## 1 Introduction  
Data collection is a fundamental application of wireless sensor networks (WSNs). For example, in the environmental monitoring scenario, the physical quantities, such as temperature, humidity and light illumination, _etc._ are usually sensed and transmitted from sensor nodes to sink nodes through multi-hop routing [1]. Since the sensor nodes usually have limited computing ability and power supply, a primary goal of data collection is to obtain the sensing data at required accuracy with least energy consumption and thus to prolong the network lifetime. Since the physical quantities describing natural phenomena are generally locally consistent and WSNs nodes are often deployed redundantly to get complete environmental monitoring data, the sensing data often have high spatial or temporal correlation [2]. Thus many researchers utilize the spatial and temporal correlation to reduce the network energy consumption by delivering part of sensing data or their aggregation, such as distributed source coding techniques [3; 4; 5], in-network collaborative wavelet transform methods [6; 7] and clustered data aggregation methods [8; 9; 10; 11]. Although these methods can save certain data communication cost, they would bring extra computational overheads for sensor nodes and so demand considerable sensing cost. Recently the low rank matrix approximation (LRMA) method [12; 13] provides a new scheme for data collection in WSNs [14; 15; 16]. The main idea of the LRMA based method is that the sensing data of WSNs are regarded as a low rank matrix and recovered from a subset sensing data. As demonstrated in Figure 1, the data collection consists of two stages: the data sensing stage and the data recovering stage. At the data sensing stage, instead of using all sensor nodes, a subset of sensor nodes which are the shaded ones in the second subfigure is randomly selected to sense physical quantity and deliver the sensing data to the sink node at each data collection round. At the data recovering stage, the sink node receives these incomplete sensing data over some data collection rounds shown in the third subfigure in which the shaded entries represent the valid sensing data and the white entries are unknown. And then we could use them to recover the complete data by LRMA method. As the LRMA based data collection methods reduce either the data communication cost or the data sensing cost, high energy efficiency can be achieved even with an increased computation cost at the sink node. Figure 1: The correlated spatio-temporal data collection method based on LRMA and optimized node sampling. However, the current LRMA based data collection methods [14; 15; 16] also suffer from certain inefficiency both at the data sensing stage and the data recovering stage. At the data sensing stage, a random sampling scheme for the selection of sensing nodes is adopted in the current methods [14; 15; 16]. According to the LRMA theory [13], the uniformly random sampling method is an optimized sampling way to get good matrix complement result. However, the ideal random sampling method will result in uneven energy distribution of the network as the many-to-one data collection pattern, and thus shorten the network lifetime. At the data recovering stage, most of the current methods do not explore and utilize the spatial or temporal correlation of the sensing data. This could be the contributing factor to the certain setback of these methods, such as low data recovery accuracy, and lacking robustness to noises and outliers. Therefore, to improve the efficiency of the current LRMA based data collection methods, we always face two challenges. On the one hand, we should find out a proper way to formulate the relation between two important components in the data sensing procedure, the node sampling scheme and the scheme of estimating energy status of the network. Generally, the former has direct effect on the data recovery accuracy, and the later can determine the lifetime of the network.