Modeling Path Loss With the Growth Stages of a Rice Plantation Using Fuzzy Linear Regression

The application of a wireless sensor network (WSN) for infrastructure monitoring has become a popular method of reducing the labor cost and increasing production. However, WSN connectivity may be interrupted due to vegetative growth between communication nodes. To address this issue, this study applies models that employ fuzzy linear regression (FLR) to model the radio path loss (PL) for the WSN. The PL data in a crop field from recent works are reviewed and analyzed for the crop growth stages at 2.4 GHz frequency and selected to train the proposed FLR model. A vegetative factor (V) is defined as a membership function of a fuzzy set for the feasible PL data interval of the vegetative stage of crop growth. The obtained fuzzy set provides the upper and lower boundaries for predicting the PL in other stages, such as tiller initiation, panicle, flowering, and maturity. The PL in these stages includes the uncertainty from temperature, humidity, wind, and antenna heights. To prove the introduced concept, the proposed models were validated and compared to existing approaches.


I. INTRODUCTION
The agricultural and industrial trends worldwide follow sustainable development goals, especially for rice plantations. However, climate change may make rice production problematic; therefore, governments must support farmers with a technology that can solve and improve agricultural production. The new Internet-of-Things technology with a low-power wireless sensor network (LPWSN) is one effective solution, but may be challenging to implement. This technology can control the product quality and quantity, realizing cost reductions in precision agriculture. Accordingly, the LPWSN has been used by many studies, focusing on coverage area prediction with the path loss model (PLM) in agricultural areas. The PLMs for vegetation are generally classified into two groups: empirical PLM (E-PLM) and machine learning (ML) PLM (ML-PLM). E-PLMs have two types, namely, log-distance and exponential-decay PLMs [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], and they sometimes provide error The associate editor coordinating the review of this manuscript and approving it for publication was Yizhang Jiang . prediction caused by multi-path fast fading. ML-PLMs were created to address this issue [11], [12], [13], [14], [15]. E-PLMs are based on measurement data from real environments. They result in error when the environment changes. Therefore, a new PLM for accurate prediction is needed. E-PLM-related studies focused on the ultra-high-frequency (UHF) band (300-3000 MHz). These works include those of Pal et al., proposed PLM 2.4 GHz modeling for rice and millet and presented the path loss exponents for each growth stage [1]; Kuramoto et al. proposed exponential-decay PLMs 2.4 GHz for paddy fields at 55, 105, and 155 cm antenna heights [2]; Olasupo et al., proposed PLMs for the vertical and horizontal polarizations of natural short and long grasses at 2.4 GHz frequency [3]; Alsayyari et al., proposed a logdistance E-PLM for tall grass and sparse tree with a 16-radial measurement [4]; and Zang and Wang, verified a theoretical multi-ray PLM 400 MHz of a grass field with 0.6 m mean height compared with the LOS measurement at 3.6 m antenna height and showed 3.6 root-mean-square error [5]. Another related study is that of Tang et al., proposed a log-distance PLM with a breakpoint distance on the ground at 5 cm, 50 cm, and 1 m antenna heights above the ground at 470 MHz frequency [6]. Mahesh et al. also proposed a log-distance PLM for agricultural fields comprising corn, paddy, and groundnut that are in both the growth and maturity stages and a coconut garden with dry and wet green grasses [7]. Balachander et al. proposed a log-distance PLM for a coconut garden with green grass and an open lawn with dry and wet green grasses and compared their measurement results with those of the COST 235, ITU vegetation, and Weissberger's model at 2.4 GHz frequency [8]. Yoshimura et al. proposed an empirical vegetation attenuation model with a non-zero gradient for bush at 920 and 2400 MHz frequencies [9]. Lastly, Srisooksai et al. described the propagation characteristics of a sugarcane field using a channel sounder with 45.6 MHz bandwidth at 2.45 GHz frequency [10].
The second type of PLM is the ML-PLM that uses the training data obtained from the measurement campaign in real vegetation environments. The ML-PLM provides small-scale fast fading from multi-path large-scale fading and other conditions, as in the training data. Related studies on ML-PLMs focused on the grass family. Pal et al. proposed a multiple regression model for two medium grass vegetations: paddy and sugarcane. They optimized the regression coefficient using 2.4 MHz measured data over node height and crop cycle periodic combinations [11]. They also proposed non-dominated sorting genetic algorithm (NSGA-III) multi-objective optimization to improve the PLMs [12]. Meanwhile, Sharma et al. proposed crop pest prediction based on the WSN using a fuzzy logic system [13]. Chiroma et al. reviewed artificial intelligent models, including the support vector machine, neural network, genetic algorithm, and adaptive network-based fuzzy interference system (i.e., adaptive neuro fuzzy inference system (ANFIS)), in several types of communication environments (i.e., urban, suburban, rural) [14]. Hakim et al. developed ANFIS PLMs in forest, jungle, and open dirt road environments at 433, 868, and 920 MHz frequencies. They then compared their results with those from conventional empirical models, such as the optimized FITU-R near-ground model, the Okumura-Hata model, and the ITU-R MA FSPL model [15].
The abovementioned both E-PLMs and ML-PLMs are accurate under the conditions of different vegetative growth stages, temperatures, humidities, and antenna heights. However, the set of path loss predictions of the growth stage must be classified to maintain the crops. The abovementioned PLMs are multiple independent variables and deal with complex details (e.g., plant species and different terrains). Accordingly, this study proposes a fuzzy linear regression PLM (FLR-PLM) to predict the path loss signal in a crop plantation. The vegetative factor (V ) is defined as the membership level of the feasible path loss data in the vegetative stage with a boundary covering the other growth stages in the same fuzzy set. The uncertainties of different vegetation growth stages, temperatures, humidities, and antenna heights are included in the fuzzy set. These ensure that the FLR-PLM provides the best network planning with a suitable coverage area for first to final stages of plantation. The study contributions are summarized as follows: • An FLR-PLM with a vegetative factor (V ) of the membership function is proposed based on the vegetative stages with a support boundary for all the stages of the plantation.
• PLMs for rice, millet, and grass are compared. The remainder of this paper is structured as follows: Section II presents the related path loss models; Section III explains the crop growth stage path loss; Section IV presents the results and discusses them; and Section V concludes this study.
The relationship between the PL and the distance (d) for the radio wave propagation is expressed as follows in a mathematical linear regression form: where A 0 and A 1 are relationship coefficients. The distance and the path loss values express the regression coefficients with fuzzy numbers as follows due to the wave propagation: The fuzzy membership functions of the fuzzy path loss variables ( PL i ) are derived from the measurement uncertainties. The membership functions of the A 0 and A 1 fuzzy coefficients are evaluated using the fuzzy regression analysis based on the fuzzy extension principle [21], [22], [23], [24], [25], [26]. The left-right (L-R) presentation of the fuzzy number provides a suitable means for representing the fuzzy coefficients. Let A i be the coefficient expressed below: where a i is the central value; c L i and c R i are the left and right spreads, respectively (Fig. 1); and subscript i is the i th fuzzy coefficient. The membership function µ A i (a) of the triangular L-R fuzzy number is given as follows by (4): The maximum L-R spreads are the model boundary that occurs at membership level 0.0. We incorporated the uncertainty not captured in the available measurement data sets by using the proposed FLR-PLM and defined a variable membership level as a factor for fitting the feasible path loss data, called the vegetative factor (V ). The L-R spreads are expanded for the uncertainty at the lower membership levels as depicted in Fig. 1. According to this approach, each measured path loss must be within the boundaries around the estimated regression curves at lower V values. The spread of the membership function and the fuzziness of the regression variables are controlled by specifying the V level between 0 and 1. The V value is useful when considering quantified uncertainties, such as the maximum or minimum path loss caused by the fading electromagnetic wave from the environment. Accordingly, the left and right spreads of the L-R fuzzy numbers A 0 and A 1 are expressed as follows: The spread of the PL fuzzy number PL i is obtained from the measurement uncertainty analysis and is expressed as: The uncertainty PL measurement shows that the fuzzy membership function of the output has a triangular shape (Fig. 1).  (2) can be combined, such that the lower bound of the fuzzy data curve intersects at the left spread boundary, and the upper bound intersects at that of the right spread. The lower and upper bounds of the fuzzy regression curve are expressed in the following forms: and The minimum spread criteria are considered to evaluate the fuzziness output. The minimum spread of the fuzzy numbers is obtained as follows by minimizing the output support for the n measurements: The set of (7a)-(b) and (8) provides a mathematical formulation of the fuzzy regression analysis problem using the fuzzy form of the input and output variables. This formulation leads to an optimization problem for the coefficient evaluation in (7a) and (b) in terms of the central value, the left and right spreads.

III. CROP GROWTH STAGE PL
In this work, the paddy rice crops were classified as medium grass vegetation. The crops were planted at 22 and 28 cm row intervals for millet and rice, respectively [1].     [1]. The theoretical measured path loss data were obtained from (9) as follows to train the proposed FLR-PLM and compare the models: where P t is the output power of the transmitter (dBm); G t and G r are the antenna gains of the transmitter and the receiver, respectively; and P r (d) is the RSSI at distance d(dBm). The transceiver output power was 7 dBm. The antennas were omni-directional ones with 3 dB gain. Figs. 4 and 5 depict the path loss data of (9) for the crop growth stages for rice and millet, respectively. The PLE values of the tiller initiation and emergence stage curves were 1.92, confirming the LOS environment in the first plantation stage, which needed a huge amount of fertilizer to grow to the next phase. The Fresnel zone region theory [26] states that the breakpoint distance, d BP , at 2.4 GHz frequency and 52 cm antenna height is calculated as follows: where h t and h r are the antenna heights of the transmitter and the receiver (m), respectively. For this measured path loss, d BP between two nodes was 8.65 m. The measurements were started at 10 m. Therefore, a single slope or PLE can be used for modeling. The PLE increased with 3.37 and 4.6 in the tillering and five-leaf stages, which comprised the second phase. These PLEs confirmed the NLOS environment stages before increasing with the maximum PLE values of 5.9 and 6.4 in the third phase. The full NLOS situations in the last phase occurred for the flag leaf, milk, and dough stages for millet, including the panicle, flowering, and maturity stages for rice.   both nodes in horizontal polarization [2]. The transmitter generated 4.77 dBm output power. The RSSI measurement was performed with a 0-60 m distance between the nodes. The measured path loss was obtained using (9). The second candidate measurement was long natural grass with more than 1 m height [3]. The RSSI measurement for both the vertical and horizontal polarization was performed with a 0-30 m distance between the nodes. The path loss result for the vertical polarization is as follows: That for the horizontal polarization is Equations (11)-(12) were plotted in Fig. 6 as the path losses for validation. Note that the NLOS (non-line-of-sight) stages of rice lead to the RSSI decrease or the path loss increase with the stage sequence up, except for millet. This is attributed to the last dough stage making the RSSI increase or the path loss decrease due to the physical maturity stage of the millet seed and leaf (yellow dots, Figs. 3 and 5).

IV. RESULTS AND DISCUSSION
The RSSI data of the semi-NLOS were selected for fitting because of their very large swing and main membership in the fuzzy set. Accordingly, FLR-PLM was applied. The individual RSSI measurements with all the growth stages were fuzzified and aggregated into a combined uncertainty for the RSSI-distance relationship based on the semi-NLOS conditions. The RSSI uncertainty is expressed herein as triangular fuzzy numbers with a 0 to 1 membership level. The fuzzy aggregation of the uncertainties was used in the fuzzy LR analysis with fuzzy output variables.

A. PROPOSED FLR MODEL
Figs. 7 and 10 depict the triangular L-R fuzzy numbers for coefficients A 0 and A 1 for rice and millet, respectively. The decision variables for the fuzzy regression were six constants: central values C a , spreads L a , R a , and C b , and spreads L b and R b . The V factor was added for the modeling, consequently increasing or decreasing the fuzzy regression curve and output spreads. The path loss measurement uncertainty was expressed by a wide spread of their fuzzy numbers. Thus, V can be varied for the optimizations. The fuzzy regression analysis result using the minimum spread criteria with the h-certain factor is expressed below: Rice had an h-certain factor of 0.5, Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.     (Fig. 8). Meanwhile, those between the lower 0.0 and upper 0.0 bounds represented a feasible path loss in the tiller initiation, panicle, flowering, and maturity stages (Fig. 9). Note that the tiller initiation stage was in the lower bound with the LOS scenario (PLE: approximately 1.94). Similarly, for millet, 0.3 V provided the boundary curves between the lower 0.3 and upper 0.3 bounds, covering the feasible path loss in the available data set of the five-leaf stage (Fig. 11). By contrast, the boundary curves between the lower 0.0 and upper 0.0 bounds covered all the feasible path losses of the other stages (i.e., emergence, flag leaf, milk, and dough stages) as shown in Fig. 12. Almost all path loss data were in the boundaries between the central line 1.0 and the lower boundary (i.e., both lower 0.0 and upper 0.0). The V factor of the fuzzy set of millet was lower than that of the fuzzy set of rice because the feasible path loss in the five-leaf stage of the 93594 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   millet plantation took a longer time than the tillering stage of the rice plantation. This is the nature of crop plantation that can be determined using the V factor.Note that the dough stage in millet plantation was the only stage, in which the path loss decreased from last stage (i.e., milk stage) due to the physical dimension of millet. Note, however, that this stage was still in the limit line of the upper 0.3 boundary as shown in Fig. 11.

B. MODEL VALIDATION
The proposed FLR-PLMs were validated using the other measured path losses for the paddy from [2] and the long grass from [3] at 2.4 GHz frequency as shown in Figs. 13 and 14. The PL data in the paddy field, in which the rice grown were 105 cm tall, were in the boundary between the upper [0.0] and upper [0.5] bounds at 55 and 105 cm antenna heights. This confirmed that the proposed FLR-PLM provided boundary support in the fully-grown stage of the rice crops (Fig. 13). The outer path loss caused by fast fading near the transmitter was similar to that found in the two-ray model at a distance before the breakpoint distance in (10).  case of the proposed FLR model for millet, the path loss data of the paddy were almost in the upper [0.5] and lower [0.5] boundaries because this boundary supported the five leaves of millet, which made them bigger than the rice leaves during the tillering stage. Consequently, the path loss became larger than that observed during that stage for rice (Fig. 14). Lastly, the measured path loss on a flat soil was close in the lower  (Figs. 13 and 14). The PL of the natural grass was generally lower than that of the paddy because natural grass grew without care and maintenance. The path loss data for the horizontal polarization of the antennas (H-H) were in the same boundary, albeit with a larger path loss.

C. MODEL COMPARISON
Three popular vegetation models were used to compare the proposed FLR-PLM:

1) WEISSBERGER'S MODEL
This model is an exponential-decay model presented in (12). It is applicable when a ray path is blocked by trees [16]: 93596 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  where fis the frequency in GHz, and d is the vegetation depth in meters. The frequency range is 0.23 to 95 GHz.

2) COST 235 MODEL
The COST 235 model  proposed based on measurements made in millimeter wave frequencies (9.6-57.6 GHz) through a small grove of trees uses the out-of-leaf in-leaf approach. In the COST 235 model, the measurements are performed over two seasons, that is, when the trees are in-leaf and out-of-leaf. This model is also applicable to frequencies between 200 MHz and 95 GHz. For the COST 235 model, f is the frequency in MHz, and d is the tree depth in meters.

3) ITU-R FOLIAGE ATTENUATION MODEL
The ITU Recommendation (ITU-R) [19] was developed from measurements mainly performed at the UHF and proposed for cases where either the transmit or receive antenna is near a small grove of trees, such that the majority of the signal propagates through the trees: where f is the frequency in MHz, and d is the vegetation depth in meters. The frequency range is 0.20 to 95 GHz.

4) FITU-R FOLIAGE ATTENUATION WITH PLANE EARTH MODEL
This model considers data sets collected during two foliation states (i.e., in-leaf and out-of-leaf) at 11.2 and  20 GHz [20].
where f is the frequency in MHz, and d is the vegetation depth in meters. The frequency range is 0.20 to 95 GHz. The theoretically free-space PLM in Eq. (20) acts as a reference for the path loss estimation in (16)- (19): where f is the frequency in MHz, and d is the distance between the transmitting and receiving antennas expressed in meters. Figs boundary. In other words, there are two significant results: 1) the proposed FLR models provide a feasible path loss for general-vegetation path loss models, especially for the millet model (Fig. 16); and 2) the upper [0.0] boundary can be used only for high-density tall grass, while the lower [0.0] boundary can be used for a near-ground LOS environment (e.g., tillering initiation and emergence stages) (Figs. 15 and 16). The COST 235 model provided a good prediction during the flowering stage of rice (Fig. 15), but only predicted the path loss in near the upper [0.3] boundary in the millet model (Fig. 16).

V. CONCLUSION
This study proposed a propagation model with FLR-PLM for rice and millet plantations at 2.4 GHz frequency. The training PL data were obtained from different crop growth stages at 50 cm antenna height, while the testing PL data were obtained from two sources, that is, a 105 cm-long paddy field at 55, 105, and 155 cm antenna heights and tall grass at 50 cm antenna height both for vertical and horizontal polarizations. After training, the proposed FLR-PLM for rice yielded a V factor at the 0.3 membership function, providing boundaries that covered the tillering and other stages at the 0.0 membership function. Meanwhile, the FLR-PLM for millet showed a V factor at the 0.5 membership function, providing a boundary that covered the five-leaf and other stages at the 0.0 membership function. The validation results of the proposed models showed that the upper [0.0] boundary of the rice model could support the paddy and tall grass, showing good agreement at the distance after the breakpoint. The proposed FLR models were also compared with four vegetation models. The results depicted that the proposed models supported the vegetation model path loss within the boundary of the 0.3 and 0.5 V membership levels. Finally, the proposed models can be widely used for general vegetation and can limit the possibility of the upper and lower signals for the network planning and monitoring of the crop growth stages.