1. Covariance is just a scaled version of correlation . So, when two points are close to each other, then they are expected to be similar, and so their covariance(correlation) will be large.
2. represents the autocorrelation between the variables. Empirical covariance is the cross-product (Z(Si)-Z)(Z(Sj)-Z), where Z is the average of the data, Z(Si) is the value of location Si. If Si and Sj are correlated, Z(Si) and Z(Sj) are similar and will be both above or both below the mean Z, and so the covariance is positive. Otherwise Si and Sj are much less correlated, the covariance will be negative half the time and positive the other half of the time, so their average will be near zero.
3. an absolute measure of the relationship between returns of two series.
4. The statistical tendency of two variables of the same type, attribute, name, and so on, to vary in
ways that are related to each other. Positive covariance occurs when both variables tend to be
above their respective means together, and negative covariance occurs if one variable tends to be
above its mean when the other variable is below its mean.
See also correlation, and compare to crosscovariance