A comparison between Bayes discriminant analysis and logistic regression for prediction of debris flow in southwest Sichuan, China

Geomorphology 201 (2013) 45–51

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A comparison between Bayes discriminant analysis and logistic regression for prediction of debris flow in southwest Sichuan, China Wenbo Xu a,⁎, Shaocai Jing a, Wenjuan Yu a, Zhaoxian Wang a, Guoping Zhang b, Jianxi Huang c a b c

School of Resources and Environment, University of Electronic Science and Technology of China, 611731 Chengdu, Sichuan Province, China Public Weather Service Center of China Meteorological Administration, 100081 Beijing, China College of Information and Electrical Engineering, China Agricultural University, 100083 Beijing, China

a r t i c l e

i n f o

Article history: Received 19 January 2013 Received in revised form 4 June 2013 Accepted 7 June 2013 Available online 15 June 2013 Keywords: Debris flow Rainfall and environmental factors Prior probability Bayes discriminant analysis Logistic regression

a b s t r a c t In this study, the high risk areas of Sichuan Province with debris flow, Panzhihua and Liangshan Yi Autonomous Prefecture, were taken as the studied areas. By using rainfall and environmental factors as the predictors and based on the different prior probability combinations of debris flows, the prediction of debris flows was compared in the areas with statistical methods: logistic regression (LR) and Bayes discriminant analysis (BDA). The results through the comprehensive analysis show that (a) with the mid-range scale prior probability, the overall predicting accuracy of BDA is higher than those of LR; (b) with equal and extreme prior probabilities, the overall predicting accuracy of LR is higher than those of BDA; (c) the regional predicting models of debris flows with rainfall factors only have worse performance than those introduced environmental factors, and the predicting accuracies of occurrence and nonoccurrence of debris flows have been changed in the opposite direction as the supplemented information. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Debris flow is a sudden natural disaster specifically occurring in mountain areas, with strong carrying, lashing, and burying abilities and intense destructiveness, and has become a huge threat to the security of human life and property and an obstacle to economic development (Ma, 2010). The occurrences of debris flows owe to the interaction of geology, topography, geomorphology, hydrology, weather, and other natural factors, which can be divided into two groups: one is rainfall, which directly triggers the occurrence of debris flow, and the other is the environmental factors that are the basic conditions of the occurrence of debris flow. This disaster has caught unprecedented attention in the world; lots of researchers are continuously carrying out relevant research, mainly focusing on its prediction. In the earlier debris flow prediction, most prediction models were built by means of the investigation of relationship between rainfall and debris flow on the basis of the processing of rainfall data (Kenneth, 1987; Tan and Han, 1992; Chen et al., 2007; Shieh et al., 2009). With the deep-going research of debris flow prediction and the innovative development of data obtaining technologies, environmental factors are paid more attention. These environmental factors are comprehensively analyzed in order to conduct debris flow susceptibility evaluation and risk zoning (Lee, 2005; Liu, 2006; Pradhan and Lee, 2007; Pradhan, 2010). And these factors—along with rainfall factors—are used as independent predictor variables for debris flow ⁎ Corresponding author. Tel./fax: +86 28 61831571. E-mail address: [email protected] (W. Xu). 0169-555X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.geomorph.2013.06.003

forecasting; that is, the forecasting model is established containing these two kinds of parameters (Jomelli et al., 2003; Ohlmacher and Davis, 2003; Rupert et al., 2008). Therefore, considering the environmental factors for the prediction of debris flow is necessary. Recently, the prediction of debris flow mainly focuses on two aspects: mechanical prediction based on the disaster formation mechanism (Liu, 2002) and quantitative prediction based on mathematical statistics. The quantitative prediction models are usually applied in the research of regional debris flow, and then this kind of prediction can be divided into probabilistic prediction and deterministic prediction according to the predicting results. Probabilistic prediction is represented by the logistic regression (LR) model, which has been used to build a model based on the combination of various rainfall and environmental factors (Ohlmacher and Davis, 2003; Rupert et al., 2008). Deterministic prediction is represented by the Bayes discriminant analysis (BDA); Spiegelhalter (1986) applied the Bayes formula to build spatial forecasting models at an earlier time, and later this method was used for the prediction of debris flows and landslides (Leclerc, 1994; Graf et al., 2009). The multivariate statistical methods, BDA and LR, are widely used for analysis of data in event classification. Many researchers have used two classification methods in various practical fields (Maja et al., 2004; Alkarkhi and Easa, 2008), especially in health sciences and clinical psychology (Payne et al., 1998; Udris et al., 2001). The LR is a form of regression and uses the logit transformation to calculate the ratio of probability by using probability outcome divided by probability without it and to predict the probabilities of group memberships in relation to several variables (Worth and Cronin, 2003). Bayes discriminant analysis is derived from the linear discriminant analysis (LDA),

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W. Xu et al. / Geomorphology 201 (2013) 45–51

which distinguishes new samples and classifies them into known groups (Fan and Mei, 2002). The two methods have different basic ideas; in the whole, BDA is usually used to simulate the linear relationships between the independent variables and dependent variables under the assumptions of multivariate normality and equal covariance, while LR simulates the nonlinear relationship and makes no such assumptions (Lei and Koehly, 2003). In general, BDA will give better results when these assumptions are met, but in other cases LR will be more suitable (Efron, 1975; Harrell and Lee, 1985). However, which of these two methods will be selected is more relevant to the actual statistical application field than to the assumption satisfaction. In practice, the assumptions are nearly violated, therefore doing continuous simulation experiments to find the more suitable method is necessary (Maja et al., 2004). In addition, prior probabilities, which are the proportions of group members that exist in the populations, also affect the classification results of BDA and LR. For instance, Fan and Wang (1999) and Lei and Koehly (2003) compared the classification error rates of LDA and LR by using the Monte Carlo simulation under different prior probabilities in the binary cases. Consequently, both methods are very applicable in debris flow prediction and worthy of study. In view of the previous research, the feasibility of modeling based on the occurrence mechanism is low; and the mathematical statistical model still occupies an important position in the prediction of regional debris flow. Nevertheless, BDA is less used than LR for debris flow prediction, and the studies of comparison between the two are much rarer. On the mastery of both theoretical methods, the main objective of this study is to compare the performances of BDA and LR to predict debris flow with different combinations of debris flow prior

probabilities in terms of the historical debris flow data in the period 1981–2000, including rainfall and environmental background data. 2. Study areas and data source 2.1. Study areas According to the susceptibility of debris flows in Sichuan Province (Xu et al., 2013), Panzhihua and Liangshan Yi Autonomous Prefecture belonging to the debris-flow-prone areas, are located in the southwest of Sichuan and are bounded by longitudes of 100°15′ E. and 103°53′ E., latitudes of 26°03′ N. and 29°27′ N (Fig. 1). Obvious dry and wet seasons are to provide concentrated rainfall. Elevation has significant differences, a low-lying West High East; and the complex topography is mainly mountainous. The geological structure is also complicated, with staggered fault zones and seismic belts. Hence, debris flow is easy to outbreak in the regions. 2.2. Data sources Firstly, the corresponding debris flow material, in the period 1981– 2000, were extracted from the China Institute of Geo-Environmental Monitoring, which contain attributes about event time and accurate positioning information like latitude and longitude. And these data were based on a day as the event unit. The count was 129. In order to meet the model condition (which is that building the models needs occurrence and nonoccurrence of debris flow), the precipitation records of meteorological stations that were from the nearest disaster points

Fig. 1. Map of distribution of Liangshan Yi Autonomous Prefecture and Panzhihua Administrative Region, debris flow (1981–2000) and DEM.

W. Xu et al. / Geomorphology 201 (2013) 45–51

were chosen as the records that the debris flow did not happen as a result of the limitation of economic and natural conditions. Secondly, the daily rainfall data were collected from the China Meteorological Data Sharing Service System, and the count is 54. In the study areas, these data came from 24 meteorological stations during the period from 1981 to 2000. Two analysis predictors were the intraday rainfall that was obtained by spatial interpolation, and the 10-day previous effective rainfall. Thirdly, considering the environmental factors fully in accordance with the actual situation of the study areas, availability, reliability, and economy, is required. For large-scale studied areas such as provinces or cities, by reference to Soeter's research (Soeters and van Westen, 1996), elevation, slope, aspect, flow accumulation, vegetation coverage, soil types, and land use were selected as the environmental background predictors. In this study, the environmental data are divided into the following: the topographic data such as elevation, slope, and aspects and the hydrology data flow accumulation (which were derived from SRTM-DEM with spatial resolution of 90 × 90 m); vegetation coverage was represented by the NDVI index (which was extracted from the AVHRR data product in the period of 1981–2000); the land use data (which came from the Sichuan Environmental Monitoring Centre in the scale of 1:100,000); and the soil types data (which came from the soil thematic image sublibrary of China in the scale of 1:1,000,000). And then the qualitative data were turned into quantitative data. 3. Methods The computational process is as shown in Fig. 2. First, the relevant predictors were preprocessed: daily rainfall was obtained by the co-kriging spatial interpolation method considering the environmental factors; and unified projection and quantitative processing of the environmental factor data layer were performed. Then the predicting factors were divided into two groups: rainfall-oriented factors and rainfall- and environmental-oriented factors. Second, the processing of the training samples was organized by eliminating the missing data samples and multiple collinear samples. In reference to the research of Fan and Wang (1999) and Lei and Koehly (2003), three combinations about the prior probabilities of occurrence and

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nonoccurrence of debris flow are equal prior probability (0.5:0.5), mid-range prior probabilities (0.67:0.33 or 0.75:0.25), and extreme prior probability (0.9:0.1). The LR and BDA methods were to model for predicting under these prior probability combinations. Finally, the results were discussed by model test, accuracy comparisons and receiver operating characteristic curve (ROC). 3.1. Spatial interpolation Because debris flows mostly occurred in remote areas with complex terrain and the limited number of meteorological stations, the daily rainfall data is difficult to obtain. Thus, in practice, spatial interpolation methods are used to interpolate the stations in the adjacent areas to obtain the daily rainfall data (Tao et al., 2009). The co-kriging (as one of them, with relevant environmental factors introduced, such as elevation, slope, and aspect) is a method based on variogram theory and structural analysis and is used to perform unbiased optimal estimation with regionalized variables in a limited area — it is a part of the major content of geostatistics. The corresponding intraday rainfall of disaster sites was extracted from the above interpolation. And then, effective antecedent rainfall was calculated by a formula that is extensively used (Senoo, 1985): n

Ra ¼ ∑i¼1 K  Ri

ð1Þ

where Ra is the effective rainfall of previous n days; Ri is the rainfall of the i-th day before debris flow occurs, and K is the attenuation coefficient. The value of attenuation coefficient K is usually an empirical value, which lacks practical relevance. The attenuation coefficient is calculated by Eqs. (2) and (3) (proposed by Xu et al., 2012), the effective rainfall coefficients of short-time-heavy rainfall and long-timelight rainfall, respectively; x indicates the number of days; f(x) is the attenuation coefficient of x-th. f ðxÞ ¼ 50:8351  expð−1:0093xÞ

ð2Þ

f ðxÞ ¼ 30:6558  expð−0:4131xÞ:

ð3Þ

3.2. Bayes discriminant analysis Daily rainfall data set

Environment background data set

The general form of linear discriminant function is Y ¼ C1 X1 þ C2 X2 þ ⋯ þ Cm Xm þ C0

Spatial interpolation

Intraday rainfall

Quantitative processing

10-day previous effective rainfall

Different combinations of debris flow prior probabilities

Bayes discriminant analysis

Logistic regression

Model test

where Y is the discriminant index, which may be probability or coordinate value or score value in different situations; X1, X2,⋯, Xm are the variables of the research object; C0 is the constant; and C1,C2,⋯,Cm are the relevant discriminant coefficients. Bayes discriminant analysis (BDA) is a kind of commonly used classification method based on linear discriminant analysis ideas, and under the premise of considering the prior probability, Bayes formula is used to build each category of discriminant function in accordance with certain criteria. Bayes discriminant function is as follows (Zheng et al., 2009): 8 qj f j ðX Þ > > > P ðj=X Þ ¼ j ¼ 1; ⋯; k > > k > X < qi f i ðX Þ > i¼1  >  −1=2    > > 1  −1 −p=2  > > ⋅ exp − X−μ ðjÞ ′∑ðjÞ X−μ ðjÞ ΣðjÞ  : f j ¼ ð2πÞ 2 j

Accuracy comparison

Fig. 2. Flow chart of computational process.

ð4Þ

ð5Þ

qj f j ðX Þ → max

ð6Þ

P ðjjX Þ ¼ ln qj þ C 0ðjÞ þ C jðjÞ X i

ð7Þ

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W. Xu et al. / Geomorphology 201 (2013) 45–51

Table 1 Wilks' λ statisticsa.

Table 3 Prediction accuracy of equal prior probabilitya.

Prior probability

Functions

Wilks' λ

Ndf

Sig.

Prior probability

Functions

POverall

Plarge-group

Psmall-group

0.5:0.5

FBDA1 FBDA2 FBDA1 FBDA2 FBDA1 FBDA2 FBDA1 FBDA2

0.724 0.952 0.669 0.824 0.741 0.887 0.885 0.941

9 2 9 2 9 2 9 2

1.020E-009 0.008 1.966E-012 1.126E-008 1.397E-007 4.096E-005 0.007 0.003

0.5:0.5

FBDA1 FLR1 FBDA2 FLR2

67.5% 75.8% 56.2% 56.7%

62.9% 75.3% 68.0% 68.0%

72.2% 76.3% 44.3% 45.4%

0.67:0.33 0.75:0.25 0.9:0.1

a BDA1 is a Bayes discriminant analysis model with rainfall and environment as predictors; BDA2 is a Bayes discriminant analysis model with rainfall as predictors; df is degree of freedom that is the number of independent variables in the model.

−1 where, qj = nj/n, C0(j) = − (1/2)(μ(j))T ∑ −1 (j) μ(j), Cj(j) = ∑ (j) μ(j), P is the posterior probability, fj(X) is the probability density function, qj is the prior probability of the j-th population, n is the total sample size, nj is the sample size of the j-th population, μ(j) is the mean vector of the j-th population, and ∑ (j) is the covariance matrix of the j-th population. The goal of BDA is to distinguish the presence or absence of debris flow, a binary event (Y = 1 indicates the debris-flow event occurs; Y = 0 indicates the debris-flow event does not occur; Y is a binary outcome variable), according to a set of independent predictor variables such as daily rainfall, topography, geomorphology, and so on. The classification rule of BDA used in this study is the following: Xi is classified into the presence of debris flow, if P(Y = 1|Xi) > P(Y = 0|Xi); otherwise, Xi is classified into the absence of debris flow, which is the maximum posterior probability rule.

3.3. Logistic regression We know from the above recommendation of BDA that the BDA is a part of the general linear model (Fan, 1996), but LR is not because it models the nonlinear probabilistic function (Neter et al., 1989). Given a binary outcome variable Y, with a set of independent predictor variables, the probability of an event is calculated by the logistic function:  logitðP Þ ¼ ln

P 1−P

 ¼ β0 þ β1 X 1 þ β2 X 2 þ ⋯ þ βm X m

ð8Þ

B

P¼

e 1 þ eB

ð9Þ

B ¼ β0 þ β1 X 1 þ β2 X 2 þ ⋯ þ βm X m

ð10Þ

where P is the probability of an event occurrence; X1,X2,⋯,Xm are independent variables; β0 is the constant of the equation; and β1,β2,⋯,βm are the coefficients of independent variables. In addition, maximum likelihood estimation is usually used to estimate these parameters. Table 2 Logistic regression model statisticsa. Prior probability 0.5:0.5 0.67:0.33 0.75:0.25 0.9:0.1 a

Functions

Hosmer and Lemeshow Sig.

FLR1 FLR2 FLR1 FLR2 FLR1 FLR2 FLR1 FLR2

0.305 0.007 0.944 0.348 0.983 0.572 0.899 0.768

a Overall, large group and small group are representative of the overall prediction accuracy, the prediction accuracies of occurrence and nonoccurrence of debris flow in samples, respectively.

As discussed earlier, the classification criterion of BDA is the maximum posterior probability rule. The classification rule in LR is deeB scribed in the present study: if P ðY ¼ 1jX i Þ ¼ 1þe B > 0:5 then Xi is classified into the presence of debris flow; otherwise, Xi is classified into the absence of debris flow. 4. Results and discussion 4.1. Prediction model results 4.1.1. BDA model test In the practical application, the regional debris flow data do not fulfill the assumptions of multivariate normal distribution and equal covariance. Hence, Wilks' λ statistics were chosen to test the validity of discriminant functions of BDA. The value of Wilks' λ should be as small as possible when variables have a more significant impact on the model; significant parameter value (Sig.) is b 0.05, illustrating that discriminant functions are effective to carry out discriminant analysis. The results of Wilks' λ statistics are shown in Table 1. In this table, Wilks' λ statistics of BDA, taking the rainfall and environment factors as predictors, are smaller than those of BDA considering the rainfall factors as predictors only, indicating that the variables of the former have a more significant influence than that of the latter; their Sig. values are b 0.05, stating that both Bayes discriminant functions are effective. 4.1.2. LR model test The goodness of fit of the logistic regression model is usually assessed by coefficients of determination R2 (Cox and Snell R2 (1968) and Nagelkerke R2 (1991)) or the Hosmer–Lemeshow statistic (1989): the former reflects the variation percentage of dependent variables explained using all the independent variables in the model, and the closer its value is to 1, the better it is; the latter is used for the significant test of the regression model, and a significant value is > 0.05, which manifests that no significant difference exists in the observed frequency and the expected frequency obtained by prediction probability, namely the regression model is satisfactory. Logistic regression statistics are shown in Table 2.

Table 4 Prediction accuracy of mid-range prior probabilitiesa. Cox and Snell R2

Nagelkerke R2

Prior probability 0.67:0.33

0.369 0.050 0.445 0.260 0.369 0.183 0.202 0.105

0.491 0.066 0.620 0.362 0.548 0.272 0.413 0.215

LR1 is a logistic regression model with rainfall and environment as predictors; LR2 is a logistic regression model with rainfall as predictors.

0.75:0.25

Functions BDA1

F FLR1 FBDA2 FLR2 FBDA1 FLR1 FBDA2 FLR2

POverall

Plarge-group

Psmall-group

83.9% 80.7% 74.5% 69.8% 83.6% 83.6% 78.9% 78.4%

95.3% 88.7% 96.9% 84.5% 96.9% 92.2% 99.2% 97.7%

60.3% 65.1% 28.6% 39.7% 42.9% 57.1% 16.7% 19.0%

a Overall, large group and small group are representative of the overall prediction accuracy, the prediction accuracies of occurrence and nonoccurrence of debris flow in samples, respectively.

W. Xu et al. / Geomorphology 201 (2013) 45–51 Table 5 Prediction accuracy of extreme prior probabilitya. Prior probability

Functions

POverall

Plarge-group

Psmall-group

0.9:0.1

FBDA1 FLR1 FBDA2 FLR2

90.5% 91.1% 90.0% 90.5%

100.0% 98.8% 100.0% 100.0%

10.0% 25.0% 5.0% 5.3%

a Overall, large group and small group are representative of the overall prediction accuracy, the prediction accuracies of occurrence and nonoccurrence of debris flow in samples, respectively.

We see from Table 2 that in the goodness-of-fit testing of Hosmer and Lemeshow (1989), the significant value of the model considering rainfall only (priors 0.5:0.5) is b0.05, and the significant values of the rest are > 0.05, which indicate the models fit well; the values of two determination coefficients R2 also show that the LR models with rainfall and the environment factors as predictors are ideal relatively.

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4.1.3. Comparison between BDA and LR Owing to sample size limitations, this paper made use of their respective training samples to perform prediction accuracy tests, and the overall prediction accuracy is given priority, supplemented by the prediction accuracies of occurrence and nonoccurrence of debris flow (large group and small group). The results in training samples are shown in Tables 3, 4, and 5. And the corresponding receiver operating characteristic (ROC) plots of the training data were obtained to make the model capacities clear (Figs. 3, 4, and 5). According to Table 3, in the equal prior probability, the overall prediction accuracy of BDA is 7.3% lower than that of LR with rainfall and environment as predictors and is 0.5% lower than that of LR with rainfall only as predictors. In addition to the overall prediction accuracies, the area under curve of ROC was obtained (Fig. 3). What can be observed from Table 4, in the mid-range prior probability. The overall prediction accuracies of BDA are 3.2% higher (priors 0.67:0.33) and 0% higher (priors 0.75:0.25) than those of the LR based on rainfall and environment factors and are 4.7% higher (priors 0.67:0.33) and 0.5% higher (priors 0.75:0.25) based on rainfall factor. In addition to the overall prediction accuracies, the area under curve of ROC was obtained (Fig. 4). Table 5 shows that the overall prediction accuracy of BDA is 0.6% lower than that of LR in view of rainfall and environment factors and is 0.5% lower than that of LR in the extreme prior probability. In addition to the overall prediction accuracies, the area under curve of ROC was obtained (Fig. 5). From the Plarge-group and Psmall-group columns of three tables, prediction accuracies of the occurrence of debris flows gradually increase, while those of the nonoccurrence of debris flows gradually decrease. 4.2. Discussion

Fig. 3. ROC curve and AUC evaluation with respect to equal prior probability.

The above results show that BDA and LR have different performance under the various combinations of prior probability and predictors in the study area. In general, the more the predicting factors are in the model, the higher the model's accuracy is. Therefore, the prediction models introduced environmental factors on the basis of rainfall have varying degrees of improvement in predicting accuracy. As far as the two models are concerned, certain conditions have added to the uncertainty of their results. The same condition is that both Bayes discriminant and logistic regression need independent

Fig. 4. ROC curve and AUC evaluation with respect to mid-range prior probabilities: (a) is 0.67:0.33; (b) is 0.75:0.25.

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W. Xu et al. / Geomorphology 201 (2013) 45–51

discriminant analysis and logistic regression for researchers interested in debris flow prediction. Acknowledgments This study is supported by the National Natural Science Foundation of China (Grant Nos. 40971016 and 11102124), the important National Science and Technology Specific Projects of China (No.2012ZX10004-901001), the Program for New Century Excellent Talents in University, Ministry of Education of China (NCET-10-0604), the Provincial Key Science and Technology Research and Development Program of Sichuan, China (2013SZ0002), and the Fundamental Research Funds for the Central Universities (Grant Nos. ZYGX2012J152, A03008023401011, and A03008023401021). References

Fig. 5. ROC curve and AUC evaluation with respect to extreme prior probability.

variables to build the prediction models. The difference is that Bayes requires some variable assumptions; but these assumptions are difficult to be completely met in practice, which does not necessarily indicate that Bayes discriminant is poorer than logistic regression or vice versa, that all the assumptions fulfilled does not mean that Bayes is necessarily better than logistic regression. The selection of methods should be analyzed and discussed at length according to the actual situation. In this study, none of Bayes discriminant's assumptions are met; however, through Wilks' λ statistic, Bayes discriminant functions can still effectively be used for the forecast. The major disadvantage of the logistic regression model is the estimation method of model parameters: maximum likelihood estimation, which always requires large enough samples for the iterative calculation to obtain stable model parameters. In the study, the prior probabilities have some effect on relationship equations of BDA and LR. With equal and extreme prior probability, the relations among the predictors tend toward the nonlinear model (LR). When the prior probability is mid-range, the relations are approximately linear (BDA) with the priors of 0.67 and 0.33; with the priors of 0.75 and 0.25, BDA and LR have little difference; and these priors may be in the boundary line of mid-range priors and extreme priors. Hence, the prior probabilities impact the prediction accuracies. For the occurrence and nonoccurrence of debris flow, both accuracies change in opposite directions and their difference also becomes bigger and bigger, which are as supplementary information of model evaluation. In addition, the quality and robustness of selected sample data are both very important, and the errors existing in the data processing and system error will all affect the accuracy of results.

5. Conclusions Aimed at the Panzhihua and Liangshan Yi Autonomous Prefecture areas, this study compared Bayes discriminant analysis and logistic regression for predicting debris flow with different priors and different predictors. Through the comprehensive consideration the results indicated that Bayes discriminant analysis had good prediction with the mid-range prior probability while logistic regression had good prediction with equal and extreme prior probability, based on rainfall factors or the combination of rainfall and environmental factors. This paper provides a reference about the statistical model of Bayes

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