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Template:Expert-subject In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ^[1]) which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and probability measures which are a subset of classical measures.

Definitions

Let $X$ be a universe of discourse, $𝒞$ be a class of subsets of $X$ , and $E, F \in 𝒞$ . A function $g : 𝒞 \to ℝ$ where

is called a fuzzy measure. A fuzzy measure is called normalized or regular if $g (X) = 1$ .

Properties of fuzzy measures

For any $E, F \in 𝒞$ , a fuzzy measure is:

additive if $g (E \cup F) = g (E) + g (F) .$ for all $E \cap F = \emptyset$ ;
supermodular if $g (E \cup F) + g (E \cap F) \geq g (E) + g (F)$ ;
submodular if $g (E \cup F) + g (E \cap F) \leq g (E) + g (F)$ ;
superadditive if $g (E \cup F) + g (E \cap F) \geq g (E) + g (F)$ for all $E \cap F = \emptyset$ ;
subadditive if $g (E \cup F) + g (E \cap F) \leq g (E) + g (F)$ for all $E \cap F = \emptyset$ ;
symmetric if $| E | = | F |$ implies $g (E) = g (F)$ ;
Boolean if $g (E) = 0$ or $g (E) = 1$ .

Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.

Möbius representation

Let g be a fuzzy measure, the Möbius representation of g is given by the set function M, where for every $E, F \subseteq X$ ,

M (E) = \sum_{F \subseteq E} (- 1)^{| E ∖ F |} g (F) .

The equivalent axioms in Möbius representation are:

$M (\emptyset) = 0$ .
$\sum_{F \subseteq E | i \in F} M (F) \geq 0$ , for all $E \subseteq X$ and all $i \in E$

A fuzzy measure in Möbius representation M is called normalized if $\sum_{E \subseteq X} M (E) = 1 .$

Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure g in standard representation can be recovered from the Möbius form using the Zeta transform:

g (E) = \sum_{F \subseteq E} M (F), \forall E \subseteq X .

Simplification assumptions for fuzzy measures

Fuzzy measures are defined on a semiring of sets or monotone class which may be as granular as the power set of X, and even in discrete cases the number of variables can as large as 2^|X|. For this reason, in the context of multi-criteria decision analysis and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that $g (E) = \sum_{i \in E} g ({i})$ and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or $λ$ -fuzzy measure and k-additive measures, introduced by Sugeno^[2] and Grabisch^[3] respectively.

Sugeno λ-measure

The Sugeno $λ$ -measure is a special case of fuzzy measures defined iteratively. It has the following definition:

Definition

Let $X = {x_{1}, \dots, x_{n}}$ be a finite set and let $λ \in (- 1, + \infty)$ . A Sugeno $λ$ -measure is a function $g : 2^{X} \to [0, 1]$ such that

$g (X) = 1$ .
if $A, B \subseteq X$ (alternatively $A, B \in 2^{X}$ ) with $A \cap B = \emptyset$ then $g (A \cup B) = g (A) + g (B) + λ g (A) g (B)$ .

As a convention, the value of g at a singleton set ${x_{i}}$ is called a density and is denoted by $g_{i} = g ({x_{i}})$ . In addition, we have that $λ$ satisﬁes the property

λ + 1 = \prod_{i = 1}^{n} (1 + λ g_{i})

.

Tahani and Keller ^[4] as well as Wang and Klir have showed that once the densities are known, it is possible to use the previous polynomial to obtain the values of $λ$ uniquely.

k-additive fuzzy measure

The k-additive fuzzy measure limits the interaction between the subsets $E \subseteq X$ to size $| E | = k$ . This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to X, it allows for a compromise between modelling ability and simplicity.

Definition

A discrete fuzzy measure g on a set X is called k-additive ( $1 \leq k \leq | X |$ ) if its Möbius representation verifies $M (E) = 0$ , whenever $| E | > k$ for any $E \subseteq X$ , and there exists a subset F with k elements such that $M (F) \neq 0$ .

Shapley and interaction indices

In game theory, the Shapley value or Shapley index is used to indicate the weight of a game. Shapley values can calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton.

For a given fuzzy measure g, and $| X | = n$ , the Shapley index for every $i, \dots, n \in X$ is:

ϕ (i) = \sum_{E \subseteq X ∖ {i}} \frac{(n - | E | - 1)! | E |!}{n!} [g (E \cup {i}) - g (E)] .

The Shapley value is the vector $ϕ (g) = (ψ (1), \dots, ψ (n)) .$

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.

Beliakov, Pradera and Calvo, Aggregation Functions: A Guide for Practitioners, Springer, New York 2007.
Wang, Zhenyuan, and, George J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1991.

External links

Fuzzy Measure Theory at Fuzzy Image Processing

↑ 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
↑ 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

[2] 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

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+{{Expert-subject|Mathematics|date=November 2008}}
+In [[mathematics]], '''fuzzy measure theory''' considers generalized [[measure (mathematics)|measure]]s in which the additive property is replaced by the weaker property of monotonicity.  The central concept of fuzzy measure theory is the fuzzy measure (also ''capacity'', see <ref>{{cite journal
+|author         = [[Gustave Choquet]]
+|title          = Theory of Capacities
+|journal        = Annales de l'Institut Fourier
+|volume         = 5
+|pages          = 131–295
+|year           = 1953
+}}</ref>) which was introduced by [[Gustave Choquet|Choquet]] in 1953 and independently defined by Sugeno in 1974 in the context of [[Sugeno integral|fuzzy integrals]].  There exists a number of different classes of fuzzy measures including [[Dempster-Shafer theory|plausibility/belief]] measures; [[Possibility theory|possibility/necessity]] measures; and [[probability measure|probability]] measures which are a subset of [[measure (mathematics)|classical]] measures.
+==Definitions==
+Let <math>\mathbf{X}</math> be a [[universe of discourse]], <math>\mathcal{C}</math> be a [[Class (mathematics)|class]] of [[subset]]s of <math>\mathbf{X}</math>, and <math>E,F\in\mathcal{C}</math>.  A [[function (mathematics)|function]] <math>g:\mathcal{C}\to\mathbb{R}</math> where
+# <math>\emptyset \in \mathcal{C} \Rightarrow g(\emptyset)=0</math>
+# <math>E \subseteq F \Rightarrow g(E)\leq g(F)</math>
+is called a ''fuzzy measure''.
+A fuzzy measure is called ''normalized'' or ''regular'' if <math>g(\mathbf{X})=1</math>.
+<!--
+Adding some examples of fuzzy measures-->
+==Properties of fuzzy measures==
+For any <math> E,F \in \mathcal{C} </math>, a fuzzy measure is:
+* '''additive''' if  <math> g(E \cup F) = g(E) + g(F). </math>  for all <math>  E \cap F = \emptyset </math>;
+* '''supermodular''' if <math>g(E \cup F) + g(E \cap F) \geq g(E) + g(F)</math>;
+*  '''[[submodular]]''' if  <math>g(E \cup F) + g(E \cap F) \leq g(E) + g(F)</math>;
+*  '''superadditive''' if  <math>g(E \cup F) + g(E \cap F) \geq g(E) + g(F)</math> for all <math>E \cap F = \emptyset</math>;
+*  '''subadditive''' if  <math>g(E \cup F) + g(E \cap F) \leq g(E) + g(F)</math> for all <math>E \cap F = \emptyset</math>;
+*  '''symmetric''' if <math>|E| = |F|</math> implies <math> g(E) = g(F)</math>;
+*  '''Boolean''' if <math> g(E) = 0 </math> or <math> g(E) = 1 </math>.
+Understanding the properties of fuzzy measures is useful in application.  When a fuzzy measure is used to define a function such as the [[Sugeno integral]] or [[Choquet integral]], these properties will be crucial in understanding the function's behavior.  For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the [[Lebesgue integral]].  In discrete cases, a symmetric fuzzy measure will result in the [[Ordered weighted averaging aggregation operator|ordered weighted averaging]] (OWA) operator.  Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
+==Möbius representation==
+Let ''g'' be a fuzzy measure, the Möbius representation of ''g'' is given by the set function ''M'', where for every <math> E,F \subseteq X </math>,
+:<math>M(E) = \sum_{F \subseteq E} (-1)^{|E \backslash F|} g(F).</math>
+The equivalent axioms in Möbius representation are:
+# <math> M(\emptyset)=0</math>.
+# <math> \sum_{F \subseteq E|i \in F} M(F) \geq 0</math>, for all <math> E \subseteq \mathbf{X} </math> and all <math> i \in E </math>
+A fuzzy measure in Möbius representation  ''M'' is called ''normalized''
+if <math>\sum_{E \subseteq \mathbf{X}}M(E)=1.</math>
+Möbius representation can be used to give an indication of which subsets of '''X''' interact with one another.  For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons.    The fuzzy measure ''g'' in standard representation can be recovered from the Möbius form using the Zeta transform:
+:<math>  g(E) = \sum_{F \subseteq E} M(F), \forall E \subseteq \mathbf{X} .</math>
+==Simplification assumptions for fuzzy measures==
+Fuzzy measures are defined on a [[semiring|semiring of sets]] or [[monotone class]] which may be as granular as the [[power set]] of '''X''', and even in discrete cases the number of variables can as large as 2<sup>|'''X'''|</sup>.  For this reason, in the context of [[multi-criteria decision analysis]] and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use.  For instance, when it is assumed the fuzzy measure is ''additive'', it will hold that <math> g(E) = \sum_{i \in E} g(\{i\}) </math> and the values of the fuzzy measure can be evaluated from the values on '''X'''.  Similarly, a ''symmetric'' fuzzy measure is defined uniquely by |'''X'''| values.  Two important fuzzy measures that can be used are the Sugeno- or <math>\lambda</math>-fuzzy measure and ''k''-additive measures, introduced by Sugeno<ref>{{cite journal
+|author         = M. Sugeno
+|title          = Theory of fuzzy integrals and its applications. Ph.D. thesis
+|journal        = Tokyo Institute of Technology, Tokyo, Japan
+|year           = 1974
+}}</ref> and Grabisch<ref>{{cite journal
+|author         = M. Grabisch
+|title          = ''k''-order additive discrete fuzzy measures and their representation
+|journal        = Fuzzy Sets and Systems
+|volume         = 92
+|pages          = 167–189
+|year           = 1997
+|issue         = 2
+|doi         = 10.1016/S0165-0114(97)00168-1
+}}</ref> respectively.
+===Sugeno ''&lambda;''-measure===
+The Sugeno <math>\lambda</math>-measure is a special case of fuzzy measures defined iteratively. It has the following definition:
+====Definition====
+Let <math>\mathbf{X} = \left\lbrace x_1,\dots,x_n  \right\rbrace </math>  be a finite set and let <math>\lambda \in (-1,+\infty)</math>. A '''Sugeno <math>\lambda</math>-measure''' is a function <math>g:2^X\to[0,1]</math> such that
+# <math>g(X) = 1</math>.
+# if <math>A, B\subseteq \mathbf{X}</math> (alternatively <math>A, B\in 2^{\mathbf{X}}</math>) with <math>A \cap B = \emptyset</math> then <math>g(A \cup B) =g(A)+g(B)+\lambda g(A)g(B)</math>.
+As a convention, the value of g at a singleton set <math>\left\lbrace x_i \right\rbrace </math>
+is called a density and is denoted by <math>g_i = g(\left\lbrace x_i \right\rbrace)</math>. In addition, we have that <math>\lambda</math> satisﬁes the property
+:<math> \lambda +1 = \prod_{i=1}^n (1+\lambda g_i) </math>.
+Tahani and Keller <ref>{{cite journal
+|author         = H. Tahani and J. Keller
+|title          = Information Fusion in Computer Vision Using the Fuzzy Integral
+|journal        = IEEE Transactions on Systems, Man and Cybernetic
+|volume         = 20
+|pages          = 733–741
+|year           = 1990
+|doi          = 10.1109/21.57289
+|issue         = 3
+}}</ref> as well as Wang and Klir have showed that once the densities are known, it is possible to use the previous [[polynomial]] to obtain the values of <math>\lambda</math> uniquely.
+===''k''-additive fuzzy measure===
+The ''k''-additive fuzzy measure limits the interaction between the subsets <math> E \subseteq X </math> to size <math>|E|=k</math>.  This drastically reduces the number of variables needed to define the fuzzy measure, and as ''k'' can be anything from 1 (in which case the fuzzy measure is additive) to '''X''', it allows for a compromise between modelling ability and simplicity.
+====Definition====
+A discrete fuzzy measure ''g'' on a set '''X''' is called ''k-additive'' (<math> 1 \leq k \leq |\mathbf{X}|</math>) if its Möbius representation verifies <math>M(E) = 0 </math>, whenever <math> |E| > k </math> for any <math> E \subseteq \mathbf{X} </math>, and there exists a subset ''F'' with ''k'' elements such that <math> M(F) \neq 0 </math>.
+==Shapley and interaction indices==
+In [[game]] theory, the [[Shapley value]] or Shapley index is used to indicate the weight of a game.  Shapley values can calculated for fuzzy measures in order to give some indication of the importance of each singleton.  In the case of additive fuzzy measures, the Shapley value will be the same as each singleton.
+For a given fuzzy measure ''g'', and <math>|\mathbf{X}|=n</math>, the Shapley index for every <math> i,\dots,n \in X </math> is:
+:<math> \phi (i) = \sum_{E \subseteq \mathbf{X} \backslash \{i\}} \frac{(n-|E|-1)!|E|!}{n!} [g(E \cup \{i\}) - g(E)]. </math>
+The Shapley value is the vector <math> \mathbf{\phi}(g) = (\psi(1),\dots,\psi(n)).</math>
+==See also==
+*[[Probability theory]]
+*[[Possibility theory]]
+==References==
+{{reflist}}
+* Beliakov, Pradera and Calvo, ''Aggregation Functions: A Guide for Practitioners'', Springer, New York 2007.
+* Wang, Zhenyuan, and, [[George J. Klir]], ''Fuzzy Measure Theory'', Plenum Press, New York, 1991.
+==External links==
+*[http://pami.uwaterloo.ca/tizhoosh/measure.htm Fuzzy Measure Theory at Fuzzy Image Processing]
+[[Category:Exotic probabilities]]
+[[Category:Measure theory]]
+[[Category:Fuzzy logic]]

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Revision as of 23:54, 24 January 2014

Contents

Definitions

Properties of fuzzy measures

Möbius representation

Simplification assumptions for fuzzy measures

Sugeno λ-measure

Definition

k-additive fuzzy measure

Definition

Shapley and interaction indices

See also

References

External links

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Time-lapse photography: Difference between revisions

Revision as of 23:54, 24 January 2014

Definitions

Properties of fuzzy measures

Möbius representation

Simplification assumptions for fuzzy measures

Sugeno λ-measure

Definition

k-additive fuzzy measure

Definition

Shapley and interaction indices

See also

References

External links

Navigation menu

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