Introduction

As the strongest greenhouse gas in the atmosphere and the main component of the water cycle, the greenhouse contribution of water vapor is several times that of carbon dioxide (Jones et al. 2007). It in the lower troposphere is the main source of precipitation for all weather systems. Water vapor absorbs radiation through the formation and evolution of clouds, and affects the changes of other variables in the climate system (Nian et al. 2018). Relative Humidity (RH) refers to the ratio of the maximum amount of water vapor in the atmosphere to the amount of water vapor that the air can contain at a certain temperature (Xie et al. 2011)). It is a physical quantity measuring atmospheric water vapor content. Predicting RH is of great importance in weather, climate, industrial production, crops, human health, and disease transmission, since it is helpful in making critical decisions. RH plays a vital role in driving electricity demand during the warm months (June–September) (Xie et al. 2018). Negative temperature and high RH are important conditions in the prediction of aircraft icing area (Ivanova 2009). The study of Duan et al. (2019) demonstrated that the encountering high and low RH, the daily allergic rhinitis outpatients increased. Humans are more susceptible to respiratory novel coronavirus (COVID-19) when the RH decreases (Mangla et al. 2021). In crops, RH is crucial in regulating root hydraulic characteristics (Calvo-Polanco et al. 2017). Models for dust storm predicting may be improved by utilizing RH and wind speed as main drivers for dust generation and transport (Csavina et al. 2014). Kwon et al. (2019) uses public weather forecast information about temperature, RH, dew point, and sky coverage as a training set in the naive Bayes classifier classification of hourly resolution for global horizontal irradiance prediction.

Quite a few methods have been utilized to predict RH. Yu (2009) used correlation analysis with the index station and RH reference value for predicting precipitation with RH. In addition, Lu and Viljanen (2009) used external input nonlinear autoregressive (NNARX) model and genetic algorithm to establish a neural network to achieve the purpose of prediction. Practice of Kuzugudenli (2018) has proved that the artificial neural network method had greater predictive power than the model developed with multiple linear regression. However, Tkacz (2001) has found that artificial neural networks are not able to improve on an autoregressive model. Although the regression model, the correlation analysis, and back propagation (BP) neural network method have their own advantages, such non-parametric methods have a great dependence on the choice of variables (Li et al. 2019b). In recent years, Long Short-Term Memory (LSTM) network has performed well in predicting meteorological variables with dynamic characteristics, such as temperature (TEMP), RH, and precipitation (PRCP) due to its special network (Gao et al. 2021; Hutapea et al. 2020; Casallas et al. 2021).

Since RH is a time series recorded at intervals of time, there may be a certain trend and periodicity between the series. Autoregressive moving average method (ARIMA) is one of the commonly used prediction methods in parametric methods (Eymen and Köylü 2019; Rathod et al. 2017; Fernández-González et al. 2016). For dealing with seasonal time series, such as RH, seasonal autoregressive integrated moving average (SARIMA) had a great effect for forecasting as shown by (Valipour 2015; Bas Cerdá et al. 2017; Fang and Lahdelma 2016; Qiu et al. 2021; Murthy et al. 2018; Cong et al. 2019; Shad et al. 2022).

To overcome the problems of non-stationary of the time series, Engle and Granger (1987) provides the cointegration theory. If there is a cointegration relationship between non-stationary time series, there will be no pseudo-regression problem. Cointegration theory does not require all sequences to be stable, only their regression residual sequence is stable. The cointegration model performs well in measuring the long-term equilibrium relationship of the series (Granger and Swanson 2010; Zhang et al. 2015; Abdi et al. 2022). While the error correction model (ECM) as a complementary model performs well in explaining the short-term fluctuation relationship of the series as indicated in (Li et al. 2013; Ma et al. 2015; Abdi et al. 2022). Meanwhile, some researchers started to introduce the cointegration theory into the meteorological field and also found many valuable results. Statistical analysis was performed on water level, temperature, and humidity using cointegrated vector autoregression models by Appiah (2017). Htet (2017) proposed the Airline Error Correction Model (AECM), and forecast CO using traffic, precipitation, and air temperature as extrinsic variables. A novel multi-step forecasting method of hourly PM2.5 concentration is proposed with ECM using for correcting the prediction error according to studies of Yin et al. (2021). However, it is relative rare in the relationship for RH. The purpose of this study was to combine SARIMA with cointegration theory to form the SARIMA-EG-ECM (SEE) model, and to use the SEE model to predict RH at the Agricultural Ecological Experimental Station of the Chinese Academy of Science. This paper utilizes the cointegration model on the basis of SARIMA to establish a dynamic model with air temperature, dew point temperature, precipitation, and other meteorological variables as covariates, and use the ECM model to discuss the effect of the current fluctuation of covariates on the fluctuation of RH. This paper will verify the performance of the SEE model by comparing SARIMA model (including the multiplicative seasonality model and the additive seasonality model), LSTM model, and SEE model.

Materials and methods

SARIMA model

The SARIMA method can be used to model series with seasonal effects and periodic fluctuations. According to the difficulty of extracting seasonal effects, it is divided into additive seasonality model and multiplicative seasonality model (Danhui 2019).

Additive seasonality model

In the additive seasonality model, the seasonal change \(S_{t}\), the trend \(T_{t}\), and the immediate \(I_{t}\) in the time series are in the additive relationship shown in the formula (1), namely

$$\begin{aligned} x_{t} =S_{t}+T_{t}+I_{t}. \end{aligned}$$
(1)

The series can be smoothed by the trend difference and the seasonal difference, and the smoothed series can be fitted by ARIMA model. The structure of the additive season model is

$$\begin{aligned} \nabla _{D}\nabla ^{d}x_{t}=\frac{\Theta \, (B)}{\Phi \,(B)}\varepsilon _{t}, \end{aligned}$$
(2)

where D is the step size of the seasonal period, d is the order of the difference, and \(\Theta (B)=1-\theta _1B-\cdots -\theta _qB^q\) is the q-order AR coefficient polynomial. \(\Phi (B)=1-\phi _1B-\cdots -\phi _pB^p\) is the MA coefficient polynomial of order p. \(\left\{ \varepsilon _{t}\right\}\) is a white noise series, and \(E(\varepsilon _t)=0\), \(Var(\varepsilon _t)=\sigma ^{2}_\varepsilon\).

Multiplicative seasonality model

Usually, the long-term trend effect, seasonal effect, and random fluctuation of time series are not easy to be separate like the previous subsection because of the complex interaction between them. At this time, the additive seasonality model cannot fully extract their interaction. The multiplicative seasonality model needs to be adopted. The construction principle of the multiplicative seasonal model is shown in Fig. 1. In fact, due to the multiplicative relationship between the short-term correlation of the series and the seasonal effect, the multiplicative seasonality model is the product of ARMA(p, q) and ARMA(P, Q), denoted as ARIMA\((p,d,q)\times (P,D,Q)_S\). The structure is

$$\begin{aligned} \nabla ^d\nabla ^D_Sx_t=\frac{\Theta \, (B)\Theta _S\,(B)}{\Phi \,(B)\Phi _{S}\,(B)}\,\varepsilon _t, \end{aligned}$$
(3)
Fig. 1
figure 1

Construction principle of the multiplicative seasonality model

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where \(\Theta (B)=1-\theta _1B-\cdots -\theta _qB^q\) is the non-seasonal q-order AR coefficient polynomial. \(\Phi (B)=1-\phi _1B-\cdots -\phi _pB^p\) is the non-seasonal MA coefficient polynomial of order p. \(\Theta _S(B)=1-\theta _1B^S-\cdots -\theta _QB^{QS}\) is the seasonal Q-order AR coefficient polynomial. \(\Phi _S(B)=1-\phi _1B^S-\cdots -\phi _PB^{PS}\) is the seasonal MA coefficient polynomial of order P.

Cointegration model

Some series themselves change unevenly, but there are close long-term equilibrium relationships between the series. Cointegration model can measure whether there are long-term equilibrium relationships between the series effectively. Assuming that series of independent variables are \(\left\{ x_1\right\} , \left\{ x_2\right\} , \cdots , \left\{ x_n\right\}\). And the series of response variable is \(\left\{ y_t \right\}\). We can construct a regression model

$$\begin{aligned} y_t=\beta _0+\sum _{i=1}^k\beta _ix_{it}. \end{aligned}$$
(4)

If the residual series \(\left\{ \varepsilon _t\right\}\) in the regression model is stationary, it is said that there is a cointegration relationship between the series of response variable \(\left\{ y_t \right\}\) and the series of independent variables \(\left\{ x_1\right\} , \left\{ x_2\right\} , \cdots , \left\{ x_n\right\} .\)

Error correction model

As a supplementary model of the cointegration model, ECM was originally proposed by Hendry and Anderson (1977), which can explain the short-term fluctuation relationship of the series.

If there is a cointegration relationship among the series of response variable \(\left\{ y_t \right\}\) and the series of independent variables \(\left\{ x_1\right\} , \left\{ x_2\right\} , \cdots ,\left\{ x_n\right\}\), that is

$$\begin{aligned} \begin{aligned}y_t=\beta x_t+\varepsilon _t,\\\varepsilon _t=y_t-\beta {x_t}{\sim }\,I(0). \end{aligned} \end{aligned}$$
(5)

According to Eq. (5), there is

$$\begin{aligned} y_t-y_{t-1}=\beta x_t-y_{t-1}+\varepsilon _t. \end{aligned}$$
(6)

Combine Eq. (6) with \(y_{t-1}=\beta x_{t-1}+\varepsilon _{t-1}\), there is

$$\begin{aligned} y_t-y_{t-1}=\beta x_t-\beta x_{t-1}+\varepsilon _{t-1}+\varepsilon _t. \end{aligned}$$
(7)

Let the least square estimate of \(\beta\) be \(\hat{\beta }\). Then, \(\hat{\varepsilon }_{t-1}=y_{t-1}-\hat{beta}_{t-1}\) stands for the error from the previous period, denoted as \(ECM_{t-1}\). Equation (7) can be written as

$$\begin{aligned} \nabla y_t=\beta \nabla x_t-ECM_{t-1}+\varepsilon _t. \end{aligned}$$
(8)

According to Eq. (8), there are three main types of short-term fluctuations that will influence the current fluctuations (\(\nabla y_t\)) of the response series. They are:

1. \(\nabla x_t\): Current fluctuation of the input series;

2. \(ECM_{t-1}\): Error from the previous period;

3. \(\varepsilon _t\): Random fluctuations in the current period.

In summary, the structure of the model is

$$\begin{aligned} \nabla y_t=\beta _0\nabla x_t+\beta _1 ECM_{t-1}+\varepsilon _t. \end{aligned}$$
(9)

Among them, \(\beta _1(\beta _1<0)\) is the coefficient of error correction, indicating the extent to which the error correction term can correct the current fluctuation.

Long short-term memory networks


Hochreiter and Schmidhuber (1997) proposed the LSTM, which is an improved Recurrent Neutral Network (RNN) model. The LSTM unit consists of input gate \(i_t\), forgetting gate \(f_t\), and output gate \(o_t\). The Forget Gate \(f_t\) controls how much information is forgotten by the internal state \(c_t-1\) at the previous moment, the input gate \(i_t\) controls how much information is saved by the candidate state \(c_t\) at the current moment, and the output gate \(o_t\) controls how much information is output by the internal state \(c_t\) at the current moment to the external state \(h_t\). LSTM structure is shown in Fig. 2, where ‘\(\times\)’ and ‘+’ represent the multiplication and addition operations of the matrix, respectively. \(\sigma\) and tanh are activation functions. The mathematical definitions are as follows:

$$\begin{aligned} \begin{aligned}f_t=\sigma \, (W_f[h_{t-1}, x_t]+b_f),\\ i_t=\sigma \,(W_i[h_{t-1}, x_t]+b_i),\\ \widetilde{C_t}=\tan {h}(W_c[h_{t-1}, x_t]+b_c),\\ C_t=f_t{\times }C_{t-1}+i_t\times {\widetilde{C_t}},\\ \sigma _t=\sigma \,(W_0[h_{t-1}, x_t]+b_0),\\ h_t=\sigma _t\times \tan \sigma (C_t), \end{aligned} \end{aligned}$$
(10)

where \(W_f, W_i, W_0\) is the weight parameter, and \(b_i\), \(b_f\), \(b_0\) is the deviation parameter. The mathematical formula mentioned above is for a unit. The work of an LSTM network has layers, and each layer has several units. In this paper, the network configuration is characterized by the following parameters: batch size=1, epochs=3000, and neurons=6.

Fig. 2
figure 2

The unit structure of LSTM network

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SARIMA-EG-ECM hybrid model

Previous studies have used the dynamic ARIMA model with covariates (Li et al. 2021). Our research focuses on the seasonality of RH. Therefore, based on the SARIMA model, the SEE model is established. The methodology used for the determination of the SEE model includes three steps. First, a cointegration test is performed on RH and other meteorological variables to consider the long-term equilibrium relationship among the series. Second, a cointegration model based on the SARIMA model is fitted using the meteorological variables that have a cointegration relationship with RH. Third, an ECM is established as a supplement to the cointegration model to explore the impact of the current fluctuation of meteorological variables on the current fluctuation of RH, so as to describe the short-term fluctuation relationship among the series. Figure 3 describes the procedure.

Fig. 3
figure 3

Modeling flowchart of the hybrid model

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Forecast accuracy measures

The choice of the fitted model should be considered from two aspects: on the one hand, the likelihood function is maximized, and on the other hand, the number of unknown parameters in the model is minimized. The larger the likelihood function value, the better the model fitting effect. The more unknown parameters in the model, the more independent variables, the more flexible the model changes, and the higher the accuracy of model fitting. However, only measuring the pros and cons of the model by fitting accuracy will result in an increasing number of unknown parameters in the model and an increase in unknown risks. Correspondingly, the model becomes more and more complex, and the estimation of parameter becomes more and more difficult (Danhui 2019). Therefore, when selecting a fitting model, it is necessary to choose a comprehensive optimal configuration of fitting accuracy and the number of unknown parameters.

1. Akaike Information Criterion


Akaike Information Criterion (AIC) was proposed by Japanese statistician Akaike (Akaike 1973). AIC is a weighted function of fitting accuracy and the number of parameters: the calculation method is as follows:

$$\begin{aligned} \begin{aligned} AIC=2N_1-2ln \, (L_1), \end{aligned} \end{aligned}$$
(11)

where \(N_1\) is the number of model parameters and \(L_1\) is the maximum likelihood function of the model.

2. Bayesian Information Criterion


Although, the AIC criterion provides an effective criterion for the choice of fitting model. When faced with a complex model containing multiple independent variables, the information provided by the fitting error in the AIC criterion will be amplified by the sample size and the number of parameters. The penalty factor of the number is always 2, which has nothing to do with the sample size. Therefore, when the sample size is large, the fitting model selected using the AIC criterion contains more unknown parameters than the real model, and does not converge to the real model. Bayesian Information Criterion (SBC) was proposed by Schwarz (1978) based on Bayes theory. The penalty weight for the number of unknown parameters was changed from a constant 2 to the logarithmic function of the sample size ln(n), which made up for the deficiency of the AIC criterion in the case of large sample size. The calculation method of SBC is

$$\begin{aligned} \begin{aligned} SBC=ln\,(n)\,N_2-2ln\,(L_2), \end{aligned} \end{aligned}$$
(12)

where \(N_2\) is the number of parameters in the model and \(L_2\) is the value of maximum-likelihood function.

When selecting the fitting model in this paper, using the AIC criterion and the SBC criterion helps us find the relative optimum fitting model within a limited range of orders. The model that minimizes the AIC or SBC function is the relatively optimal model.

In addition to AIC criterion and SBC criterion, the performance of the hybrid model can be evaluated by various statistical metrics including Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Residual Standard Error (RSE), and Coefficient of Determination (\(R^2\)).

MSE and RMSE are used in detecting the deviation between the predicted value of the model and the true value. When their value equals 0, the model used for prediction is the optimal model, and accordingly, the larger the error, the larger the value.

RSE describes the average offset between the target and the real regression line, which is used in estimating the standard deviation of the residual. Values of RSE close to 0 represent the optimal performances.

\(R^2\) is the proportion that reflects the total variance of the dependent variable that can be explained by the independent variable through the regression relationship. \(R^2\) provides a method to evaluate the performance of the same model on different data. Its value ranges from 0 to 1, with 0 that indicates the optimal performances.


The formulas can be defined as follows:

$$\begin{aligned}& RMSE=\sqrt{\frac{\sum_{i=1}^n(f_i-y_i)^2}{n}}; \\ & MSE=\frac{1}{n}\sum_{i=1}^n(f_i-y_i)^2; \\ & RSE=\sqrt{\frac{1}{n-2}{\sum_{i=1}^n(f_i-y_i)^2}};\\ & R^2=1-\frac{\sum_{i=1}^n(f_i-y_i)^2}{\sum_{i=1}^n(f_i-y_i)^2}, \end{aligned} $$
(13)

where n is the total number of time series data, \(f_i\) is the predicted value of the i-th data, and \(y_i\) is the measured value of the i-th data.

Results

Data preparation

The meteorological data used in this paper are derived from the meteorological data set observed by Hailun Agricultural Ecology Experimental Station, China Academic of Science, and a total of data set take the month as the scale to collate 10-year meteorological data published from 2009 to 2018 including 17 meteorological variables (Li et al. 2019a). Average RH for 108 months from January, 2009 to December, 2017 in Hailun Agricultural Ecology Experimental Station were selected to establish predictive models. The 12-month data from January, 2018 to December, 2018 are utilized in evaluating. In this study, we used the SEE model to predict RH for 6 and 12 months. It is found that due to voltage instability or other unknown reasons, part of the data contained missing values. To ensure the effectiveness of data analysis and prediction, we require imputing the 17 time series of meteorological variables in the data set with the multiple imputation method.

Take the RH time series as an example: as shown in Table 1, the number of multiple imputation is 5, and the relative efficiency is as high as 0.999520. P value of the parameter estimation for imputing the vacancies is less than 0.05, which passed the hypothesis test, as shown in Table 2.

Table 1 Variance information of the multiple imputation procedure
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Table 2 Parameter estimates (\(H_0:parameter=\theta _0\))
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Model fitting and prediction based on multiplicative seasonality model

The Phillips–Perron unit root test is performed on the RH time series, and it can be seen from Table 3 that the autoregressive process of the drift-free term of the series is non-stationary.

Table 3 Phillips–Perron unit root test of the original series
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We drew a timing diagram of the RH as Fig. 4. It can be seen that the series has a periodic effect with a year as a period. We performed the Phillips–Perron unit root test. As shown in Table 4, not all P values are less than 0.05, which indicates that the series after the difference is a non-white noise series. It contains relevant information worthy of being extracted. Therefore, we performed first-order 12-step difference on the original series to extract the information contained in the original series. The red part in Fig. 5 illustrates differential series of RH. To judge whether the differential series is stationary, we performed the Phillips–Perron unit root test. P values less than 0.05 in Table 5 show that the differential series is significantly stationary.

Fig. 4
figure 4

Timing diagram of RH

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Fig. 5
figure 5

Timing diagram of RH after first-order 12-step difference

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Table 4 White noise test of the original series
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Table 5 Phillips–Perron unit root test of the series after first-order 12-step difference
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Figure 6 illustrates the trend and correlation analysis for RH after the first-order 12-step difference. Combined with the characteristics of the autocorrelation coefficient and partial autocorrelation coefficient mentioned in Fig. 6, the multiplicative seasonality model (ARIMA\((0,1,1)\times (0,1,1)_{12}\)) with \(AIC=669.8517\) and \(SBC=674.9594\) performs better than additive seasonality model (ARIMA\((0,1,1)\times (0,1,0)_{12}\)) with \(AIC=683.5248\) and \(SBC=688.6325\). We use the least square method to estimate the parameters. From Table 6, we can see that the parameters are significant and pass the test with P values less than 0.05. The model is

$$\begin{aligned} \nabla \nabla _{12}x_t=(1-0.80061*B)*(1-0.50315*B^{12})\varepsilon _t,Var \, (\varepsilon _t)=66.17914. \end{aligned}$$
(14)
Fig. 6
figure 6

Autocorrelations and partial autocorrelations of RH after first-order 12-step difference

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Table 6 Conditional least-squares estimation of parameters of multiplicative seasonality model
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From Table 7, it can be seen that the residual series has passed the test with P values larger than 0.05. This confirms that the model has fully extracted the seasonal effect and short-term correlation of the series, and fits well.

Table 7 Residual autocorrelation test of multiplicative seasonality model
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According to the multiplicative seasonality model, we made prediction (Fig. 7), where the black asterisk is the actual measured value of RH, the red line is the predicted value of RH, and the blue line is the 95% confidence upper and lower limit of the series.

Fig. 7
figure 7

RH prediction map based on multiplicative seasonality model

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RH forecast based on SEE model

In fact, changes in RH are not only affected by the changes in the series itself, but also by other meteorological conditions, such as temperature, precipitation, etc. Therefore, we also took into account changes in other meteorological conditions to obtain more accurate results. The cointegration test was performed on other meteorological condition series and RH series in the processed data set, and the results are shown in Table 8. It shows that RH has a cointegration relationship with TEMP, dew point temperature (DEWP), PRCP, atmospheric pressure (ATMO), sea-level pressure (SLP), and 40 cm soil temperature (40ST), which reveals the long-term equilibrium relationship between sequences. These meteorological variables play a significant role in the model of predictions. To fit the interaction between RH and other meteorological conditions, a dynamic regression model can be constructed

$$\begin{aligned} \begin{aligned} \nabla _{12}y_{RHt}=&-3.60887*x_{TEMPt}+3.40734*x_{DEWPt}+0.0047355*x_{PRCPt}\\&+3.05656*x_{ATMOt}-3.08589*x_{SLPt}+0.20930*x_{40STt}\\&+(1-0.74194B)(1-0.66551B^{12})\varepsilon _t, \\&Var(\varepsilon _t)=1.459924. \end{aligned} \end{aligned}$$
(15)
Table 8 Parameter estimation results of cointegration model
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The prediction results of the SEE model are illustrated in Fig. 8.

Fig. 8
figure 8

RH forecast map based on SEE model

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We used the differential series of meteorological conditions that have a cointegration relationship with the RH and previous error series to construct an ECM model

$$\begin{aligned} \begin{aligned} \nabla {{\nabla }_{12}{y_{RHt}}}= -3.57290\nabla \nabla _{12}x_{TEMPt}+3.47467\nabla \nabla _{12}x_{DEWPt}\\+0.00383\nabla \nabla _{12}x_{PRCPt}+2.04517\nabla \nabla _{12}x_{ATMOt}\\-2.08302\nabla \nabla _{12}x_{SLPt}+0.09009\nabla \nabla _{12}x_{40STt}\\-0.00011056ECM_{t-1}+\varepsilon _t, \end{aligned} \end{aligned}$$
(16)

where \(\nabla \nabla _{12}x_{TEMPt}\) is the first-order 12-step difference series of air temperature, \(\nabla \nabla _{12}x_{DEWPt}\) is the first-order 12-step difference series of dew point temperature, \(\nabla \nabla _{12}x_{PRCPt}\)is the first-order 12-step difference series of precipitation, \(\nabla \nabla _{12}x_{ATMOt}\) is the first-order 12-step difference series of atmospheric pressure, \(\nabla \nabla _{12}x_{SLPt}\)is the first-order 12-step difference series of sea-level presssure, \(\nabla \nabla _{12}x_{40STt}\) is the first-order 12-step difference series of 40 cm soil temperature, \(ECM_{t-1}\)is the previous error series, and \(\varepsilon _t\) is the residual sequence of regression.

The results of the analysis of variance shown in Table 9 indicates that the equation was significantly linearly correlated, and the value of \(R^2\) was 0.8889. Parameter estimations of ECM model shown in Table 10 indicate that the current fluctuations of TEMP, DEWP, ATMO, and SLP have a significant impact on the current fluctuations of RH, and the adjustment range of RH fluctuations is large. Their adjustments are, respectively, \(-\)3.57290, 3.47467, 2.04517, \(-\)2.08302 for a unit, which explains the short-term volatility relationship between the series. While, PRCP, 40ST, and previous period errors have no significant impact on current fluctuations, and the adjustment range of current fluctuations of RH is not large. Their adjustments are, respectively, 0.00383, 0.09009, and \(-\)0.00011056 for a unit.

Table 9 The results of the analysis
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Table 10 Parameter estimation of ECM model
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The forecast evaluations of the SEE model, the additive seasonality model, the multiplicative seasonality model, and the LSTM model are shown in Table 11. For simple, we denote the additive seasonality model as ASM and the multiplicative seasonality model as MSM in the table. The model with minimum AIC, SBC, RMSE, RSE, and maximum \(R^2\) is the optimal model. The optimal results have been boldly marked in Table 11. These results are discussed in the following section.

Table 11 Performance metrics of four models
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Discussion

This section discusses the modeling results. Figure 9 shows prediction of SEE model, the additive seasonality model, the multiplicative seasonality model, and the LSTM model for 6 months and 12 months. The radar charts shown in Fig. 10 summarize the performance of several methods in predicting RH for different periods and prepare for further discussion.

Fig. 9
figure 9

Predicted and measured RH

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Fig. 10
figure 10

Comparation based on individual metrics

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First, the SEE model performs better than the other models with minimal RMSE, RSE, and maximal \(R^2\) (RMSE=3.1776, RSE=0.1111, \(R^2\)=0.8889, AIC=309.9675, and SBC=330.3138 for 6-month predicting; RMSE=5.946, RSE=0.3136, \(R^2\)=0.6864, AIC=309.9675, and SBC=330.3138 for 12-month predicting). Compared with the multiplicative seasonality model, the SEE model performs better in fitting and predicting RH, resulting in 53.73% reduction in AIC, 51.06% reduction in SBC, 6.83% reduction in RMSE, 13.14% reduction in RSE, and 1.93% increase in \(R^2\) for 6-month predicting; 10.89% reduction in RMSE, 20.59% reduction in RSE, and 13.44% increase in \(R^2\) for 12-month predicting. The comparation between SEE model and additive seasonality model indicates that the SEE model results in 54.65% reduction in AIC, 52.03% reduction in SBC, 9.79% reduction in RMSE, 18.61% reduction in RSE, and 2.94% increase in \(R^2\) for 6-month predicting; 9.64% reduction in RMSE, 18.35% reduction in RSE, and 11.45% increase in \(R^2\) for 12-month predicting. AS for the LSTM model, the SEE model results in 47.87% reduction in RMSE, 72.82% reduction in RSE, and 50.33% increase in \(R^2\) for 6-month predicting; 4.72% reduction in RMSE, 9.21% reduction in RSE, and 4.86% increase in \(R^2\) for 12-month predicting. Moreover, the study confirms that when the prediction horizon is 6 months, the SARIMA model performs better than the artificial intelligence method with smaller MSE, RSE, and larger \(R^2\), which is consistent with the results in the research of Aghelpour et al. (2021).

Second, incorporating EG theory and ECM into SARIMA model is able to increase the forecasting accuracy. SEE introduces EG theory and ECM based on SARIMA model. We perform cointegration tests on RH and other meteorological conditions, as shown in Table 8, and establish a cointegration model as indicated in formula (15). It shows that RH has a cointegration relationship with TEMP, DEWP, PRCP, ATMO, SLP, and 40ST, which reveals the long-term equilibrium relationship among series; Table 10 indicates that the current fluctuations of TEMP, DEWP, ATMO, and SLP have a significant impact on the current fluctuations of RH. Their adjustments are, respectively, \(-\)3.57290, 3.47467, 2.04517, \(-\)2.08302 for a unit, which explains the short-term volatility relationship between the series. In contrast, the performance of the SEE model is better than the SARIMA model including the multiplicative seasonality model and the additive seasonality model according to the value of AIC, SBC, RMSE, RSE, and \(R^2\). The time series modeling of Li et al. (2021) also reveals that adding covariates can improve the prediction performance of ARIMA model.

Third, increasing the prediction horizon from 6 months to 12 months results in a decrease in the accuracy of the SEE model. According to Table 11, it can be calculated that the increase in the prediction horizon results in 22.78 %reduction in \(R^2\) for 6-month predicting. Nevertheless, the SEE model still performs better than the other models in predicting RH.

Fourth, Fig. 9 illustrates that RH will decrease from January to April and from September to October. It will increase from May to August and from November to December. The RH will reach the minimum in April and the maximum in August. Studies of Shad et al. (2022) have similar discussions. The spread of respiratory diseases, such as COVID-19, is enhanced when the RH decreases (Mangla et al. 2021). Therefore, the prediction method proposed in this paper is helpful to prepare for the transmission and prevention of diseases that may occur in the future.

Conclusions

This paper proposes an SARIMA-EG-ECM model suitable for RH prediction. The accuracy of the model is evaluated by various statistical metrics. Monthly predictions of the RH in Hailun Agricultural Ecological Experimental Station in the next 6 months and 12 months have been carried out. It demonstrates that the SEE model performs better than the multiplicative seasonality, the additive seasonality model, and the LSTM model with minimal RMSE, RSE, and maximal \(R^2\). The SEE model has the following characteristics: it takes full account of seasonality of the time series; it shows that RH has a cointegration relationship with TEMP, DEWP, PRCP, ATMO, SLP, and 40ST, which reveals the long-term equilibrium relationship among series; it indicated that the current fluctuations of TEMP, DEWP, ATMO, and SLP have a significant impact on the current fluctuations of RH. Their adjustments are respectively: \(-\)3.57290, 3.47467, 2.04517, \(-\)2.08302 for a unit, which explains the short-term volatility relationship between the series.

The accuracy of the SEE model decreased slightly when the prediction horizon was increased from 6 to 12 months. Nevertheless, the SEE model still performs better than the other models in predicting RH. We can observe from the prediction results of the SEE model that there will be a decrease in the RH from January to April and from September to October in the next year. There will be an increase in the RH from May to August and from November to December. The RH will reach the minimum in April and reach the maximum in August. The results will help to evaluate the applicability of SEE model in predicting RH in the future development of this study.