Probabilistic Forecast Kiyoharu Takano Climate Prediction Division JMA

Probabilistic Forecast Kiyoharu Takano Climate Prediction Division JMA

3 -month(Dec. -Feb. ) forecast issued by JMA at 25, Nov.

Contents (1) Predictability of Seasonal Forecast (2) Example of Probability Forecast (3) Quality of Probability Forecast Reliability Resolution (4) Economical Benefit of Forecast Deterministic forecast Probabilistic forecast

(1) Predictability of Seasonal Forecast cf. Dr. Sugi’s presentation < Predictability of 1 st kind > Originates from Initial condition Deterministic forecast fails beyond a few weeks due to the growth of errors contained in the initial states (Lorenz, 1963; 1965). < Predictability of 2 nd kind > Lorenz (1975) Originates from boundary condition Effective for longer time scale; Month to season There remains internal variability which is not controlled by boundary conditions

Growth of forecast error (Predictability of 1 st kind) 1. Error growth due to imperfectness of numerical prediction ⇒Improvement of numerical model 2. Growth of initial condition error ⇒ Improvement of objective analysis However………. . . (1) There remains finite (non zero) error in initial condition, (although it will be reduced as improvement of objective analysis). We cannot know ‘true’ initial condition (2)Small initial error(difference) grows fast(exponentially) as time progresses and the magnitude of error becomes the same order as natural variability after a certain time. ⇒Chaos Deterministic forecast becomes meaningless after sufficiently long time

Chaos Lorenz, E. N. , 1963, J. A. S. 20, 130 - Equations for simplified role type convection. ‘The Lorenz system’ X: Fourier component of stream function Y, Z：Fourier component of temperature Nonlinear terms : XZ, XY r：Stability parameter

Findings of Lorenz X ‘The solution (x(t), y(t), z(t)) behaves strangely’ • The solution (x(t), y(t), z(t)) is bound. • Non-periodic • Time Z Slight initial difference causes large difference. Chaos Y Observation X Time steps (x 0. 01)) Numerical prediction

What can we do on the predictability limit? Predictability limit varies depending on flow pattern.

What can we do on the predictability limit? • Predictability limit varies depending on atmospheric flow pattern and initial condition • Prediction of uncertainty of forecast are important 0 : 100 slow initial errors growth warm cold fast initial errors growth 70 : 30 50 : 50

The Ensemble Prediction a forecast +prediction of uncertainty of forecast Temperature anomalies at 850 h. Pa over the Northern Japan (7 day running mean) 6/ 5 Initial： 5 June 2003 7/24 Initial： 24 July 2003

Longer than one month forecast One-week forecast One-month forecast Seasonal forecast Longer than one month forecast based on initial condition is impossible at least generally!

Predictability of second kind In seasonal time scale, forecast based on initial condition is impossible at least generally The forecast is based on the influences of boundary conditions such as SSTs or soil wetness. Prediction of second kind Regression OLR map with Nino 3 SSTs (DJF) However ………. .

Atmospheric variation is not fully controlled by variation of boundary condition such as SSTs but there are internal variation. . Examples of internal variations are baroclinic instability , typhoon, Madden-Julian oscillation e. t. c. and these can be predicted as initial value problem in short time scale but they are unpredictable in seasonal time scale. Then a variation X is written as X=Xext+Xin Xext : variation controlled by boundary conditions(Signal) Xin : internal variation(Noise) (cf. Mr. Sugi’s presentation)

30 day mean 850 h. Pa Temperature prediction at the Nansei Islands Noise Signal Individual numerical prediction Average of individual predictions (ensemble mean) Observation

Reduction of noise Since the internal variation can be reduced by time mean but the signal is not reduced, Time-mean is taken in seasonal forecast. predictable Unpredictable reduced by time mean This time mean is effective especially in tropics. In addition, main SST signals such as ENSO are in tropics. Seasonal forecast in tropics is easier than in mid-latitude.

The noise reduction effect of time mean 5 day mean Individual numerical prediction Average of individual predictions (ensemble mean) Observation 30 day mean 90 day mean

Though time mean is effective to reduce noise, the noise is not removed completely. Therefore there remains uncertainty from internal variation and again probabilistic forecast is necessary. Individual numerical prediction Average of individual predictions (ensemble mean) Observation 90 day mean

(2) Example of probabilistic forecast (a) One-month forecast Temperature at 850 h. Pa NJ WJ EJ Nansei Surface Temperature Forecast 1 month Forecast Period 11. 22 -12. 21 1 st week 2 nd week 11. 22 -11. 28 11. 29 -12. 5 3 rd-4 th week 12. 6 -12. 19 category - 0 + Northern Japan 20 40 40 20 50 30 20 40 40 Eastern Japan 20 30 50 20 40 40 Western Japan 10 40 50 20 30 50 20 40 40 Nansei Islands 10 40 50 10 30 60 20 30 50 20 40 40 ( category : below normal, 0 : near normal, + : above normal, Unit : %)

(2) 3 month forecast Temperature Forecast Period 3 months 1 st month 2 nd month 3 rd month Nov. -Jan. Nov. Dec. Jan. Category - 0 + Northern Japan 30 50 20 20 50 30 Eastern Japan 20 50 30 20 40 40 Western Japan 20 40 40 20 50 30 20 40 40 Nansei Islands 20 30 50 20 40 40 20 30 50 ( category ? : below normal, 0 : near normal, + : above normal, Unit : %)

Precipitation Forecast (3 month forecast) Period 3 months 1 st month 2 nd month 3 rd month Nov. -Jan. Nov. Dec. Jan. category - 0 + Northern Japan Sea side Pacific side 20 50 30 30 40 30 30 50 20 30 40 30 Eastern Japan Sea side Pacific side 20 50 30 30 50 20 20 50 30 20 40 40 Western Japan Sea side Pacific side 20 50 30 30 40 30 20 50 30 20 40 40 Nansei Islands 20 50 30 40 40 20 20 40 40 ( category 30 40 30 -: below normal, 0 : near normal, + : above normal, Unit : %)

Seasonal forecast of U. S.

Seasonal forecast by I. R. I.

Examples of probabilistic forecast excluding seasonal forecast at JMA uses probabilistic expression not only in seasonal forecast but in many forecasts where there is uncertainty of forecast. (1) Short-range forecast Tokyo District Today North-easterly wind, fine, occasionally cloudy, Wave 0. 5 m Probability of Precipitation 12 -18 10% 18 -00 0% Temperature forecast today’s maximum in Tokyo 14 degrees centigrade

(2) One-week forecast Date 24 Mon Max. , Min. T(℃） Probability of Precipitation 25 Tue １ 0 / 16 70 26 Wed 11/15 50 27 Thu 7/15 40 28 Fri 9/16 30 29 Sat 30 Sun 9/17 30 11/18 40 Issued at 23, NOV

(3) Typhoon Forecast The probability that a district will be in storm warming area is also being issued.

Quality of Probabilistic Forecast What is good probabilistic forecast? ‘A good probabilistic forecast must express the uncertainty of forecast exactly and have large dispersion from ‘climatic proportion of frequency’ (a) Reliability (b) Resolution

(a) Reliability Probability forecast P was issued ‘M(P)’ times for a event ‘E’. In M(P) times, event ‘E’ occurred N(P) times. If probability forecast is reliable for large number of M(P). Ex. Probability 30% was issued 50 times. Event ‘E’ is expected to occur about 15 times in them.

The reliability diagram Event E Relative frequency Forecast probability Ideal reliability diagram

Reliability Diagram Monthly mean surface temperature forecast in Japan Relative frequency Forecast probability

Reliability diagram Example of probability by ensemble one month forecast monthly mean Psea anomalies >0

(b) Resolution Climatorogical relative frequency for a event is perfectly reliable so far as there is no climatic Change. ex. It is known that relative frequency of rainy day is about 30% at Tokyo from historical data set. Can we issue probability of 30% as a tomorrow's probability of precipitation at Tokyo every day? When the reliability is perfect, dispersion of probability from climatorogical relative frequency is another measure of probabilistic forecast quality ‘resolution’. The best resolution probabilistic forecasts are those of 100% or 0% provided that reliability is perfect. =perfect forecast.

Resolution Monthly mean surface temperature forecast in Japan Relative frequency Forecast probability Frequency

Resolution of probability forecast 7 day mean Z 500 anomalies>0 Reliability diagram 1 st week Relative frequency(resolution) 1 st week probabilistic forecast is better than 2 nd week in ‘resolution’ measure From one-month ensemble forecast

Quantitative evaluation of probabilistic forecast -------The Brier score is defined as pi : forecast probability vi=1: if event E occurred =0 : if not occurred N: total number of forecast b is mean square error of probability

After some algebraic transformations, b can be rewritten as brel bres bunc where Nt : frequency of forecast probability pt Mt: frequency of occurrence of event E within Nt

represents ‘Reliability’ represents ‘Resolution’ represents ‘Uncertainty’ (not depends on the forecast. shows difficulty of forecast) Murphy’s decomposition(1973)

Brier skill score The absolute value of brier score is difficult to understand except perfect reliability and resolution case (=0). Then usually following Brier skill score is used where bc is the brier score of climatorogical relative frequency( probabilistic climate forecast). B=1 when the forecast is perfect. Additionally, following skill score is also used. Brel=Bres=1 when the forecast is perfect.

Example of Brier skill score Ensemble one-month forecast 1 st week 7 day mean Z 500 anomalies>0 All scores are expressed in % 2 nd week

(4) Economical value of forecast ----a consideration with Cost/Loss model----The Cost/Loss model an event E ① If E occurs, the loss is L due to a damage ② To protect form the damage, a action costs C (<L) ex. If the temperature exceeds a threshold, a crop is damaged by a pest. The damage is L. To protect the crop from a pest , spraying agrochemical is necessary and it costs C.

The cost/benefit model a Event E if the event E occurs , a benefit is B. If not occurs, we lose C. Example、 If it is fine, a benefit B is earned by selling lunch boxes, but if it rains, no benefit is earned. The cost to make lunch boxes is C. The discussion of Cost/Loss is almost the same as that of Cost/Benefit model.

The case without forecast (Cost/Loss model) Considering D times operations. The climatorogical proportion of occurrence of event E is R. When always taking action to protect from damage, the expense is , If no action is done, the expense is Then, when R>C/L If the event E occurs frequently, it is better to take action always.

The case with perfect forecast (Cost/Loss model) If forecast is perfect, we take action only when event E is forecasted. Then the expense is of course, Perfect forecast always reduces the expense

The case with actual deterministic forecast Actual forecast sometimes fails. Then we make following contingency matrix Occurs Forecast ( ) shows in case of perfect forecast If we use these forecasts, the expenses corresponding individual boxes above are, Occurs Take Action

The relationships with pre-defined variables are Occurs Forecast We newly define following variables which express forecast skill Hit rate False-alarm rate The lager H is and the smaller F is , the better forecast is. H=1 and F=0, for perfect forecast

The expense with these forecasts is If forecast is worse, Mp sometimes become lager than Mclim 2 or Mclim 1. IF H=0 and F=1(the worst forecast!) Mp=DLR+D(1 -R)C>Mclim 1, Mclim 2

It should be also noted that Mp depends not only on H and F but C/L and R. An imperfect deterministic forecast is not always useful for all users.

The case with probabilistic forecast (Cost/Loss model) How do we use probabilistic forecast? Consider to take action always when the probabilistic forecast is P 0 for the occurrence of event E. The expense is, Mp 0=Dp. C where Dp is total frequency that the event E was forecasted with probability P 0 If probabilistic forecast is perfectly reliable, the frequency that event E occurred is Dp. P 0 and then the expense without taking action is Mc=Dp. P 0 L

Probabilistic forecast is useful when, Mp 0=Dp. C < Mc=Dp. P 0 L Therefore, we should take action when P 0>C/L This is the simplest way to use probabilistic forecast for decision making. Note threshold probabilities are different for individual users with different C/L and all users can get some gain with their own threshold. In case of Cost-benefit model, with similar calculations, the criterion is, P 0 >C/(B+C)

Ex. 1(Cost-Loss model) If the temperature exceeds a threshold =33℃, a crop is damaged by a pest. The damage is L=10, 000$ To protect the crop from a pest , spraying agrochemical is necessary and it costs C＝ 3, 000$ C/L=3000/10000=0. 3 Consider 10 times forecast of above 33℃ with probability 20%(30%, 40%). When we take action the expense is 3000 x 10=30, 000$ When we do not take action, the expense is 10000 x(10 x 0. 2)=20, 000 $ for probability 20% 10000 x(10 x 0. 3)=30, 000 $ for probability 30% 10000 x(10 x 0. 4)=40, 000 $ for probability 40% We had better take action when Probability>30%=C/L

Example for cost-benefit model If it is fine, a benefit B is earned by selling lunch boxes, but if it rains, no benefit is earned. The cost to make lunch boxes is C , which is the loss when it rains. Price of lunch box=10$ and 100 lunch boxes are sold in a fine day. The cost to make one lunch box=5$ The benefit in a fine day is B=(10 -5)x 100=500$ The cost is C=5 x 100=500$ , which is the loss when it rains C/(B+C)=0. 5 １０ times forecast of fine with probability 40%(50%, 60%) When we sell the lunch boxes, The expected cost is 500 x(1 -0. 4))=30, 000 $ for probability 40% 500 x(1 -0. 5))=25, 000 $ for probability 50% 10000 x(1 -0. 6)))=20, 000 $ for probability 60% The expected benefit is 500 x(10 x 0. 4)=20, 000 $ for probability 40% 500 x(10 x 0. 5)=25, 000 $ for probability 50% 500 x(10 x 0. 6)=30, 000 $ for probability 60% We had better sell when Probability>50%=C/(B+C)

Verification of probabilistic forecast -----How to use an actual probabilistic forecast----We used a important assumption to derive threshold probability to take action in the previous section. Assumption probabilistic forecast is perfectly reliable. Although this condition would be satisfied approximately in most practical probabilistic forecasts, there is no guarantee that it is always satisfied. In addition, the expense reduction (Mp-Mclim) with probabilistic forecast also depends on the ‘resolution’ of probabilistic forecast and we cannot know how much it is without verification. Therefore verification is important to use probabilistic forecast actually.

A Verification of probabilistic forecast from the economical view point We assume E will occur when P>Pt and E will not occur when P<Pt where Pt is a threshold probability. And again we make contingency matrix as follows. Occurs Forecast

As similar to before, the expense is, The expense for perfect forecast is, That for climatic forecast is,

We define the ‘Value of forecast’ as the reduction in Mp over Mclim normalized by the maximum possible reduction. That is, V(Pt)=1 for the perfect forecast and negative for bad forecast.

We calculate V for various threshold probabilities and C/L. For a given C/L, and a event E, the optimal value is,

Probability of T 850 anomalies >0 From Palmer(2000) 60% line 10% 20% 40% 30% From this graph, 50% 60%70% 80% 90% For a user with C/L=0. 6, V=12% and best threshold probability is 70%, although threshold 60%(=C/L) gives some benefit(7%). • The user can choose threshold which brings maximum benefit( Note reliability is not completely perfect). • A user with a C/L can estimate maximum benefit

Vopt for Probability forecast V for deterministic forecast The graph of Vopt for probabilistic forecast is higher and wider than that of V for the deterministic forecast. Because, for deterministic forecast, only one contingency matrix is made and then only one graph of V is drawn. On the other hand, Vopt is the maximum of the graphs of V with various probability threshold.

A verification example of monthly surface temperature anomaly probability forecast in Japan (Statistical down scaling from ensemble forecast)) Surface temperature (below normal) Surface temperature (above normal) Surface temperature Reliability diagram for above normal Reliability diagram for below normal

A verification of dymamical one-month ensemble forecast Anom. (Z 500)>0 28 day mean ( winter 2001)

1 st week 2 nd week

3 -4 th week

Conclusions • Seasonal forecast has uncertainty due to chaos of atmospheric flow. The probabilistic forecast is the best method to express this uncertainty. • In probabilistic forecast, a user can use his/her own threshold probability to take action depending on his/her own C/L in Cost-Loss model. For a sufficiently ‘reliable’ probabilistic forecast, the threshold probability is equal to C/L. If ‘reliability’ is not enough, user can know the best threshold probability by a verification. • In general, the probabilistic forecast is superior to the deterministic forecast at least from the economical point of view so far as the forecast is not perfect although it seems difficult in some degree. More dissemination of probabilistic forecast is necessary.

Please remember the words ‘uncertainty of forecast’ and ‘cost-loss ratio C/L’. Thank you!

Monthly mean surface temperature forecast at Japan Surface temperature above normal Vopt