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Location estimation in wireless sensor networks is the problem of estimating the location of an object from a set of noisy measurements, when the measurements are acquired in a distributed manner by a set of sensors.

Motivation

Many civilian and military applications require monitoring that can identify objects in a specific area, such as monitoring the front entrance of a private house by a single camera. Monitored areas that are large relative to objects of interest often require multiple sensors (e.g., infra-red detectors) at multiple locations. A centralized observer or computer application monitors the sensors. The communication to Power and bandwidth requirements call for efficient design of the sensor, transmission, and processing.

The CodeBlue system[1] of Harvard university is an example where a vast number of sensors distributed among hospital facilities allow staff to locate a patient in distress. In addition, the sensor array enables online recording of medical information while allowing the patient to move around. Military applications (e.g. locating an intruder into a secured area) are also good candidates for setting a wireless sensor network.

Setting

File:LocationEstimation WSN.JPG

Let $θ$ denote the position of interest. A set of $N$ sensors acquire measurements $x_{n} = θ + w_{n}$ contaminated by an additive noise $w_{n}$ owing some known or unknown probability density function (PDF). The sensors transmit measurements to a central processor. The $n$ th sensor encodes $x_{n}$ by a function $m_{n} (x_{n})$ . The application processing the data applies a pre-defined estimation rule $\hat{θ} = f (m_{1} (x_{1}), \cdot, m_{N} (x_{N}))$ . The set of message functions $m_{n}, 1 \leq n \leq N$ and the fusion rule $f (m_{1} (x_{1}), \cdot, m_{N} (x_{N}))$ are designed to minimize estimation error. For example: minimizing the mean squared error (MSE), $𝔼 ‖ θ - \hat{θ} ‖^{2}$ .

Ideally, sensors transmit their measurements $x_{n}$ exactly to the processing center, that is $m_{n} (x_{n}) = x_{n}$ . In this settings, the maximum likelihood estimator (MLE) $\hat{θ} = \frac{1}{N} \sum_{n = 1}^{N} x_{n}$ is an unbiased estimator whose MSE is $𝔼 ‖ θ - \hat{θ} ‖^{2} = var (\hat{θ}) = \frac{σ^{2}}{N}$ assuming a white Gaussian noise $w_{n} \sim 𝒩 (0, σ^{2})$ . The next sections suggest alternative designs when the sensors are bandwidth constrained to 1 bit transmission, that is $m_{n} (x_{n})$ =0 or 1.

Known noise PDF

We begin with an example of a Gaussian noise $w_{n} \sim 𝒩 (0, σ^{2})$ , in which a suggestion for a system design is as follows

^[1]

m_{n} (x_{n}) = I (x_{n} - τ) = {\begin{cases} 1 & x_{n} > τ \\ 0 & x_{n} \leq τ \end{cases}

\hat{θ} = τ - F^{- 1} (\frac{1}{N} \sum_{n = 1}^{N} m_{n} (x_{n})), F (x) = \frac{1}{\sqrt{2 π} σ} \int_{x}^{\infty} e^{- w^{2} / 2 σ^{2}} d w

Here $τ$ is a parameter leveraging our prior knowledge of the approximate location of $θ$ . In this design, the random value of $m_{n} (x_{n})$ is distributed Bernoulli~ $(q = F (τ - θ))$ . The processing center averages the received bits to form an estimate $\hat{q}$ of $q$ , which is then used to find an estimate of $θ$ . It can be verified that for the optimal (and infeasible) choice of $τ = θ$ the variance of this estimator is $\frac{π σ^{2}}{4}$ which is only $π / 2$ times the variance of MLE without bandwidth constraint. The variance increases as $τ$ deviates from the real value of $θ$ , but it can be shown that as long as $| τ - θ | \sim σ$ the factor in the MSE remains approximately 2. Choosing a suitable value for $τ$ is a major disadvantage of this method since our model does not assume prior knowledge about the approximated location of $θ$ . A coarse estimation can be used to overcome this limitation. However, it requires additional hardware in each of the sensors.

A system design with arbitrary (but known) noise PDF can be found in.^[2] In this setting it is assumed that both $θ$ and the noise $w_{n}$ are confined to some known interval $[- U, U]$ . The estimator of ^[2] also reaches an MSE which is a constant factor times $\frac{σ^{2}}{N}$ . In this method, the prior knowledge of $U$ replaces the parameter $τ$ of the previous approach.

Unknown noise parameters

A noise model may be sometimes available while the exact PDF parameters are unknown (e.g. a Gaussian PDF with unknown $σ$ ). The idea proposed in ^[3] for this setting is to use two thresholds $τ_{1}, τ_{2}$ , such that $N / 2$ sensors are designed with $m_{A} (x) = I (x - τ_{1})$ , and the other $N / 2$ sensors use $m_{B} (x) = I (x - τ_{2})$ . The processing center estimation rule is generated as follows:

{\hat{q}}_{1} = \frac{2}{N} \sum_{n = 1}^{N / 2} m_{A} (x_{n}), {\hat{q}}_{2} = \frac{2}{N} \sum_{n = 1 + N / 2}^{N} m_{B} (x_{n})

\hat{θ} = \frac{F^{- 1} ({\hat{q}}_{2}) τ_{1} - F^{- 1} ({\hat{q}}_{1}) τ_{2}}{F^{- 1} ({\hat{q}}_{2}) - F^{- 1} ({\hat{q}}_{1})}, F (x) = \frac{1}{\sqrt{2 π}} \int_{x}^{\infty} e^{- v^{2} / 2} d w

As before, prior knowledge is necessary to set values for $τ_{1}, τ_{2}$ to have an MSE with a reasonable factor of the unconstrained MLE variance.

Unknown noise PDF

We now describe the system design of ^[2] for the case that the structure of the noise PDF is unknown. The following model is considered for this scenario:

x_{n} = θ + w_{n}, n = 1, \dots, N

θ \in [- U, U]

w_{n} \in 𝒫, that is : w_{n} is bounded to [- U, U], 𝔼 (w_{n}) = 0

In addition, the message functions are limited to have the form

m_{n} (x_{n}) = {\begin{cases} 1 & x \in S_{n} \\ 0 & x \notin S_{n} \end{cases}

where each $S_{n}$ is a subset of $[- 2 U, 2 U]$ . The fusion estimator is also restricted to be linear, i.e. $\hat{θ} = \sum_{n = 1}^{N} α_{n} m_{n} (x_{n})$ .

The design should set the decision intervals $S_{n}$ and the coefficients $α_{n}$ . Intuitively, we would allocate $N / 2$ sensors to encode the first bit of $θ$ by setting their decision interval to be $[0, 2 U]$ , then $N / 4$ sensors would encode the second bit by setting their decision interval to $[- U, 0] \cup [U, 2 U]$ and so on. It can be shown that these decision intervals and the corresponding set of coefficients $α_{n}$ produce a universal $δ$ -unbiased estimator, which is an estimator satisfying $| 𝔼 (θ - \hat{θ}) | < δ$ for every possible value of $θ \in [- U, U]$ and for every realization of $w_{n} \in 𝒫$ . In fact, this intuitive design of the decision intervals is also optimal in the following sense. The above design requires $N \geq ⌈ \log \frac{8 U}{δ} ⌉$ to satisfy the universal $δ$ -unbiased property while theoretical arguments show that an optimal (and a more complex) design of the decision intervals would require $N \geq ⌈ \log \frac{2 U}{δ} ⌉$ , that is: the number of sensors is nearly optimal. It is also argued in ^[2] that if the targeted MSE $𝔼 ‖ θ - \hat{θ} ‖ \leq ϵ^{2}$ uses a small enough $ϵ$ , then this design requires a factor of 4 in the number of sensors to achieve the same variance of the MLE in the unconstrained bandwidth settings.

Additional information

The design of the sensor array requires optimizing the power allocation as well as minimizing the communication traffic of the entire system. The design suggested in ^[4] incorporates probabilistic quantization in sensors and a simple optimization program that is solved in the fusion center only once. The fusion center then broadcasts a set of parameters to the sensors that allows them to finalize their design of messaging functions $m_{n} (\cdot)$ as to meet the energy constraints. Another work employs a similar approach to address distributed detection in wireless sensor arrays.^[5]

External links

CodeBlue Harvard group working on wireless sensor network technology to a range of medical applications.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Template:Wireless Sensor Network

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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[1] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[Luo-2] 2.0 ^2.1 ^2.2 ^2.3 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[3] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[4] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[5] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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BTZ black hole: Difference between revisions

Revision as of 01:48, 20 December 2013

Contents

Motivation

Setting

Known noise PDF

Unknown noise parameters

Unknown noise PDF

Additional information

External links

References

Navigation menu

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+{{cleanup|date=July 2009}}
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+'''Location estimation''' in '''[[wireless sensor networks]]''' is the problem of [[Estimation theory|estimating]] the location of an object from a set of noisy measurements, when the measurements are acquired in a distributed
+manner by a set of sensors.
+==Motivation==
+Many civilian and military applications require monitoring that can identify objects in a specific area, such as monitoring the front entrance of a private house by a single camera. Monitored areas that are large relative to objects of interest often require multiple sensors (e.g., infra-red detectors) at multiple locations. A centralized observer or computer application monitors the sensors. The communication to Power and bandwidth requirements call for efficient design of the sensor, transmission, and processing.
+The ''[[CodeBlue system]]''[http://www.eecs.harvard.edu/~mdw/proj/codeblue/] of [[Harvard university]] is an example where a
+vast number of sensors distributed among hospital facilities
+allow staff to locate a patient in distress. In addition, the sensor
+array enables online recording of medical information while
+allowing the patient to move around. Military applications (e.g.
+locating an intruder into a secured area) are also good candidates
+for setting a wireless sensor network.
+==Setting==
+[[Image:LocationEstimation WSN.JPG|right|370px]]
+Let <math>\theta</math> denote the position of interest. A set of <math>N</math> sensors
+acquire measurements <math>x_n = \theta + w_n</math> contaminated by an
+additive noise <math>w_n</math> owing some known or unknown [[probability density function]] (PDF). The sensors transmit measurements to a central processor. The <math>n</math>th sensor encodes
+<math>x_n</math> by a function <math>m_n(x_n)</math>. The application processing the data applies a pre-defined estimation rule
+<math>\hat{\theta}=f(m_1(x_1),\cdot,m_N(x_N))</math>. The set of message functions
+<math>m_n,\, 1\leq n\leq N</math> and the fusion rule <math>f(m_1(x_1),\cdot,m_N(x_N))</math> are
+designed to minimize estimation error.
+For example: minimizing the [[mean squared error]] (MSE),
+<math>\mathbb{E}\|\theta-\hat{\theta}\|^2</math>.
+Ideally, sensors transmit their measurements <math>x_n</math>
+exactly to the processing center, that is <math>m_n(x_n)=x_n</math>. In this
+settings, the [[maximum likelihood estimator]] (MLE) <math>\hat{\theta} =
+\frac{1}{N}\sum_{n=1}^N x_n</math> is an [[unbiased estimator]] whose MSE is
+<math>\mathbb{E}\|\theta-\hat{\theta}\|^2 = \text{var}(\hat{\theta}) =
+\frac{\sigma^2}{N}</math> assuming a white [[Normal distribution|Gaussian]] noise
+<math>w_n\sim\mathcal{N}(0,\sigma^2)</math>. The next sections suggest
+alternative designs when the sensors are bandwidth constrained to
+bit transmission, that is <math>m_n(x_n)</math>=0 or 1.
+==Known noise PDF==
+We begin with an example of a Gaussian noise
+<math>w_n\sim\mathcal{N}(0,\sigma^2)</math>, in which a suggestion for a
+system design is as follows
+:<ref>{{cite journal
+  | last = Ribeiro
+  | first = Alejandro
+  | coauthors = [[Georgios B. Giannakis]]
+  | title = Bandwidth-constrained distributed estimation for wireless sensor Networks-part I: Gaussian case
+  | journal = IEEE Trans. On Sig. Proc.
+  | date = March 2006}}
+</ref>
+: <math>
+m_n(x_n)=I(x_n-\tau)=
+\begin{cases}
+& x_n > \tau  \\
+& x_n\leq \tau
+\end{cases}
+</math>
+: <math>
+\hat{\theta}=\tau-F^{-1}\left(\frac{1}{N}\sum\limits_{n=1}^{N}m_n(x_n)\right),\quad
+F(x)=\frac{1}{\sqrt{2\pi}\sigma} \int\limits_{x}^{\infty}
+e^{-w^2/2\sigma^2} \, dw
+</math>
+Here <math>\tau</math> is a parameter leveraging our prior knowledge of the
+approximate location of <math>\theta</math>. In this design, the random value
+of <math>m_n(x_n)</math> is distributed [[Bernoulli distribution|Bernoulli]]~<math>(q=F(\tau-\theta))</math>. The
+processing center averages the received bits to form an estimate
+<math>\hat{q}</math> of <math>q</math>, which is then used to find an estimate of <math>\theta</math>. It can be verified that for the optimal (and
+infeasible) choice of <math>\tau=\theta</math> the variance of this estimator
+is <math>\frac{\pi\sigma^2}{4}</math> which is only <math>\pi/2</math> times the
+variance of MLE without bandwidth constraint. The variance
+increases as <math>\tau</math> deviates from the real value of <math>\theta</math>, but it can be shown that as long as <math>|\tau-\theta|\sim\sigma</math> the factor in the MSE remains approximately 2. Choosing a suitable value for <math>\tau</math> is a major disadvantage of this method since our model does not assume prior knowledge about the approximated location of <math>\theta</math>. A coarse estimation can be used to overcome this limitation. However, it requires additional hardware in each of
+the sensors.
+A system design with arbitrary (but known) noise PDF can be found in.<ref name="Luo">{{cite journal
+  | last = Luo
+  | first = Zhi-Quan
+  | title = Universal decentralized estimation in a bandwidth constrained sensor network
+  | journal = IEEE Trans. On Inf. Th.
+  | date = June 2005}}
+</ref> In this setting it is assumed that both <math>\theta</math> and
+the noise <math>w_n</math> are confined to some known interval <math>[-U,U]</math>. The
+estimator of <ref name="Luo"/> also reaches an MSE which is a constant factor
+times <math>\frac{\sigma^2}{N}</math>. In this method, the prior knowledge of <math>U</math> replaces
+the parameter <math>\tau</math> of the previous approach.
+==Unknown noise parameters==
+A noise model may be sometimes available while the exact PDF parameters are unknown (e.g. a Gaussian PDF with unknown <math>\sigma</math>). The idea proposed in <ref>{{cite journal
+  | last = Ribeiro
+  | first = Alejandro
+  | coauthors = [[Georgios B. Giannakis]]
+  | title = Bandwidth-constrained distributed estimation for wireless sensor networks-part II: unknown probability density function
+  | journal = IEEE Trans. On Sig. Proc.
+  | date = July 2006}}
+</ref> for this setting is to use two
+thresholds <math>\tau_1,\tau_2</math>, such that <math>N/2</math> sensors are designed
+with <math>m_A(x)=I(x-\tau_1)</math>, and the other <math>N/2</math> sensors use
+<math>m_B(x)=I(x-\tau_2)</math>. The processing center estimation rule is generated as follows:
+: <math>
+\hat{q}_1=\frac{2}{N}\sum\limits_{n=1}^{N/2}m_A(x_n), \quad
+\hat{q}_2=\frac{2}{N}\sum\limits_{n=1+N/2}^{N}m_B(x_n)
+</math>
+: <math>
+\hat{\theta}=\frac{F^{-1}(\hat{q}_2)\tau_1-F^{-1}(\hat{q}_1)\tau_2}{F^{-1}(\hat{q}_2)-F^{-1}(\hat{q}_1)},\quad
+F(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{x}^{\infty}e^{-v^2/2}dw
+</math>
+As before, prior knowledge is necessary to set values for
+<math>\tau_1,\tau_2</math> to have an MSE with a reasonable factor
+of the unconstrained MLE variance.
+==Unknown noise PDF==
+We now describe the system design of <ref name="Luo"/> for the case that the structure of the noise
+PDF is unknown. The following model is considered for this scenario:
+: <math>
+x_n=\theta+w_n,\quad n=1,\dots,N
+</math>
+: <math>
+\theta\in[-U,U]
+</math>
+: <math>
+w_n\in\mathcal{P}, \text{ that is }: w_n \text{ is bounded to }
+[-U,U], \mathbb{E}(w_n)=0
+</math>
+In addition, the message functions are limited to have the form
+: <math>
+m_n(x_n)=
+\begin{cases}
+& x\in S_n  \\
+& x \notin S_n
+\end{cases}
+</math>
+where each <math>S_n</math> is a subset of <math>[-2U,2U]</math>. The fusion estimator is also restricted to be linear, i.e.
+<math>\hat{\theta}=\sum\limits_{n=1}^{N}\alpha_n m_n(x_n)</math>.
+The design should set the decision intervals <math>S_n</math> and the
+coefficients <math>\alpha_n</math>. Intuitively, we would allocate <math>N/2</math> sensors to encode the first bit of <math>\theta</math> by setting their decision interval to be <math>[0,2U]</math>, then <math>N/4</math> sensors would encode the second bit by setting their decision interval to
+<math>[-U,0]\cup[U,2U]</math> and so on. It can be shown that these decision
+intervals and the corresponding set of coefficients <math>\alpha_n</math>
+produce a universal <math>\delta</math>-unbiased estimator, which is an
+estimator satisfying <math>|\mathbb{E}(\theta-\hat{\theta})|<\delta</math>
+for every possible value of <math>\theta\in[-U,U]</math> and for every realization of <math>w_n\in\mathcal{P}</math>. In fact, this intuitive
+design of the decision intervals is also optimal in the following
+sense. The above design requires
+<math>N\geq\lceil\log\frac{8U}{\delta}\rceil</math> to satisfy the universal
+<math>\delta</math>-unbiased property while theoretical arguments show that
+an optimal (and a more complex) design of the decision intervals
+would require <math>N\geq\lceil\log\frac{2U}{\delta}\rceil</math>, that is:
+the number of sensors is nearly optimal. It is also argued in <ref name="Luo"/>
+that if the targeted MSE
+<math>\mathbb{E}\|\theta-\hat{\theta}\|\leq\epsilon^2</math> uses a small
+enough <math>\epsilon</math>, then this design requires a factor of 4 in the
+number of sensors to achieve the same variance of the MLE in
+the unconstrained bandwidth settings.
+==Additional information==
+The design of the sensor array requires optimizing the power
+allocation as well as minimizing the communication traffic of the
+entire system. The design suggested in <ref>{{cite journal
+  | last = Xiao
+  | first = Jin-Jun
+  | coauthors = Andrea J. Goldsmith
+  | title = Joint estimation in sensor networks under energy constraint
+  | journal = IEEE Trans. On Sig. Proc.
+  | date = June 2005}}
+</ref> incorporates probabilistic quantization in
+sensors and a simple optimization program that is solved in the
+fusion center only once. The fusion center then broadcasts a set
+of parameters to the sensors that allows them to finalize their
+design of messaging functions <math>m_n(\cdot)</math> as to meet the energy
+constraints. Another work employs a similar approach to address
+distributed detection in wireless sensor arrays.<ref>{{cite journal
+  | last = Xiao
+  | first = Jin-Jun
+  | coauthors = Zhi-Quan Luo
+  | title = Universal decentralized detection in a bandwidth-constrained sensor network
+  | journal = IEEE Trans. On Sig. Proc.
+  | date = August 2005}}</ref>
+== External links ==
+* [http://www.eecs.harvard.edu/~mdw/proj/codeblue/ CodeBlue] Harvard group working on wireless sensor network technology to a range of medical applications.
+== References ==
+{{reflist}}
+{{Wireless Sensor Network}}
+[[Category:Estimation theory]]
+[[Category:Detection theory]]
+[[Category:Wireless sensor network]]

BTZ black hole: Difference between revisions

Revision as of 01:48, 20 December 2013

Motivation

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