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| In the [[decision theory|mathematical theory of decisions]], '''decision-theoretic rough sets (DTRS)''' is a probabilistic extension of [[rough set]] classification. First created in 1990 by Dr. Yiyu Yao,<ref>{{cite journal|last=Yao|first=Y.Y.|coauthors=Wong, S.K.M. and Lingras, P.|year=1990|title=A decision-theoretic rough set model|journal=Methodologies for Intelligent Systems, 5, Proceedings of the 5th International Symposium on Methodologies for Intelligent Systems|publisher=North-Holland|location=Knoxville, Tennessee, USA|pages=17–25}}</ref> the extension makes use of loss functions to derive '''<math>\textstyle \alpha</math>''' and '''<math>\textstyle \beta</math>''' region parameters. Like rough sets, the lower and upper approximations of a set are used.
| | These fuzzies may be a small annoyance, but they can make your gym seem unkempt. Macaskill is tackling the Cuillin Ridge in Scotland, not far from where this two-wheeling genius hails from on the Isle of Skye. As a result of this various protective gear manufacturers have also ventured into production of improved design of padded shorts. Riding position is typically more vertical than that of a road bike. When shopping for different bike accessories, parts, or whatever it is you are looking for, being a smart shopper is always in order. <br><br> |
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| ==Definitions==
| | As hardtails have front-only suspension, they have less pedal bob and increased pedal stroke efficiency when compared to dual suspension MTB bikes. After their devastating loss to the Dallas Mavericks, there has been a lot of question as to whether they are really capable of winning a title or not. David Moffat founded Winter Park as a workers camp during construction of the Moffat Tunnel while building the transcontinental railroad from Denver to the Pacific. Cost shouldn't be a factor as a good helmet can be acquired for around $25-$45. However, that being said, many cross country riders prefer hardtail ones because the lighter and stiffer rear ends are better for acceleration and sprinting. <br><br>Given below is a list of essential items that you should carry along whenever you set off on a this ride. It is just much easier to be a participant in a team and enter the race as a whole unit. There are great coaches, who will train you to tweet like a bird in spring or Link in, befriend and get a fan page. If you can't ride from home, park on either side of Galvin Parkway Road at the first intersection north of Van Buren Street (a. The bike shop personnel should be able to advise you on how to choose the right frame size. <br><br>The tracks are mostly dirt paths on outdoor scenery. In between, the biker can perform whatever new trick he or she has learnt up and this trick is known as the gap jump. As the tracks are very rough it is far more difficult on a hardtail bike. People like to ride the mountain bike to those terrains because they enjoy the thrill and the adventure. If you beloved this report and you would like to obtain extra facts concerning [http://www.kandengue.com/profile/quscollen Womens mountain bike sizing.] kindly check out our site. You should be selective in choosing mountain bike tires since mountain bike is already made heavy and if you choose heavy tires you should consider yourself about how you will handle the bike and the terrain at the same time. <br><br>(many commute bikes come with front suspension and disc brakes these days). The bike should be examined to check it is working correctly. Trying out various sizes is the first step in choosing the correct folding bike. Whether it's a testament to South Dakota's commitment to recycling, or just whimsy, Bone Collector embodies the features of the best bike trails in the world: Sedona, Moab, Flagstaff, Whistler and Tahoe, without the long drive or expensive plane ticket. From there you can connect to the west switchback trail simply call "Switchbacks" or "The Old Switchbacks". |
| The following contains the basic principles of decision-theoretic rough sets.
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| ===Conditional risk===
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| Using the Bayesian decision procedure, the decision-theoretic rough set (DTRS) approach allows for minimum-risk decision making based on observed evidence. Let <math>\textstyle A=\{a_1,\ldots,a_m\}</math> be a finite set of <math>\textstyle m</math>
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| possible actions and let <math>\textstyle \Omega=\{w_1,\ldots, w_s\}</math> be a finite set of <math>s</math> states. <math>\textstyle P(w_j\mid[x])</math> is
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| calculated as the conditional probability of an object <math>\textstyle x</math> being in state <math>\textstyle w_j</math> given the object description
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| <math>\textstyle [x]</math>. <math>\textstyle \lambda(a_i\mid w_j)</math> denotes the loss, or cost, for performing action <math>\textstyle a_i</math> when the state is <math>\textstyle w_j</math>.
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| The expected loss (conditional risk) associated with taking action <math>\textstyle a_i</math> is given
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| by:
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| : <math>
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| R(a_i\mid [x]) = \sum_{j=1}^s \lambda(a_i\mid w_j)P(w_j\mid[x]).
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| </math>
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| Object classification with the approximation operators can be fitted into the Bayesian decision framework. The
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| set of actions is given by <math>\textstyle A=\{a_P,a_N,a_B\}</math>, where <math>\textstyle a_P</math>, <math>\textstyle a_N</math>, and <math>\textstyle a_B</math> represent the three
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| actions in classifying an object into POS(<math>\textstyle A</math>), NEG(<math>\textstyle A</math>), and BND(<math>\textstyle A</math>) respectively. To indicate whether an
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| element is in <math>\textstyle A</math> or not in <math>\textstyle A</math>, the set of states is given by <math>\textstyle \Omega=\{A,A^c\}</math>. Let
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| <math>\textstyle \lambda(a_\diamond\mid A)</math> denote the loss incurred by taking action <math>\textstyle a_\diamond</math> when an object belongs to
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| <math>\textstyle A</math>, and let <math>\textstyle \lambda(a_\diamond\mid A^c)</math> denote the loss incurred by take the same action when the object
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| belongs to <math>\textstyle A^c</math>.
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| ===Loss functions===
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| Let <math>\textstyle \lambda_{PP}</math> denote the loss function for classifying an object in <math>\textstyle A</math> into the POS region, <math>\textstyle \lambda_{BP}</math> denote the loss function for classifying an object in <math>\textstyle A</math> into the BND region, and let <math>\textstyle \lambda_{NP}</math> denote the loss function for classifying an object in <math>\textstyle A</math> into the NEG region. A loss function <math>\textstyle \lambda_{\diamond N}</math> denotes the loss of classifying an object that does not belong to <math>\textstyle A</math> into the regions specified by <math>\textstyle \diamond</math>.
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| Taking individual can be associated with the expected loss <math>\textstyle R(a_\diamond\mid[x])</math>actions and can be expressed as:
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| : <math>\textstyle R(a_P\mid[x]) = \lambda_{PP}P(A\mid[x]) + \lambda_{PN}P(A^c\mid[x]),</math>
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| : <math>\textstyle R(a_N\mid[x]) = \lambda_{NP}P(A\mid[x]) + \lambda_{NN}P(A^c\mid[x]),</math> | |
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| : <math>\textstyle R(a_B\mid[x]) = \lambda_{BP}P(A\mid[x]) + \lambda_{BN}P(A^c\mid[x]),</math>
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| where <math>\textstyle \lambda_{\diamond P}=\lambda(a_\diamond\mid A)</math>, <math>\textstyle \lambda_{\diamond N}=\lambda(a_\diamond\mid A^c)</math>, and <math>\textstyle \diamond=P</math>, <math>\textstyle N</math>, or <math>\textstyle B</math>.
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| ===Minimum-risk decision rules===
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| If we consider the loss functions <math>\textstyle \lambda_{PP} \leq \lambda_{BP} < \lambda_{NP}</math> and <math>\textstyle \lambda_{NN} \leq \lambda_{BN} < \lambda_{PN}</math>, the following decision rules are formulated (''P'', ''N'', ''B''):
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| * '''P''': If <math>\textstyle P(A\mid[x]) \geq \gamma</math> and <math>\textstyle P(A\mid[x]) \geq \alpha</math>, decide POS(<math>\textstyle A</math>);
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| * '''N''': If <math>\textstyle P(A\mid[x]) \leq \beta</math> and <math>\textstyle P(A\mid[x]) \leq \gamma</math>, decide NEG(<math>\textstyle A</math>);
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| * '''B''': If <math>\textstyle \beta \leq P(A\mid[x]) \leq \alpha</math>, decide BND(<math>\textstyle A</math>);
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| where,
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| : <math>\alpha = \frac{\lambda_{PN} - \lambda_{BN}}{(\lambda_{BP} - \lambda_{BN}) - (\lambda_{PP}-\lambda_{PN})},</math>
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| : <math>\gamma = \frac{\lambda_{PN} - \lambda_{NN}}{(\lambda_{NP} - \lambda_{NN}) - (\lambda_{PP}-\lambda_{PN})},</math>
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| : <math>\beta = \frac{\lambda_{BN} - \lambda_{NN}}{(\lambda_{NP} - \lambda_{NN}) - (\lambda_{BP}-\lambda_{BN})}.</math>
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| The <math>\textstyle \alpha</math>, <math>\textstyle \beta</math>, and <math>\textstyle \gamma</math> values define the three different regions, giving us an associated risk for classifying an object. When <math>\textstyle \alpha > \beta</math>, we get <math>\textstyle \alpha > \gamma > \beta</math> and can simplify (''P'', ''N'', ''B'') into (''P''1, ''N''1, ''B''1): | |
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| * '''P1''': If <math>\textstyle P(A\mid [x]) \geq \alpha</math>, decide POS(<math>\textstyle A</math>);
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| * '''N1''': If <math>\textstyle P(A\mid[x]) \leq \beta</math>, decide NEG(<math>\textstyle A</math>);
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| * '''B1''': If <math>\textstyle \beta < P(A\mid[x]) < \alpha</math>, decide BND(<math>\textstyle A</math>).
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| When <math>\textstyle \alpha = \beta = \gamma</math>, we can simplify the rules (P-B) into (P2-B2), which divide the regions based solely on <math>\textstyle \alpha</math>:
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| * '''P2''': If <math>\textstyle P(A\mid[x]) > \alpha</math>, decide POS(<math>\textstyle A</math>);
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| * '''N2''': If <math>\textstyle P(A\mid[x]) < \alpha</math>, decide NEG(<math>\textstyle A</math>);
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| * '''B2''': If <math>\textstyle P(A\mid[x]) = \alpha</math>, decide BND(<math>\textstyle A</math>).
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| [[Data mining]], [[feature selection]], [[information retrieval]], and [[Classification (machine learning)|classifications]] are just some of the applications in which the DTRS approach has been successfully used.
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| ==See also==
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| * [[Rough sets]]
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| * [[Granular computing]]
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| * [[Soft computing]]
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| * [[Fuzzy set theory]]
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| ==References==
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| <references/>
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| ==External links==
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| * [http://roughsets.home.pl/www/ The International Rough Set Society]
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| * [http://www.roughsets.com The Decision-theoretic Rough Set Portal]
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| {{DEFAULTSORT:Decision-Theoretic Rough Sets}}
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| [[Category:Decision theory]]
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These fuzzies may be a small annoyance, but they can make your gym seem unkempt. Macaskill is tackling the Cuillin Ridge in Scotland, not far from where this two-wheeling genius hails from on the Isle of Skye. As a result of this various protective gear manufacturers have also ventured into production of improved design of padded shorts. Riding position is typically more vertical than that of a road bike. When shopping for different bike accessories, parts, or whatever it is you are looking for, being a smart shopper is always in order.
As hardtails have front-only suspension, they have less pedal bob and increased pedal stroke efficiency when compared to dual suspension MTB bikes. After their devastating loss to the Dallas Mavericks, there has been a lot of question as to whether they are really capable of winning a title or not. David Moffat founded Winter Park as a workers camp during construction of the Moffat Tunnel while building the transcontinental railroad from Denver to the Pacific. Cost shouldn't be a factor as a good helmet can be acquired for around $25-$45. However, that being said, many cross country riders prefer hardtail ones because the lighter and stiffer rear ends are better for acceleration and sprinting.
Given below is a list of essential items that you should carry along whenever you set off on a this ride. It is just much easier to be a participant in a team and enter the race as a whole unit. There are great coaches, who will train you to tweet like a bird in spring or Link in, befriend and get a fan page. If you can't ride from home, park on either side of Galvin Parkway Road at the first intersection north of Van Buren Street (a. The bike shop personnel should be able to advise you on how to choose the right frame size.
The tracks are mostly dirt paths on outdoor scenery. In between, the biker can perform whatever new trick he or she has learnt up and this trick is known as the gap jump. As the tracks are very rough it is far more difficult on a hardtail bike. People like to ride the mountain bike to those terrains because they enjoy the thrill and the adventure. If you beloved this report and you would like to obtain extra facts concerning Womens mountain bike sizing. kindly check out our site. You should be selective in choosing mountain bike tires since mountain bike is already made heavy and if you choose heavy tires you should consider yourself about how you will handle the bike and the terrain at the same time.
(many commute bikes come with front suspension and disc brakes these days). The bike should be examined to check it is working correctly. Trying out various sizes is the first step in choosing the correct folding bike. Whether it's a testament to South Dakota's commitment to recycling, or just whimsy, Bone Collector embodies the features of the best bike trails in the world: Sedona, Moab, Flagstaff, Whistler and Tahoe, without the long drive or expensive plane ticket. From there you can connect to the west switchback trail simply call "Switchbacks" or "The Old Switchbacks".