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{{one source|date=November 2010}}
There are many types of bike brakes for different applications. Various means of transportation come second - from yachts or boats and cars to motorcycles. The trail then follows the side of the glacial Lake Pukaki through to Twizel, the town build in the 60s only to house workers from the Waitaki Hydroelectricity Scheme. Mountain bike frames spend a lot of time on roads these days, too. But except few exceptions; their speed is not as fast as young runners. <br><br>


In [[statistical theory]], the '''Huber loss function''' is a function used in [[robust statistics|robust]] estimation that allows construction of an estimate which allows the effect of [[outlier]]s to be reduced, while treating non-outliers in a more standard way.
As hardtails have front-only suspension, they have less pedal bob and increased pedal stroke efficiency when compared to dual suspension MTB bikes. You can install an electric motor on your regular bike in about an hour or hire a bike mechanic to do it. Experts recommend that runners drink only when thirsty. Since you are reading this article on the Cannondale Mountain bike I will assume that you already know about the exquisite design and reliability of this mountain bike. He is known for his expertise on the subject and on other Business and Finance related articles. <br><br>Mountain bikes in Arizona have two great rides in Sedona, Highline and the Dry Creek Loop.  If you cherished this article and you would like to acquire much more details relating to [http://igrice.square7.net/profile/fawolfe Cannondale mountain bike sizing.] kindly check out our own web-page. There are two main types of mountain and road bike stems. Please remember the stem lengths can impact just how reactive the bike is. Just like a rectangle you have 2 long sides and 2 short sides. These are stable forks whose weights are directly in proportion to their durability. <br><br>Most people will increase the power and size of their electric motor with the purpose of traveling at a higher speed and increasing the torque of the bike. It is important for the rider to note that not all frames are compatible with every type of forks. Maybe you like to go out; maybe you like to mountain bike. I'd been slowly coming to that conclusion over the course of about a year or so and that mutant nose hair pretty much confirmed it. In order to find the best triathlon bikes, the following are a few places you can look. <br><br>(many commute bikes come with front suspension and disc brakes these days). It's usually best to buy from your bike store to begin with, as they can help you fit the bike and give you advice. And the cyclist may certainly sense the distinction. The ratio is chosen based on the terrain in which the bike will be ridden, the size of the bike and the strength of the rides. Part of Park City's distinction as the International Mountain Bike Association's first and only gold-level Ride Center is its abundance of beginner trails.
 
==Definition==
The '''Huber loss function''' describes the penalty incurred by an [[estimator|estimation procedure]].  [[Peter J. Huber|Huber]] (1964<ref>{{Citation|last=Huber|first=Peter J.|year=1964|title=Robust Estimation of a Location Parameter|journal=Annals of Statistics|volume=53|pages=73–101}}</ref>) defines the loss function piecewise by
 
:<math> L_\delta (a) = (1/2){a^2} \qquad \qquad \text{  for } |a| \le \delta ,</math>
:<math> L_\delta (a)            = \delta (|a| - \delta/2 ), \qquad \text{otherwise}. </math>
 
This function is quadratic for small values of ''a'', and linear for large values, with equal values and slopes of the different sections at the two points where |''a''|&nbsp;=&nbsp;''δ''. In use, the variable <math>a</math> often refers to the residuals, that is to the difference between the observed and predicted values, i.e. <math> a = y - \hat{y} </math>.
 
==Motivation==
For [[parameter estimation|estimating parameters]], it is desirable for a loss function to have the following properties (for all values of <math>a</math> of the [[parameter space]]):
 
# It is greater than or equal to the [[0-1 loss function]] (which is defined as <math>L(a)=0</math> if <math>a=0</math> and <math>L(a)=1</math> otherwise).
# It is continuous (or lower [[semi-continuity|semicontinuous]]).
 
Two very commonly used loss functions are the [[mean squared error|squared loss]], <math>L(a) = a^2</math>, and the [[absolute deviation|absolute loss]], <math>L(a)=|a|</math>. While the absolute loss is not differentiable at exactly one point, <math>a=0</math>, where it is [[subdifferential|subdifferentiable]] with its [[convex analysis|convex]] [[subdifferential]] equal to the interval  <math>[-1+1]</math>; the absolute-value loss function results in a median-unbiased estimator, which can be evaluated for particular data sets by [[linear programming]]. The squared loss has the disadvantage that it has the tendency to be dominated by outliers---when summing over a set of <math>a</math>'s (as in <math>\sum_{i=1}^n L(a_i) </math> ), the sample mean is influenced too much by a few particularly large a-values when the distribution is heavy tailed: in terms of [[estimation theory]], the asymptotic relative efficiency of the mean  is poor for heavy-tailed distributions
 
As defined above, the Huber loss function is [[convex function|convex]] in a uniform neighborhood of its minimum <math>a=0</math>, at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points <math> a=-\delta </math> and <math> a = \delta </math>. These properties allow it to combine much of the sensitivity of the mean-unbiased, minimum-variance estimator of the mean (using the quadratic loss function) and the robustness of the median-unbiased estimor (using the absolute value function).
 
The [[log cosh loss function]], which is defined as <math> L(a) = \log(\cosh(a)) </math> has a behavior like that of the Huber loss function.
 
==Pseudo-Huber loss function==
 
The '''Pseudo-Huber loss function''' can be used as a smooth approximation of the Huber loss function, and ensures that derivatives are continuous for all degrees. It is defined as{{citation needed|date=February 2012}}
 
:<math>L_\delta (a) = \delta^2(\sqrt{1+(a/\delta)^2}-1).</math>
 
As such, this function approximates <math>a^2/2</math> for small values of <math>a</math>, and is parallel with slope <math>\delta</math> for large values of <math>a</math>.
 
==Applications==
The Huber loss function is used in [[robust statistics]],  [[M-estimator|M-estimation]] and [[additive model]]ling.<ref>Friedman, J. H. (2001), "Greedy Function Approximation: A Gradient Boosting Machine", The Annals of Statistics, Vol. 26, No.5 (Oct. 2001), 1189-1232</ref>
 
==See also==
* [[Robust regression]]
* [[M-estimator]]
* [[Robust statistics#M-estimators|Visual comparison of different M-estimators]]
 
{{More footnotes|date=November 2010}}
 
==References==
<references/>
 
{{DEFAULTSORT:Huber Loss Function}}
[[Category:Robust statistics]]
[[Category:M-estimators]]
[[Category:Loss functions]]

Revision as of 15:52, 2 March 2014

There are many types of bike brakes for different applications. Various means of transportation come second - from yachts or boats and cars to motorcycles. The trail then follows the side of the glacial Lake Pukaki through to Twizel, the town build in the 60s only to house workers from the Waitaki Hydroelectricity Scheme. Mountain bike frames spend a lot of time on roads these days, too. But except few exceptions; their speed is not as fast as young runners.

As hardtails have front-only suspension, they have less pedal bob and increased pedal stroke efficiency when compared to dual suspension MTB bikes. You can install an electric motor on your regular bike in about an hour or hire a bike mechanic to do it. Experts recommend that runners drink only when thirsty. Since you are reading this article on the Cannondale Mountain bike I will assume that you already know about the exquisite design and reliability of this mountain bike. He is known for his expertise on the subject and on other Business and Finance related articles.

Mountain bikes in Arizona have two great rides in Sedona, Highline and the Dry Creek Loop. If you cherished this article and you would like to acquire much more details relating to Cannondale mountain bike sizing. kindly check out our own web-page. There are two main types of mountain and road bike stems. Please remember the stem lengths can impact just how reactive the bike is. Just like a rectangle you have 2 long sides and 2 short sides. These are stable forks whose weights are directly in proportion to their durability.

Most people will increase the power and size of their electric motor with the purpose of traveling at a higher speed and increasing the torque of the bike. It is important for the rider to note that not all frames are compatible with every type of forks. Maybe you like to go out; maybe you like to mountain bike. I'd been slowly coming to that conclusion over the course of about a year or so and that mutant nose hair pretty much confirmed it. In order to find the best triathlon bikes, the following are a few places you can look.

(many commute bikes come with front suspension and disc brakes these days). It's usually best to buy from your bike store to begin with, as they can help you fit the bike and give you advice. And the cyclist may certainly sense the distinction. The ratio is chosen based on the terrain in which the bike will be ridden, the size of the bike and the strength of the rides. Part of Park City's distinction as the International Mountain Bike Association's first and only gold-level Ride Center is its abundance of beginner trails.

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