AI Data Usage

AI Data Usage — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

• 

  Control-flow integrity

  Control-flow integrity (CFI) is a general term for computer security techniques that prevent a wide variety of malware attacks from redirecting the flow of execution (the control flow) of a program. == Background == A computer program commonly changes its control flow to make decisions and use different parts of the code. Such transfers may be direct, in that the target address is written in the code itself, or indirect, in that the target address itself is a variable in memory or a CPU register. In a typical function call, the program performs a direct call, but returns to the caller function using the stack – an indirect backward-edge transfer. When a function pointer is called, such as from a virtual table, we say there is an indirect forward-edge transfer. Attackers seek to inject code into a program to make use of its privileges or to extract data from its memory space. Before executable code was commonly made read-only, an attacker could arbitrarily change the code as it is run, targeting direct transfers or even do with no transfers at all. After W^X became widespread, an attacker wants to instead redirect execution to a separate, unprotected area containing the code to be run, making use of indirect transfers: one could overwrite the virtual table for a forward-edge attack or change the call stack for a backward-edge attack (return-oriented programming). CFI is designed to protect indirect transfers from going to unintended locations. == Techniques == Associated techniques include code-pointer separation (CPS), code-pointer integrity (CPI), stack canaries, shadow stacks (SS), and vtable pointer verification. These protections can be classified into either coarse-grained or fine-grained based on the number of targets restricted. A coarse-grained forward-edge CFI implementation, could, for example, restrict the set of indirect call targets to any function that may be indirectly called in the program, while a fine-grained one would restrict each indirect call site to functions that have the same type as the function to be called. Similarly, for a backward edge scheme protecting returns, a coarse-grained implementation would only allow the procedure to return to a function of the same type (of which there could be many, especially for common prototypes), while a fine-grained one would enforce precise return matching (so it can return only to the function that called it). == Implementations == Related implementations are available in Clang (LLVM front-end),, GNU Compiler Collection, Microsoft's Control Flow Guard and Return Flow Guard, Google's Indirect Function-Call Checks and Reuse Attack Protector (RAP). === LLVM/Clang === The LLVM compiler's C/C++ front-end Clang provides a number of "CFI" schemes that works on the forward edge by checking for errors in virtual tables and type casts. Not all of the schemes are supported on all platforms and most of them, the exception being two "kcfi" schemes intended for low-level kernel software, depends on link-time optimization (LTO) to know what functions are supposed to be called in normal cases. Also provided is a separate "shadow call stack" (SCS) instrumentation pass that defends on the backward edge by checking for call stack modifications, available only for the aarch64 and RISC-V ISAs. And due to use of a shared processor register SCS is only enforceable on certain ABIs or if in other ways it is ensured that any other software using the register set (thread/processor) does not interfere with this use. Google has shipped Android with the Linux kernel compiled by Clang with link-time optimization (LTO) and CFI enabled since 2018. Even though SCS is available for the Linux kernel as an option, and support is also available for Android's system components it is recommended only to enable it for components for which it can be ensured that no third party code is loaded. === GCC === The GNU Compiler Collection implemented a "shadow call stack" compatible with Clang for aarch64 in v12 released in 2022. This feature is primarily intended for building the Linux kernel as support is missing from GCC user space libraries. === Intel Control-flow Enforcement Technology === Intel Control-flow Enforcement Technology (CET) detects compromises to control flow integrity with a shadow stack (SS) and indirect branch tracking (IBT). The kernel must map a region of memory for the shadow stack not writable to user space programs except by special instructions. The shadow stack stores a copy of the return address of each CALL. On a RET, the processor checks if the return address stored in the normal stack and shadow stack are equal. If the addresses are not equal, the processor generates an INT #21 (Control Flow Protection Fault). Indirect branch tracking detects indirect JMP or CALL instructions to unauthorized targets. It is implemented by adding a new internal state machine in the processor. The behavior of indirect JMP and CALL instructions is changed so that they switch the state machine from IDLE to WAIT_FOR_ENDBRANCH. In the WAIT_FOR_ENDBRANCH state, the next instruction to be executed is required to be the new ENDBRANCH instruction (ENDBR32 in 32-bit mode or ENDBR64 in 64-bit mode), which changes the internal state machine from WAIT_FOR_ENDBRANCH back to IDLE. Thus every authorized target of an indirect JMP or CALL must begin with ENDBRANCH. If the processor is in a WAIT_FOR_ENDBRANCH state (meaning, the previous instruction was an indirect JMP or CALL), and the next instruction is not an ENDBRANCH instruction, the processor generates an INT #21 (Control Flow Protection Fault). On processors not supporting CET indirect branch tracking, ENDBRANCH instructions are interpreted as NOPs and have no effect. === Microsoft Control Flow Guard === Control Flow Guard (CFG) was first released for Windows 8.1 Update 3 (KB3000850) in November 2014. Developers can add CFG to their programs by adding the /guard:cf linker flag before program linking in Visual Studio 2015 or newer. As of Windows 10 Creators Update (Windows 10 version 1703), the Windows kernel is compiled with CFG. The Windows kernel uses Hyper-V to prevent malicious kernel code from overwriting the CFG bitmap. CFG operates by creating a per-process bitmap, where a set bit indicates that the address is a valid destination. Before performing each indirect function call, the application checks if the destination address is in the bitmap. If the destination address is not in the bitmap, the program terminates. This makes it more difficult for an attacker to exploit a use-after-free by replacing an object's contents and then using an indirect function call to execute a payload. ==== Implementation details ==== For all protected indirect function calls, the _guard_check_icall function is called, which performs the following steps: Convert the target address to an offset and bit number in the bitmap. The highest 3 bytes are the byte offset in the bitmap The bit offset is a 5-bit value. The first four bits are the 4th through 8th low-order bits of the address. The 5th bit of the bit offset is set to 0 if the destination address is aligned with 0x10 (last four bits are 0), and 1 if it is not. Examine the target's address value in the bitmap If the target address is in the bitmap, return without an error. If the target address is not in the bitmap, terminate the program. ==== Bypass techniques ==== There are several generic techniques for bypassing CFG: Set the destination to code located in a non-CFG module loaded in the same process. Find an indirect call that was not protected by CFG (either CALL or JMP). Use a function call with a different number of arguments than the call is designed for, causing a stack misalignment, and code execution after the function returns (patched in Windows 10). Use a function call with the same number of arguments, but one of pointers passed is treated as an object and writes to a pointer-based offset, allowing overwriting a return address. Overwrite the function call used by the CFG to validate the address (patched in March 2015) Set the CFG bitmap to all 1's, allowing all indirect function calls Use a controlled-write primitive to overwrite an address on the stack (since the stack is not protected by CFG) === Microsoft eXtended Flow Guard === eXtended Flow Guard (XFG) has not been officially released yet, but is available in the Windows Insider preview and was publicly presented at Bluehat Shanghai in 2019. XFG extends CFG by validating function call signatures to ensure that indirect function calls are only to the subset of functions with the same signature. Function call signature validation is implemented by adding instructions to store the target function's hash in register r10 immediately prior to the indirect call and storing the calculated function hash in the memory immediately preceding the target address's code. When the indirect call is made, the XFG validation function compares the value in r10 to the target

  Read more →
• 

  Softplus

  In mathematics and machine learning, the softplus function is f ( x ) = ln ⁡ ( 1 + e x ) . {\displaystyle f(x)=\ln(1+e^{x}).} It is a smooth approximation (in fact, an analytic function) to the ramp function, which is known as the rectifier or ReLU (rectified linear unit) in machine learning. For large negative x {\displaystyle x} it is ln ⁡ ( 1 + e x ) = ln ⁡ ( 1 + ϵ ) ⪆ ln ⁡ 1 = 0 {\displaystyle \ln(1+e^{x})=\ln(1+\epsilon )\gtrapprox \ln 1=0} , so just above 0, while for large positive x {\displaystyle x} it is ln ⁡ ( 1 + e x ) ⪆ ln ⁡ ( e x ) = x {\displaystyle \ln(1+e^{x})\gtrapprox \ln(e^{x})=x} , so just above x {\displaystyle x} . The names softplus and SmoothReLU are used in machine learning. The name "softplus" (2000), by analogy with the earlier softmax (1989) is presumably because it is a smooth (soft) approximation of the positive part of x, which is sometimes denoted with a superscript plus, x + := max ( 0 , x ) {\displaystyle x^{+}:=\max(0,x)} . == Alternative forms == This function can be approximated as: ln ⁡ ( 1 + e x ) ≈ { ln ⁡ 2 , x = 0 , x 1 − e − x / ln ⁡ 2 , x ≠ 0 {\displaystyle \ln \left(1+e^{x}\right)\approx {\begin{cases}\ln 2,&x=0,\\[6pt]{\frac {x}{1-e^{-x/\ln 2}}},&x\neq 0\end{cases}}} By making the change of variables x = y ln ⁡ ( 2 ) {\displaystyle x=y\ln(2)} , this is equivalent to log 2 ⁡ ( 1 + 2 y ) ≈ { 1 , y = 0 , y 1 − e − y , y ≠ 0. {\displaystyle \log _{2}(1+2^{y})\approx {\begin{cases}1,&y=0,\\[6pt]{\frac {y}{1-e^{-y}}},&y\neq 0.\end{cases}}} A sharpness parameter k {\displaystyle k} may be included: f ( x ) = ln ⁡ ( 1 + e k x ) k , f ′ ( x ) = e k x 1 + e k x = 1 1 + e − k x . {\displaystyle f(x)={\frac {\ln(1+e^{kx})}{k}},\qquad \qquad f'(x)={\frac {e^{kx}}{1+e^{kx}}}={\frac {1}{1+e^{-kx}}}.} Additionally, the softplus function is equivalent to the log of the sigmoid function in the following way: − ln ⁡ ( sigmoid ( − x ) ) = − ln ⁡ ( 1 1 + e x ) = ln ⁡ ( 1 + e x ) = softplus ( x ) {\displaystyle -\ln({\text{sigmoid}}(-x))=-\ln \left({\frac {1}{1+e^{x}}}\right)=\ln \left(1+e^{x}\right)={\text{softplus}}(x)} == Related functions == The derivative of softplus is the standard logistic function: f ′ ( x ) = e x 1 + e x = 1 1 + e − x {\displaystyle f'(x)={\frac {e^{x}}{1+e^{x}}}={\frac {1}{1+e^{-x}}}} The logistic function or the sigmoid function is a smooth approximation of the rectifier, the Heaviside step function. === LogSumExp === The multivariable generalization of single-variable softplus is the LogSumExp with the first argument set to zero: L S E 0 + ⁡ ( x 1 , … , x n ) := LSE ⁡ ( 0 , x 1 , … , x n ) = ln ⁡ ( 1 + e x 1 + ⋯ + e x n ) . {\displaystyle \operatorname {LSE_{0}} ^{+}(x_{1},\dots ,x_{n}):=\operatorname {LSE} (0,x_{1},\dots ,x_{n})=\ln(1+e^{x_{1}}+\cdots +e^{x_{n}}).} The LogSumExp function is LSE ⁡ ( x 1 , … , x n ) = ln ⁡ ( e x 1 + ⋯ + e x n ) , {\displaystyle \operatorname {LSE} (x_{1},\dots ,x_{n})=\ln(e^{x_{1}}+\cdots +e^{x_{n}}),} and its gradient is the softmax; the softmax with the first argument set to zero is the multivariable generalization of the logistic function. Both LogSumExp and softmax are used in machine learning. === Convex conjugate === The convex conjugate (specifically, the Legendre transformation) of the softplus function is the negative binary entropy function (with base e). This is because (following the definition of the Legendre transformation: the derivatives are inverse functions) the derivative of softplus is the logistic function, whose inverse function is the logit, which is the derivative of negative binary entropy. Softplus can be interpreted as logistic loss (as a positive number), so, by duality, minimizing logistic loss corresponds to maximizing entropy. This justifies the principle of maximum entropy as loss minimization.

  Read more →
• 

  Quadratic classifier

  In statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier. == The classification problem == Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be. For a quadratic classifier, the correct solution is assumed to be quadratic in the measurements, so y will be decided based on x T A x + b T x + c {\displaystyle \mathbf {x^{T}Ax} +\mathbf {b^{T}x} +c} In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (i.e., either a line, a circle or ellipse, a parabola or a hyperbola). In this sense, we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces. == Quadratic discriminant analysis == Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical. When the normality assumption is true, the best possible test for the hypothesis that a given measurement is from a given class is the likelihood ratio test. Suppose there are only two groups, with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and covariance matrices Σ 0 , Σ 1 {\displaystyle \Sigma _{0},\Sigma _{1}} corresponding to y = 0 {\displaystyle y=0} and y = 1 {\displaystyle y=1} respectively. Then the likelihood ratio is given by Likelihood ratio = | 2 π Σ 1 | − 1 exp ⁡ ( − 1 2 ( x − μ 1 ) T Σ 1 − 1 ( x − μ 1 ) ) | 2 π Σ 0 | − 1 exp ⁡ ( − 1 2 ( x − μ 0 ) T Σ 0 − 1 ( x − μ 0 ) ) < t {\displaystyle {\text{Likelihood ratio}}={\frac {{\sqrt {|2\pi \Sigma _{1}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{1})^{T}\Sigma _{1}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{1})\right)}{{\sqrt {|2\pi \Sigma _{0}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{0})^{T}\Sigma _{0}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{0})\right)}} Read more →

• 

  Latent Dirichlet allocation

  In natural language processing, latent Dirichlet allocation (LDA) is a generative statistical model that explains how a collection of text documents can be described by a set of unobserved "topics." For example, given a set of news articles, LDA might discover that one topic is characterized by words like "president", "government", and "election", while another is characterized by "team", "game", and "score". It is one of the most common topic models. The LDA model was first presented as a graphical model for population genetics by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. The model was subsequently applied to machine learning by David Blei, Andrew Ng, and Michael I. Jordan in 2003. Although its most frequent application is in modeling text corpora, it has also been used for other problems, such as in clinical psychology, social science, and computational musicology. The core assumption of LDA is that documents are represented as a random mixture of latent topics, and each topic is characterized by a probability distribution over words. The model is a generalization of probabilistic latent semantic analysis (pLSA), differing primarily in that LDA treats the topic mixture as a Dirichlet prior, leading to more reasonable mixtures and less susceptibility to overfitting. Learning the latent topics and their associated probabilities from a corpus is typically done using Bayesian inference, often with methods like Gibbs sampling or variational Bayes. == History == In the context of population genetics, LDA was proposed by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. LDA was applied in machine learning by David Blei, Andrew Ng and Michael I. Jordan in 2003. == Overview == === Population genetics === In population genetics, the model is used to detect the presence of structured genetic variation in a group of individuals. The model assumes that alleles carried by individuals under study have origin in various extant or past populations. The model and various inference algorithms allow scientists to estimate the allele frequencies in those source populations and the origin of alleles carried by individuals under study. The source populations can be interpreted ex-post in terms of various evolutionary scenarios. In association studies, detecting the presence of genetic structure is considered a necessary preliminary step to avoid confounding. === Clinical psychology, mental health, and social science === In clinical psychology research, LDA has been used to identify common themes of self-images experienced by young people in social situations. Other social scientists have used LDA to examine large sets of topical data from discussions on social media (e.g., tweets about prescription drugs). Additionally, supervised Latent Dirichlet Allocation with covariates (SLDAX) has been specifically developed to combine latent topics identified in texts with other manifest variables. This approach allows for the integration of text data as predictors in statistical regression analyses, improving the accuracy of mental health predictions. One of the main advantages of SLDAX over traditional two-stage approaches is its ability to avoid biased estimates and incorrect standard errors, allowing for a more accurate analysis of psychological texts. In the field of social sciences, LDA has proven to be useful for analyzing large datasets, such as social media discussions. For instance, researchers have used LDA to investigate tweets discussing socially relevant topics, like the use of prescription drugs and cultural differences in China. By analyzing these large text corpora, it is possible to uncover patterns and themes that might otherwise go unnoticed, offering valuable insights into public discourse and perception in real time. === Musicology === In the context of computational musicology, LDA has been used to discover tonal structures in different corpora. === Machine learning === One application of LDA in machine learning – specifically, topic discovery, a subproblem in natural language processing – is to discover topics in a collection of documents, and then automatically classify any individual document within the collection in terms of how "relevant" it is to each of the discovered topics. A topic is considered to be a set of terms (i.e., individual words or phrases) that, taken together, suggest a shared theme. For example, in a document collection related to pet animals, the terms dog, spaniel, beagle, golden retriever, puppy, bark, and woof would suggest a DOG_related theme, while the terms cat, siamese, Maine coon, tabby, manx, meow, purr, and kitten would suggest a CAT_related theme. There may be many more topics in the collection – e.g., related to diet, grooming, healthcare, behavior, etc. that we do not discuss for simplicity's sake. (Very common, so called stop words in a language – e.g., "the", "an", "that", "are", "is", etc., – would not discriminate between topics and are usually filtered out by pre-processing before LDA is performed. Pre-processing also converts terms to their "root" lexical forms – e.g., "barks", "barking", and "barked" would be converted to "bark".) If the document collection is sufficiently large, LDA will discover such sets of terms (i.e., topics) based upon the co-occurrence of individual terms, though the task of assigning a meaningful label to an individual topic (i.e., that all the terms are DOG_related) is up to the user, and often requires specialized knowledge (e.g., for collection of technical documents). The LDA approach assumes that: The semantic content of a document is composed by combining one or more terms from one or more topics. Certain terms are ambiguous, belonging to more than one topic, with different probability. (For example, the term training can apply to both dogs and cats, but are more likely to refer to dogs, which are used as work animals or participate in obedience or skill competitions.) However, in a document, the accompanying presence of specific neighboring terms (which belong to only one topic) will disambiguate their usage. Most documents will contain only a relatively small number of topics. In the collection, e.g., individual topics will occur with differing frequencies. That is, they have a probability distribution, so that a given document is more likely to contain some topics than others. Within a topic, certain terms will be used much more frequently than others. In other words, the terms within a topic will also have their own probability distribution. When LDA machine learning is employed, both sets of probabilities are computed during the training phase, using Bayesian methods and an expectation–maximization algorithm. LDA is a generalization of older approach of probabilistic latent semantic analysis (pLSA), The pLSA model is equivalent to LDA under a uniform Dirichlet prior distribution. pLSA relies on only the first two assumptions above and does not care about the remainder. While both methods are similar in principle and require the user to specify the number of topics to be discovered before the start of training (as with k-means clustering) LDA has the following advantages over pLSA: LDA yields better disambiguation of words and a more precise assignment of documents to topics. Computing probabilities allows a "generative" process by which a collection of new "synthetic documents" can be generated that would closely reflect the statistical characteristics of the original collection. Unlike LDA, pLSA is vulnerable to overfitting especially when the size of corpus increases. The LDA algorithm is more readily amenable to scaling up for large data sets using the MapReduce approach on a computing cluster. == Model == With plate notation, which is often used to represent probabilistic graphical models (PGMs), the dependencies among the many variables can be captured concisely. The boxes are "plates" representing replicates, which are repeated entities. The outer plate represents documents, while the inner plate represents the repeated word positions in a given document; each position is associated with a choice of topic and word. The variable names are defined as follows: M denotes the number of documents N is number of words in a given document (document i has N i {\displaystyle N_{i}} words) α is the parameter of the Dirichlet prior on the per-document topic distributions β is the parameter of the Dirichlet prior on the per-topic word distribution θ i {\displaystyle \theta _{i}} is the topic distribution for document i φ k {\displaystyle \varphi _{k}} is the word distribution for topic k z i j {\displaystyle z_{ij}} is the topic for the j-th word in document i w i j {\displaystyle w_{ij}} is the specific word. The fact that W is grayed out means that words w i j {\displaystyle w_{ij}} are the only observable variables, and the other variables are latent variables. As proposed in the original paper, a sparse Dirichlet prior can be used to model the to

  Read more →
• 

  WebGPU Shading Language

  WebGPU Shading Language (WGSL, internet media type: text/wgsl) is a high-level shading language and the normative shader language for the WebGPU API on the web. WGSL's syntax is influenced by Rust and is designed with strong static validation, explicit resource binding, and portability in mind for secure execution in browsers. In web contexts, WebGPU implementations accept WGSL source and perform compilation to platform-specific intermediate forms (for example, to SPIR‑V, DXIL, or MSL via the user agent), but such backends are not exposed to web content. == History and background == Graphics on the web historically used WebGL, with shaders written in GLSL ES. As applications demanded more modern GPU features and finer control over compute and graphics pipelines, the W3C's GPU for the Web Community Group and Working Group created WebGPU and its companion shading language, WGSL, to provide a secure, portable model suitable for the web platform. WGSL was developed to be human-readable, avoid undefined behavior common in legacy shading languages, and align closely with WebGPU's resource and validation model. == Design goals == WGSL's design emphasizes: Safety and determinism suitable for web security constraints (extensive static validation and well-defined semantics). Portability across diverse GPU backends via an abstract resource model shared with WebGPU. Readability and explicitness (no preprocessor, minimal implicit conversions, explicit address spaces and bindings). Alignment with modern GPU features (compute, storage buffers, textures, atomics) while retaining a familiar C/Rust-like syntax. == Language overview == === Types and values === Core scalar types include bool, i32, u32, and f32. Vectors (e.g., vec2, vec3, vec4) and matrices (up to 4×4) are available for floating-point element types. Optional f16 (half precision) may be enabled via a WebGPU feature; availability is implementation-dependent. Atomic types (atomic, atomic) support limited atomic operations in qualified address spaces. === Variables and address spaces === Variables are declared with let (immutable), var (mutable), or const (compile-time constant). Storage classes (address spaces) include function, private, workgroup, uniform, and storage with read or read_write access as applicable. WGSL defines explicit layout and alignment rules; attributes such as @align, @size, and @stride control data layout for buffer interoperability. === Functions and control flow === Functions use explicit parameter and return types. Control flow includes if, switch, for, while, and loop constructs, with break/continue. Recursion is disallowed; entry-point call graphs must be acyclic. === Entry points and attributes === Shaders define stage entry points with @vertex, @fragment, or @compute. Attributes annotate bindings and interfaces, including @group, @binding (resource binding), @location (user-defined I/O), @builtin (stage built-ins such as position or global_invocation_id), @interpolate, and @workgroup_size. === Resources === WGSL exposes buffers (uniform, storage), textures (sampled, storage, and multisampled variants), and samplers (filtering/non-filtering/comparison). The binding model is explicit via descriptor sets called groups and bindings, matching WebGPU's pipeline layout model. == Compilation and validation == Browsers compile WGSL to platform-appropriate representations and native driver formats; the specific compilation pipeline is not observable by web content. WGSL source undergoes strict parsing and static validation, and WebGPU enforces robust resource access rules to avoid out-of-bounds memory hazards, contributing to predictable behavior across implementations. == Shader stages == WGSL supports three pipeline stages: vertex, fragment, and compute. === Vertex shaders === Vertex shaders transform per-vertex inputs and produce values for rasterization, including a clip-space position written to the position builtin. ==== Example ==== === Fragment shaders === Fragment shaders run per-fragment and compute color (and optionally depth) outputs written to color attachments. ==== Example ==== If half-precision (vec4h, shorthand for vec4) is desired, the code must be prefaced with a enable f16; statement. === Compute shaders === Compute shaders run in workgroups and are used for general-purpose GPU computations. ==== Example ==== == Differences from GLSL and HLSL == Compared with legacy shading languages, WGSL: Omits a preprocessor and requires explicit types and conversions. Uses explicit address spaces and binding annotations aligned with WebGPU's model. Enforces strict validation to avoid undefined behavior common in other shading languages. Defines a portable, web-focused feature set; 16-bit types and other features are opt-in and may depend on device capabilities.

  Read more →
• 

  FastICA

  FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology. Like most ICA algorithms, FastICA seeks an orthogonal rotation of prewhitened data, through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. Non-gaussianity serves as a proxy for statistical independence, which is a very strong condition and requires infinite data to verify. FastICA can also be alternatively derived as an approximative Newton iteration. == Algorithm == === Prewhitening the data === Let the X := ( x i j ) ∈ R N × M {\displaystyle \mathbf {X} :=(x_{ij})\in \mathbb {R} ^{N\times M}} denote the input data matrix, M {\displaystyle M} the number of columns corresponding with the number of samples of mixed signals and N {\displaystyle N} the number of rows corresponding with the number of independent source signals. The input data matrix X {\displaystyle \mathbf {X} } must be prewhitened, or centered and whitened, before applying the FastICA algorithm to it. Centering the data entails demeaning each component of the input data X {\displaystyle \mathbf {X} } , that is, for each i = 1 , … , N {\displaystyle i=1,\ldots ,N} and j = 1 , … , M {\displaystyle j=1,\ldots ,M} . After centering, each row of X {\displaystyle \mathbf {X} } has an expected value of 0 {\displaystyle 0} . Whitening the data requires a linear transformation L : R N × M → R N × M {\displaystyle \mathbf {L} :\mathbb {R} ^{N\times M}\to \mathbb {R} ^{N\times M}} of the centered data so that the components of L ( X ) {\displaystyle \mathbf {L} (\mathbf {X} )} are uncorrelated and have variance one. More precisely, if X {\displaystyle \mathbf {X} } is a centered data matrix, the covariance of L x := L ( X ) {\displaystyle \mathbf {L} _{\mathbf {x} }:=\mathbf {L} (\mathbf {X} )} is the ( N × N ) {\displaystyle (N\times N)} -dimensional identity matrix, that is, A common method for whitening is by performing an eigenvalue decomposition on the covariance matrix of the centered data X {\displaystyle \mathbf {X} } , E { X X T } = E D E T {\displaystyle E\left\{\mathbf {X} \mathbf {X} ^{T}\right\}=\mathbf {E} \mathbf {D} \mathbf {E} ^{T}} , where E {\displaystyle \mathbf {E} } is the matrix of eigenvectors and D {\displaystyle \mathbf {D} } is the diagonal matrix of eigenvalues. The whitened data matrix is defined thus by === Single component extraction === The iterative algorithm finds the direction for the weight vector w ∈ R N {\displaystyle \mathbf {w} \in \mathbb {R} ^{N}} that maximizes a measure of non-Gaussianity of the projection w T X {\displaystyle \mathbf {w} ^{T}\mathbf {X} } , with X ∈ R N × M {\displaystyle \mathbf {X} \in \mathbb {R} ^{N\times M}} denoting a prewhitened data matrix as described above. Note that w {\displaystyle \mathbf {w} } is a column vector. To measure non-Gaussianity, FastICA relies on a nonquadratic nonlinear function f ( u ) {\displaystyle f(u)} , its first derivative g ( u ) {\displaystyle g(u)} , and its second derivative g ′ ( u ) {\displaystyle g^{\prime }(u)} . Hyvärinen states that the functions are useful for general purposes, while may be highly robust. The steps for extracting the weight vector w {\displaystyle \mathbf {w} } for single component in FastICA are the following: Randomize the initial weight vector w {\displaystyle \mathbf {w} } Let w + ← E { X g ( w T X ) T } − E { g ′ ( w T X ) } w {\displaystyle \mathbf {w} ^{+}\leftarrow E\left\{\mathbf {X} g(\mathbf {w} ^{T}\mathbf {X} )^{T}\right\}-E\left\{g'(\mathbf {w} ^{T}\mathbf {X} )\right\}\mathbf {w} } , where E { . . . } {\displaystyle E\left\{...\right\}} means averaging over all column-vectors of matrix X {\displaystyle \mathbf {X} } Let w ← w + / ‖ w + ‖ {\displaystyle \mathbf {w} \leftarrow \mathbf {w} ^{+}/\|\mathbf {w} ^{+}\|} If not converged, go back to 2 === Multiple component extraction === The single unit iterative algorithm estimates only one weight vector which extracts a single component. Estimating additional components that are mutually "independent" requires repeating the algorithm to obtain linearly independent projection vectors - note that the notion of independence here refers to maximizing non-Gaussianity in the estimated components. Hyvärinen provides several ways of extracting multiple components with the simplest being the following. Here, 1 M {\displaystyle \mathbf {1_{M}} } is a column vector of 1's of dimension M {\displaystyle M} . Algorithm FastICA Input: C {\displaystyle C} Number of desired components Input: X ∈ R N × M {\displaystyle \mathbf {X} \in \mathbb {R} ^{N\times M}} Prewhitened matrix, where each column represents an N {\displaystyle N} -dimensional sample, where C <= N {\displaystyle C<=N} Output: W ∈ R N × C {\displaystyle \mathbf {W} \in \mathbb {R} ^{N\times C}} Un-mixing matrix where each column projects X {\displaystyle \mathbf {X} } onto independent component. Output: S ∈ R C × M {\displaystyle \mathbf {S} \in \mathbb {R} ^{C\times M}} Independent components matrix, with M {\displaystyle M} columns representing a sample with C {\displaystyle C} dimensions. for p in 1 to C: w p ← {\displaystyle \mathbf {w_{p}} \leftarrow } Random vector of length N while w p {\displaystyle \mathbf {w_{p}} } changes w p ← 1 M X g ( w p T X ) T − 1 M g ′ ( w p T X ) 1 M w p {\displaystyle \mathbf {w_{p}} \leftarrow {\frac {1}{M}}\mathbf {X} g(\mathbf {w_{p}} ^{T}\mathbf {X} )^{T}-{\frac {1}{M}}g'(\mathbf {w_{p}} ^{T}\mathbf {X} )\mathbf {1_{M}} \mathbf {w_{p}} } w p ← w p − ∑ j = 1 p − 1 ( w p T w j ) w j {\displaystyle \mathbf {w_{p}} \leftarrow \mathbf {w_{p}} -\sum _{j=1}^{p-1}(\mathbf {w_{p}} ^{T}\mathbf {w_{j}} )\mathbf {w_{j}} } w p ← w p ‖ w p ‖ {\displaystyle \mathbf {w_{p}} \leftarrow {\frac {\mathbf {w_{p}} }{\|\mathbf {w_{p}} \|}}} output W ← [ w 1 , … , w C ] {\displaystyle \mathbf {W} \leftarrow {\begin{bmatrix}\mathbf {w_{1}} ,\dots ,\mathbf {w_{C}} \end{bmatrix}}} output S ← W T X {\displaystyle \mathbf {S} \leftarrow \mathbf {W^{T}} \mathbf {X} }

  Read more →
• 

  Ground truth

  Ground truth is information that is known to be real or true, provided by direct observation and measurement (i.e. empirical evidence) as opposed to information provided by inference. The term ground truth appeared in remote sensing literature as early as 1972, when NASA described it as essential "data about ... materials on the earth's surface" used to calibrate measurements. It was later adopted by the statistical modeling and machine learning communities. == Etymology == The Oxford English Dictionary (s.v. ground truth) records the use of the word Groundtruth in the sense of 'fundamental truth' from Henry Ellison's poem "The Siberian Exile's Tale", published in 1833. == Usage == The term "ground truth" can be used as a noun, adjective, and verb. Noun: "ground truth" (no hyphen). Example: "The ground truth is essential for training accurate models." Adjective: "ground-truth" (hyphenated compound adjective). Example: "We need to use ground-truth data to validate the model." Verb: "to ground-truth" or "to groundtruth" (compound verb,). Example: "We need to ground-truth the results to ensure their accuracy." == Statistics and machine learning == In statistics and machine learning, ground truth is the ideal expected result, used in statistical models to prove or disprove research hypotheses. "Ground truthing" is the process of gathering the good data for this test. Ground truth is typically included in labeled data. In machine learning, "ground truth" is not necessarily objectively correct or true. For example, in training AI models or relevance rankers, it may be a set of judgments made by people or inferred from user behavior, which may depend on context. For example, in Bayesian spam filtering, a supervised learning system is typically trained by examples labeled as spam and non-spam. Although these labels may be subjective or inaccurate, they are considered ground truth. True ground truth in machine learning is objective data. For example, suppose we are testing a stereo vision system to see how well it can estimate 3D positions. A calibrated laser rangefinder may provide accurate distances as ground truth. == Remote sensing == In remote sensing, "ground truth" refers to information collected at the imaged location. Ground truth allows image data to be related to real features and materials on the ground. The collection of ground truth data enables calibration of remote-sensing data, and aids in the interpretation and analysis of what is being sensed. Examples include cartography, meteorology, analysis of aerial photographs, satellite imagery and other techniques in which data are gathered at a distance. More specifically, ground truth may refer to a process in which "pixels" on a satellite image are compared to what is imaged (at the time of capture) in order to verify the contents of the "pixels" in the image (noting that the concept of "pixel" is imaging-system-dependent). In the case of a classified image, supervised classification can help to determine the accuracy of the classification by the remote sensing system which can minimize error in the classification. Ground truth is usually done on site, correlating what is known with surface observations and measurements of various properties of the features of the ground resolution cells under study in the remotely sensed digital image. The process also involves taking geographic coordinates of the ground resolution cell with GPS technology and comparing those with the coordinates of the "pixel" being studied provided by the remote sensing software to understand and analyze the location errors and how it may affect a particular study. Ground truth is important in the initial supervised classification of an image. When the identity and location of land cover types are known through a combination of field work, maps, and personal experience these areas are known as training sites. The spectral characteristics of these areas are used to train the remote sensing software using decision rules for classifying the rest of the image. These decision rules such as Maximum Likelihood Classification, Parallelopiped Classification, and Minimum Distance Classification offer different techniques to classify an image. Additional ground truth sites allow the remote sensor to establish an error matrix that validates the accuracy of the classification method used. Different classification methods may have different percentages of error for a given classification project. It is important that the remote sensor chooses a classification method that works best with the number of classifications used while providing the least amount of error. Ground truth also helps with atmospheric correction. Since images from satellites have to pass through the atmosphere, they can get distorted because of absorption in the atmosphere. So ground truth can help fully identify objects in satellite photos. === Errors of commission === An example of an error of commission is when a pixel reports the presence of a feature (such a tree) that, in reality, is absent (no tree is actually present). Ground truthing ensures that the error matrices have a higher accuracy percentage than would be the case if no pixels were ground-truthed. This value is the complement of the user's accuracy, i.e. Commission Error = 1 - user's accuracy. === Errors of omission === An example of an error of omission is when pixels of a certain type, for example, maple trees, are not classified as maple trees. The process of ground-truthing helps to ensure that the pixel is classified correctly and the error matrices are more accurate. This value is the complement of the producer's accuracy, i.e. Omission Error = 1 - producer's accuracy == Geographical information systems == In GIS the spatial data is modeled as field (like in remote sensing raster images) or as object (like in vectorial map representation). They are modeled from the real world (also named geographical reality), typically by a cartographic process (illustrated). Geographic information systems such as GIS, GPS, and GNSS, have become so widespread that the term "ground truth" has taken on special meaning in that context. If the location coordinates returned by a location method such as GPS are an estimate of a location, then the "ground truth" is the actual location on Earth. A smart phone might return a set of estimated location coordinates such as 43.87870, −103.45901. The ground truth being estimated by those coordinates is the tip of George Washington's nose on Mount Rushmore. The accuracy of the estimate is the maximum distance between the location coordinates and the ground truth. We could say in this case that the estimate accuracy is 10 meters, meaning that the point on Earth represented by the location coordinates is thought to be within 10 meters of George's nose—the ground truth. In slang, the coordinates indicate where we think George Washington's nose is located, and the ground truth is where it really is. In practice a smart phone or hand-held GPS unit is routinely able to estimate the ground truth within 6–10 meters. Specialized instruments can reduce GPS measurement error to under a centimeter. == Military usage == US military slang uses "ground truth" to refer to the facts comprising a tactical situation—as opposed to intelligence reports, mission plans, and other descriptions reflecting the conative or policy-based projections of the industrial·military complex. The term appears in the title of the Iraq War documentary film The Ground Truth (2006), and also in military publications, for example Stars and Stripes saying: "Stripes decided to figure out what the ground truth was in Iraq."

  Read more →
• 

  Oja's rule

  Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja (Finnish pronunciation: [ˈojɑ], AW-yuh), is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons. == Theory == Oja's rule requires a number of simplifications to derive, but in its final form it is demonstrably stable, unlike Hebb's rule. It is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability and success. === Formula === Consider a simplified model of a neuron y {\displaystyle y} that returns a linear combination of its inputs x using presynaptic weights w: y ( x ) = ∑ j = 1 m x j w j {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}} Oja's rule defines the change in presynaptic weights w given the output response y {\displaystyle y} of a neuron to its inputs x to be Δ w = w n + 1 − w n = η y n ( x n − y n w n ) , {\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),} where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as d w d t = η y ( t ) ( x ( t ) − y ( t ) w ( t ) ) . {\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).} === Derivation === The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written Δ w = η y ( x n ) x n {\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}} , or in scalar form with implicit n-dependence, w i ( n + 1 ) = w i ( n ) + η y ( x ) x i {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}} , where y(xn) is again the output, this time explicitly dependent on its input vector x. Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one: w i ( n + 1 ) = w i ( n ) + η y ( x ) x i ( ∑ j = 1 m [ w j ( n ) + η y ( x ) x j ] p ) 1 / p {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}} . Note that in Oja's original paper, p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality. For a small learning rate | η | ≪ 1 {\displaystyle |\eta |\ll 1} the equation can be expanded as a Power series in η {\displaystyle \eta } . w i ( n + 1 ) = w i ( n ) ( ∑ j w j p ( n ) ) 1 / p + η ( y x i ( ∑ j w j p ( n ) ) 1 / p − w i ( n ) ∑ j y x j w j p − 1 ( n ) ( ∑ j w j p ( n ) ) ( 1 + 1 / p ) ) + O ( η 2 ) {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})} . For small η, our higher-order terms O(η2) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or y ( x ) = ∑ j = 1 m x j w j p − 1 {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}} . We also specify that our weights normalize to 1, which will be a necessary condition for stability, so | w | = ( ∑ j = 1 m w j p ) 1 / p = 1 {\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1} , which, when substituted into our expansion, gives Oja's rule, or w i ( n + 1 ) = w i ( n ) + η y ( x i − w i ( n ) y ) {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)} . === Stability and PCA === In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis. A principal component aj is extracted from a dataset x through some associated vector qj, or aj = qj⋅x, and we can restore our original dataset by taking x = ∑ j a j q j {\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}} . In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q1, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors Xi, that its correlation matrix Rij = XiXj has an associated eigenvector given by qj with eigenvalue λj. The variance of outputs of our Oja neuron σ2(n) = ⟨y2(n)⟩ then converges with time iterations to the principal eigenvalue, or lim n → ∞ σ 2 ( n ) = λ 1 {\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}} . These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is ∑ n = 1 ∞ η ( n ) = ∞ , ∑ n = 1 ∞ η ( n ) p < ∞ , p > 1 {\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1} . Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time. == Applications == Oja's rule was originally described in Oja's 1982 paper, but the principle of self-organization to which it is applied is first attributed to Alan Turing in 1952. PCA has also had a long history of use before Oja's rule formalized its use in network computation in 1989. The model can thus be applied to any problem of self-organizing mapping, in particular those in which feature extraction is of primary interest. Therefore, Oja's rule has an important place in image and speech processing. It is also useful as it expands easily to higher dimensions of processing, thus being able to integrate multiple outputs quickly. A canonical example is its use in binocular vision. === Biology and Oja's subspace rule === There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of Δ w i j ∝ ⟨ x i y j ⟩ − ϵ ⟨ ( c p r e ∗ ∑ k w i k y k ) ⋅ ( c p o s t ∗ y j ) ⟩ , {\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }y_{j}\right)\right\rangle ,} where as before wij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and cpre and cpost are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace, namely Δ w = C x ⋅ w − w ⋅ C y . {\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}

  Read more →
• 

  Psychology of reasoning

  The psychology of reasoning (also known as the cognitive science of reasoning) is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory. Psychological experiments on how humans and other animals reason have been carried out for over 100 years. An enduring question is whether or not people have the capacity to be rational. Current research in this area addresses various questions about reasoning, rationality, judgments, intelligence, relationships between emotion and reasoning, and development. == Everyday reasoning == One of the most obvious areas in which people employ reasoning is with sentences in everyday language. Most experimentation on deduction has been carried out on hypothetical thought, in particular, examining how people reason about conditionals, e.g., If A then B. Participants in experiments make the modus ponens inference, given the indicative conditional If A then B, and given the premise A, they conclude B. However, given the indicative conditional and the minor premise for the modus tollens inference, not-B, about half of the participants in experiments conclude not-A and the remainder concludes that nothing follows. The ease with which people make conditional inferences is affected by context, as demonstrated in the well-known selection task developed by Peter Wason. Participants are better able to test a conditional in an ecologically relevant context, e.g., if the envelope is sealed then it must have a 50 cent stamp on it compared to one that contains symbolic content, e.g., if the letter is a vowel then the number is even. Background knowledge can also lead to the suppression of even the simple modus ponens inference Participants given the conditional if Lisa has an essay to write then she studies late in the library and the premise Lisa has an essay to write make the modus ponens inference 'she studies late in the library', but the inference is suppressed when they are also given a second conditional if the library stays open then she studies late in the library. Interpretations of the suppression effect are controversial Other investigations of propositional inference examine how people think about disjunctive alternatives, e.g., A or else B, and how they reason about negation, e.g., It is not the case that A and B. Many experiments have been carried out to examine how people make relational inferences, including comparisons, e.g., A is better than B. Such investigations also concern spatial inferences, e.g. A is in front of B and temporal inferences, e.g. A occurs before B. Other common tasks include categorical syllogisms, used to examine how people reason about quantifiers such as All or Some, e.g., Some of the A are not B. For example if all A are B and some B are C, what (if anything) follows? == Theories of reasoning == There are several alternative theories of the cognitive processes that human reasoning is based on. One view is that people rely on a mental logic consisting of formal (abstract or syntactic) inference rules similar to those developed by logicians in the propositional calculus. Another view is that people rely on domain-specific or content-sensitive rules of inference. A third view is that people rely on mental models, that is, mental representations that correspond to imagined possibilities. A fourth view is that people compute probabilities. One controversial theoretical issue is the identification of an appropriate competence model, or a standard against which to compare human reasoning. Initially classical logic was chosen as a competence model. Subsequently, some researchers opted for non-monotonic logic and Bayesian probability. Research on mental models and reasoning has led to the suggestion that people are rational in principle but err in practice. Connectionist approaches towards reasoning have also been proposed. Despite the ongoing debate about the cognitive processes involved in human reasoning, recent research has shown that multiple approaches can be useful in modeling human thinking. For instance, studies have found that people's reasoning is often influenced by their prior beliefs, which can be modeled using Bayesian probability theory. Additionally, research on mental models has shown that people tend to reason about problems by constructing multiple mental representations of the situation, which can help them to identify relevant features and make inferences based on their understanding of the problem. Moreover, connectionist approaches to reasoning have also gained attention, which focus on the neural network models that can learn from data and generalize to new situations. == Development of reasoning == It is an active question in psychology how, why, and when the ability to reason develops from infancy to adulthood. Jean Piaget's theory of cognitive development posited general mechanisms and stages in the development of reasoning from infancy to adulthood. According to the neo-Piagetian theories of cognitive development, changes in reasoning with development come from increasing working memory capacity, increasing speed of processing, and enhanced executive functions and control. Increasing self-awareness is also an important factor. In their book The Enigma of Reason, the cognitive scientists Hugo Mercier and Dan Sperber put forward an "argumentative" theory of reasoning, claiming that humans evolved to reason primarily to justify our beliefs and actions and to convince others in a social environment. Key evidence for their theory includes the errors in reasoning that solitary individuals are prone to when their arguments are not criticized, such as logical fallacies, and how groups become much better at performing cognitive reasoning tasks when they communicate with one another and can evaluate each other's arguments. Sperber and Mercier offer one attempt to resolve the apparent paradox that the confirmation bias is so strong despite the function of reasoning naively appearing to be to come to veridical conclusions about the world. The study of the development of reasoning abilities is an ongoing area of research in psychology, and multiple factors have been proposed to explain how, why, and when reasoning develops from infancy to adulthood. Recent research has suggested that early experiences and social interactions play a critical role in the development of reasoning abilities. For example, studies have shown that infants as young as six months old can engage in basic logical reasoning, such as reasoning about the relationship between objects and their properties. Furthermore, research has highlighted the importance of parental interaction and cognitive stimulation in the development of children's reasoning abilities. Additionally, studies have suggested that cultural factors, such as educational practices and the emphasis on critical thinking, can also influence the development of reasoning skills across different populations. == Different sorts of reasoning == Philip Johnson-Laird trying to taxonomize thought, distinguished between goal-directed thinking and thinking without goal, noting that association was involved in unrelated reading. He argues that goal directed reasoning can be classified based on the problem space involved in a solution, citing Allen Newell and Herbert A. Simon. Inductive reasoning makes broad generalizations from specific cases or observations. In this process of reasoning, general assertions are made based on past specific pieces of evidence. This kind of reasoning allows the conclusion to be false even if the original statement is true. For example, if one observes a college athlete, one makes predictions and assumptions about other college athletes based on that one observation. Scientists use inductive reasoning to create theories and hypotheses. Philip Johnson-Laird distinguished inductive from deductive reasoning, in that the former creates semantic information while the later does not . In opposition, deductive reasoning is a basic form of valid reasoning. In this reasoning process a person starts with a known claim or a general belief and from there asks what follows from these foundations or how will these premises influence other beliefs. In other words, deduction starts with a hypothesis and examines the possibilities to reach a conclusion. Deduction helps people understand why their predictions are wrong and indicates that their prior knowledge or beliefs are off track. An example of deduction can be seen in the scientific method when testing hypotheses and theories. Although the conclusion usually corresponds and therefore proves the hypothesis, there are some cases where the conclusion is logical, but the generalization is not. For example, the argument, "All young girls wear skirts; Julie is a young

  Read more →
• 

  ARKA descriptors in QSAR

  In computational chemistry and cheminformatics, ARKA descriptors in QSAR are a class of molecular descriptors used in quantitative structure–activity relationship (QSAR) modeling (or related approaches such as QSPR and QSTR), a computational method for predicting the biological activity or toxicity of chemical compounds based on their molecular structure. Molecular descriptors are numerical values that summarize information about a molecule's structure, topology, geometry, or physicochemical properties in a form suitable for machine learning or statistical modeling. ARKA (Arithmetic Residuals in K-Groups Analysis) descriptors differ from traditional descriptors by encoding atomic-level information through recursive autoregression techniques, which aim to capture subtle structural patterns and improve predictive accuracy. They are designed to be both interpretable and well-suited to modeling nonlinear relationships in QSAR studies. == Comparisons == While QSAR is essentially a similarity-based approach, the occurrence of activity/property cliffs may greatly reduce the predictive accuracy of the developed models. The novel Arithmetic Residuals in K-groups Analysis (ARKA) approach is a supervised dimensionality reduction technique developed by the DTC Laboratory, Jadavpur University that can easily identify activity cliffs in a data set. Activity cliffs are similar in their structures but differ considerably in their activity. The basic idea of the ARKA descriptors is to group the conventional QSAR descriptors based on a predefined criterion and then assign weightage to each descriptor in each group. ARKA descriptors have also been used to develop classification-based and regression-based QSAR models with acceptable quality statistics. The ARKA descriptors have been used for the identification of activity cliffs in QSAR studies and/or model development by multiple researchers. A tutorial presentation on the ARKA descriptors is available. Recently a multi-class ARKA framework has been proposed for improved q-RASAR model generation.

  Read more →
• 

  Jackknife variance estimates for random forest

  In statistics, jackknife variance estimates for random forest are a way to estimate the variance in random forest models, in order to eliminate the bootstrap effects. == Jackknife variance estimates == The sampling variance of bagged learners is: V ( x ) = V a r [ θ ^ ∞ ( x ) ] {\displaystyle V(x)=Var[{\hat {\theta }}^{\infty }(x)]} Jackknife estimates can be considered to eliminate the bootstrap effects. The jackknife variance estimator is defined as: V ^ j = n − 1 n ∑ i = 1 n ( θ ^ ( − i ) − θ ¯ ) 2 {\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\hat {\theta }}_{(-i)}-{\overline {\theta }})^{2}} In some classification problems, when random forest is used to fit models, jackknife estimated variance is defined as: V ^ j = n − 1 n ∑ i = 1 n ( t ¯ ( − i ) ⋆ ( x ) − t ¯ ⋆ ( x ) ) 2 {\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\overline {t}}_{(-i)}^{\star }(x)-{\overline {t}}^{\star }(x))^{2}} Here, t ⋆ {\displaystyle t^{\star }} denotes a decision tree after training, t ( − i ) ⋆ {\displaystyle t_{(-i)}^{\star }} denotes the result based on samples without i t h {\displaystyle ith} observation. == Examples == E-mail spam problem is a common classification problem, in this problem, 57 features are used to classify spam e-mail and non-spam e-mail. Applying IJ-U variance formula to evaluate the accuracy of models with m=15,19 and 57. The results shows in paper( Confidence Intervals for Random Forests: The jackknife and the Infinitesimal Jackknife ) that m = 57 random forest appears to be quite unstable, while predictions made by m=5 random forest appear to be quite stable, this results is corresponding to the evaluation made by error percentage, in which the accuracy of model with m=5 is high and m=57 is low. Here, accuracy is measured by error rate, which is defined as: E r r o r R a t e = 1 N ∑ i = 1 N ∑ j = 1 M y i j , {\displaystyle ErrorRate={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij},} Here N is also the number of samples, M is the number of classes, y i j {\displaystyle y_{ij}} is the indicator function which equals 1 when i t h {\displaystyle ith} observation is in class j, equals 0 when in other classes. No probability is considered here. There is another method which is similar to error rate to measure accuracy: l o g l o s s = 1 N ∑ i = 1 N ∑ j = 1 M y i j l o g ( p i j ) {\displaystyle logloss={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij}log(p_{ij})} Here N is the number of samples, M is the number of classes, y i j {\displaystyle y_{ij}} is the indicator function which equals 1 when i t h {\displaystyle ith} observation is in class j, equals 0 when in other classes. p i j {\displaystyle p_{ij}} is the predicted probability of i t h {\displaystyle ith} observation in class j {\displaystyle j} .This method is used in Kaggle These two methods are very similar. == Modification for bias == When using Monte Carlo MSEs for estimating V I J ∞ {\displaystyle V_{IJ}^{\infty }} and V J ∞ {\displaystyle V_{J}^{\infty }} , a problem about the Monte Carlo bias should be considered, especially when n is large, the bias is getting large: E [ V ^ I J B ] − V ^ I J ∞ ≈ n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle E[{\hat {V}}_{IJ}^{B}]-{\hat {V}}_{IJ}^{\infty }\approx {\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}} To eliminate this influence, bias-corrected modifications are suggested: V ^ I J − U B = V ^ I J B − n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle {\hat {V}}_{IJ-U}^{B}={\hat {V}}_{IJ}^{B}-{\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}} V ^ J − U B = V ^ J B − ( e − 1 ) n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle {\hat {V}}_{J-U}^{B}={\hat {V}}_{J}^{B}-(e-1){\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}}

  Read more →
• 

  Ordinal regression

  In statistics, ordinal regression, also called ordinal classification, is a type of regression analysis used for predicting an ordinal variable, i.e. a variable whose value exists on an arbitrary scale where only the relative ordering between different values is significant. It can be considered an intermediate problem between regression and classification. Examples of ordinal regression are ordered logit and ordered probit. Ordinal regression turns up often in the social sciences, for example in the modeling of human levels of preference (on a scale from, say, 1–5 for "very poor" through "excellent"), as well as in information retrieval. In machine learning, ordinal regression may also be called ranking learning. == Linear models for ordinal regression == Ordinal regression can be performed using a generalized linear model (GLM) that fits both a coefficient vector and a set of thresholds to a dataset. Suppose one has a set of observations, represented by length-p vectors x1 through xn, with associated responses y1 through yn, where each yi is an ordinal variable on a scale 1, ..., K. For simplicity, and without loss of generality, we assume y is a non-decreasing vector, that is, yi ≤ {\displaystyle \leq } yi+1. To this data, one fits a length-p coefficient vector w and a set of thresholds θ1, ..., θK−1 with the property that θ1 < θ2 < ... < θK−1. This set of thresholds divides the real number line into K disjoint segments, corresponding to the K response levels. The model can now be formulated as Pr ( y ≤ i ∣ x ) = σ ( θ i − w ⋅ x ) {\displaystyle \Pr(y\leq i\mid \mathbf {x} )=\sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )} or, the cumulative probability of the response y being at most i is given by a function σ (the inverse link function) applied to a linear function of x. Several choices exist for σ; the logistic function σ ( θ i − w ⋅ x ) = 1 1 + e − ( θ i − w ⋅ x ) {\displaystyle \sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )={\frac {1}{1+e^{-(\theta _{i}-\mathbf {w} \cdot \mathbf {x} )}}}} gives the ordered logit model, while using the CDF of the standard normal distribution gives the ordered probit model. A third option is to use an exponential function σ ( θ i − w ⋅ x ) = 1 − exp ⁡ ( − exp ⁡ ( θ i − w ⋅ x ) ) {\displaystyle \sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )=1-\exp(-\exp(\theta _{i}-\mathbf {w} \cdot \mathbf {x} ))} which gives the proportional hazards model. === Latent variable model === The probit version of the above model can be justified by assuming the existence of a real-valued latent variable (unobserved quantity) y, determined by y ∗ = w ⋅ x + ε {\displaystyle y^{}=\mathbf {w} \cdot \mathbf {x} +\varepsilon } where ε is normally distributed with zero mean and unit variance, conditioned on x. The response variable y results from an "incomplete measurement" of y, where one only determines the interval into which y falls: y = { 1 if y ∗ ≤ θ 1 , 2 if θ 1 < y ∗ ≤ θ 2 , 3 if θ 2 < y ∗ ≤ θ 3 ⋮ K if θ K − 1 < y ∗ . {\displaystyle y={\begin{cases}1&{\text{if}}~~y^{}\leq \theta _{1},\\2&{\text{if}}~~\theta _{1} Read more →

• 

  Lexalytics

  Lexalytics, Inc. provides sentiment and intent analysis to an array of companies using SaaS and cloud based technology. Salience 6, the engine behind Lexalytics, was built as an on-premises, multi-lingual text analysis engine. It is leased to other companies who use it to power filtering and reputation management programs. In July, 2015 Lexalytics acquired Semantria to be used as a cloud option for its technology. In September, 2021 Lexalytics was acquired by CX company InMoment. == History == Lexalytics spun into existence in January 2003 out of a content management startup called Lightspeed. Lightspeed consolidated on America's West Coast. Jeff Catlin, a Lightspeed General Manager, and Mike Marshall, a Lighstpeed Principal Engineer, convinced investors to give them the East Coast company so as to avoid shutdown costs. Catlin and Marshall renamed the operation Lexalytics. Catlin took on the role of chief executive officer with Marshall working as Chief Technology Officer. Lexalytics opted to not accept venture cash. Instead, the company initially shared sales and marketing expenses with U.K. based document management company Infonic. The partner companies soon formed a joint venture in July 2008, which was later dissolved. Since then, Lexalytics has worked with many other companies, like Bottlenose, Salesforce, Thomson Reuters, Oracle and DataSift. Relationships with social media monitoring companies like Datasift tend to find Lexalytics’ Salience engine baked into the product itself. Lexalytics is used similarly to monitor sentiment as it relates to stock trading. In December 2014, Lexalytics announced the latest iteration to its sentiment analysis engine, Salience 6. Earlier that year Lexalytics acquired Semantria in a bid to appeal to a wider variety of business models. Created by former Lexalytics Marketing Director Oleg Rogynskyy, Semantria is a SaaS text mining service offered as an API and Excel based plugin that measures sentiment. The goal of the acquisition, which cost Lexalytics less than US$10 million, was to expand the customer base both within the United States and abroad with multilingual support. The engine that powers Semantria, Salience, is grounded in its deep learning ability. An example of this is its concept matrix, which allows Salience an understanding of concepts and relationship between concepts based on a detailed reading of the entire repository of Wikipedia. This matrix allows Salience to use Wikipedia for automatic categorization. Along with features like the concept matrix, Salience supports 16 international languages. The engine has earned Lexalytics a spot on EContent's “Top 100 Companies in the Digital Content Industry” List for 2014–2015. In September 2018, Lexalytics launched document data extraction market using natural language processing (NLP).

  Read more →
• 

  T-distributed stochastic neighbor embedding

  t-distributed stochastic neighbor embedding (t-SNE) is a statistical method for visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. It is based on Stochastic Neighbor Embedding originally developed by Geoffrey Hinton and Sam Roweis, where Laurens van der Maaten and Hinton proposed the t-distributed variant. It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability. The t-SNE algorithm comprises two main stages. First, t-SNE constructs a probability distribution over pairs of high-dimensional objects in such a way that similar objects are assigned a higher probability while dissimilar points are assigned a lower probability. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the Kullback–Leibler divergence (KL divergence) between the two distributions with respect to the locations of the points in the map. While the original algorithm uses the Euclidean distance between objects as the base of its similarity metric, this can be changed as appropriate. A Riemannian variant is UMAP. t-SNE has been used for visualization in a wide range of applications, including genomics, computer security research, natural language processing, music analysis, cancer research, bioinformatics, geological domain interpretation, and biomedical signal processing. For a data set with n {\displaystyle n} elements, t-SNE runs in O ( n 2 ) {\displaystyle O(n^{2})} time and requires O ( n 2 ) {\displaystyle O(n^{2})} space. == Details == Given a set of N {\displaystyle N} high-dimensional objects x 1 , … , x N {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{N}} , t-SNE first computes probabilities p i j {\displaystyle p_{ij}} that are proportional to the similarity of objects x i {\displaystyle \mathbf {x} _{i}} and x j {\displaystyle \mathbf {x} _{j}} , as follows. For i ≠ j {\displaystyle i\neq j} , define p j ∣ i = exp ⁡ ( − ‖ x i − x j ‖ 2 / 2 σ i 2 ) ∑ k ≠ i exp ⁡ ( − ‖ x i − x k ‖ 2 / 2 σ i 2 ) {\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}}} and set p i ∣ i = 0 {\displaystyle p_{i\mid i}=0} . Note the above denominator ensures ∑ j p j ∣ i = 1 {\displaystyle \sum _{j}p_{j\mid i}=1} for all i {\displaystyle i} . As van der Maaten and Hinton explained: "The similarity of datapoint x j {\displaystyle x_{j}} to datapoint x i {\displaystyle x_{i}} is the conditional probability, p j | i {\displaystyle p_{j|i}} , that x i {\displaystyle x_{i}} would pick x j {\displaystyle x_{j}} as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at x i {\displaystyle x_{i}} ." Now define p i j = p j ∣ i + p i ∣ j 2 N {\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}} This is motivated because p i {\displaystyle p_{i}} and p j {\displaystyle p_{j}} from the N samples are estimated as 1/N, so the conditional probability can be written as p i ∣ j = N p i j {\displaystyle p_{i\mid j}=Np_{ij}} and p j ∣ i = N p j i {\displaystyle p_{j\mid i}=Np_{ji}} . Since p i j = p j i {\displaystyle p_{ij}=p_{ji}} , you can obtain previous formula. Also note that p i i = 0 {\displaystyle p_{ii}=0} and ∑ i , j p i j = 1 {\displaystyle \sum _{i,j}p_{ij}=1} . The bandwidth of the Gaussian kernels σ i {\displaystyle \sigma _{i}} is set in such a way that the entropy of the conditional distribution equals a predefined entropy using the bisection method. As a result, the bandwidth is adapted to the density of the data: smaller values of σ i {\displaystyle \sigma _{i}} are used in denser parts of the data space. The entropy increases with the perplexity of this distribution P i {\displaystyle P_{i}} ; this relation is seen as P e r p ( P i ) = 2 H ( P i ) {\displaystyle Perp(P_{i})=2^{H(P_{i})}} where H ( P i ) {\displaystyle H(P_{i})} is the Shannon entropy H ( P i ) = − ∑ j p j | i log 2 ⁡ p j | i . {\displaystyle H(P_{i})=-\sum _{j}p_{j|i}\log _{2}p_{j|i}.} The perplexity is a hand-chosen parameter of t-SNE, and as the authors state, "perplexity can be interpreted as a smooth measure of the effective number of neighbors. The performance of SNE is fairly robust to changes in the perplexity, and typical values are between 5 and 50.". Since the Gaussian kernel uses the Euclidean distance ‖ x i − x j ‖ {\displaystyle \lVert x_{i}-x_{j}\rVert } , it is affected by the curse of dimensionality, and in high dimensional data when distances lose the ability to discriminate, the p i j {\displaystyle p_{ij}} become too similar (asymptotically, they would converge to a constant). It has been proposed to adjust the distances with a power transform, based on the intrinsic dimension of each point, to alleviate this. t-SNE aims to learn a d {\displaystyle d} -dimensional map y 1 , … , y N {\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{N}} (with y i ∈ R d {\displaystyle \mathbf {y} _{i}\in \mathbb {R} ^{d}} and d {\displaystyle d} typically chosen as 2 or 3) that reflects the similarities p i j {\displaystyle p_{ij}} as well as possible. To this end, it measures similarities q i j {\displaystyle q_{ij}} between two points in the map y i {\displaystyle \mathbf {y} _{i}} and y j {\displaystyle \mathbf {y} _{j}} , using a very similar approach. Specifically, for i ≠ j {\displaystyle i\neq j} , define q i j {\displaystyle q_{ij}} as q i j = ( 1 + ‖ y i − y j ‖ 2 ) − 1 ∑ k ∑ l ≠ k ( 1 + ‖ y k − y l ‖ 2 ) − 1 {\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k}\sum _{l\neq k}(1+\lVert \mathbf {y} _{k}-\mathbf {y} _{l}\rVert ^{2})^{-1}}}} and set q i i = 0 {\displaystyle q_{ii}=0} . Herein a heavy-tailed Student t-distribution (with one-degree of freedom, which is the same as a Cauchy distribution) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map. The locations of the points y i {\displaystyle \mathbf {y} _{i}} in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution P {\displaystyle P} from the distribution Q {\displaystyle Q} , that is: K L ( P ∥ Q ) = ∑ i ≠ j p i j log ⁡ p i j q i j {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}} The minimization of the Kullback–Leibler divergence with respect to the points y i {\displaystyle \mathbf {y} _{i}} is performed using gradient descent. The result of this optimization is a map that reflects the similarities between the high-dimensional inputs. == Output == While t-SNE plots often seem to display clusters, the visual clusters can be strongly influenced by the chosen parameterization (especially the perplexity) and so a good understanding of the parameters for t-SNE is needed. Such "clusters" can be shown to even appear in structured data with no clear clustering, and so may be false findings. Similarly, the size of clusters produced by t-SNE is not informative, and neither is the distance between clusters. Thus, interactive exploration may be needed to choose parameters and validate results. It has been shown that t-SNE can often recover well-separated clusters, and with special parameter choices, approximates a simple form of spectral clustering. == Software == A C++ implementation of Barnes-Hut is available on the github account of one of the original authors. The R package Rtsne implements t-SNE in R. ELKI contains tSNE, also with Barnes-Hut approximation scikit-learn, a popular machine learning library in Python implements t-SNE with both exact solutions and the Barnes-Hut approximation. Tensorboard, the visualization kit associated with TensorFlow, also implements t-SNE (online version) The Julia package TSne implements t-SNE

  Read more →
• 

  Preference regression

  Preference regression is a statistical technique used by marketers to determine consumers’ preferred core benefits. It usually supplements product positioning techniques like multi dimensional scaling or factor analysis and is used to create ideal vectors on perceptual maps. == Application == Starting with raw data from surveys, researchers apply positioning techniques to determine important dimensions and plot the position of competing products on these dimensions. Next they regress the survey data against the dimensions. The independent variables are the data collected in the survey. The dependent variable is the preference datum. Like all regression methods, the computer fits weights to best predict data. The resultant regression line is referred to as an ideal vector because the slope of the vector is the ratio of the preferences for the two dimensions. If all the data is used in the regression, the program will derive a single equation and hence a single ideal vector. This tends to be a blunt instrument so researchers refine the process with cluster analysis. This creates clusters that reflect market segments. Separate preference regressions are then done on the data within each segment. This provides an ideal vector for each segment. == Alternative methods == Self-stated importance method is an alternative method in which direct survey data is used to determine the weightings rather than statistical imputations. A third method is conjoint analysis in which an additive method is used.

  Read more →