Krzysztof Wołk (born 16 August 1986) is a Polish IT researcher who specializes in artificial intelligence, machine learning, mobile applications, linguistic engineering, multimedia, NLP and graphic applications. His research works have been cited in more than 70 international research journals, books and research papers. He is member of scientific committee at the Health and Social Care Information Systems and Technologies (HCist), an international conference which brings in new ideas, new technologies, academic scientists, healthcare IT professionals, managers and solution providers from all over the world. His research in statistical machine learning has been recognized as one of the most cited researches in the world. He is the member of Scientific Committee-Reviewers at Research Conference in Technical Disciplines (RCITD), based in Slovakia, which brings together the academic scientists and researchers from all around the world. == Biography == He obtained the doctorate degree in 2016 from the Polish-Japanese Academy of Information and Technology in Warsaw, Poland. He is currently working as researcher and assistant professor at the Polish-Japanese Computer Science Academy (PJATK) in Warsaw, Poland. == Achievements == He has published three books: Biblia Windows Server 2012, Administrator's Guide, Mac OS X Server 10.8, and MAC OS X Server 10.6 and 10.7 Practical Guide has been cited by many researchers in the scholarly books, research journals and articles. His research work on the Polish-English statistical machine translation has been featured in the book New Research in Multimedia and Internet System. Similarly, his works regarding the machine translation system have been featured in the books New Perspective in Information System and Technologies Volume 1, Multimedia and Network Information System, and Recent Advances in Information Systems and Technologies, Volume 1.
Esdat
ESdat is a data management, analysis and reporting software for environmental and groundwater data, developed by EarthScience Information Systems (EScIS). It is used to manage many types of environmental data including laboratory chemistry (analytical results, QA data, lab sample planning, and electronic Chain of Custody), field chemistry (water, gas, and soil), hydrogeological data (groundwater, borehole and well construction, lithological, geotechnical and stratigraphic, and LNAPL), meteorological data (rain, wind, and temperature), emission data (dust deposition, HiVol, air quality, and noise) and logger data. Data can be compared against environmental standards or site-specific trigger levels to generate exceedence tables, time series graphs, maps, statistics, and other outputs. ESdat integrates with Power BI and ArcGIS and data can also be exported in a range of other database formats, including USEPA Regions 2,4 & 5, and NYS DEC. ESdat is used by environmental consultants, government, mining and industry for validation, interrogation, and reporting of data derived from complex environmental programs, such as contaminated sites, groundwater investigations, and regulatory compliance for landfills or mining operations.
Multiple kernel learning
Multiple kernel learning refers to a set of machine learning methods that use a predefined set of kernels and learn an optimal linear or non-linear combination of kernels as part of the algorithm. Reasons to use multiple kernel learning include a) the ability to select for an optimal kernel and parameters from a larger set of kernels, reducing bias due to kernel selection while allowing for more automated machine learning methods, and b) combining data from different sources (e.g. sound and images from a video) that have different notions of similarity and thus require different kernels. Instead of creating a new kernel, multiple kernel algorithms can be used to combine kernels already established for each individual data source. Multiple kernel learning approaches have been used in many applications, such as event recognition in video, object recognition in images, and biomedical data fusion. == Algorithms == Multiple kernel learning algorithms have been developed for supervised, semi-supervised, as well as unsupervised learning. Most work has been done on the supervised learning case with linear combinations of kernels, however, many algorithms have been developed. The basic idea behind multiple kernel learning algorithms is to add an extra parameter to the minimization problem of the learning algorithm. As an example, consider the case of supervised learning of a linear combination of a set of n {\displaystyle n} kernels K {\displaystyle K} . We introduce a new kernel K ′ = ∑ i = 1 n β i K i {\displaystyle K'=\sum _{i=1}^{n}\beta _{i}K_{i}} , where β {\displaystyle \beta } is a vector of coefficients for each kernel. Because the kernels are additive (due to properties of reproducing kernel Hilbert spaces), this new function is still a kernel. For a set of data X {\displaystyle X} with labels Y {\displaystyle Y} , the minimization problem can then be written as min β , c E ( Y , K ′ c ) + R ( K , c ) {\displaystyle \min _{\beta ,c}\mathrm {E} (Y,K'c)+R(K,c)} where E {\displaystyle \mathrm {E} } is an error function and R {\displaystyle R} is a regularization term. E {\displaystyle \mathrm {E} } is typically the square loss function (Tikhonov regularization) or the hinge loss function (for SVM algorithms), and R {\displaystyle R} is usually an ℓ n {\displaystyle \ell _{n}} norm or some combination of the norms (i.e. elastic net regularization). This optimization problem can then be solved by standard optimization methods. Adaptations of existing techniques such as the Sequential Minimal Optimization have also been developed for multiple kernel SVM-based methods. === Supervised learning === For supervised learning, there are many other algorithms that use different methods to learn the form of the kernel. The following categorization has been proposed by Gonen and Alpaydın (2011) ==== Fixed rules approaches ==== Fixed rules approaches such as the linear combination algorithm described above use rules to set the combination of the kernels. These do not require parameterization and use rules like summation and multiplication to combine the kernels. The weighting is learned in the algorithm. Other examples of fixed rules include pairwise kernels, which are of the form k ( ( x 1 i , x 1 j ) , ( x 2 i , x 2 j ) ) = k ( x 1 i , x 2 i ) k ( x 1 j , x 2 j ) + k ( x 1 i , x 2 j ) k ( x 1 j , x 2 i ) {\displaystyle k((x_{1i},x_{1j}),(x_{2i},x_{2j}))=k(x_{1i},x_{2i})k(x_{1j},x_{2j})+k(x_{1i},x_{2j})k(x_{1j},x_{2i})} . These pairwise approaches have been used in predicting protein-protein interactions. ==== Heuristic approaches ==== These algorithms use a combination function that is parameterized. The parameters are generally defined for each individual kernel based on single-kernel performance or some computation from the kernel matrix. Examples of these include the kernel from Tenabe et al. (2008). Letting π m {\displaystyle \pi _{m}} be the accuracy obtained using only K m {\displaystyle K_{m}} , and letting δ {\displaystyle \delta } be a threshold less than the minimum of the single-kernel accuracies, we can define β m = π m − δ ∑ h = 1 n ( π h − δ ) {\displaystyle \beta _{m}={\frac {\pi _{m}-\delta }{\sum _{h=1}^{n}(\pi _{h}-\delta )}}} Other approaches use a definition of kernel similarity, such as A ( K 1 , K 2 ) = ⟨ K 1 , K 2 ⟩ ⟨ K 1 , K 1 ⟩ ⟨ K 2 , K 2 ⟩ {\displaystyle A(K_{1},K_{2})={\frac {\langle K_{1},K_{2}\rangle }{\sqrt {\langle K_{1},K_{1}\rangle \langle K_{2},K_{2}\rangle }}}} Using this measure, Qui and Lane (2009) used the following heuristic to define β m = A ( K m , Y Y T ) ∑ h = 1 n A ( K h , Y Y T ) {\displaystyle \beta _{m}={\frac {A(K_{m},YY^{T})}{\sum _{h=1}^{n}A(K_{h},YY^{T})}}} ==== Optimization approaches ==== These approaches solve an optimization problem to determine parameters for the kernel combination function. This has been done with similarity measures and structural risk minimization approaches. For similarity measures such as the one defined above, the problem can be formulated as follows: max β , tr ( K t r a ′ ) = 1 , K ′ ≥ 0 A ( K t r a ′ , Y Y T ) . {\displaystyle \max _{\beta ,\operatorname {tr} (K'_{tra})=1,K'\geq 0}A(K'_{tra},YY^{T}).} where K t r a ′ {\displaystyle K'_{tra}} is the kernel of the training set. Structural risk minimization approaches that have been used include linear approaches, such as that used by Lanckriet et al. (2002). We can define the implausibility of a kernel ω ( K ) {\displaystyle \omega (K)} to be the value of the objective function after solving a canonical SVM problem. We can then solve the following minimization problem: min tr ( K t r a ′ ) = c ω ( K t r a ′ ) {\displaystyle \min _{\operatorname {tr} (K'_{tra})=c}\omega (K'_{tra})} where c {\displaystyle c} is a positive constant. Many other variations exist on the same idea, with different methods of refining and solving the problem, e.g. with nonnegative weights for individual kernels and using non-linear combinations of kernels. ==== Bayesian approaches ==== Bayesian approaches put priors on the kernel parameters and learn the parameter values from the priors and the base algorithm. For example, the decision function can be written as f ( x ) = ∑ i = 0 n α i ∑ m = 1 p η m K m ( x i m , x m ) {\displaystyle f(x)=\sum _{i=0}^{n}\alpha _{i}\sum _{m=1}^{p}\eta _{m}K_{m}(x_{i}^{m},x^{m})} η {\displaystyle \eta } can be modeled with a Dirichlet prior and α {\displaystyle \alpha } can be modeled with a zero-mean Gaussian and an inverse gamma variance prior. This model is then optimized using a customized multinomial probit approach with a Gibbs sampler. These methods have been used successfully in applications such as protein fold recognition and protein homology problems ==== Boosting approaches ==== Boosting approaches add new kernels iteratively until some stopping criteria that is a function of performance is reached. An example of this is the MARK model developed by Bennett et al. (2002) f ( x ) = ∑ i = 1 N ∑ m = 1 P α i m K m ( x i m , x m ) + b {\displaystyle f(x)=\sum _{i=1}^{N}\sum _{m=1}^{P}\alpha _{i}^{m}K_{m}(x_{i}^{m},x^{m})+b} The parameters α i m {\displaystyle \alpha _{i}^{m}} and b {\displaystyle b} are learned by gradient descent on a coordinate basis. In this way, each iteration of the descent algorithm identifies the best kernel column to choose at each particular iteration and adds that to the combined kernel. The model is then rerun to generate the optimal weights α i {\displaystyle \alpha _{i}} and b {\displaystyle b} . === Semisupervised learning === Semisupervised learning approaches to multiple kernel learning are similar to other extensions of supervised learning approaches. An inductive procedure has been developed that uses a log-likelihood empirical loss and group LASSO regularization with conditional expectation consensus on unlabeled data for image categorization. We can define the problem as follows. Let L = ( x i , y i ) {\displaystyle L={(x_{i},y_{i})}} be the labeled data, and let U = x i {\displaystyle U={x_{i}}} be the set of unlabeled data. Then, we can write the decision function as follows. f ( x ) = α 0 + ∑ i = 1 | L | α i K i ( x ) {\displaystyle f(x)=\alpha _{0}+\sum _{i=1}^{|L|}\alpha _{i}K_{i}(x)} The problem can be written as min f L ( f ) + λ R ( f ) + γ Θ ( f ) {\displaystyle \min _{f}L(f)+\lambda R(f)+\gamma \Theta (f)} where L {\displaystyle L} is the loss function (weighted negative log-likelihood in this case), R {\displaystyle R} is the regularization parameter (Group LASSO in this case), and Θ {\displaystyle \Theta } is the conditional expectation consensus (CEC) penalty on unlabeled data. The CEC penalty is defined as follows. Let the marginal kernel density for all the data be g m π ( x ) = ⟨ ϕ m π , ψ m ( x ) ⟩ {\displaystyle g_{m}^{\pi }(x)=\langle \phi _{m}^{\pi },\psi _{m}(x)\rangle } where ψ m ( x ) = [ K m ( x 1 , x ) , … , K m ( x L , x ) ] T {\displaystyle \psi _{m}(x)=[K_{m}(x_{1},x),\ldots ,K_{m}(x_{L},x)]^{T}} (the kernel distance between the labe
Softmax function
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes. == Definition == The softmax function takes as input a tuple z of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some tuple components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval ( 0 , 1 ) {\displaystyle (0,1)} , and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities. Formally, the standard (unit) softmax function σ : R K → ( 0 , 1 ) K {\displaystyle \sigma :\mathbb {R} ^{K}\to (0,1)^{K}} , where K > 1 {\displaystyle K>1} , takes a tuple z = ( z 1 , … , z K ) ∈ R K {\displaystyle \mathbf {z} =(z_{1},\dotsc ,z_{K})\in \mathbb {R} ^{K}} and computes each component of vector σ ( z ) ∈ ( 0 , 1 ) K {\displaystyle \sigma (\mathbf {z} )\in (0,1)^{K}} with σ ( z ) i = e z i ∑ j = 1 K e z j . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{z_{i}}}{\sum _{j=1}^{K}e^{z_{j}}}}\,.} In words, the softmax applies the standard exponential function to each element z i {\displaystyle z_{i}} of the input tuple z {\displaystyle \mathbf {z} } (consisting of K {\displaystyle K} real numbers), and normalizes these values by dividing by the sum of all these exponentials. The normalization ensures that the sum of the components of the output vector σ ( z ) {\displaystyle \sigma (\mathbf {z} )} is 1. The term "softmax" derives from the amplifying effects of the exponential on any maxima in the input tuple. For example, the standard softmax of ( 1 , 2 , 8 ) {\displaystyle (1,2,8)} is approximately ( 0.001 , 0.002 , 0.997 ) {\displaystyle (0.001,0.002,0.997)} , which amounts to assigning almost all of the total unit weight in the result to the position of the tuple's maximal element (of 8). In general, instead of e a different base b > 0 can be used. As above, if b > 1 then larger input components will result in larger output probabilities, and increasing the value of b will create probability distributions that are more concentrated around the positions of the largest input values. Conversely, if 0 < b < 1 then smaller input components will result in larger output probabilities, and decreasing the value of b will create probability distributions that are more concentrated around the positions of the smallest input values. Writing b = e β {\displaystyle b=e^{\beta }} or b = e − β {\displaystyle b=e^{-\beta }} (for real β) yields the expressions: σ ( z ) i = e β z i ∑ j = 1 K e β z j or σ ( z ) i = e − β z i ∑ j = 1 K e − β z j for i = 1 , … , K . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{\beta z_{i}}}{\sum _{j=1}^{K}e^{\beta z_{j}}}}{\text{ or }}\sigma (\mathbf {z} )_{i}={\frac {e^{-\beta z_{i}}}{\sum _{j=1}^{K}e^{-\beta z_{j}}}}{\text{ for }}i=1,\dotsc ,K.} A value proportional to the reciprocal of β is sometimes referred to as the temperature: β = 1 / k T {\textstyle \beta =1/kT} , where k is typically 1 or the Boltzmann constant and T is the temperature. A higher temperature results in a more uniform output distribution (i.e. with higher entropy; it is "more random"), while a lower temperature results in a sharper output distribution, with one value dominating. In some fields, the base is fixed, corresponding to a fixed scale, while in others the parameter β (or T) is varied. The softmax function is a multiple-variable generalization of the logistic function. == Interpretations == === Smooth arg max === The Softmax function is a smooth approximation to the arg max function: the function whose value is the index of a tuple's largest element. The name "softmax" may be misleading. Softmax is not a smooth maximum (that is, a smooth approximation to the maximum function). The term "softmax" is also used for the closely related LogSumExp function, which is a smooth maximum. For this reason, some prefer the more accurate term "softargmax", though the term "softmax" is conventional in machine learning. This section uses the term "softargmax" for clarity. Formally, instead of considering the arg max as a function with categorical output 1 , … , n {\displaystyle 1,\dots ,n} (corresponding to the index), consider the arg max function with one-hot representation of the output (assuming there is a unique maximum arg): a r g m a x ( z 1 , … , z n ) = ( y 1 , … , y n ) = ( 0 , … , 0 , 1 , 0 , … , 0 ) , {\displaystyle \operatorname {arg\,max} (z_{1},\,\dots ,\,z_{n})=(y_{1},\,\dots ,\,y_{n})=(0,\,\dots ,\,0,\,1,\,0,\,\dots ,\,0),} where the output coordinate y i = 1 {\displaystyle y_{i}=1} if and only if i {\displaystyle i} is the arg max of ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} , meaning z i {\displaystyle z_{i}} is the unique maximum value of ( z 1 , … , z n ) {\displaystyle (z_{1},\,\dots ,\,z_{n})} . For example, in this encoding a r g m a x ( 1 , 5 , 10 ) = ( 0 , 0 , 1 ) , {\displaystyle \operatorname {arg\,max} (1,5,10)=(0,0,1),} since the third argument is the maximum. This can be generalized to multiple arg max values (multiple equal z i {\displaystyle z_{i}} being the maximum) by dividing the 1 between all max args; formally 1/k where k is the number of arguments assuming the maximum. For example, a r g m a x ( 1 , 5 , 5 ) = ( 0 , 1 / 2 , 1 / 2 ) , {\displaystyle \operatorname {arg\,max} (1,\,5,\,5)=(0,\,1/2,\,1/2),} since the second and third argument are both the maximum. In case all arguments are equal, this is simply a r g m a x ( z , … , z ) = ( 1 / n , … , 1 / n ) . {\displaystyle \operatorname {arg\,max} (z,\dots ,z)=(1/n,\dots ,1/n).} Points z with multiple arg max values are singular points (or singularities, and form the singular set) – these are the points where arg max is discontinuous (with a jump discontinuity) – while points with a single arg max are known as non-singular or regular points. With the last expression given in the introduction, softargmax is now a smooth approximation of arg max: as β → ∞ {\displaystyle \beta \to \infty } , softargmax converges to arg max. There are various notions of convergence of a function; softargmax converges to arg max pointwise, meaning for each fixed input z as β → ∞ {\displaystyle \beta \to \infty } , σ β ( z ) → a r g m a x ( z ) . {\displaystyle \sigma _{\beta }(\mathbf {z} )\to \operatorname {arg\,max} (\mathbf {z} ).} However, softargmax does not converge uniformly to arg max, meaning intuitively that different points converge at different rates, and may converge arbitrarily slowly. In fact, softargmax is continuous, but arg max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The reason it fails to converge uniformly is that for inputs where two coordinates are almost equal (and one is the maximum), the arg max is the index of one or the other, so a small change in input yields a large change in output. For example, σ β ( 1 , 1.0001 ) → ( 0 , 1 ) , {\displaystyle \sigma _{\beta }(1,\,1.0001)\to (0,1),} but σ β ( 1 , 0.9999 ) → ( 1 , 0 ) , {\displaystyle \sigma _{\beta }(1,\,0.9999)\to (1,\,0),} and σ β ( 1 , 1 ) = 1 / 2 {\displaystyle \sigma _{\beta }(1,\,1)=1/2} for all inputs: the closer the points are to the singular set ( x , x ) {\displaystyle (x,x)} , the slower they converge. However, softargmax does converge compactly on the non-singular set. Conversely, as β → − ∞ {\displaystyle \beta \to -\infty } , softargmax converges to arg min in the same way, where here the singular set is points with two arg min values. In the language of tropical analysis, the softmax is a deformation or "quantization" of arg max and arg min, corresponding to using the log semiring instead of the max-plus semiring (respectively min-plus semiring), and recovering the arg max or arg min by taking the limit is called "tropicalization" or "dequantization". It is also the case that, for any fixed β, if one input z i {\displaystyle z_{i}} is much larger than the others relative to the temperature, T = 1 / β {\displaystyle T=1/\beta } , the output is approximately the arg max. For example, a difference of 10 is large relative to a temperature of 1: σ ( 0 , 10 ) := σ 1 ( 0 , 10 ) = ( 1 / ( 1 + e 10 ) , e 10 / ( 1 + e 10 ) ) ≈ ( 0.00005 , 0.99995 ) {\displaystyle \sigma (0,\,10):=\sigma _{1}(0,\,10)=\left(1/\left(1+e^{10}\right),\,e^{10}/\left(1+e^{10}\right)\right)\approx (0.00005
Homogeneity blockmodeling
In mathematics applied to analysis of social structures, homogeneity blockmodeling is an approach in blockmodeling, which is best suited for a preliminary or main approach to valued networks, when a prior knowledge about these networks is not available. This is because homogeneity blockmodeling emphasizes the similarity of link (tie) strengths within the blocks over the pattern of links. In this approach, tie (link) values (or statistical data computed on them) are assumed to be equal (homogenous) within blocks. This approach to the generalized blockmodeling of valued networks was first proposed by Aleš Žiberna in 2007 with the basic idea, "that the inconsistency of an empirical block with its ideal block can be measured by within block variability of appropriate values". The newly–formed ideal blocks, which are appropriate for blockmodeling of valued networks, are then presented together with the definitions of their block inconsistencies. Similar approach to the homogeneity blockmodeling, dealing with direct approach for structural equivalence, was previously suggested by Stephen P. Borgatti and Martin G. Everett (1992).
Bitstrips
Bitstrips, Inc. was a Canadian media and technology company based in Toronto, founded in 2007 by Jacob Blackstock, David Kennedy, Shahan Panth, Dorian Baldwin, and Jesse Brown. The company created and offered a web application, Bitstrips.com, which allowed users to create comic strips using personalized avatars, and preset templates and poses. Brown and Blackstock explained that the service was meant to enable self-expression without the need to have artistic skills. Bitstrips was first presented in 2008 at South by Southwest in Austin, Texas, and the service later piloted and launched a version designed for use as educational software. The service achieved increasing prominence following the launch of versions for Facebook and mobile platforms. In 2014, Bitstrips launched a spin-off app known as Bitmoji, which allows users to create personalized stickers for use in instant messaging. In July 2016, Snapchat Inc. announced that it had acquired the company; the Bitstrips comic service was shut down, but Bitmoji remains operational, and has subsequently been given greater prominence within Snapchat's overall platform. == History == Bitstrips was co-developed by Toronto-based comic artist Jacob Blackstock and his high school friend, journalist Jesse Brown. The service was originally envisioned as a means to allow anyone to create their own comic strip without needing artistic skills. Brown explained that "it's so difficult and time-consuming to tell a story in comic book form, drawing the same characters again and again in these tiny little panels, and just the amount of craftsmanship required. And even if you can do it well, which I never could, it takes years to make a story." Brown stated that the service would be "groundwork for a whole new way to communicate", and went as far as describing the service as being a "YouTube for comics". Blackstock explained that the concept of Bitstrips was influenced by his own use of comics as a form of socialization; a student, Blackstock and his friends drew comics featuring each other and shared them during classes. He felt that Bitstrips was a "medium for self-expression", stating that "It's not just about you making the comics, but since you and your friends star in these comics, it's like you're the medium. The visual nature of comics just speaks so much louder than text." The service was publicly unveiled at South by Southwest in 2008. In 2009, the service introduced a version oriented towards the educational market, Bitstrips for Schools, which was initially piloted at a number of schools in Ontario. The service was praised by educators for being engaging to students, especially within language classes. Brown noted that students were using the service to create comics outside of class as well, stating that it was "so gratifying and shocking what people do with your tool to make their own stories in ways that you never would have anticipated. Some of them are just brilliant." In December 2012, Bitstrips launched a version for Facebook; by July 2013, Bitstrips had 10 million unique users on Facebook, having created over 50 million comics. In October 2013, Bitstrips launched a mobile app; in two months, Bitstrips became a top-downloaded app in 40 countries, and over 30 million avatars had been created with it. In November 2013, Bitstrips secured a round of funding from Horizons Ventures and Li Ka-shing. In October 2014, Bitstrips launched Bitmoji, a spin-off app that allows users to create stickers featuring Bitstrips characters in various templates. In July 2016, following unconfirmed reports earlier in the year, Snapchat Inc. announced that it had acquired Bitstrips. The company's staff continue to operate out of Toronto, but the original Bitstrips comic service was shut down in favour of focusing exclusively on Bitmoji, leaving many Bitstrips users to call for a reboot of the comic service.
Loss function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss. == Examples == === Regret === Leonard J. Savage argued that using non-Bayesian methods such as minimax, the loss function should be based on the idea of regret, i.e., the loss associated with a decision should be the difference between the consequences of the best decision that could have been made under circumstances will be known and the decision that was in fact taken before they were known. === Quadratic loss function === The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is t {\displaystyle t} , then a quadratic loss function is λ ( x ) = C ( t − x ) 2 {\displaystyle \lambda (x)=C(t-x)^{2}\;} for some constant C {\displaystyle C} ; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the squared error loss (SEL). Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratic loss function. The quadratic loss function is also used in linear-quadratic optimal control problems. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions. In the context of stochastic control, the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the Huber, log-cosh and SMAE losses are used when the data has many large outliers. === 0-1 loss function === In statistics and decision theory, a frequently used loss function is the 0-1 loss function L ( y ^ , y ) = { 0 if y = y ^ 1 if y ≠ y ^ {\displaystyle L({\hat {y}},y)={\begin{cases}0&{\text{if }}y={\hat {y}}\\1&{\text{if }}y\neq {\hat {y}}\end{cases}}} In information theory, this loss function is known as Hamming distortion. == Constructing loss and objective functions == In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. The existing methods for constructing objective functions are collected in the proceedings of two dedicated conferences. In particular, Andranik Tangian showed that the most usable objective functions — quadratic and additive — are determined by a few indifference points. He used this property in the models for constructing these objective functions from either ordinal or cardinal data that were elicited through computer-assisted interviews with decision makers. Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities and the European subsidies for equalizing unemployment rates among 271 German regions. == Expected loss == In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X {\displaystyle X} . === Statistics === Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under the two paradigms. ==== Frequentist expected loss ==== We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the probability distribution, P θ {\displaystyle P_{\theta }} , of the observed data, X {\displaystyle X} . This is also referred to as the risk function of the decision rule δ {\displaystyle \delta } and the parameter θ {\displaystyle \theta } . Here the decision rule depends on the outcome of X {\displaystyle X} . The risk function is given by: R ( θ , δ ) = E θ L ( θ , δ ( X ) ) = ∫ X L ( θ , δ ( x ) ) d P θ ( x ) . {\displaystyle R(\theta ,\delta )=\operatorname {E} _{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{X}L{\big (}\theta ,\delta (x){\big )}\,\mathrm {d} P_{\theta }(x).} Here, θ {\displaystyle \theta } is a fixed but possibly unknown state of nature, X {\displaystyle X} is a vector of observations stochastically drawn from a population, E θ {\displaystyle \operatorname {E} _{\theta }} is the expectation over all population values of X {\displaystyle X} , d P θ {\displaystyle \mathrm {d} P_{\theta }} is a probability measure over the event space of X {\displaystyle X} (parametrized by θ {\displaystyle \theta } ) and the integral is evaluated over the entire support of X {\displaystyle X} . ==== Bayes Risk ==== In a Bayesian approach, the expectation is calculated using the prior distribution π ∗ {\displaystyle \pi ^{}} of the parameter θ {\displaystyle \theta } : ρ ( π ∗ , a ) = ∫ Θ ∫ X L ( θ , a ( x ) ) d P ( x | θ ) d π ∗ ( θ ) = ∫ X ∫ Θ L ( θ , a ( x ) ) d π ∗ ( θ | x ) d M ( x ) {\displaystyle \rho (\pi ^{},a)=\int _{\Theta }\int _{\mathbf {X}}L(\theta ,a({\mathbf {x}}))\,\mathrm {d} P({\mathbf {x}}\vert \theta )\,\mathrm {d} \pi ^{}(\theta )=\int _{\mathbf {X}}\int _{\Theta }L(\theta ,a({\mathbf {x}}))\,\mathrm {d} \pi ^{}(\theta \vert {\mathbf {x}})\,\mathrm {d} M({\mathbf {x}})} where M ( x ) {\displaystyle M(\mathbf {x} )} is known as the predictive likelihood wherein θ {\displaystyle \theta } has been "integrated out," π ∗ ( θ | x ) {\displaystyle \pi ^{}(\theta |\mathbf {x} )} is the posterior distribution, and the order of integration has been changed. One then should choose the action a ∗ {\displaystyle a^{}} which minimises this expected loss, which is referred to as Bayes Risk. In the latter equation, the integrand inside d x {\displaystyle \mathrm {d} x} is known as the Posterior Risk, and minimising it with respect to decision a {\displaystyle a} also minimizes the overall Bayes Risk. This optimal decision, a ∗ {\displaystyle a^{}} is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ {\displaystyle \theta } , over all possible (probability-weighted) data outcomes. One advantage of the Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one, whereas choosing the actual frequentist optimal decision rule as a function of all possible observations, is a much more difficult problem. Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ {\displaystyle \theta } . ==== Examples in statistics ==== For a scalar parameter θ {\displaystyle \theta } , a decision function whose output θ ^ {\displaystyle {\hat {\theta }}} is an estimate of θ {\displaystyle \theta } , and a quadratic loss function (squared error loss) L ( θ , θ ^ ) = ( θ − θ ^ ) 2 , {\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat {\theta }})^{2},} the risk function becomes the mean squared error of the estimate, R ( θ , θ ^ ) = E θ [ ( θ − θ ^ ) 2 ] . {\displaystyle R(\theta ,{\hat {\thet