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)].

# Regressogram: a commonly used simple non-parametric method in science

Recently, I work a lot with scientists and notice that they are often using a simple method to visualize the data and make basic analysis.

This method is called “regressogram” in statistics. Simply put, Regressogram = Regression + Histogram. Here’s an example for regressogram:

This is an analysis for astronomy data. On the X-axis is the galaxy distance to some cosmological structure and on the Y-axis is the correlation for some features of this galaxy. We binned the data according to galaxy distance and take the mean within each bin as a landmark (or summary) and show how this landmark changes along galaxy distance. This visualizes the trend of the data in an obvious way.

In fact, the original data is very ugly! Here’s the scatter plot for the original data:

Note that now the range of Y is (0,1) while in the regressogram, the range is (0.7, 0.8). If you want to visualize the data, I don’t think this scatter plot is very helpful. The regressogram, however, is a simple approach to visualize hidden structure within this complicated data.

Here’s the steps for constructing regressogram. First we bin the data according to the X-axis (shown by red lines):

Then we compute the mean within each bin (shown by the blue points):

We can show only the blue points (and blue curves, which just connects each points) so that the result looks much more concise:

However, since the range for Y-axis is too large, this does not show the trend. So we zoom-in and compute the error for estimating the mean within each bin. This gives the first plot we have seen:

The advantage for regressogram is its simplicity. Since we’re summarizing the whole data by points representing the mean within each bin, the interpretation is very straight-forward. One can easily understand regressogram without any deep knowledge of statistics. Also, it shows the trend (and error bars) for the data so that we have rough idea what’s going on. Moreover, no matter how complicated the original plot is, the regressogram uses only a few of statistics (the mean within each bin) to summarize the whole data. Notice that we do not make any assumption on distribution (like normally distributed) of the data; thus, regressogram is a non-parametric method.

However, in statistics, regressogram is barely mentioned and very few people actually use it in research. The main reason is that the regressogram is a method for non-parametric regression and is not optimal. The predicted value of Y given X using regressogram is to find the bin where X lays in and use the sample mean within that bin as a predictor for Y. This prediction is suboptimal and there’re many alternative method such as local regression and kernel regression that has much better prediction accuracy. (The main problem for regressogram is the huge bias. We predict the same value within the same bin which makes it inflexible in prediction.)

Despite not being an optimal method for prediction, regressogram is still very attractive due to its simplicity. Especially for the case when we just want to grasp a rough trend of the data, some loss of accuracy to trade simplicity is always preferred. Hence, regressogram is still very popular in science.

I end this article with an interesting observation. Although regressogram is widely used in scientific research, very few scientists know the name of this method. Maybe regressogram is too intuitive for most scientists so that they do not care about its name.