Monthly Temperature Forecasts by Using a Complex Autoregressive Model

In order to find a new method to improve the skill of short-rang climate prediction, a complex autoregressive model is established based on mathematic derivation of the complex least-square, in which the conventional least-square formula is extended from the real number domain into the complex number domain.This complex least-square solution is an exact analytic formula, and the conventional way is corrected that the real number and the imaginary number are separately calculated to reserve the least-square in the complex number domain.With a spatial expansion of Fourier series on monthly temperature fields in mainland China, the applications of this complex autoregressive model (M1) to monthly temperature forecasts show a high skill comparing with other conventional statistical models in predicting monthly temperature anomalies for July and most other months at 160 meteorological stations in mainland China.The conventional statistical models include an autoregressive model in the complex number domain that the real number and the imaginary number are separately disposed (M2), an autoregressive model in the real number domain (M3), and a persistence-forecast model (M4). For example, the anomaly correlation coefficient and root mean square error prediction for July by the M1 reaches up to 0.185 and 1.079 ℃ comparing with 0.089 and 1.113 ℃ by the M2, 0.061 and 1.147 ℃ by the M3, and 0.064 and 1.449 ℃ by the M 4 respectively, although the M2 does somewhat higher skill than the M3 and M4. It is expected that a better method of spatial expansion should improve further the forecast skill.The complex least-square derived in this study is an exact solution comparing with the conventional method that the real part and the imaginary part are separately calculated.In fact, the conventional method does not reach the actual least square in a complex number domain.The forecast experiments suggest that the complex least-square is an effective technique to dispose a complex number series, and may be applied to the linear and non-linear regression and similar statistic methods that are based on the least-square method.Developments of complex statistical models could be a perspective way to improve sim ulation and forecast skill in complex number fields in meteorology and relative disciplines.