Goodman and Kruskal's lambda

Goodman and Kruskal described several measures of association between two variables [CastellanCastellan1979,Siegel Castellan JrSiegel Castellan Jr1988]. The question behind their investigation is the following: "How much does the knowledge of the classification of one variable improves the prediction of the classification of the other variable?" Transposed in terms of time series analysis it may be reformulated this way: "How much does the knowledge of the state of the system at time $t$ improves the prediction of state of the system at time $t+1$?"

Suppose a researcher wants to predict the next state of a system. Without the transitional matrix information, the best guest is to predict the state that has the highest probability of occurrence; or more precisely, the state having the largest marginal total, $max(n_{1+}, n_{2+}, n_{m+})$. But when one knows the state at time $t$, the best guess to predict the state with the highest probability given this knowledge, taking into account the transitional frequencies. It thus reduces the probability of error.