What is the intuition of covariance?

If two variables tend to vary together (that is, when one of them is above its expected value, then the other variable tends to be above its expected value too), then the covariance between the two variables will be positive.

What should I infer from the covariance matrix?

You can use the covariance to determine the direction of a linear relationship between two variables as follows:

• If both variables tend to increase or decrease together, the coefficient is positive.
• If one variable tends to increase as the other decreases, the coefficient is negative.

What do eigenvalues of a covariance matrix tell us?

The eigenvalues still represent the variance magnitude in the direction of the largest spread of the data, and the variance components of the covariance matrix still represent the variance magnitude in the direction of the x-axis and y-axis.

What does the value of covariance mean?

Covariance indicates the relationship of two variables whenever one variable changes. If an increase in one variable results in an increase in the other variable, both variables are said to have a positive covariance. When a positive number is used to indicate the magnitude of covariance, the covariance is positive.

Is covariance a correlation?

Covariance indicates the direction of the linear relationship between variables while correlation measures both the strength and direction of the linear relationship between two variables. Correlation is a function of the covariance.

How do you interpret covariance?

Covariance gives you a positive number if the variables are positively related. You’ll get a negative number if they are negatively related. A high covariance basically indicates there is a strong relationship between the variables. A low value means there is a weak relationship.

What does it mean when variance is 1?

The mean (and expected value) of a standard normal distribution is zero. Unit variance means that the standard deviation of a sample as well as the variance will tend towards 1 as the sample size tends towards infinity.

What is a co-variance matrix?

The co-variance represent a transformation that rotate any vector into the direction of the greatest variance of the data. This property is the foundation of PCA. Ready to see some magic? The data is simulated so that the direction of greatest variance is the red dash line. The co-variance matrix of this set of data (M) is:

What are the eigenvectors of the empirical covariance matrix?

Eigenvectors of the empirical covariance matrix are directions where data has maximal variance. We know that the eigenvector basis of a linear operator is the basis where the operator has diagonal representation.

How do you find the covariance matrix with a zero mean?

Following from this equation, the covariance matrix can be computed for a data set with zero mean with C = XXT n−1 C = X X T n − 1 by using the semi-definite matrix XXT X X T. In this article, we will focus on the two-dimensional case, but it can be easily generalized to more dimensional data.

Is the covariance matrix symmetric?

NOTE: Covariance matrix is always symmetric. Since the non-diagonal elements of S are positive, we can see that x and y are postively correlated which is what we can see from the diagram as well. i.e as x increases y also increases.