If a particle is dropped at time t = 0, physical theory indicates that the relationship between the distance traveled r and the time elapsed t is r = gtk for some positive constants g and k. A transformation to linearity can be obtained by taking logarithms: log r = log g + k log t.
By letting y = log r, A = log g, and x = log t, this relation becomes y = A + kx. Due to random error in measurement, however, it can be stated only that E(Y׀x) = A + kx. Assume that Y is normally distributed with mean A + kx and variance σ2.
A physicist who wishes to estimate k and g performs the following experiment: At time 0 the particle is dropped. At time t the distance r is measured. He performs this experiment five times, obtaining the following data (where all logarithms are to base 10).
(a) Obtain least-squares estimates for k and log g, and forecast the distance traveled when log t = +3.0.
(b) Starting with a forecast for log r when log t = 0, use the exponential smoothing method with an initial estimate of log r = –3.95 and α = 0.1, that is,
Forecast of log r (when log t = 0) = 0.1(–2.12) + 0.9(3.95), to forecast each log r for all integer log t through log t = +3.0.
(c) Repeat part (b), except adjust the exponential smoothing method to incorporate a trend factor into the underlying model as described in Sec. 27.6. Use an initial estimate of trend equal to the slope found in part (a). Let β = 0.1.