

message( "Coefficient of Determination:" )
print( coefficientOfDetermination )

Output:^{[lower-alpha 2]}

Coefficient of Determination:
[1] 1

Single plot

This R program will display a scatter plot with an embedded regression line and regression equation illustrating the relationship between ${\displaystyle x}$ and ${\displaystyle y}$:

x <- c( 30, 27, 31, 28 )
y <- c( 86.0, 80.6, 87.8, 82.4 )

linearModel <- lm( y ~ x )
coefficients <- coefficients( linearModel )
slope <- coefficients[["x"]]
intercept <- coefficients[["(Intercept)"]]

# Execute paste() to build the regression equation string
regressionEquation <- paste( "y =", intercept, "+", slope, "x" )

# Display a scatter plot with the regression line and equation embedded
plot(
    x,
	y,
	main = "Fahrenheit Depends Upon Celsius",
	sub = regressionEquation,
	xlab = "Degress Celsius",
	ylab = "Degress Fahrenheit",
	abline( linearModel ) )

Output:

Multi plot

This R program will generate a multi-plot and a table of residuals.

# The independent variable is a list of numbers 1 to 6
x <- 1:6

# The dependent variable is a list of each independent variable squared
y <- x^2

# Linear regression model where y depends upon x
linearModel <- lm(y ~ x)

# Display the residuals
summary( linearModel )

# Create a 2 by 2 multi-plot
par(mfrow = c(2, 2))

# Output the multi-plot
plot( linearModel )

Output:

Residuals:
      1       2       3       4       5       6       7       8      9      10
 3.3333 -0.6667 -2.6667 -2.6667 -0.6667  3.3333

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  -9.3333     2.8441  -3.282 0.030453 * 
x             7.0000     0.7303   9.585 0.000662 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.055 on 4 degrees of freedom
Multiple R-squared:  0.9583, Adjusted R-squared:  0.9478
F-statistic: 91.88 on 1 and 4 DF,  p-value: 0.000662

Diagnostic plots from plotting “linearModel” (q.v. “plot.lm()” function). Notice the mathematical notation allowed in labels (lower left plot).

Mandelbrot graphic

This Mandelbrot set example highlights the use of complex numbers. It models the first 20 iterations of the equation z = z2 + c, where c represents different complex constants.

Install the package that provides the write.gif() function beforehand:

install.packages("caTools")

R program:

library(caTools)

jet.colors <-
    colorRampPalette(
        c("green", "pink", "#007FFF", "cyan", "#7FFF7F",
          "white", "#FF7F00", "red", "#7F0000"))

dx <- 1500 # define width
dy <- 1400 # define height

C  <-
    complex(
        real =
            rep(
                seq(-2.2, 1.0, length.out = dx), each = dy),
                imag = rep(seq(-1.2, 1.2, length.out = dy),
                dx))

# reshape as matrix of complex numbers
C <- matrix(C, dy, dx)

# initialize output 3D array
X <- array(0, c(dy, dx, 20))

Z <- 0

# loop with 20 iterations
for (k in 1:20) {

  # the central difference equation
  Z <- Z^2 + C

  # capture the results
  X[, , k] <- exp(-abs(Z))
}

write.gif(
    X,
    "Mandelbrot.gif",
    col = jet.colors,
    delay = 100)

Output:

"Mandelbrot.gif" graphic created in R

Object

R stores data inside an object. An object is assigned a name which the computer program uses to set and retrieve the data.^[30] An object is created by placing its name to the left of the symbol pair <-.^[31]

To create an object named x and assign it the integer 82:

x <- 82L
print( x )

Output:

[1] 82

The [1] displayed before the number is a subscript. It shows the container for this integer is index one of an array.

Vector

The most primitive R object is the vector.^[32] A vector is a one dimensional array of data. To assign multiple elements to the array, use the c() function to "combine" the elements. The elements must be the same data type.^[33] R lacks scalar data types, which are placeholders for a single word — usually an integer. Instead, a single integer is stored into the first element of an array. The single integer is retrieved using the index subscript of [1].

R program to store and retrieve a single integer:

store <- 82L
retrieve <- store[1]
print( retrieve[1] )

Output:

[1] 82

x <- 1:3
print( x )
print( x + 1 )

[1] 1 2 3
[1] 2 3 4

x <- 82L
print( x[1] )
message( "Data type:" )
typeof( x )

[1] 82
Data type:
[1] "integer"

x <- c( 1L, 2L, 3L )
print( x )
message( "Data type:" )
typeof( x )

[1] 1 2 3
Data type:
[1] "integer"

x <- 1:3
print( x )
message( "Data type:" )
typeof( x )

[1] 1 2 3
Data type:
[1] "integer"

x <- 82
print( x[1] )
message( "Data type:" )
typeof( x )

[1] 82
Data type:
[1] "double"

x <- c( 1, 2, 3 )
print( x )
message( "Data type:" )
typeof( x )

[1] 1 2 3
Data type:
[1] "double"

x <- 3 < 4
print( x[1] )
message( "Data type:" )
typeof( x )

[1] TRUE
Data type:
[1] "logical"

x <- "hello world"
print( x[1] )
message( "Data type:" )
typeof( x )

[1] "hello world"
Data type:
[1] "character"

x <- c( "hello", "world" )
print( x )
message( "Data type:" )
typeof( x )

[1] "hello" "world"
Data type:
[1] "character"

Data frame

A data frame stores a two dimensional array of vectors. It is the most useful structure for data analysis.^[38] Data frames are created using the data.frame() function. The input is a list of vectors (of any data type). Each vector becomes a column in a table. The elements in each vector are aligned to form the rows in the table.

R program:

integer <- c( 82L, 83L )
string <- c( "hello", "world" )
data.frame <- data.frame( integer, string )
print( data.frame )
message( "Data type:" )
class( data.frame )

Output:

  integer string
1      82  hello
2      83  world
Data type:
[1] "data.frame"

Data frames can be deconstructed by providing a vector's name between double brackets. This returns the original vector. Each element in the returned vector can be accessed by its index number.

R program to extract the word "world". It is stored in the second element of the "string" vector:

integer <- c( 82L, 83L )
string <- c( "hello", "world" )
data.frame <- data.frame( integer, string )
vector <- data.frame[["string"]]
print( vector[2] )
message( "Data type:" )
typeof( vector )

Output:

[1] "world"
Data type:
[1] "character"

Functions

A function is an object that stores computer code instead of data.^[39] The purpose of storing code inside a function is to be able to reuse it in another context.^[39] To instantiate a function object, execute the function() statement and assign the results to a name.^[40]

A function receives input both from global variables and input parameters (often called arguments). Objects created within the function body remain local to the function.

R code to create a function:

# The input parameters are x and y.
# The return value is a numeric double vector.
f <- function(x, y)
{
    first_expression <- x * 2
    second_expression <- y * 3
    first_expression + second_expression

    # The return statement may be omitted
    # if the last expression is unassigned.
    # This will save a few clock cycles.
}

Usage output:

> f(1, 2)
[1] 8

Function arguments are passed in by value.

R supports generic functions. They act differently depending on the classes of the arguments passed in. The process is to dispatch the method specific to the set of classes. A common implementation is R's print() function. It can print almost every class of object. For example, print(objectName).^[41]