Let us understand the correlation matrix and covariance matrix
(1) Correlation is a scaled version of covariance that takes on values in of the relationship between the two variables, while correlation does. The first and foremost difference between covariance and correlation is that the value of correlation takes place between -1 and +1. Conversely. What is the relationship between the variance/covariance matrix of a Covariance · Variance . Effective method to estimate multidimensional Gaussian states.
Covariance versus Correlation As we see from the formula of covariance, it assumes the units from the product of the units of the two variables.
On the other hand, correlation is dimensionless. It is a unit-free measure of the relationship between variables. This is because we divide the value of covariance by the product of standard deviations which have the same units. The value of covariance is affected by the change in scale of the variables.
If all the values of the given variable are multiplied by a constant and all the values of another variable are multiplied, by a similar or different constant, then the value of covariance also changes. However, on doing the same, the value of correlation is not influenced by the change in scale of the values.
Another difference between covariance and correlation is the range of values that they can assume. Application in Analytics Now that we are done with mathematical theory, let us explore how and where it can be applied in the field of data analytics. Correlation analysis, as a lot of analysts would know is a vital tool for feature selection and multivariate analysis in data preprocessing and exploration. Correlation helps us investigate and establish relationships between variables.
This is employed in feature selection before any kind of statistical modelling or data analysis. So how do we decide what to use? Correlation matrix or the covariance matrix?
In simple words, you are advised to use the covariance matrix when the variable are on similar scales and the correlation matrix when the scales of the variables differ. To help you with implementation if needed, I shall be covering examples in both R and Python. Let us see the first example where we see how PCA results differ when computed with the correlation matrix and the covariance matrix respectively.
From the above image, we see that all the columns are numerical and hence, we can go ahead with the analysis. So, prcomp returns 5 key measures: Let us briefly go through all the measures here: The center and scale provide the respective means and standard deviation of the variables that were used for normalization before implementing PCA.
In other words, it shows the square roots of the eigenvalues. The rotation matrix contains the principal component loading. This is the most important result of the function. Each column of the rotation matrix contains the principal component loading vector.
correlation - What is covariance in plain language? - Cross Validated
The component loading can be represented as the correlation of a particular variable on the respective PC principal component. It can assume both positive or negative. Higher the loading value, higher is the correlation. Let us now look at the principal component loading vectors: To help with the interpretation, let us plot these results.
To read this chart, one has to look at the extreme ends top, down, left and right. The second principal component PC2 does not seem to have a strong measure. We can finish this analysis with a summary of the PCA with the covariance matrix: This is in line with our observations from the rotation matrix and the plot above. As a conclusion, not a lot of significant insights can be driven from the Principal Component Analysis on the basis of the covariance matrix. With the same definitions of all the measures above, we now see that the scale measure has values corresponding to each variable.
The rotation matrix can be observed in a similar way along with the plot. This plot looks more informative. Let us try to look at the summary of this analysis. One significant change we see is the drop in the contribution of PC1 to the total variation. It has dropped from Furthermore, the component loading values show that the relationship between the variables in the data-set is way more structured and distributed.
Another significant difference can be observed if you look at the standard deviation values in both the results above. The values from PCA done using the correlation matrix are closer to each other and more uniform as compared to the analysis done using the covariance matrix.
This analysis with the correlation matrix definitely, uncovers some better structure in the data and relationships between variables.
The above example can be used to conclude that the results significantly differ when one tries to define variable relationships using covariance and correlation. This in turn, affects the importance of the variables computed for any further analyses. Selection of predictors and independent variables is one prominent application of such exercises. Now, let us take another example to check if standardizing the data-set before performing PCA actually gives us the same results. To showcase agility of implementation across technologies, I shall execute this example in Python.
For example, you might hear that as economic growth increases, stock market returns tend to increase as well. These variables are said to be positively related because they move in the same direction.
You may also hear that as world oil production increases, gasoline prices fall. These variables are said to be negatively, or inversely, related because they move in opposite directions.
The relationship between two variables can be illustrated in a graph. In the examples below, the graph on the left illustrates how the positive relationship between economic growth and market returns might appear.
The graph indicates that as economic growth increases, stock market returns also increase. The graph on the right is an example of how the inverse relationship between oil production and gasoline prices might appear.
It illustrates that as oil production increases, gas prices fall. To determine the actual relationships of these variables, you would use the formulas for covariance and correlation. Covariance Covariance indicates how two variables are related. A positive covariance means the variables are positively related, while a negative covariance means the variables are inversely related. The formula for calculating covariance of sample data is shown below.
Before you compute the covariance, calculate the mean of x and y. The Summary Measures topic of the Discrete Probability Distributions section explains the mean formula in detail.
Now you can identify the variables for the covariance formula as follows. Since the covariance is positive, the variables are positively related—they move together in the same direction.
Difference Between Covariance and Correlation
Correlation Correlation is another way to determine how two variables are related. In addition to telling you whether variables are positively or inversely related, correlation also tells you the degree to which the variables tend to move together.
As stated above, covariance measures variables that have different units of measurement. Using covariance, you could determine whether units were increasing or decreasing, but it was impossible to measure the degree to which the variables moved together because covariance does not use one standard unit of measurement. To measure the degree to which variables move together, you must use correlation. Correlation standardizes the measure of interdependence between two variables and, consequently, tells you how closely the two variables move.
The correlation measurement, called a correlation coefficient, will always take on a value between 1 and — 1: If the correlation coefficient is one, the variables have a perfect positive correlation.
Difference Between Covariance and Correlation (with Comparison Chart) - Key Differences
This means that if one variable moves a given amount, the second moves proportionally in the same direction. A positive correlation coefficient less than one indicates a less than perfect positive correlation, with the strength of the correlation growing as the number approaches one. If correlation coefficient is zero, no relationship exists between the variables.