发布时间 : 星期日 文章2016美赛E题参考答案 - 图文更新完毕开始阅读9f36d621ba0d4a7303763aaa
Team #52766 Page 6of 42
tdQ??2??Yi??a?bXi???0??dai?1 tdQ??2???Yi??a?bXi????Xi?0dbi?1
b1???Xi?1ti?1ti?XYi?YXi?X?????X?i?1tti?XYi????2??i?1Xi?X?2
a?Y?bX
According to the industrial water consumption in each year, we can fit the industrial water consumptionB(t).
5.1.3Model of total water consumption of a region
Since it is the same for the agricultural consumption C(t)and ecological consumption D(t)to calculate as it does in industrial consumptionB(t), we can determine the model of total water consumption.Based on the model of personal consumption, industrial consumption, agricultural consumption, and ecological consumption, we can determine the model of total water consumption of a region
E(t) as follows:
atatE(t)?W*N0*exp[?(bt0?0)]*exp(bt?)?B(t)?C(t)?D(t)(6)
22225.2Model of the Gray Metabolism Model GM (1, 1)
5.2.1 Principle of common Gray GM (1, 1) Model
Gray System theory holds the view that all the random quantities are gray
variables and process within certain range and time interval. The model is established after processing these data in certain ways and ranking into regular time series. Gray System Prediction Model GM(1,1) is a first order differential equation with one variable，it is fit for prediction to the development of systematic behavior eigenvalue. Gray System Prediction Model GM (1, 1) produces random number and transforms them into ordered data, and then establishes differential equation, later, it seeks for the
Team #52766 Page 7of 42
regulation of producing the data and then restore the operating results. The specific steps are following:
We
accumulate
the
variable
x(0)?{x(0)(1),x(0)(2),?,x(0)(N)} toget
x(1)?{x(1)(1),x(1)(2),?,x(1)(N)}
hereinto，we getx(t)??x(0)(i)?x(1)(t?1),(t?1,2,...,n).
(1)itSo we can establish a differential equation in the form of an albinoas fellow:
dx(1)?ax(1)?udt
The bleaching solution of differential equation are as follows（disperse
response）:
uux(1)(t)?[x(0)(1)?]e?a(t?1)?aa
Parameter k denotes time series, can be year, season or month.
?a?Mark parameter sequences asU, U???.
?b?We obtain U from these equations:
???aU????(BTB)?1BTy
??b?While Brepresents data matrix，y denotes data column.
(1)(1)??1?x(0)(2)?1?2[x(2)?x(1)]?1(1)??(0)?(1)?[x(3)?x(2)]1x(3)??B??2y???????1??1(1)??(0)?(1)?[x(N)?x(N?1)]1x(N)?????2?
Because we get cumulative amount by GM Model is for once, and it is the
?(1)(t?1)predicted value whent?{n?1,n?2,...}, we must restore the obtained data x?(0)(t?1)(orx?(0)(t)) through repeatedly minus withdetermining ?(1)(t)) to x(or?x(I—AGO):
?(t)??x?(0)(i) x(1)i?1t?(0)(i)?x?(0)(t) ??xi?1t?1
Team #52766 Page 8of 42
?(t)?x?(k)??x?(0)(i) x(1)(1)i?1t?1?(1)(t?1)?x?(1)(t?1)?x?(1)(t),(t?0,1,2,...). ?(t?1)??x?(0)(i), we getxSincex(1)i?1t?15.2.2 Principle of Gray Metabolism Model GM (1, 1)[2]
After making a gray prediction and getting the latest information，it adds this
information into the original data series and wipes off the oldest information at the same time.Then, using the new one as original series, it repeats the above step 1.1 to set up GM（1,1） Model，so on and so forth, until the fulfillment of all the prediction objectives, and that is the Gray Metabolism Model we wanted。
5.2.3 Accuracy testing
Relative error and posterior difference ratio C are two most commonly used way to test the model, and its basic process is following:
?(0) is the series simulated by GM Model and?is residual x(0) is original series，x?(0)(t), the relative error sequence isp?1??, sequence。Within it is ?(t)?x(0)(t)?xand thus the total water resource amount in t?1 year??(?1,?2,...,?n) could be obtained. Hereinto we have?t?|?(t)x0(k)|, and ?t is the simulated relative error of the
1npoint, and????kis the average relative error.p?1??is defined as the
nt?1prediction accuracy, which is displayed in percentage.
S1??(?i??)2n?1(0),????i n(0)?(x(0)(t)?x)2?x(0)(k)S2?，x?
n?1nC?S1 S2Where S1is the mean variance of residual；S2 is the mean variance of original series;Cis the posterior difference ratio.
Here is a reference table attached that illustrates the model accuracy
Team #52766 Page 9of 42
classification in details:
Table 2: The accuracy of the model
Accuracyof model Relative error/% Averagerelative accuracy (p, %) First class Second class Third class 1 5 10 Value of C ?95 80?p?95 70?p?80 ?0.35 0.35?C?0.5 0.5?C?0.65 ?70 ?0.65 disqualification 20 Thus, the method of predicting water resource amount in one place and checking the accuracy could be gained.
5.3Model of water supply capacity of a region
We define water scarcity F(t) as:
2?????at0at2?F?t??Y?t??W?N0?exp???bt0????exp?bt???B?t??C?t??D?t?22??????
In order to estimate the situation of the region[3], we lead in variable water lacking rate u:
u?F(t)*100%E(t)
For measuring the degree of water lacking in this region, we set 4th level
evaluation on the basis of water shortage rateuthe standard see table 3:
Table 3: The classificatory standard
Scarce situation Sufficient Light Middle Heavy shortage rate <5 5~10 10~20 >20 Main problems Achieves the balance between supply and demand of water resources Normal years in balance, but in the average year, the water supply may occur. Not only in design level year of water shortage occurs, also in normal times Water supply gap is extremely serious. Wading activities are severely restricted