Efficient and Real-Time Compression Schemes of Multi-Dimensional Data from Ocean Buoys Using Golomb-Rice Coding

Abstract

The energy supply of ocean monitoring buoys is a major challenge, especially for long-term, low-power applications. Data compression can reduce transmission energy and extend system lifespan. In particular, the algorithm cannot introduce delays to ensure real-time monitoring. In this scenario, we propose an efficient real-time compression scheme for lossless data compression (ERCS_Lossless) based on Golomb-Rice coding to efficiently compress each dimensional data independently. Additionally, we propose an efficient real-time compression scheme for lossy data compression with a flag mechanism (ERCS_Lossy_Flag), which incorporates a flag bit for each dimension, indicating if the prediction error exceeds a threshold, followed by further compression using Golomb-Rice coding. We conducted experiments on 24-dimensional weather and wave element data from a single buoy, and the results show that ERCS_Lossless achieves an average compression rate of 47.40%. In real communication scenarios, splicing and byte alignment operations are performed on multidimensional data, and the results show that the variance of the payload increases but the mean decreases after compression, realizing a 38.60% transmission energy saving, which is better than existing real-time lossless compression methods. In addition, ERCS_Lossy_Flag further reduces the amount of data and improves energy efficiency when lower data accuracy is acceptable.

1. Introduction

The Internet of Things (IoT) has become one of the core technologies of modern society, profoundly affecting all aspects of individuals, businesses, and society. From smart homes and smart buildings to smart cities, smart transportation, smart healthcare, and many other fields, IoT is driving the digital transformation and intelligent development of society [1]. The wide application of IoT not only improves the convenience of life but also brings unprecedented efficiency gains and innovation opportunities to various industries.

With the continuous progress of IoT, marine scientific research is also deepening, especially in the field of marine environmental monitoring. In recent years, multifunctional ocean buoys have become increasingly prominent in marine monitoring and data collection. These buoys usually integrate multiple sensors and perform data processing and transmission through industrial controllers or embedded computing devices [2]. They are capable of accurately measuring key environmental parameters, including seawater temperature, humidity, wind speed, wave height, etc., thus providing valuable data for climate research, marine pollution monitoring, fishery resource assessment, and marine disaster warning. The multifunctional ocean buoys are becoming increasingly important in modern marine scientific research and resource management.

Each ocean buoy can be regarded as an independent sensing node in a wireless sensor network (WSN) [3]. The buoy collects multidimensional environmental data through embedded sensors and transmits the data to relay nodes or shore-based devices for further analysis and processing using wireless communication technologies (e.g., LoRa, NB-IoT, Wi-Fi, etc.) to ensure that the data can be monitored and analyzed in real time and remotely [4]. Through these technologies, the buoys can form a dynamic, collaborative, large-scale monitoring network in geographically widely distributed sea areas, greatly improving the coverage and accuracy of the monitoring system.

The issue of energy consumption has always been a major challenge that cannot be ignored in buoy design [5]. Buoys are typically deployed in remote or deep-water environments and often require costly maintenance, such as battery replacement using specialized equipment or vessels. This makes energy efficiency essential to minimizing operational costs and enhancing system sustainability. The wireless communication module, being the most energy-intensive component, consumes substantial power during data transmission. To extend system longevity, strategies such as reducing transmission frequency and parallel data transmission from multiple sensors are employed. Additionally, data compression techniques play a pivotal role in optimizing energy consumption by significantly reducing the volume of transmitted data [6]. Research has shown that reducing transmitted data by even one byte can offset the energy cost of thousands to millions of computation cycles, depending on the transceiver used [7].

Researchers have proposed various data compression algorithms to cope with the need for efficient transmission of sensed data. Ref. [8] proposed the Lightweight Temporal Compression (LTC). This method allows some error for each reading, with higher error bounds yielding greater storage savings. However, LTC is unsuitable for real-time compression systems as it only reconstructs the midpoint between two values at the receiver. Ref. [9] introduced the Lossless Entropy Compression (LEC) algorithm, which uses a very small dictionary capable of compressing sensed data in real time. LEC can compress most sensed data efficiently, but because the dictionary is static, LEC has difficulty adapting to changes in the correlation of sensor data. Ref. [10] developed a lossy compression method based on differential pulse-coded modulation that quantizes the differences between consecutive samples, but the parameters of the quantizer are computed offline using a training set, and the compression algorithm is not adaptable to changes in the data model. Ref. [11] proposed an adaptive Huffman coding method for sensing data compression. However, given the small data volumes transmitted by buoys, additional bits needed to communicate the Huffman tree significantly reduce compression efficiency. Ref. [12] presented Adaptive Lossless Data Compression (ALDC), which segments data into chunks and uses various code tables for optimal performance. Ref. [13] proposed a Fast and Efficient Lossless Adaptive Compression Scheme (FELACS), which applies Golomb-Rice coding to data chunks. Both methods rely on chunking, requiring input sequences to reach a certain length, making them unsuitable for real-time monitoring. Ref. [14] proposed a zero-latency compression algorithm, which transmits data only if the difference between the measured and predicted values exceeds a specified tolerance threshold. The algorithm uses modified Golomb code (MOD), which has a different prefix than LEC coding, making the coding more compressive for sensed data with larger fluctuating values. Ref. [15] introduced a lightweight lossless compression method for multidimensional data, employing a common prefix code across dimensions to reduce coding length. However, when signals from a multisensor system exhibit high heterogeneity, the compression performance declines significantly due to varying statistical distributions. Ref. [16] proposed the D-LZW compression algorithm, which first differentiates the data and then applies dictionary-based LZW compression. While effective at replacing duplicates with dictionary indexes, this method performs poorly for buoys due to limited data transmission volumes and low redundancy [17]. Ref. [18] introduced a lossless compression method combining Delta coding and T-RLE. T-RLE optimizes the RLE algorithm and performs well when sensors measure consecutive identical values. However, for buoy sensors such as wind speed and direction, similar consecutive measurements are rare, limiting the method’s effectiveness.

To meet the need for real-time monitoring of buoys and to further improve compression efficiency to reduce transmission energy consumption, we propose an efficient real-time compression scheme for lossless data compression (ERCS_Lossless) and an efficient real-time compression scheme for lossy data compression with a flag mechanism (ERCS_Lossy_Flag). The proposed ERCS_Lossless algorithm modifies the parameter update strategy for Golomb-Rice coding in [13] to enable real-time coding of the collected data. For the method in [13], the data must be accumulated to block size before compression and encoding operations are performed. For example, setting the block size parameter to 12 when the sampling interval is 1 h means that compression and transmission can only be performed every 12 h. On the receiver side, this results in a batch of data that can only be received every 12 h, introducing an unacceptable delay. In contrast, the proposed method predicts the current coding parameters of the data by using historical data; this allows for immediate compression and transmission of multidimensional data, ensuring real-time monitoring without the delays inherent in the approach described in [13]. Meanwhile, the ERCS_Lossy_Flag algorithm allows a trade-off between accuracy and efficiency, and the method extends the single-dimensional lossy compression proposed in [14] to multiple dimensions. The lossy compression in [14] involves only one dimension. When the prediction error of the data are less than the maximum absolute error (MAE), the data can simply be left untransmitted, and the communication protocol only needs to implement the acknowledgment mechanism. When applying this compression method to multidimensional data, a 1-bit flag is needed to distinguish whether the prediction error of each dimension is less than the MAE or not; if the prediction error is less than the MAE, no additional information is added after the flag bit; on the contrary, the prediction error will be transmitted.

Experimental results show that the proposed ERCS_Lossless algorithm achieves an average compression rate of 47.40%. This compression rate is significantly better than the existing real-time lossless compression algorithms, which proves the advantage of this method in compressing data efficiently. In real communication scenarios, we perform splicing and byte alignment operations on multidimensional data. Although the variance of the compressed payload increases, the mean value decreases, and the overall data volume is effectively compressed, ultimately realizing an energy saving of 38.60% in transmission. In addition, the proposed ERCS_Lossy_Flag algorithm further compresses the data volume and improves energy efficiency with acceptable lower data accuracy. Overall, these proposed compression schemes exhibit strong performance and can effectively meet the practical requirements of real-time monitoring applications.

The remainder of this paper is structured as follows: Section 2 describes the buoy observation dataset and presents the proposed compression methods and comparison methods. Section 3 evaluates the algorithm in terms of compression rate and energy savings and discusses the effect of parameter selection on compression rate. Finally, Section 4 concludes the paper.

2. Materials and Methods

In this section, we describe the dataset used for experiments and detail the proposed ERCS_Lossless and ERCS_Lossy_Flag algorithms. Finally, we introduce two coding schemes, Lossless Entropy Code (LEC) [9] and Modified Golomb Code (MOD) [14], used for comparison.

2.1. Dataset Description

The 2020 Dongying Offshore Buoy Observation Dataset (Yijun Hou, Ze Liu, Dongxue Mo, et al. Institute of Oceanography, Chinese Academy of Sciences, 2020) is sourced from the National Key Research and Development Program of China Research on the Mechanisms, Risk Assessment, Response Technologies, and Demonstration Applications of Major Marine Dynamic Disasters (2016YFC1402000). It comprises in situ observations recorded by a single buoy, including meteorological parameters (wind speed, air temperature, humidity, and barometric pressure) and wave parameters (wave height, wave period, and wave direction). The dataset contains 3697 time points, with a sampling interval of 1 h, and each time point includes measurements across 24 dimensions.

Each measurement obtained by the sensor is converted by an Analog-to-Digital Converter (ADC) to a binary representation of $R$ bits, with $R$ being the sensor resolution. The statistical characteristics of each dimension, along with the corresponding sensor resolution, are summarized in Table 1. We calculated several metrics, including the sample mean $\overline{r}$ and standard deviation ${\sigma }_{\overline{r}}$, the mean $\overline{d}$ and standard deviation ${\sigma }_{\overline{d}}$ of the differences between consecutive samples, the information entropy $H = -{\sum }_{n=1}^{N}p(r_n){\log }_{2}p(r_n)$, where $N$ is the number of possible measurements $r_n$, $p\left(r_n\right)$ is the probability density function of $r_n$, and the information entropy $H_d = -{\sum }_{n=1}^{N}p(d_n){\log }_{2}p(d_n)$ of the differential signal.

It is worth noting that the information entropy decreases after differencing the data in all dimensions except the maximum wave period (21st dimension). The differencing operation better utilizes the correlation between the data by capturing the amount of variation between the data rather than storing the absolute value of the data directly. Especially when the original data has a strong smoothness or trend, differencing can significantly reduce redundancy, resulting in a more centralized distribution of data in a simpler, more refined form. The decrease in information entropy also means that the potential complexity of the data are reduced and becomes easier to compress.

2.2. The Proposed Data Compression Schemes

2.2.1. Golomb-Rice Coding

Golomb-Rice coding [19] is a variant of Golomb coding, which is a simple variable-length coding. Golomb-Rice coding is a method of coding based on division remainders and quotients. It has a parameter $k$, where $k$ is a non-negative integer. For any non-negative integer $I$ to be encoded, the prefix code $q$ of the Rice Golomb encoding is a unary code with quotient ${\displaystyle \frac{I}{2^k}}$ (i.e., ${\displaystyle \frac{I}{2^k}}$ ones followed by a zero), and the suffix code $r$ is the remainder $I\%2^k$, expressed using $k$-bit binary numbers, e.g., when $k=3$ and $I=14$, the prefix code $q$ is encoded as ‘10’, and the suffix code $r$ is encoded as ‘110’, then the result of the Golomb-Rice coding is ‘10110’. Decoding is then done by first determining the number of 1s before the first 0, which is the value of $q$. The $k$ bits after the first 0 are the binary value of $r$. Then $I=q\times 2^k+r=1\times 2^3+6=14$.

Table 2 presents the minimum code lengths for integers ranging from 0 to 255 encoded using Golomb-Rice coding, along with their corresponding optimal parameter $k$. The choice of the optimal parameter $k$ is not always unique for integers $I$ (excluding $I=0$), since different parameter values may produce the same minimum code length, which suggests that Golomb-Rice coding has some flexibility. Our proposed parameter $k$ updating strategy can select the appropriate $k$ value for the data to be encoded at the current moment in most cases so that the encoding result is represented as a set of shorter binary numbers and the encoding efficiency is improved.

2.2.2. ERCS_Lossless

Existing Golomb-Rice-based coding methods usually encode a block of input sequences at once, which cannot meet the demand of real-time compression [13,20]. To overcome this limitation, we propose an improved parameter $k$ updating strategy, the basic idea of which is to use historical data to predict the parameter $k$ of the current data to be encoded, so that the compression algorithm can dynamically adjust the parameters during real-time processing, thus realizing real-time compression of sensor data.

Algorithm 1 shows the flow of the proposed ERCS_Lossless algorithm. The proposed method has two pre-set parameters: the amount of historical data used for predicting $N$ and the initial Golomb-Rice coding parameter $k=kFixed$.

Algorithm 1. ERCS_Lossless

Input: samples acquired by ADC $x\left(t_n\right)$, amount of historical data used for predicting $N$, initial Golomb-Rice coding parameter $kFixed$, sensor resolution $R$.

Output: coding result $code$.

1: BEGIN

2: if $n=1$ then

3:     code ← binary($x\left(t_1\right),R$); # Direct output using $R$-bit binary.

4: else

5:     $\delta \left(t_n\right)$ ← Equations (1) and (2); # Calculate the prediction error.

6:     $z\left(t_n\right)$ ← Equation (3); # Calculate the non-negative integer mapping.

7:     if $n\le N+1$ then

8:    $code$ ← GolombRice($z\left(t_n\right)$, $kFixed$); # Golomb-Rice coding of $z\left(t_n\right)$ using $k=kFixed$.

9:    Update $history$; # The array $history$ stores the last $N$ historical data.

10:     else

11:    $kpredicted$ ← Equation (4); # Calculate the predicted parameter $k$ using historical data array $history$.

12:    $code$ ← GolombRice($z\left(t_n\right)$, $kpredicted$); # Golomb-Rice coding of $z\left(t_n\right)$ using $k=kpredicted$.

13:    Update $history$;

14:     end if

15: end if

16: return $code$;

17: END

For $n=1$, the binary representation of $x\left(t_1\right)$ is directly output using the resolution bits of the corresponding sensor.

Otherwise, the prediction error is calculated for the output data of each ADC, with the predicted value being the data from the previous moment:

(1)$\tilde{x}\left(t_n\right)=x\left(t_{n-1}\right),$

(2)$\delta \left(t_n\right)=x\left(t_n\right)-\tilde{x}\left(t_n\right),$

where $\tilde{x}\left(t_n\right)$ is the predicted value at moment $t_n,$ and $\delta \left(t_n\right)$ is the prediction error at moment $t_n$.

Since the coding scheme based on Golomb-Rice coding uses only non-negative integers as inputs, each $\delta$ is mapped to a non-negative integer interval with the mapping equation:

(3)$z\left(t_n\right)=\left\{\begin{array}{c}2\delta \left(t_n\right) \mathrm{if} \delta \left(t_n\right)\ge 0\hfill \\ -2\delta \left(t_n\right)-1 \mathrm{if} \delta \left(t_n\right)<0\hfill \end{array}\right.,$

where $z\left(t_n\right)$ is a non-negative integer mapping of $\delta \left(t_n\right)$.

For $2\le n\le N+1$, since there is not enough historical data for predicting the parameter $k$, $k=kFixed$ (here $kFixed=3$) is used for Golomb-Rice coding of $z\left(t_n\right)$.

For $n\ge 2+N$, Golomb-Rice coding of $z\left(t_n\right)$ is performed using the predicted parameter $k\left(t_n\right)$. The predicted parameter $k\left(t_n\right)$ is calculated as follows:

(4)$k\left(t_n\right)=\left\{\begin{array}{c}0 \hspace{0.17em}\hspace{0.17em} \mathrm{if} {\displaystyle \frac{{{\displaystyle \sum }}_{m=n-N}^{n-1}z\left(t_m\right)}{N}}<1\hfill \\ \lfloor {\log }_2{\displaystyle \frac{{{\displaystyle \sum }}_{m=n-N}^{n-1}z\left(t_m\right)}{N}}\rfloor \mathrm{if} {\displaystyle \frac{{{\displaystyle \sum }}_{m=n-N}^{n-1}z\left(t_m\right)}{N}}\ge 1\hfill \end{array}\right..$

The proposed updating strategy for parameter $k$ takes full advantage of the high degree of autocorrelation inherent in sensor data and has a certain degree of adaptive capability. In sensor networks, data are usually collected continuously, and there is a strong correlation or regularity between the data of neighboring moments. In Golomb-Rice coding, the choice of parameter $k$ is crucial for coding efficiency, and parameter $k$ directly affects the coding length. The proposed method uses the neighboring $N$ historical data to predict the coding parameter $k$ for the data at the current moment, and the adaptive updating mechanism of $k$ enables the coding strategy to be dynamically adjusted to the changes in the data stream; for example, with the fluctuation of the sensor data, the distribution of prediction error may also change, and the method can achieve the timely adjustment of the parameter $k$, which can optimize the coding process and improve the compression efficiency. This adaptive ability enables the coding method to always maintain high compression performance and accurate prediction ability when facing different environments and different types of data and realize efficient data coding. The parameter $k$ is predicted using the latest samples from the historical data, which must be stored locally during the encoding process.

2.2.3. ERCS_Lossy_Flag

For lossy compression of unidimensional data, ref. [14] proposed a method that transmits only when the measured value differs from the predicted value by more than a given tolerance threshold $\epsilon$. The threshold $\epsilon$ is defined as the MAE and is calculated as a percentage of the ADC input range. Notably, when $\epsilon =0$, the compression becomes lossless. In this method, the MAE serves as a tolerance threshold that balances the compression efficiency and reconstruction quality. Specifically, the MAE threshold determines the error range that the system can accept when compressing data. If the threshold is set higher, the system will tolerate a larger error, and the compression effect will be better, but the quality of the reconstructed signal may be degraded. Conversely, when the threshold is set lower, the error is smaller and the reconstruction quality is higher, but the compression effect will be worse. Therefore, an ideal balance between compression rate and reconstruction quality can be found by choosing an appropriate MAE threshold.

Figure 1 and Figure 2 show example reconstructions of the 7th-dimensional average air temperature and the 13th-dimensional average barometric pressure, respectively. The measured signals are depicted by the black lines, while the reconstructed signals on the receiver side are depicted by the red lines.

Let $J$ represent the set $J=\left\{j:\left|\delta \left(t_j\right)\right|>\epsilon \right\}$, satisfying the specified condition. Equation (1) can be expressed as follows:

(5)$\tilde{x}\left(t_n\right)=x\left(t_{j^{*}}\right)$

where $j^{*}=\max \left(j\in J\right)$. At the protocol level, where acknowledgment of each transmitted data is required without retransmission of lost packets, the communication protocol must implement an acknowledgment mechanism. The reconstructed signal $\widehat{x}\left(t_n\right),$ at the receiver is as follows:

(6)$\widehat{x}\left(t_n\right)=\left\{\begin{array}{c}\delta \left(t_n\right)+\tilde{x}\left(t_n\right) \mathrm{new} \mathrm{sample} \mathrm{received} \hfill \\ \tilde{x}\left(t_n\right) \mathrm{otherwise}\hfill \end{array}.\right.$

Modify the mapping Equation (3) as Equations (7) and (8):

(7)$w\left(t_n\right)=\left\{\begin{array}{c}\delta \left(t_n\right)-\epsilon \mathrm{if} \delta \left(t_n\right)>\epsilon \hfill \\ \delta \left(t_n\right)+\epsilon \mathrm{if} \delta \left(t_n\right)<-\epsilon \hfill \end{array}\right.,$

(8)$z\left(t_n\right)=\left\{\begin{array}{c}2w\left(t_n\right) \mathrm{if} w\left(t_n\right)\ge 0\hfill \\ -2w\left(t_n\right)-1 \mathrm{if} w\left(t_n\right)<0\hfill \end{array}\right.,$

where $w\left(t_n\right)$ is the new to-be-transmitted error and $z\left(t_n\right)$ is a non-negative integer mapping of $w\left(t_n\right)$.

At the receiver side, the prediction error $\delta \left(t_n\right)$ is reconstructed by applying an inverse mapping function:

(9)$\delta \left(t_n\right)=\left\{\begin{array}{c}w\left(t_n\right)+\epsilon \mathrm{if} w\left(t_n\right)>0\\ w\left(t_n\right)-\epsilon \mathrm{if} w\left(t_n\right)<0\end{array}\right..$

For multidimensional data, it is uncommon for all prediction errors to fall below the tolerance threshold. To address this, a 1-bit flag is introduced for each dimension. A flag bit of ‘0’ indicates that the prediction error is within the tolerance threshold, and no further encoding is performed. Conversely, a flag bit of ‘1’ signifies that the prediction error exceeds the tolerance threshold, prompting subsequent encoding using Golomb-Rice coding with the predicted parameter $k$.

Algorithm 2 shows the flow of the proposed ERCS_Lossy_Flag algorithm. The proposed method also has two pre-set parameters: the amount of historical data used for predicting $N$ and the initial Golomb-Rice coding parameter $k=kFixed$.

Algorithm 2. ERCS_Lossy_Flag

Input: samples acquired by ADC $x\left(t_n\right)$, amount of historical data used for predicting $N$, initial Golomb-Rice coding parameter $kFixed$, sensor resolution $R$.

Output: coding result $code$.

1: BEGIN

2: if $n=1$ then

3:     code ← binary($x\left(t_1\right),R$); # Direct output using $R$-bit binary.

4: else

5:     $\delta \left(t_n\right)$ ← Equations (2) and (5); # Calculate the prediction error.

6:     if $\delta \left(t_n\right)\le \epsilon$ then

7:      Flag ← ‘0’;

8:      $code$ ← Flag;

9:     else

10:    Flag ← ‘1’;

11:    $z\left(t_n\right)$ ← Equations (7) and (8); # Calculate the non-negative integer mapping.

12:    if the number of elements in the set $J<N$ then

13:      $GRcode$ ← GolombRice($z\left(t_n\right)$, $kFixed$); # Golomb-Rice coding of $z\left(t_n\right)$ using $k=kFixed$.

14:      Update $history$; # The array $history$ stores the last $N$ historical data.

15:    else

16:      $kpredicted$ ← Equation (8); # Calculate the predicted parameter $k$ using historical data array $history$.

17:       $GRcode$ ← GolombRice($z\left(t_n\right)$, $kpredicted$); # Golomb-Rice coding of $z\left(t_n\right)$ using $k=kpredicted$.

18:       Update $history$;

19:    end if

20:    $code$ = concat(Flag, $GRcode$);

21:     end if

22: end if

23: return $code$;

24: END

For $n=1$, the binary representation of $x\left(t_1\right)$ is directly output using the resolution bits of the corresponding sensor.

When $\left|\delta \left(t_n\right)\right|\le \epsilon$, encoded as ‘0’.

If the number of elements in the set $J$ is less than $N$ and $\left|\delta \left(t_n\right)\right|>\epsilon$, encode as ‘1’ followed by Golomb-Rice coding of $z\left(t_n\right)$ using $k=kFixed$, where $kFixed=3$.

If the number of elements in the set $J$ reaches $N$ and $\left|\delta \left(t_n\right)\right|>\epsilon$, encode as ‘1’ followed by Golomb-Rice coding of $z\left(t_n\right)$ using the predicted parameter $k\left(t_n\right)$. The predicted parameter $k\left(t_n\right)$ is calculated as follows:

(10)$k\left(t_n\right)=\left\{\begin{array}{c}0 \mathrm{if} {\displaystyle \frac{{{\displaystyle \sum }}_{j\in J_{\max }^{\left(N\right)}}z\left(t_j\right)}{N}}<1\hfill \\ \lfloor {\log }_2{\displaystyle \frac{{{\displaystyle \sum }}_{j\in J_{\max }^{\left(N\right)}}z\left(t_j\right)}{N}}\rfloor \mathrm{if} {\displaystyle \frac{{{\displaystyle \sum }}_{j\in J_{\max }^{\left(N\right)}}z\left(t_j\right)}{N}}\ge 1\hfill \end{array}\right..$

ERCS_Lossy_Flag combines a lossy compression strategy for multidimensional data with Golomb-Rice coding based on Golomb-Rice coding with the predicted parameter $k$, which enables more efficient real-time compression of data with unrestricted data accuracy. As with ERCS_Lossless, some of the historical data samples must be stored locally during the encoding process.

2.3. Overview of LEC and MOD

The proposed coding scheme is compared with two other real-time lossless data compression methods: LEC and MOD. Both encoding methods consist of a prefix and a suffix. LEC is a modified version of Exp-Golomb coding, with its code table presented in Table 3. MOD is prefixed differently than the LEC to achieve better performance on datasets with large difference values $\delta$, as shown in Table 4.

3. Experimental Results and Discussion

In this section, we evaluate the compression rates and energy savings achieved by ERCS_Lossless and ERCS_Lossy_Flag. We then discuss the effect of the pre-set parameter on compression performance.

3.1. Performance Evaluation of ERCS_Lossless

The performance of a compression algorithm is typically evaluated using the compression rate, which is defined by the following formula:

(11)$\mathrm{compression} \mathrm{rate}=\left({\displaystyle \frac{\mathrm{compressed} \mathrm{size}}{\mathrm{original} \mathrm{size}}}\right)\times 100\%,$

where compressed size refers to the number of bits after compression, whereas original size represents the number of bits in the raw data, calculated by multiplying the sensor resolution by the number of time points.

Table 5 presents the compression rates of LEC, MOD, and ERCS_Lossless for each data dimension, along with the overall compression rate. The predetermined parameter $N$ for ERCS_Lossless is set to 15.

For all 24-dimensional data, the proposed compression scheme, ERCS_Lossless, achieves the best compression performance, followed by LEC, with MOD performing the worst. The difference lies in the prefix encoding of MOD and LEC. MOD typically performs better on larger values, but since the mapped $z$ are usually smaller, the compression results are suboptimal. ERCS_Lossless encodes $z$ using the predicted parameter $k$. Although the predicted $k\left(t_n\right)$ may not always be the optimal parameter for Golomb-Rice coding of $z\left(t_n\right)$ at the current time, the increase in code length due to errors in the predicted parameter $k$ is relatively low. As a result, ERCS_Lossless still demonstrates excellent compression performance.

In practical scenarios, the transmitted payload at each time is formed by the bitwise concatenation of the encoded results for all dimensions at that time, followed by byte alignment. The calculation formula is as follows:

(12)$N_{\mathrm{payload} \mathrm{byte}}\left(t_n\right)=\lceil {\displaystyle \frac{N_{\mathrm{bitstream}}\left(t_n\right)}{8}}\rceil ,$

where $N_{\mathrm{bitstream}}\left(t_n\right)$ is the number of bits after splicing of all dimensional data at moment $t_n$, and $N_{\mathrm{payload} \mathrm{byte}}\left(t_n\right)$ is the number of payload bytes after byte alignment at moment $t_n$.

We use packets in the same format as in the paper [9], adopting the send and receive primitives of TinyOS [21]. Each data packet header occupies 9 bytes, and the maximum payload capacity is 29 bytes. The structural diagram is shown in Figure 3.

When the payload of the transmitted data does not exceed 29 bytes, only one packet is used for transmission; otherwise, two or more packets need to be transmitted. The number of packets transmitted at moment $t_n$ is calculated as follows:

(13)$N_{\mathrm{package}}\left(t_n\right)=\lceil {\displaystyle \frac{N_{\mathrm{payload} \mathrm{byte}}\left(t_n\right)}{29}}\rceil .$

The sum of the resolutions of all dimensions (aligned to the nearest byte) represents the payload of the transmitted packet at each moment of the original, where the resolution is shown in $R$ of Table 1. This can be expressed as: $N_{\mathrm{payload} \mathrm{byte}}\left(t_n\right)=\lceil {\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{24}{R}_i}{8}}\rceil =28\left(\mathrm{byte}\right)$.

The bar charts illustrating the payload distribution after compression using different methods are presented in Figure 4, Figure 5 and Figure 6. When employing ERCS_Lossless compression, the encoded results at most time steps are less than 28 bytes, with only a few instances exceeding this threshold. This anomaly arises because, at a few points in time, the predicted parameter $k$ for certain variables significantly deviates from the actual optimal parameter $k$, leading to encoding lengths longer than those of the original encoding. Consequently, the encoding efficiency is reduced, representing a limitation of this compression method. However, such instances are rare in buoy observation data. It is uncommon for individual variables to remain stable over extended periods and then exhibit abrupt changes at subsequent time steps. Moreover, these rare occurrences have a negligible impact on the overall compression rate.

Under typical operating conditions, the energy consumed by the wireless communication module for data transmission and reception is significantly greater than the energy required for data processing. The proposed compression algorithm relies on simple operations such as bit shifts, comparisons, and masking to execute the encoding process. Consequently, the computational time and resources required for the algorithm are minimal compared to the energy demands of the communication module, rendering the energy consumption of the compression algorithm negligible.

The energy saving is calculated using the following formula:

(14)$\mathrm{Energy} \mathrm{saving}\approx \left(1-{\displaystyle \frac{N_{\mathrm{transffered} \mathrm{byte} \mathrm{compressed}}}{N_{\mathrm{transffered} \mathrm{byte} \mathrm{original}} }}\right)\times 100\%,$

where $N_{\mathrm{transffered} \mathrm{byte} \mathrm{original}}$ represents the total number of bytes transmitted in the original data. It is calculated as $\left(9+28\right)\times 3697=136,789\left(\mathrm{byte}\right)$. $N_{\mathrm{transffered} \mathrm{byte} \mathrm{compressed}}$ represents the total number of bytes transmitted after compression, which is calculated using the following formula:

(15)$N_{\mathrm{transffered} \mathrm{byte} \mathrm{compressed}}={{\displaystyle \sum }}_{\mathrm{n}=1}^{3697}\left(9\times N_{\mathrm{package}}\left(t_n\right)+N_{\mathrm{payload} \mathrm{byte}}\left(t_n\right)\right).$

The energy-saving results for each method are presented in Table 6.

The proposed ERCS_Lossless method effectively reduces the total number of transmitted bytes, resulting in significant energy savings. When applied to the buoy dataset, the ERCS_Lossless algorithm has an energy saving of 38.60%. However, in real-world applications, the actual energy savings may be slightly lower due to the computational overhead associated with implementing the compression algorithm.

3.2. Performance Evaluation of ERCS_Lossy_Flag

In the lossy compression strategy, a 1-bit flag is introduced for encoding each dimension. The mapping $z\left(t_n\right)$, implemented by Equations (7) and (8), is then encoded using Golomb-Rice coding with the predicted parameter $k$. To validate the effectiveness of the flag, the proposed ERCS_Lossy_Flag is compared with a lossy compression scheme that does not introduce a flag. Specifically, when $\left|\delta \left(t_n\right)\right|\le \epsilon$, $z\left(t_n\right)$ is set to 0, and all mappings $z\left(t_n\right)$, implemented by Equation (3), are encoded using Golomb-Rice coding with the predicted parameter $k$. The data encoded by these two methods is reconstructed with the same signal in the receiver. Refer to the method of compression without flags as ERCS_Lossy_Orig. Similarly, there are LEC_Lossy_Flag, LEC_Lossy_Orig, MOD_Lossy_Flag, and MOD_Lossy_Orig.

The relationship between the overall compression rate and the MAE for each method is depicted in Figure 7, The MAE ranges from 0% to 1.0%. Similarly, Figure 8 illustrates the relationship between energy saving and the MAE. In ERCS_Lossy_Flag and ERCS_Lossy_Orig, the parameters $N$ are all set to 15. The experimental results of the ERCS_Lossy_Flag algorithm for MAE equal to 0%, 0.2%, 0.4%, 0.6%, 0.8%, and 1% are labeled in the figure.

The results show that when $\mathrm{MAE}=0.2\%$, the ERCS_Lossy_Flag algorithm has a compression rate of 45.75% and an energy saving of 39.82%. When $\mathrm{MAE}\ge 0.2\%$, ERCS_Lossy_Flag outperforms other methods. Although adding a 1-bit flag initially introduces redundancy, the compression gain increases as the MAE threshold rises.

Golomb-Rice coding is highly sensitive to the parameter $k$. When employing the ERCS_Lossy_Orig algorithm, the predicted parameter $k$ will often be equal to 0. For data exceeding the tolerance threshold in subsequent moments, the Golomb-Rice coding becomes inefficient, resulting in a compression rate that remains relatively stable despite increasing MAE. In contrast, the other methods have significantly lower compression rates with higher MAE thresholds.

In the experiments, a uniform MAE was used for all dimensions. However, in practical applications, this uniform MAE setting is not always optimal. In order to achieve the best balance between accuracy and compression rate in practical applications, different MAE values can be customized for each dimension. In practical scenarios of multidimensional data compression, different dimensions of data usually have different characteristics. For example, some dimensions usually have low prediction errors, while others may have high prediction errors. Therefore, a uniform MAE value may not be able to effectively adapt to the changing patterns of all dimensions.

In addition, it is possible to use the ERCS_Lossy_Flag algorithm for some dimensions and the ERCS_Lossless algorithm for others. This avoids the redundant overhead of 1-bit flags for dimensions that use the ERCS_Lossless algorithm. Although these 1-bit flag bits are used to indicate whether the data exceeds the prediction error threshold and to facilitate the system’s judgment of whether or not to transmit the data, they also introduce additional redundancy overhead. When flag bits are transmitted frequently for dimensions with large data variations or small MAE settings, this can lead to a degradation of the overall compression effect because each flag bit takes up an additional bit.

Nevertheless, the ERCS_Lossy_Flag algorithm provides a viable solution for the lossy compression of multidimensional real-time data. The method effectively reduces the need to transmit redundant data by introducing a flag bit mechanism that reduces the amount of transmission when the data falls below the tolerance error.

3.3. Effect of Pre-Set Parameters

Both the proposed ERCS_Lossless and ERCS_Lossy_Flag have two pre-set parameters: the amount of historical data used for predicting $N$ and the initial Golomb-Rice coding parameter $k=kFixed$.

For $kFixed$, it plays a role primarily in the encoding of the initial data points within each dimension. Its impact is limited to the early stages of the encoding process and has a negligible effect on the overall compression rate. Therefore, the choice of $kFixed$ does not need to be too precise. In our experiments, $kFixed$ is set to 3. This value shows a more stable effect in each dimension.

As for $N$, it will affect the accuracy of the predicted parameter $k$ and thus the overall compression rate. Figure 9 and Figure 10 show the effect of the pre-set parameter $N$ on the overall compression rate in ERCS_Lossless and ERCS_Lossy_Flag, respectively. When choosing $N$, it is common to choose a value that stabilizes the compression rate to avoid unnecessary computational overhead caused by too large $N$ value. As shown in Figure 9, where the decrease in compression rate is no longer as pronounced when $N$ is greater than 5, but there is still a certain downward trend. We finally chose $N=15$ because at this point the compression rate has been almost completely stabilized and will no longer change with increasing $N$ values.

4. Conclusions

Energy remains a critical limiting factor in buoy systems, driving the need for innovative methods to reduce energy consumption and optimize energy distribution. Data compression provides an effective means to minimize the data transmitted, thereby reducing the energy required by communication units for transmission. Moreover, real-time compression is crucial for scenarios that require real-time monitoring. In this study, we introduce ERCS_Lossless, which is based on Golomb-Rice coding and uses historical data to predict the Golomb-Rice coding parameter $k$ of the current data. Evaluation of the buoy dataset shows that ERCS_Lossless provides excellent compression for all dimensions of sensing data, and outperforms LEC and MOD in all data dimensions, with an overall compression rate of 47.40% and energy saving of 38.60%. In addition, we propose ERCS_Lossy_Flag, which is able to further reduce the amount of transmitted data in scenarios with unrestricted data accuracy requirements. These algorithms have great potential for practical application in buoy hardware and other applications with similar energy and data optimization requirements.

Author Contributions

Conceptualization, Q.L. and K.C.; methodology, Z.H.; software, Z.H.; validation, Z.H. and J.X.; formal analysis, Z.H. and J.X.; investigation, Z.H.; resources, Q.L.; data curation, Z.H. and J.X.; writing—original draft preparation, Z.H.; writing—review and editing, Q.L. and K.C.; visualization, J.X.; supervision, Q.L. and K.C.; project administration, Q.L.; funding acquisition, Q.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are openly available from the https://msdc.qdio.ac.cn/data/metadata-special-detail?id=1462616378971381762&otherId=1462616379004936194 (accessed on 1 January 2025)).

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Examples of measured and reconstructed signals for average air temperature.

Figure 2. Examples of measured and reconstructed signals for average barometric pressure.

Figure 3. Data packets in practical scenarios.

Figure 4. The bar chart of the payload distribution after LEC.

Figure 5. The bar chart of the payload distribution after MOD.

Figure 6. The bar chart of the payload distribution after ERCS_Lossless ([Forumla omitted. See PDF.]).

Figure 7. The relationship between the overall compression rate and the MAE.

Figure 8. The relationship between energy saving and the MAE.

Figure 9. Effect of the pre-set parameter [Forumla omitted. See PDF.] on overall compression rate in ERCS_Lossless.

Figure 10. Effect of the pre-set parameter [Forumla omitted. See PDF.] on overall compression rate in ERCS_Lossy_Flag.

Table 1

Statistical characterization and resolution information for 24 dimensions in the dataset.

Element	$\overline{\mathit{r}}\mathbf{\pm }{\mathit{\sigma }}_{\overline{\mathit{r}}}$	$\overline{\mathit{d}}\mathbf{\pm }{\mathit{\sigma }}_{\overline{\mathit{d}}}$	$\mathit{H}$	${\mathit{H}}_{\mathit{d}}$	$\mathit{R}$
1. Average wind speed (m/s)	4.03 $\pm$ 2.24	−2.71 × 10−4 $\pm$ 1.16	6.40	5.40	10
2. Wind direction of the average wind speed (°)	173.25 $\pm$ 88.07	3.33 × 10−2 $\pm$ 65.08	8.29	6.51	9
3. Maximum wind speed (m/s)	4.48 $\pm$ 2.40	−2.4 × 10−4 $\pm$ 1.24	6.49	5.48	10
4. Wind direction of the maximum wind speed (°)	173.80 $\pm$ 87.84	3.22 × 10−2 $\pm$ 67.00	8.30	6.57	9
5. Extreme wind speed (m/s)	5.14 $\pm$ 2.79	−2.71 × 10−4 $\pm$ 1.42	6.67	5.66	10
6. Wind direction of the extreme wind speed (°)	172.82 $\pm$ 86.29	3.49 × 10−2 $\pm$ 65.36	8.28	6.72	9
7. Average air temperature (°C)	21.50 $\pm$ 3.83	2.44 × 10−4 $\pm$ 0.49	7.08	4.02	10
8. Maximum air temperature (°C)	21.94 $\pm$ 3.84	3.25 × 10−4 $\pm$ 0.85	7.11	4.76	10
9. Minimum air temperature (°C)	21.25 $\pm$ 3.87	4.06 × 10−4 $\pm$ 0.70	7.09	4.54	10
10. Average humidity (%)	83.47 $\pm$ 16.53	−1.62 × 10−3 $\pm$ 4.21	5.42	3.84	7
11. Maximum humidity (%)	85.92 $\pm$ 14.51	−1.35 × 10−3 $\pm$ 3.88	5.18	3.71	7
12. Minimum humidity (%)	82.12 $\pm$ 18.14	−1.62 × 10−3 $\pm$ 4.62	5.54	3.98	7
13. Average barometric pressure (hPa)	1007.95 $\pm$ 7.32	3.63 × 10−3 $\pm$ 0.45	7.67	4.10	13
14. Maximum barometric pressure (hPa)	1008.14 $\pm$ 7.32	3.63 × 10−3 $\pm$ 0.45	7.40	4.07	13
15. Minimum barometric pressure (hPa)	1007.84 $\pm$ 7.31	3.65 × 10−3 $\pm$ 0.46	7.45	4.10	13
16. 1/10 wave height (m)	0.57 $\pm$ 0.44	−7.21 × 10−19 $\pm$ 0.11	3.62	1.93	8
17. 1/10 wave period (s)	3.64 $\pm$ 0.93	2.4 × 10−4 $\pm$ 0.36	5.07	3.75	9
18. Average wave height (m)	0.31 $\pm$ 0.24	−1.07 × 10−18 $\pm$ 0.07	2.78	1.29	8
19. Average wave period (s)	3.49 $\pm$ 0.61	8.12 × 10−5 $\pm$ 0.24	4.48	3.24	9
20. Maximum wave height (m)	0.70 $\pm$ 0.54	2.71 × 10−5 $\pm$ 0.16	3.91	2.42	8
21. Maximum wave period (s)	3.62 $\pm$ 1.07	1.35 × 10−4 $\pm$ 0.76	3.93	4.04	9
22. Significant wave height (m)	0.46 $\pm$ 0.36	−3.60 × 10−19 $\pm$ 0.09	3.36	1.67	8
23. Significant wave period (s)	3.65 $\pm$ 0.88	1.62 × 10−4 $\pm$ 0.31	4.97	3.55	9
24. Main wave direction (°)	144.60 $\pm$ 100.87	−3.25 × 10−3 $\pm$ 80.18	7.87	6.55	9

Table 2

Minimum code length and the corresponding optimum parameter $k$ for integers ranging from 0 to 255 using Golomb-Rice coding.

$\mathit{I}$	Minimum Code Length [bit]	$\mathbf{Optimal} \mathbf{Parameter} \mathit{k}$
0	1	0
1	2	0, 1
2	3	0, 1, 2
3	3	1, 2
4–5	4	1, 2, 3
6–7	4	2, 3
8–11	5	2, 3, 4
12–15	5	3, 4
16–23	6	3, 4, 5
24–31	6	4, 5
32–47	7	4, 5, 6
48–63	7	5, 6
64–95	8	5, 6, 7
96–127	8	6, 7
128–191	9	6, 7, 8
192–255	9	7, 8

Table 3

LEC codeword.

$\mathit{\delta }$	${\mathit{n}}_{\mathit{i}}$	Prefix	Suffix	Length [bit]
0	0	00	-	2
−1,+1	1	010	0:1	4
−3,−2,+2,+3	2	011	00:11	5
−7:−4,+4:+7	3	100	000:111	6
−15:−8,+8,+15	4	110	0000:1111	7
−31:−16,+16:+32	5	1110	00000:11111	9
−63:−32,+32:+63	6	11110	000000:111111	11
−127:−64,+64:+127	7	111110	0000000:1111111	13
…	…	…	…	…

Table 4

MOD codeword.

$\mathit{\delta }$	${\mathit{n}}_{\mathit{i}}$	$\mathbf{Resolution}<8$		$\mathbf{RResolution}\ge 8$
		Prefix	Length [bit]	Prefix	Length [bit]
0	0	000	3	0000	4
−1,+1	1	001	4	0001	5
−3,-2,+2,+3	2	010	5	0010	6
−7:−4,+4:+7	3	011	6	0011	7
−15:−8,+8,+15	4	100	7	0100	8
−31:−16,+16:+32	5	101	8	0101	9
−63:−32,+32:+63	6	110	9	0110	10
−127:−64,+64:+127	7	111	10	0111	11
…	…	-	-	…	…

Table 5

The compression rates of LEC, MOD, and ERCS_Lossless.

Dimension	LEC	MOD	ERCS_Lossless (N = 15)
1	68.08%	78.31%	55.08%
2	88.42%	97.29%	75.11%
3	68.67%	78.87%	55.89%
4	89.05%	97.81%	75.83%
5	70.50%	80.52%	57.80%
6	91.58%	99.68%	77.97%
7	52.79%	64.25%	41.43%
8	59.36%	70.18%	49.76%
9	57.15%	68.26%	47.35%
10	71.64%	74.37%	55.77%
11	68.81%	72.38%	53.66%
12	73.36%	75.87%	57.81%
13	42.28%	50.99%	32.63%
14	41.58%	50.60%	32.67%
15	41.80%	50.76%	32.79%
16	40.62%	59.87%	24.61%
17	57.32%	69.98%	43.22%
18	34.11%	55.57%	18.83%
19	51.73%	65.05%	37.57%
20	46.34%	63.97%	30.28%
21	66.03%	79.25%	55.30%
22	37.92%	58.02%	22.07%
23	55.12%	68.00%	41.09%
24	88.55%	97.24%	75.23%
Overall	60.08%	71.03%	47.40%

Table 6

Energy saving results for each method.

LEC	MOD	$\mathbf{ERCS}\mathbf{\_}\mathbf{Lossless} \mathbf{(}\mathit{N}\mathbf{=}15\mathbf{)}$
29.01%	20.74%	38.60%

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