In statistics and probability theory, independence is a power condition for two random variables. For two random variables X and Y, they are independent if knowing X will not change the probability distribution of Y (and vice versa).

A related but distinct concept is correlation, which is a quantity between -1 and 1 that quantifies linear relationship between two random variables. Independence implies 0 correlation BUT not the other way around. Here’s an epic example for 0 correlation but no independence. Assume X might be -1, 0 ,or 1 with equal (1/3) probability. Let Y=X^2. Then it is easy to see that X and Y has 0 correlation but they are dependent to each other. One may interpret correlation as the linear dependence between two random variables. Thus, 0 correlation only shows that there is no the linear dependence but we might have higher order dependence. In the previous example, X and Y are in the quadratic relationship so that correlation–the linear dependence–cannot capture it.

Although independence is a stronger statement than correlation, they are equivalent if two random variables are jointly Gaussian. Namely, for two normal random variables, 0 correlation implies independence. This is an appealing feature for Gaussian and is one of the main reasons some researchers would like to make Gaussian assumption.

A more mathematical description for independence is as follows. Let F(x) and F(y) denotes the marginal (cumulative) distribution of random variables X, Y, respectively. And let F(x,y) denote the joint distribution. If X and Y are independent, then F(x,y) = F(x)F(y). Namely, the joint distribution is the product of marginals. If X and Y has densities p(x) and p(y), then independence is equivalent to say p(x,y) = p(x)p(y) (again, joint density equals to the product of marginals).

The independence is a power feature in probability and statistics. Some people even claim that independence is the key feature that makes “probability theory” a distinct field from measure theory. Indeed, independence makes it possible for the concentration of measure for a function of many random variables (concentration of measure: probability concentrates around a certain value; just think of law of large number).

Now let’s talk about something about statistics–test for independence. Given pairs of observations, say (X_1, Y_1), … ,(X_n, Y_n) that are random sample (IID). To goal is to test if the pair (X, Y) are independent to each other.

When two random variables are discrete, a well-known method is the “Pearson’s chi-squared test for independence”
(https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test#Test_of_independence). Pearson’s method is to compare the theoretical behavior of joint distribution under independence case (marginals are still based on sample) to the real observed joint distribution.

When two variables are continuous, things are more complicated. However, if we assume the joint distribution is Gaussian, we can simply test the correlation. If the correlation is non-zero, then due to the powerful feature of Gaussian–independence is equivalent to 0 correlation–we can claim they are dependent.

If we do not want to assume Gaussian, there are many nonparametric tests for independence. Here I list four famous approaches.

1. Hoeffding’s Test. Hoeffding, Wassily. “A non-parametric test of independence.” The Annals of Mathematical Statistics (1948): 546-557
(http://projecteuclid.org/euclid.aoms/1177730150). This test basically use empirical cumulative distribution (EDF) of F(x,y) versus products of marginal EDF and then take supreme over both x and y. This is like a KS-test for independence.
2. Kernel Test. Rosenblatt, Murray. “A quadratic measure of deviation of two-dimensional density estimates and a test of independence.” The Annals of Statistics (1975): 1-14 (https://projecteuclid.org/euclid.aos/1176342996). Kernel test is very similar to Hoeffding’s test but we now use the kernel density estimator to estimate both joint density and marginals and then compare joint KDE to the product of marginal KDEs.
3. Energy Test (distance covariance/correlation). An introduction is in [https://en.wikipedia.org/wiki/Distance_correlation]. Székely, Gábor J., Maria L. Rizzo, and Nail K. Bakirov. “Measuring and testing dependence by correlation of distances.” The Annals of Statistics 35, no. 6 (2007): 2769-2794 (http://projecteuclid.org/euclid.aos/1201012979). They derive a quantity called distance correlation and then show that the scaled version of this distance correlation converges to some finite value when two random variables are independent and diverges under dependence.
4. RKHS Test (RKHS: Reproducing Kernel Hilbert Space). Gretton, Arthur, and László Györfi. “Consistent nonparametric tests of independence.” The Journal of Machine Learning Research 11 (2010): 1391-1423
(http://www.jmlr.org/papers/volume11/gretton10a/gretton10a.pdf). This test uses the fact that independence implies that under all functional transformations for the two random variables, they are still uncorrelated (0 correlation). Then they use the reproducing property for the RKHS and kernel trick (note: this kernel is not the same as the kernel in kernel test) to construct a test for independence.

As a concluding remark, I would like to point out that there are some ways to “quantify” dependence. For instance, alpha-mixing, beta-mixing are some definitions for dependence among sequence of random variables that are commonly used in Time-Series. Wikipedia has a short introduction about these dependence measure [https://en.wikipedia.org/wiki/Mixing_(mathematics)].

Data Visualization and Statistics

Data visualization (and information visualization) is now a hot and important topic in both academics and industry.

Data visualization is concerned with how to visualize complicated data. One of the major goal for data visualization is to display important features in an intuitive and clear ways so that people without professional knowledge are able to understand. Without conducting a sophisticated analysis, some clear patterns can be directly observed after visualization. This is particularly useful for scientists to promote their work to general audience and potential collaborators.

Moreover, data visualization serves as a tool for exploratory data analysis. That is, we can visualize the data first and according to the structure we observe from the visualization, we choose how to analyze the data. When the dimension of the data (number of variables) is greater than 3, this technique is very useful.

Statistics and data visualization can have more interplay. With proper cooperation, statistics and data visualization can help solving problems from each other.

In data visualization, a problem is that we discard part of the information when we visualize the data. If the information we throw away is critical to our research, we will get into trouble. Thus, there is a need to study the information that each visualization approach discards and statisticians are perfect to do this job. Many visualization tools use some summary statistics and keep track of these features when visualizing; statistical analysis for these summaries allows us to understand what kind of information the summary provide and what type of information is ignored.

For statistics, a common problem for statistical analysis is that we cannot see the result we have analyzed. For instance, when estimating a multivariate function or a “set” in high dimensions, like a region of interest, we cannot see the result. A more concrete example is “clustering” at dimension greater than 3; it is hard to really see clusters in high dimensions. This problem is especially severe in nonparametric statistics; the “parameter of interest” is often infinite dimensional and it’s hard for statistician to “see” the estimator. However, tools from data visualization may provide helps for this problem. We can use the approaches from data visualization to display our result. Despite the fact that we may loss some information, we can have a rough idea how does our estimator look like and we can fine-tune our analysis accordingly.

The following two papers are examples for combining data visualization and statistics:

1. Gerber, Samuel, and Kristin Potter. “Data analysis with the morse-smale complex: The msr package for r.” Journal of Statistical Software (2011). URL: http://www.jstatsoft.org/v50/i02/paper

2. Chen, Yen-Chi, Christopher R. Genovese, and Larry Wasserman. “Enhanced mode clustering.” arXiv preprint arXiv:1406.1780 (2014). URL: http://arxiv.org/abs/1406.1780

Here are some useful links about data visualization (thanks to Yen-Chia Hsu@CMU – Robotic Institute):