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their activity levels despite fear of injury and discomfort associated with the use of deconditioned muscles (Dolce, Crocker, Moletteire, & Doleys, 1986). Operant Conditioning Contingencies of Reinforcement The effects of environmental factors in shaping the experience of people suffering with pain was acknowledged long ago (Collie, 1913). A new era in thinking about pain began with Fordyce s (1976) extension of operant conditioning to chronic pain. The main focus of operant learning is modi cation in frequency of a given behavior. The fundamental principle is that if the consequence of a given behavior is rewarding, its occurrence increases; whereas if the consequence is aversive, the likelihood of its occurrence decreases. When a person is exposed to a stimulus that causes tissue damage, the immediate behavioral response is withdrawal in an attempt to escape from noxious sensations. Such re exive behaviors are adaptive and appropriate. Behaviors associated with pain, such as limping and moaning, are called pain behaviors. Pain behaviors include overt expressions of pain, distress, and suffering. According to Fordyce (1976), these behaviors can become subjected to the principles of operant conditioning. These behaviors may be positively reinforced directly, for example, by attention from a family member, acquaintance, or health care provider. The principles of operant learning suggest that behaviors that are positively reinforced will be reported more frequently. Pain behaviors may also be maintained by the escape from noxious stimulation by the use of drugs or rest, or the avoidance of undesirable activities. In addition well behaviors (e.g., activity, working) may not be positively reinforced, and the more rewarded pain behaviors may therefore be maintained. The following example illustrates the role of operant conditioning: When back pain ares up, the sufferer may lie down and hold her back. Her husband may observe her behavior and infer that she is experiencing pain. He may respond by offering to rub her back. This response may positively reward the woman and her pain behaviors (i.e., lying down) may be repeated even in the absence of pain. In other words, her pain behaviors are being maintained by the learned consequences. The woman s pain behaviors are reinforced by allowing the person to avoid undesirable activities. When observing his wife lying on the oor, her husband may suggest that they cancel the evening plans with his brother, an activity that she may have preferred to avoid anyway. In this situation, her
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The principal mathematical operation of the gradient method is called multilateration. Multilateration is a great deal like triangulation, except that multilateration can incorporate ranges from more than three reference points. Formally, given m beacons with known Cartesian positions bi , i 1 m and possibly noisy range measurements ri from the known nodes to an unknown sensor node s, multilateration nds the most likely position of s. The mathematics of multilateration are outlined in Appendix 1.9. Using gradients to compute ranges for multilateration has been proposed by a number of researchers [11,16 19]. These algorithms all assume that there are at least three beacon nodes somewhere in the network (though probably more). Each of these beacon nodes propagates a gradient through the network, which is the distributed equivalent of computing the shortest path distance between all the beacons and all of the unlocalized nodes. The gradient propagation is as follows: Step 1. For each node j and beacon k, let djk (the distance from j to k) be 0 if j k, and 1 otherwise. Step 2. On each node j, perform the following steps repeatedly: Step 2a. For each beacon k and neighbor i, retrieve dik from i. Step 2b. For each beacon k and neighbor i, apply the following update formula: ^ d jk min (dik rij , d jk ) ^ where rij is the estimated distance between nodes i and j. These internode distance estimates can be either unweighted (one if there is connectivity, zero otherwise) or measured distances (e.g., using RSSI or TDoA). After some amount of settling time, each value djk will be the length of the shortest path between node j and beacon k. Figure 9.6 shows the results of running the gradient propagation algorithm with one beacon. The gradient-based distance estimate to a beacon must be adjusted, since even given perfect internode distance estimates, gradient distance estimates will always be longer than (or exactly equal to) corresponding straight-line distances. Of course, given imperfect internode distance estimates, gradient-based distance estimate can actually be shorter than straight distances. In fact, Whitehouse [3] shows that it is actually more likely that they are shorter, since underestimated internode distances skew all subsequent gradient-based estimates. Niculescu and Nath [20] suggest using a correction factor calculated by comparing the actual distance between beacons to the shortest path distances computed during gradient propagation. Each unlocalized node simply applies the correction factor from its closest beacon to its gradient distance estimate. As an alternative, Nagpal et al. [5] in their Amorphous algorithm suggest correcting this distance based on the neighborhood size nlocal, as we previously discussed in Subsection 9.2.3.
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Emerging Wireless LANs, Wireless PANs, and Wireless MANs. Edited by Y. Xiao and Y. Pan Copyright r 2009 John Wiley & Sons, Inc.
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