Extrapolation for Time-Series and Cross-Sectional Data

Extrapolation methods are reliable, objective, inexpensive, quick, and easily automated. As a result, they are widely used, especially for inventory and production forecasts, for operational planning for up to two years ahead, and for long-term forecasts in some situations, such as population forecasting. This paper provides principles for selecting and preparing data, making seasonal adjustments, extrapolating, assessing uncertainty, and identifying when to use extrapolation. The principles are based on received wisdom (i.e., experts’ commonly held opinions) and on empirical studies. Some of the more important principles are:• In selecting and preparing data, use all relevant data and adjust the data for important events that occurred in the past.• Make seasonal adjustments only when seasonal effects are expected and only if there is good evidence by which to measure them.• In extrapolating, use simple functional forms. Weight the most recent data heavily if there are small measurement errors, stable series, and short forecast horizons. Domain knowledge and forecasting expertise can help to select effective extrapolation procedures. When there is uncertainty, be conservative in forecasting trends. Update extrapolation models as new data are received.• To assess uncertainty, make empirical estimates to establish prediction intervals.• Use pure extrapolation when many forecasts are required, little is known about the situation, the situation is stable, and expert forecasts might be biased

Reduction of calcium release site models via optimized state aggregation

Background Markov chain models of calcium release sites in living cells exhibit stochastic dynamics reminiscent of the experimentally observed phenomenon of calcium puffs and sparks. Such models often take the form of stochastic automata networks in which the transition probabilities for each of a large number of intercellular channel models depend on the local calcium concentration and thus the state of nearby channels. The state-space size in such compositionally defined calcium release site models increases exponentially as the number of channels increases, which is referred to as “state-space explosion”. Methods In order to overcome the state-space explosion problem, we utilized the idea of “coarse graining” and implemented an automated procedure that reduces the state space by aggregating and lumping states of the full release site model. For a given state aggregation scheme, the transition rates between reduced states are chosen consistent with the conditional probability distribution among states within each group. A genetic algorithm-based approach is then applied to select the state aggregation schemes that lead to reduced models that approximate the observable behaviors of the full model. Results The genetic algorithm-based approach is implemented in Matlab®; and applied to two different release site models. The approach found reduced models that approximate the full model in the number of open channels, spark statistics, and the jump probability matrix as a function of time. Conclusions A novel automated genetic algorithm-based searching technique is implemented to find reduced calcium release site models that approximate observable behaviors of the full Markov chain models that possess intractable state-spaces. As compared to the full model, the reduced models produce quantitatively similar results using significantly less computational resources