Examples of discrete patterns

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Joe Song

Updated: October 25, 2018; Created October 24, 2018

We showcase applications of the functinal chi-square (FunChisq) test on several types of discrete patterns. Here we use the row to represent independent variable \(X\) and column for the dependent variable \(Y\). The FunChisq test statistically determines whether \(Y\) is a function of \(X\).

A pattern represents a perfect function if and only if function index is 1; otherwise, the pattern represents an imperfect function. A function pattern is statistically significant if the p-value from the FunChisq test is less than or equal to 0.05.

Perfect functional patterns

A significant perfect functional pattern:

require(FunChisq)
f1 <- matrix(c(5,0,0,0,0,7,0,4,0), nrow=3)
f1
#>      [,1] [,2] [,3]
#> [1,]    5    0    0
#> [2,]    0    0    4
#> [3,]    0    7    0
plot_table(f1, ylab="X (row)", xlab="Y (column)", 
           main="f1: significant perfect function")

fun.chisq.test(f1)
#> 
#>  Functional chi-squared test
#> 
#> data:  f1
#> statistic = 31.125, parameter = 4, p-value = 2.887e-06
#> sample estimates:
#> non-constant function index xi.f 
#>                                1

An significant perfect many-to-one functional pattern:

f2 <- matrix(c(7,0,3,0,6,0), nrow=3)
f2
#>      [,1] [,2]
#> [1,]    7    0
#> [2,]    0    6
#> [3,]    3    0
plot_table(f2, col="salmon", ylab="X (row)", xlab="Y (column)",
           main="f2: sigificant perfect\nmany-to-one function")

fun.chisq.test(f2)
#> 
#>  Functional chi-squared test
#> 
#> data:  f2
#> statistic = 15, parameter = 2, p-value = 0.0005531
#> sample estimates:
#> non-constant function index xi.f 
#>                                1

An insignificant perfect functional pattern:

f3 <- matrix(c(5,10,0,0,0,1), nrow=3)
f3
#>      [,1] [,2]
#> [1,]    5    0
#> [2,]   10    0
#> [3,]    0    1
plot_table(f3, col="deepskyblue4", ylab="X (row)", xlab="Y (column)",
           main="f3: insigificant perfect function")

fun.chisq.test(f3)
#> 
#>  Functional chi-squared test
#> 
#> data:  f3
#> statistic = 3.75, parameter = 2, p-value = 0.1534
#> sample estimates:
#> non-constant function index xi.f 
#>                                1

A perfect constant functional pattern:

f4 <- matrix(c(5,4,7,0,0,0,0,0,0), nrow=3)
f4
#>      [,1] [,2] [,3]
#> [1,]    5    0    0
#> [2,]    4    0    0
#> [3,]    7    0    0
plot_table(f4, col="brown", ylab="X (row)", xlab="Y (column)",
           main="f4: insignificant\nperfect constant function")

fun.chisq.test(f4)
#> 
#>  Functional chi-squared test
#> 
#> data:  f4
#> statistic = 0, parameter = 4, p-value = 1
#> sample estimates:
#> non-constant function index xi.f 
#>                                0

Imperfect patterns

We contrast four imperfect patterns to illustrate the differences in FunChisq test results. p1 and p4 represent the same non-monotonic function pattern in different sample sizes; p2 is the transpose of p1, no longer functional; and p3 is another non-functional pattern. Among the first three examples, p3 is the most statistically significant, but p1 has the highest function index \(\xi_f\). This can be explained by a larger sample size but a smaller effect in p3 than p1. However, when p1 is linearly scaled to p4 to have exactly the same sample size with p3, both the \(p\)-value and the function index \(\xi_f\) favor p4 over p3 for representing a stronger function.

p1 <- matrix(c(5,1,5,1,5,1,1,0,1), nrow=3)
p1
#>      [,1] [,2] [,3]
#> [1,]    5    1    1
#> [2,]    1    5    0
#> [3,]    5    1    1
plot_table(p1, ylab="X (row)", xlab="Y (column)", 
           main="p1: significant function pattern")

fun.chisq.test(p1)
#> 
#>  Functional chi-squared test
#> 
#> data:  p1
#> statistic = 10.043, parameter = 4, p-value = 0.03971
#> sample estimates:
#> non-constant function index xi.f 
#>                         0.544288

p2=matrix(c(5,1,1,1,5,0,5,1,1), nrow=3)
p2
#>      [,1] [,2] [,3]
#> [1,]    5    1    5
#> [2,]    1    5    1
#> [3,]    1    0    1
plot_table(p2, col="red3", ylab="X (row)", xlab="Y (column)",
           main="p2: insignificant\nfunction pattern")

fun.chisq.test(p2)
#> 
#>  Functional chi-squared test
#> 
#> data:  p2
#> statistic = 8.3805, parameter = 4, p-value = 0.07859
#> sample estimates:
#> non-constant function index xi.f 
#>                        0.4582991

p3=matrix(c(5,1,1,1,5,0,9,1,1), nrow=3)
p3
#>      [,1] [,2] [,3]
#> [1,]    5    1    9
#> [2,]    1    5    1
#> [3,]    1    0    1
plot_table(p3, col="orange", ylab="X (row)", xlab="Y (column)",
           main="p3: significant function pattern")

fun.chisq.test(p3)
#> 
#>  Functional chi-squared test
#> 
#> data:  p3
#> statistic = 10.221, parameter = 4, p-value = 0.03686
#> sample estimates:
#> non-constant function index xi.f 
#>                        0.4701105

p4=p1*sum(p3)/sum(p1)
p4
#>      [,1] [,2] [,3]
#> [1,]  6.0  1.2  1.2
#> [2,]  1.2  6.0  0.0
#> [3,]  6.0  1.2  1.2
plot_table(p4, col="purple", ylab="X (row)", xlab="Y (column)",
           main="p4: significant function pattern")

fun.chisq.test(p4)
#> 
#>  Functional chi-squared test
#> 
#> data:  p4
#> statistic = 12.051, parameter = 4, p-value = 0.01697
#> sample estimates:
#> non-constant function index xi.f 
#>                         0.544288

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