## 2.1 Overview of the study area

The study area is located in Longnan, Tianshui, Pingliang, Dingxi, and Qingyang in southeast Gansu Province. The specific location is N32.551°–37.072°, E103.363°–108.705°. As shown in Fig. 1, the study area is part of the Gansu Qinghai region, which is one of the 9 endemic regions in the northwest extreme of North China. The area has a complex terrain, cool and rainy summers, large temperature differences between day and night, and large amounts of rain and dew. The wheat fields with different altitude gradually mature from the end of June to the end of September. The self-produced wheat seedlings as well as late mature winter and spring wheat overlap and coexist, which greatly facilitates the transmission of individual pathogens. The most bacteria are present in summer, and the study area has become the main base for the pathogens of winter wheat autumn seedlings in North China and Northwest China.

Please put Fig. 1 about here

## 2.3 Data acquisition and processing

The canopy hyperspectral data collection process and disease index investigation were each performed in the flag picking stage, early filling stage, and middle filling stage.

## 2.3.1 Canopy hyperspectral data acquisition and processing

An ASD FieldSpec Pro spectrometer was used to collect the hyperspectral data of the canopy. The measurement time was 10:30 − 14:30 Beijing time. The weather was clear and cloudless. During the observation period, the probe of the spectrometer was place vertically downward and always maintained a distance of 1.6 m from the ground. The field angle of the probe was 25° and the field of view of the ground was 50 cm in diameter. The crop canopy reflectance was measured and averaged 20 times in each plot. Before and after the measurement of each cell, the reference plate was calibrated.

## 2.3.2 Disease index survey

After symptoms occurred, the disease index was investigated immediately after the canopy hyperspectral data were measured every 10 days. The five points method was used to investigate the disease index. Five symmetrical investigation points were selected for each plot; each point covered an area of approximately 2 m2, and 20 wheat plants were selected for each point to investigate the disease degree. The degree of occurrence was expressed by severity, i.e., the relative percentage of the area occupied by Fusarium on the diseased leaves and the total area of leaves. The severity was divided into 8 gradients, i.e., 1%, 5%, 10%, 20%, 40%, 60%, 80%, and 100%. The number of wheat leaves for each severity level was recorded, and the disease index (DI) was calculated according to Eq. (1).

\({\text{DI}}=\frac{{\sum {(x \times f)} }}{{n \times \sum f }} \times 100\) % (1)

where *x* is the extreme value of each gradient, *n* is the highest gradient value, and f is the number of blades of each gradient.

## 2.3.3 Meteorological data collection and processing

The daily precipitation, relative humidity, temperature, and average wind speed from March 1 to July 15 of each year were analysed as meteorological data provided by the National Meteorological Centre. The daily meteorological data was interpolated into weekly data, the relative humidity and temperature were taken as the average value, and the precipitation data was taken as the weekly total value.

The daily wind speed data was used for the calculation of the wind direction influence value. The influence value of the wind direction is defined as the cosine value of the angle between the local wind direction and other two directions. The wind direction influence value of one place to other two places is equal to the product of wind speed of one place and the wind direction influence value of other two places., The wind direction influence values of Longnan on Tianshui, Dingxi, Pingliang and Qingyang were calculated according to the time sequence and wind speed of the initial occurrence of the disease in the study area.

## 2.4 Methods

## 2.4.1 Calculation of spectral sensitivity

Kobayashi et al. (2001) proposed the concept of spectral sensitivity to analyse the different spectral responses disease index plants under ear neck blast stress; this was done to more easily compare the spectral curves of different disease indexes. When the spectral sensitivity is positive in a certain band, the spectral reflectance of the stressed plant is higher than that of the normal plant, and the higher the spectral sensitivity, the more significant the difference between the spectral reflectance of the stressed plant and that of the normal plant, and vice versa. The calculation method of spectral sensitivity is shown in Eq. (2):

$${S_{\text{S}}}=\frac{{{S_{\text{T}}} - {S_{\text{N}}}}}{{{S_{\text{N}}}}}$$

2

where *S*s represents spectral sensitivity; *S*T represents the spectral reflectance of stressed plants, and *S*N represents the spectral reflectance of normal plants.

## 2.4.2 Removal of canopy spectral continuum

Continuum removal, also known as envelope removal, uses the actual spectral band value divided by the corresponding band value on the continuum. After continuous removal of the original spectral curve, the important characteristics of the responses of various groups in the leaf tissue structure, pigment content, water, and protein to the reflection spectrum can be obtained. These absorption characteristic parameters mainly include the total area “a” of the absorption peak, the area “*A*1” of the left end of the absorption peak, the absorption depth “*D*,” and the symmetry degree “*S*” (Zheng et al.,2019).

## 2.4.3 Leave-one-out cross validation

Leave-one-out cross validation (LOOCV) can avoid the problem of having to choose between modelling samples and validation samples in an experimental design. Every sample participates in the test and training, which can address the drawbacks of using training samples with low richness. LOOCV is an effective method to evaluate the generalization ability and reliability of a regression model. Its basic principle is to assume that there are *N* samples, with each sample is regarded as a test set, and the rest of the samples (*N*-1) are regarded as a training set, which are cycled *N* times.

## 2.4.4 Adaptive network fuzzy reasoning

An adaptive network fuzzy reasoning system, also known as an adaptive network-based fuzzy inference system (ANFIS), is a type of fuzzy inference system that integrates neural network adaptability(Zhang et al.,2019). This system synthesizes the neural network learning algorithm and the concise form of fuzzy reasoning; it also generates a numerical solution by learning the training data group. The structure of ANFIS includes five layers, as shown in Fig. 2. The inference system has two inputs of *X* and *Y*, the output is *f*, and each node in the same layer has similar functions. *O*1,i is used to represent the output of the *i* node in the first layer, and so on.

Please put Fig. 2 about here

As shown in Fig. 2, the first layer represents the selection and fuzzification of input parameters, which is the first step in establishing fuzzy rules. Each node *i* in this layer is a square node represented by a node function.

\({O_{1,i}}={\mu _{Ai}}(x),i=1,2\;\;{O_{1,i}}={\mu _{B(i - 2)}}(y),i=3,4\)

*O* 1,i represents the membership function of fuzzy set A. The second layer represents the calculation of the excitation intensity of the fuzzy rule, multiplying the membership degree of the input signal; the output is:

\({O_{2,i}}={\omega _i}={\mu _{Ai}}(x){\mu _{Bi}}(y),i=1,2\)

The second layer is the calculation of the fuzzy rule excitation intensity, which multiplies the membership degree of the input signal, and the output result is as follows,

\({O_{{\text{2,}}i}}={\omega _i}={\mu _{{\text{A}}i}}(x){\mu _{{\text{B}}i}}(y)\;\;i=1,2\)

The third layer is the normalization calculation of the applicability of each rule of the node in this layer, i.e.,, the ratio of the sum of rule \({\omega _i}\) of rule I and all rule \(\omega\) of node *i* is calculated as follows:

\({O_{3,i}}=\frac{{{\omega _i}}}{{{\omega _1}+{\omega _2}}}\;\;\;i=1,2\)

The fourth layer indicates that each node *i* of this layer is an adaptive node, and its output is as follows:

\({O_{4,i}}={\bar {\omega }_i}({p_i}x+{q_i}y+{r_i})\;\;i=1,2\)

The fifth layer indicates that the single node of this layer is a fixed node, and calculates the total output of all input signals; specifically:

\({O_{5,i}}=\frac{{\sum {{\omega _i}{f_i}} }}{{\sum {{\omega _i}} }}\;\;i=1,2\)

## 2.4.5 Grey relational analysis

Grey relational analysis (GRA) is a type of grey system analysis method. According to the similarity or dissimilarity degree of development trend among factors, i.e. ‘grey relational degree’, used as a method to measure the degree of correlation among factors, GRA quantitatively describes the change trend between data(Yu et al.,2019). If the change trend in the two datasets is similar, the correlation between the data is considered high; conversely, non-similar trends indicate that the correlation between the data is low.

## 2.4.6 Partial least squares regression

Partial least squares regression (PLSR) is based on the idea of principal component regression. PLSR requires the linear function of the original independent variables \({x_1},{x_2}, \cdots ,{x_n}\); concurrently, considering its correlation with the corresponding variables, the regression selects the linear function of \({x_1},{x_2}, \cdots ,{x_n}\) as a new independent variable and the corresponding variables for regression(Jiang et al., 2003). This linear function is strongly correlated with the corresponding variables and is easy to calculate. The algorithm is based on the least squares method. In the algorithm, only the variables related to the strain are selected; not all linear functions of \({x_1},{x_2}, \cdots ,{x_n}\) are considered, and only the parts related to the strain are considered.

## 2.4.7 Akaike Information Criterion

Akaike information criterion (AIC) is an index used to measure the fitting performance of a statistical model(Xu et al.,2018), and its calculation method is shown in Eq. (3):

$${\text{AIC}}=2k - 2\ln L$$

3

where *k* is the number of model parameters and *L* is the likelihood function. The smaller AIC, the stronger the fitting effect.

## 2.4.8 Vegetation index construction

In this study, the ratio vegetation index (RVI), normalized vegetation index (NDVI), difference vegetation index (DVI), triangle vegetation index (TVI), and renormalization vegetation index (RDVI) are used to build the disease monitoring and prediction model. The calculation equation of each vegetation index is shown in Table 1(Mistele et al., 2010; Kelly et al., 2006; Klein et al., 1972; Xue et al., 2004;)

Table 1

Calculation method of vegetation index

Name | Calculation formula |

RVI | RVI=(NIR)/*R* |

NDVI | NDVI=(NIR-*R*)/(NIR + *R*) |

DVI | DVI = NIR-*R* |

TVI | TVI=(0.5)((120)(NIR-*G*)-200 (*R*-*G*)) |

Please put Table 1 about here

## 2.4.9 Correlation analysis

Correlation analysis is used to analyse the change trend of two or more groups of data for consistency to obtain whether the relationship between them is close as well as its degree; this approach often uses correlation coefficients for discriminant analysis. The correlation coefficient is obtained by dividing the covariance of two random variables by the standard deviation. If the correlation coefficient is between − 1 and 1, the greater the absolute value and the greater the correlation degree; however, when the coefficient is close to 0, this indicates that there is no correlation. The calculation method is shown in formula (4):

$${\rho _{X,Y}}=\frac{{\operatorname{cov} (X,Y)}}{{{\sigma _X}{\sigma _Y}}}=\frac{{E(XY) - E(X)E(Y)}}{{\sqrt {E({X^2}) - {E^2}(X)} \sqrt {E({Y^2}) - {E^2}(Y)} }}$$

4

where \({\rho _{X,Y}}\) represent the correlation coefficient, while \(\operatorname{cov} (X,Y)\) and \(\sigma\) represent the covariance and standard deviation, respectively.

## 2.4.10 Model accuracy evaluation

The coefficient of determination (*R*2) and root mean square error (RMSE) were selected as the model accuracy evaluation index. The calculation method is shown in equations (5) and (6).

$${R^2}=\frac{{(\sum\nolimits_{{i=1}}^{n} {{y_i} - \bar {y}{)^2}} }}{{(\sum\nolimits_{{i=1}}^{n} {{x_i} - \bar {y}{)^2}} }}$$

5

$${\text{RMSE}}=\sqrt {\frac{{\sum\nolimits_{{i=1,j=1}}^{n} {{{({x_i} - {y_i})}^2}} }}{n}}$$

6

where \({x_i}\)is the measured value of the sample, \({y_i}\)is the estimated value of the sample, \(\bar {y}\)is the mean value, and n is the number of samples. In general, the larger *R*2 is, the smaller RMSE is, indicating the superior fitting effect of the model.