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{{see also|Ensemble average}} | |||
{{more footnotes|date=June 2010}} | |||
In [[machine learning]], particularly in the creation of [[artificial neural networks]], '''ensemble averaging''' is the process of creating multiple models and combining them to produce a desired output, as opposed to creating just one model. Frequently an ensemble of models performs better than any individual model, because the various errors of the models "average out." | |||
== Overview == | |||
Ensemble averaging is one of the simplest types of [[committee machine]]s. Along with [[Boosting (meta-algorithm)|boosting]], it is one of the two major types of static committee machines.<ref name="haykin">Haykin, Simon. Neural networks : a comprehensive foundation. 2nd ed. Upper Saddle River N.J.: Prentice Hall, 1999.</ref> In contrast to standard network design in which many networks are generated but only one is kept, ensemble averaging keeps the less satisfactory networks around, but with less weight.<ref name="hashem">Hashem, S. "Optimal linear combinations of neural networks." Neural Networks 10, no. 4 (1997): 599–614.</ref> The theory of ensemble averaging relies on two properties of artificial neural networks:<ref name="naft">Naftaly, U., N. Intrator, and D. Horn. "Optimal ensemble averaging of neural networks." Network: Computation in Neural Systems 8, no. 3 (1997): 283–296.</ref> | |||
# In any network, the bias can be reduced at the cost of increased variance | |||
# In a group of networks, the variance can be reduced at no cost to bias | |||
Ensemble averaging creates a group of networks, each with low bias and high variance, then combines them to a new network with (hopefully) low bias and low variance. It is thus a resolution of the [[bias-variance dilemma]].<ref name="geman">Geman, S., E. Bienenstock, and R. Doursat. "Neural networks and the bias/variance dilemma." Neural computation 4, no. 1 (1992): 1–58.</ref> The idea of combining experts has been traced back to [[Pierre-Simon Laplace]].<ref>Clemen, R. T. "Combining forecasts: A review and annotated bibliography." International Journal of Forecasting 5, no. 4 (1989): 559–583.</ref> | |||
== Method == | |||
The theory mentioned above gives an obvious strategy: create a set of experts with low bias and high variance, and then average them. Generally, what this means is to create a set of experts with varying parameters; frequently, these are the initial synaptic weights, although other factors (such as the learning rate, momentum etc.) may be varied as well. Some authors recommend against varying weight decay and early stopping.<ref name="naft" /> The steps are therefore: | |||
# Generate ''N'' experts, each with their own initial values. (Initial values are usually chosen randomly from a distribution.) | |||
# Train each expert separately. | |||
# Combine the experts and average their values. | |||
Alternatively, [[domain knowledge]] may be used to generate several ''classes'' of experts. An expert from each class is trained, and then combined. | |||
A more complex version of ensemble average views the final result not as a mere average of all the experts, but rather as a weighted sum. If each expert is <math>y_i</math>, then the overall result <math>\tilde{y}</math> can be definied as: | |||
: <math>\tilde{y}(\bold{x}; \bold{\alpha}) = \sum_{j=1}^{p} \alpha_j y_j(\bold{x})</math> | |||
where <math>\bold{\alpha}</math> is a set of weights. The optimization problem of finding alpha is readily solved through neural networks, hence a "meta-network" where each "neuron" is in fact an entire neural network can be trained, and the synaptic weights of the final network is the weight applied to each expert. This is known as a ''linear combination of experts''.<ref name="hashem" /> | |||
It can be seen that most forms of neural networks are some subset of a linear combination: the standard neural net (where only one expert is used) is simply a linear combination with all <math>\alpha_j = 0</math> and one <math>\alpha_k=1</math>. A raw average is where all <math>\alpha_j</math> are equal to some constant value, namely one over the total number of experts.<ref name="hashem" /> | |||
A more recent ensemble averaging method is negative correlation learning,<ref>Y. Liu and X. Yao, [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T08-3XXCXC7-6&_user=10&_coverDate=12%2F31%2F1999&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1697373933&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=57475686ca0dcb592a456830d077cd1e&searchtype=a Ensemble Learning via Negative Correlation] Neural Networks, Volume 12, Issue 10, December 1999, Pages 1399-1404</ref> proposed by Y. Liu and X. Yao. Now this method has been widely used in [[evolutionary computing]]. | |||
== Benefits == | |||
* The resulting committee is almost always less complex than a single network which would achieve the same level of performance<ref>Pearlmutter, B. A., and R. Rosenfeld. "Chaitin–Kolmogorov complexity and generalization in neural networks." In Proceedings of the 1990 conference on Advances in neural information processing systems 3, 931. Morgan Kaufmann Publishers Inc., 1990.</ref> | |||
* The resulting committee can be trained more easily on smaller input sets<ref name="haykin" /> | |||
* The resulting committee often has improved performance over any single network<ref name="hashem" /> | |||
* The risk of [[overfitting]] is lessened, as there are fewer parameters (weights) which need to be set<ref name="haykin" /> | |||
== References == | |||
<references /> | |||
* {{citation | author = Perrone, M. P. | title = Improving regression estimation: Averaging methods for variance reduction with extensions to general convex measure optimization |year =1993 }} | |||
* {{citation | author=Wolpert, D. H. |title=Stacked generalization |journal=[[Neural Networks (journal)|Neural Networks]]|volume= 5|issue= 2|year= 1992|pages= 241–259}} | |||
* {{citation |author=Hashem, S.|title=Optimal linear combinations of neural networks|journal=[[Neural Networks (journal)|Neural Networks]]|volume= 10|issue= 4 |year=1997|pages=599–614}} | |||
* {{citation | author=Hashem, S. and B. Schmeiser |title=Approximating a function and its derivatives using MSE-optimal linear combinations of trained feedforward neural networks|journal=Proceedings of the Joint Conference on Neural Networks|volume= 87|pages=617–620|year= 1993}} | |||
[[Category:Artificial intelligence]] | |||
Latest revision as of 03:35, 18 January 2014
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Template:More footnotes
In machine learning, particularly in the creation of artificial neural networks, ensemble averaging is the process of creating multiple models and combining them to produce a desired output, as opposed to creating just one model. Frequently an ensemble of models performs better than any individual model, because the various errors of the models "average out."
Overview
Ensemble averaging is one of the simplest types of committee machines. Along with boosting, it is one of the two major types of static committee machines.[1] In contrast to standard network design in which many networks are generated but only one is kept, ensemble averaging keeps the less satisfactory networks around, but with less weight.[2] The theory of ensemble averaging relies on two properties of artificial neural networks:[3]
- In any network, the bias can be reduced at the cost of increased variance
- In a group of networks, the variance can be reduced at no cost to bias
Ensemble averaging creates a group of networks, each with low bias and high variance, then combines them to a new network with (hopefully) low bias and low variance. It is thus a resolution of the bias-variance dilemma.[4] The idea of combining experts has been traced back to Pierre-Simon Laplace.[5]
Method
The theory mentioned above gives an obvious strategy: create a set of experts with low bias and high variance, and then average them. Generally, what this means is to create a set of experts with varying parameters; frequently, these are the initial synaptic weights, although other factors (such as the learning rate, momentum etc.) may be varied as well. Some authors recommend against varying weight decay and early stopping.[3] The steps are therefore:
- Generate N experts, each with their own initial values. (Initial values are usually chosen randomly from a distribution.)
- Train each expert separately.
- Combine the experts and average their values.
Alternatively, domain knowledge may be used to generate several classes of experts. An expert from each class is trained, and then combined.
A more complex version of ensemble average views the final result not as a mere average of all the experts, but rather as a weighted sum. If each expert is , then the overall result can be definied as:
where is a set of weights. The optimization problem of finding alpha is readily solved through neural networks, hence a "meta-network" where each "neuron" is in fact an entire neural network can be trained, and the synaptic weights of the final network is the weight applied to each expert. This is known as a linear combination of experts.[2]
It can be seen that most forms of neural networks are some subset of a linear combination: the standard neural net (where only one expert is used) is simply a linear combination with all and one . A raw average is where all are equal to some constant value, namely one over the total number of experts.[2]
A more recent ensemble averaging method is negative correlation learning,[6] proposed by Y. Liu and X. Yao. Now this method has been widely used in evolutionary computing.
Benefits
- The resulting committee is almost always less complex than a single network which would achieve the same level of performance[7]
- The resulting committee can be trained more easily on smaller input sets[1]
- The resulting committee often has improved performance over any single network[2]
- The risk of overfitting is lessened, as there are fewer parameters (weights) which need to be set[1]
References
- ↑ 1.0 1.1 1.2 Haykin, Simon. Neural networks : a comprehensive foundation. 2nd ed. Upper Saddle River N.J.: Prentice Hall, 1999.
- ↑ 2.0 2.1 2.2 2.3 Hashem, S. "Optimal linear combinations of neural networks." Neural Networks 10, no. 4 (1997): 599–614.
- ↑ 3.0 3.1 Naftaly, U., N. Intrator, and D. Horn. "Optimal ensemble averaging of neural networks." Network: Computation in Neural Systems 8, no. 3 (1997): 283–296.
- ↑ Geman, S., E. Bienenstock, and R. Doursat. "Neural networks and the bias/variance dilemma." Neural computation 4, no. 1 (1992): 1–58.
- ↑ Clemen, R. T. "Combining forecasts: A review and annotated bibliography." International Journal of Forecasting 5, no. 4 (1989): 559–583.
- ↑ Y. Liu and X. Yao, Ensemble Learning via Negative Correlation Neural Networks, Volume 12, Issue 10, December 1999, Pages 1399-1404
- ↑ Pearlmutter, B. A., and R. Rosenfeld. "Chaitin–Kolmogorov complexity and generalization in neural networks." In Proceedings of the 1990 conference on Advances in neural information processing systems 3, 931. Morgan Kaufmann Publishers Inc., 1990.
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010