NOTE: The package is not yet officially available for public use. Please do respect our wish by not installing the package until this note is removed. Thank you!

eccficm: Expected Conditional Characteristic Function-based Independence Criterion Measures

The goal of eccficm is to provide new distance correlation/covariance type of measures for testing independence between two random vectors using the expected conditional characteristic function-based independence criterion (ECCFIC) method developed by Yin and Yuan (2019) and Ke and Yin (2019). The method is nonparametric, can detect both linear and nonlinear dependencies, and overcomes many challenges/limitations of independence measures.

Although the package is based on both ECD measures (Yin and Yuan 2019) and ECCFIC measures (Ke and Yin 2019), we chose to use only ECCFIC in the name of the package because ECD belongs to the family of generalized ECCFIC measures. However, to preserve the unique implementations of the different methods from the two papers (Yin and Yuan 2019; Ke and Yin 2019), different R functions are separately created and made available for the ECD measures as well as the ECCFIC measures. As a result, R functions based on ECD are either named “ecd” or prefixed by “ecd”, with the ECCFIC related R functions also named in a similar fashion.

Installation

Once the package becomes officially available, the following two options will be available for installation.

Installing development version

You can install the development version of eccficm from GitHub with:

# install.packages("devtools")
devtools::install_github("williamagyapong/eccficm")

Installing released version

You can install the released version of eccficm from CRAN with:

install.packages("eccficm")

Available Functions

The main functions are ecd, eccfic, ecd.test, and eccfic.test as described below.

• ecd, eccfic: compute the covariance and correlation type statistics.

• ecd.test, eccfic.test: perform a permutation test of independence based on the ECD and ECCFIC correlation and covariance type statistics, respectively.

• generateData: generates optional data for feature screening based on some specific models.

• fscreen: Performs the sure independence feature screening using correlation learning between each predictor and the response.

Although the slicing method for ECD and ECCFIC is not very sensitive to the number of slices (ns), it is suggested that the number of data points in each slice should exceed 5 but not close to n/2, where n is the sample size.

Examples

Aircraft data: Speed versus Span

In this example, we use the aircraft data available in the sm package in R. The data contains six characteristics of aircraft designs in the twentieth century. We consider two variables, wing span (m) and maximum speed (km/h) in period 3 with 230 observations. Our goal is to test the independence of log(Span) and log(Speed) using ECCFIC correlations as test statistics. Results based on Pearson’s product-moment correlatoin and distance correlation are included for comparison. 999 replicates are used for the permutation test.

We first load the sm package to get access to the aircraft data and filter the aircraft data to obtain data for just the third brand time period.

library(sm)

# get the aircraft data
data("aircraft", package = "sm")

# get data for the third brand time period
aircraft_p3 <- aircraft[aircraft$Period==3, ] 

attach(aircraft_p3) # expose variable names to the R search path

Scatter plot of log(Span) against log(Speed)

Create a scatter plot to explore the relationship between log(Speed) and log(Span).

library(ggplot2)

ggplot(aircraft_p3, aes(log(Span), log(Speed))) +
  geom_point(alpha = 0.8, size=3, shape=1) +
  geom_smooth(method = "loess", se=F) +
  theme_classic()

# plot(log(Speed), log(Span))

The above graph suggests the existence of some sort of association between Speed and Span which appears to be nonlinear. In the subsequent codes we will use the hypothesis testing functions in the eccficm package as well as one from the energy and stats packages to determine if there is enough evidence to conclude on the observed relationship.

At this point we load our software package eccficm.

library(eccficm)

ECD and ECCFIC association statistics

Let’s first take a look at the eccfic statistics.

# compute ecd statistics
ecd(log(Speed), log(Span), ns=10)
#> $ecdCov
#> [1] 0.2950304
#> 
#> $ecdCor
#> [1] 0.3286453
#> 
#> $ecdVarX
#> [1] 0.8977167

# compute eccfic statistics
eccfic(log(Speed), log(Span), est = 'slice', ns=10)
#> $eccfic
#> [1] 0.08586636
#> 
#> $eccficCor
#> [1] 0.1538209

Hypothesis Test of Independence

The alternative hypothesis of interest is that log(Speed) is a function of log(Span). In other words, we want to test whether the log(Speed) depends on log(Span).

We first test the independence of log(Speed) and log(Span) with the ecd.test function. We use 10 slices for the slicing estimation, ECD correlation as test statistic, and perform 999 permutations.

ecd.test(log(Speed), log(Span), est="slice", ns=10, ts="ecdcor",  B=999)
#> 
#>  ecdCor permutation test of independence - Estimation Method: Slicing
#> 
#> data:  X -> log(Speed) , Y -> log(Span) , Replicates = 999
#> ecdCor = 0.32865, p-value = 0.001
#> alternative hypothesis: X is not independent of Y
#> sample estimates:
#>    ecdCor    ecdCov ecdVar(X) 
#> 0.3286453 0.2950304 0.8977167

Using the ECD correlation as test statistic, we obtain a p-value of 0.001.

Similarly, we use the eccfic.test function with 10 slices for its slicing estimation, ECCFIC correlation as test statistic, and 999 total permutations. The p-value for this test is also 0.001.

eccfic.test(log(Speed), log(Span), est="slice", ns=10, ts="eccficcor",  B=999)
#> 
#>  eccficCor permutation test of independence - Estimation Method:
#>  Slicing on Y
#> 
#> data:  X -> log(Speed) , Y -> log(Span) , Replicates = 999
#> eccficCor = 0.15382, p-value = 0.001
#> alternative hypothesis: X is not independent of Y
#> sample estimates:
#>     eccfic  eccficCor 
#> 0.08586636 0.15382091

Finally, we compare the results for our independence measures with that of the distance correlation with the same number of permutations as well as the Pearson’s product-moment correlation. The DC independence testing function is available in the energy package. Interestingly, using DC as test statistic produces the same p-value of 0.001. However, the p-value associated with the test based on Pearson’s correlation is 0.8001.

# Distance correlation as test statistic
energy::dcor.test(log(Speed), log(Span), R=999)
#> 
#>  dCor independence test (permutation test)
#> 
#> data:  index 1, replicates 999
#> dCor = 0.28045, p-value = 0.001
#> sample estimates:
#>      dCov      dCor   dVar(X)   dVar(Y) 
#> 0.1218536 0.2804530 0.4872107 0.3874712

# Pearson's product-moment as test statistic
stats::cor.test(log(Speed), log(Span))
#> 
#>  Pearson's product-moment correlation
#> 
#> data:  log(Speed) and log(Span)
#> t = 0.25349, df = 228, p-value = 0.8001
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  -0.1128179  0.1458274
#> sample estimates:
#>        cor 
#> 0.01678556

From the above outputs of test results, despite the nonlinear relation between wing span and maximum speed, the ecd.test, eccfic.test and dcor.test functions returned a very small p-value of 0.001, suggesting that maximum speed of the aircraft depends on its wing span. This shows how sensitive distance correlation and ECCFIC correlations are to nonlinear associations. On the other hand, the Pearson’s correlation test suggested no evidence of association, showing how Pearson’s correlation cannot detect non-linear associations if one truly exists. Moreover, we see how comparable our measures are to the distance correlation.

References

Ke, Chenlu, and Xiangrong Yin. 2019. “Expected Conditional Characteristic Function-Based Measures for Testing Independence.” Journal of the American Statistical Association. https://doi.org/10.1080/01621459.2019.1604364.

Yin, Xiangrong, and Qingcong Yuan. 2019. “A New Class of Measures for Testing Independence.” Statistica Sinica. http://www3.stat.sinica.edu.tw/ss_newpaper/SS-2017-0538_na.pdf.