1. Statistical Methods for long-range forecast By Syunji Takahashi Climate Prediction Division JMA

2. Let’s thinking chaos of our atmosphere using Lorenz system Lorenz equations nonlinear equations : approximated equations of convection : representing the essential nature of our atmosphere (Chaotic feature) : easy to solve it by PC X : a component of stream function Y,Z: two components of temperature

3. Lorenz system making Chaos ! Trajectory of solution on X-Z plain Time series of solution X Two solution with slight different initials Features of solution Circling around two attractors (Lorenz attractor) alternatively With no certain period Small difference becoming greater soon (Chaos)

4. Predictability Problem Predicting the average value within the period Statistics of 3000 times simulations with small disturbance generated stochastically

5. Predictability of the Second Kind Solution of Lorenz system with a forcing Forcing generates bias in the solutions → Signal

6. Why Probability function becoming normal ? Central limit theory Mean of any stochastic variables becoming to be normally distributed (not strictly speaking) Central Dogma of Statistic

7. Chaotic feature of the atmosphere and long-range Forecast In both numerical and statistical prediction We can’t predict the long-term future precisely. Possible target of long-range forecast is biased state caused by boundary forcing. Possible target of long-range forecast is averaging state. 4Probabilistic forecast is essential. 5Noise can be assumed to be normally distributed.

8. History of long-range forecast at JMA and statistical methods 1942 starting long-range forecasting frequent cool summers 1943 formal issuing 1 month, 3 months, seasonal harmonic analysis criticism on the accuracy 1949 division being closed frequent unusual weather 1953 restarting long-range forecast increasing upper sounding data simple regression method 1974 establishment of long-range forecast division analog method increasing demand for climate information 1987 publish of monthly report of climate system spectral analysis multiple regression method 1996 rearranging climate prediction division 1 month numerical prediction probability forecast OCN 2003 3 months numerical prediction CCA

9. 1 Analog Method Cluster Analysis not being used How is they similar ? 2 Spectral or Harmonic Methods not being used sometimes good, but not always 3 Optimal Climate Normal (OCN) now being used simple ! 4 Multiple Regression Analysis Simple Regression not being used available technique 5 Canonical Correlation Analysis (CCA) now being used fashionable technique ! Statistical methods at JMA

10. Concept of Analog Method Basis of analog method similar states will evolve similarly and similar in the future Analog method using 500hPa patterns Searching past years which similar to the target year in 500 hPa pattern, to predict the future of the year using the past futures of these similar years Selecting 10 similar years, frequency distribution of these temperatures in the past futures is considered as a probability forecast.

11. Definitions of Similarity or Distance Similar Euclid distance Similar Correlation coefficient Forecasting year Different definition, different results!

12. Cluster Analysis grouping method Distance Space Gathering the nearest pairs of member or group Various definitions of the distance between groups

13. Example of Cluster Analysis Data: 1971-2000 sequences of monthly temp.(Feb.) Distance: unity minus correlation coefficient Group distance: Word method (It is my favorite) Dendro-gram

14. Analysis of time series Sometimes, obvious cycle appears in the sequence of meteorological element. In that case, prediction using the periodicity is very efficient.

15. Prediction by Auto-Regression Model Assuming Auto-Regression Model such as Determining the coefficients and variance of noise from past data. We can predict the future as And so on

16. Successive Case

17. Failure Case

18. Optimal Climate Normal (OCN) Normal, past 30 years mean is not always optimal ‘first estimate’ in the case of being obvious increasing or decreasing trend or climatic jump. Investigating the past data, 10 years mean is the optimal first estimate in both temp. and precip. of Japan.

19. Break Time for 10 minutes EXCELL files of Chaos, Cluster analysis, and Spectral analysis are prepared in this PC.

20. Situation of Multiple Regression Model Predictand Vector Predictor Matrix Independent Data

21. Multiple Regression Equation The multiple regression model assumes predictand vector is sum of a linear combination of predictors and a noise. This can be rewritten as Su

22. Determine Regression Coefficients The coefficient vector is usually estimated so as to minimize the sum of squared errors of predicted vector.

23. Visual Image of Multiple Regression Predictand Vector Subspace S(X) Predicted Vector Orthogonal Projection y to S(X) Residual Vector

24. Visual Image of Multiple Regression Residual Vector Projection of Error Vector True Regression Vector Error Vector

25. Property of Regression Residual

26. Detecting Trend Using Simple Regression

27. Property of Calculated Trend Confidence Interval of the estimated trend Confidence Interval of the regression line

28. Property of Calculated Trend Confidence Interval of the estimated trend Warming trend in Tokyo is significant Confidence Interval of the regression line

29. Urbanization and Warming Trend

30. Estimation of Global Warming Trend Confidence Interval of the section (constant) Global warming trend and its confidence interval can be estimated even using the data effected urbanization

31. Why is CCA Currently Used ? Disadvantages of Multiple Regression method 1 Covariance matrix being singular singular not calculated 2 Not taking accounts of the correlations among the predictands Regressions of many predictands are independently determined.

32. CCA Flow Chart

33. Transform Real Space to EOF Space

34. Determining Canonical Component

35. Summary of Multiple Regression and CCA 1Multiple regression and CCA are fashionable tools,which are available to treat bulk data. 2Selection of variable is very important for successive prediction using independent data. Stepwise and all- subsets methods are available. 3Rank deficiency problem can be avoided by transformation the real data to one in EOF space in both multiple regression and CCA case. 4Correlation between predictands can be considered in CCA. CCA is the most fashionable tool.

36. Thank you