Tip and cue, sometimes referred to as tip and que, tipping and cueing, or tipping and queing, is a method for satellite imagery and reconnaissance satellites to automatically coordinate tracking of objects across different satellites in real or near real-time. This technique ensures continuous tracking of targets as they move across different regions by handing them off between satellites, sharing satellite imagery and collateral across discrete satellites. The coordination between various satellites and their complementary sensors allows for more accurate and efficient data collection. This system is particularly useful in scenarios requiring real-time monitoring and rapid response; the method significantly improves situational awareness and operational effectiveness. Tip and cue techniques involve integrating various sensor systems, each playing a specific role in the tracking process. As a target moves, it is handed off from one satellite to another, ensuring continuous monitoring. This coordination optimizes data collection and analysis, enhancing overall tracking accuracy. The real-time information gathered by these satellites is critical for decision-making in various applications, including defense and surveillance. By leveraging multiple satellites and their sensors, it provides broader coverage and more reliable tracking, and the continuous handoff between satellites ensures there are no gaps in monitoring, essential for high-stakes applications. The real-time data provided by this system allows for timely and informed decisions, improving response times and outcomes. Tip and cue methodologies are a part of geospatial intelligence, or GEOINT. Robert Cardillo, a former director of the National Geospatial-Intelligence Agency, highlighted the importance of tip and cue methods to their data collection efforts in 2015. == Historical Development == The concept of tip and cue in satellite monitoring has its origins in early military applications designed to enhance missile detection and tracking systems. During the Cold War, advancements in infrared sensing technologies laid the groundwork for more sophisticated tip and cue techniques. The integration of different sensor types, such as radar and optical sensors, in the 1990s expanded the capabilities of tip and cue systems beyond military applications. These advancements have made tip and cue techniques essential for various civilian uses, including disaster monitoring and environmental surveillance. Significant progress was made with the advent of high-speed data processing and communication technologies in the early 2000s, further refining the method. Advanced algorithms and data fusion techniques have been introduced to better integrate information from multiple sensors. Machine learning technologies now play a crucial role in improving detection and prediction capabilities, allowing for more adaptive and efficient tracking. Richmond and Brennan of Lockheed Martin, presenting to the annual technical conference of the Maui Space Surveillance Complex (formerly the Air Force Maui Optical Station (AMOS)), discussed the algorithms needed for 'tip and cue', to facilitate "multi-phenomenology data fusion." The Space Surveillance Telescope (SST) at Naval Communication Station Harold E. Holt in Australia, operated by the United States Space Force and designed by the Massachusetts Institute of Technology Lincoln Laboratory, was reported by the Defense Advanced Research Projects Agency (DARPA) to be a leader in creating and improving tip and cue techniques, from a large library of orbital object data. == Technical overview == Tip and cue systems utilize a network of at least two satellites equipped with complementary sensor technologies to track moving objects in real-time. The method involves detecting a target with a primary sensor, such as an infrared or photographic sensor, which then cues secondary sensors on the same or other satellites for more detailed monitoring. This handoff process between discrete systems ensures continuous tracking as the target moves across different areas, leveraging each systems strengths. Data collected by these systems and sensors are rapidly processed and shared among the network, enhancing situational awareness. This coordination optimizes resource usage and improves the accuracy of tracking moving objects over large areas. The primary sensors detect initial targets based on specific signatures, such as heat or movement, and then cue secondary sensors to gather more precise data. This ensures that each sensor operates within its optimal range, maintaining high tracking accuracy and reliability. The integration of various sensor types, including optical, radar, and infrared, allows the system to function effectively under different conditions and environments. Real-time data processing and communication between satellites and ground stations are crucial for timely and accurate target tracking. Satellites using tip and cue processes may use either passive or active scanning methodoloigies. These systems may also leverage both orbital and ground-based ELINT (electronic signals intelligence). == Known use cases == Tip and cue systems have been extensively utilized in military applications, particularly for missile detection and defense. These systems enable early detection of missile launches using infrared sensors, which then cue other sensors to track the missile's trajectory more accurately. In environmental monitoring, tip and cue techniques help track natural disasters such as wildfires and hurricanes by coordinating various satellite sensors for comprehensive data collection and analysis. Surveillance and reconnaissance operations also benefit from tip and cue systems, which provide continuous and precise tracking of moving objects, enhancing situational awareness. Additionally, these systems are used in maritime surveillance to monitor ship movements and detect illegal activities such as smuggling and piracy. Tip and cue systems are used in disaster management. For instance, during wildfires, infrared sensors can detect heat signatures, prompting other sensors to gather detailed imagery and data on fire spread and intensity. This coordinated approach allows for real-time monitoring and rapid response, crucial for mitigating damage and saving lives. Similarly, in hurricane tracking, satellites equipped with various sensors can monitor storm development and progression, providing timely information for emergency management agencies. The integration of multiple sensor types ensures accurate and comprehensive coverage of these dynamic and fast-changing events. In maritime surveillance, or maritime domain awareness (MDA), tip and cue systems enhance the detection and monitoring of vessel movements, contributing to maritime security. By coordinating satellite sensors, these systems can track ships over vast ocean areas, identifying potential threats or illegal activities such as smuggling, piracy, and illegal fishing. The ability to maintain continuous surveillance and share data in real-time with maritime authorities improves response times and enforcement capabilities. This application of tip and cue systems not only aids in law enforcement but also supports environmental conservation efforts by monitoring protected marine areas. Automatic Identification System (AIS) is one of the most important sources of data for the MDA agencies. AIS is used in order for ships to know each other's whereabouts, they transmit a signal from ship to ship and to shore. Lately, the system has been developed into satellite system, so called satellite AIS, which makes the system more effective. All ocean-going vessels above 300 tons, are supposed to use and transmit via AIS according to the International Maritime Organization. The satellite constellations help facilitate this with tip and cue methodologies.
NCAA transfer portal
The NCAA transfer portal is a National Collegiate Athletic Association (NCAA) application, database, and compliance tool that facilitates student athletes' transfers between member institutions. It is intended to bring greater transparency to the transfer process and to enable student athletes to publicize their desire to transfer. The transfer portal is an NCAA-wide database covering all three NCAA divisions, although most media coverage of the transfer portal involves its use in the top-level Division I (D-I). The portal launched on October 15, 2018. Regulations adopted in 2021 allowed student-athletes in D-I football, men's and women's basketball, men's ice hockey, and baseball to transfer schools using the portal once without sitting out a year. In 2024, the NCAA authorized athletes unlimited transfers. == Process == For Divisions I and II, once an athlete desiring to transfer informs their school; the school must enter the athlete's name in the database within two business days. Then coaches and staff from other universities may contact the athlete about potentially transferring. Before the January 2026 NCAA convention, Division III schools were allowed, but not required, to enter such a student into the portal. A proposal to require use of the portal in that division was approved at the convention. The timeline for D-III members to enter athletes into the portal differs from that of the other divisions. Athletes wishing to enter the portal must first complete an educational module. Once completed, the school has seven calendar days to enter the athlete's transfer request into the portal. == Transfer windows == On August 31, 2022, the D-I board adopted a series of changes to transfer rules, introducing the concept of transfer windows, similar to those used in professional soccer worldwide. Student-athletes who wish to take advantage of the one-time transfer rule must, under normal circumstances, enter the portal within a designated window for their sport. These windows are slightly different for each NCAA sport, but are broadly grouped by the NCAA's three athletic "seasons". At that time, the windows were as follows: Fall sports – A 45-day winter window opening the day after championship selections are made in that sport, and a spring window from May 1–15. According to the NCAA, "reasonable accommodations" would be made for participants in football's FBS and FCS championship games (respectively the College Football Playoff National Championship and Division I Football Championship Game), both of which take place in early January. Participants in those games had a 14-day window opening on the day after the championship game, as well as the spring window. Winter sports – A 60-day window opening the day after championship selections are made in that sport. Spring sports – A winter window from December 1–15, and a 45-day spring window opening the day after championship selections are made in that sport. For sports included in the NCAA Emerging Sports for Women program, transfer windows are the same as those for fully recognized NCAA sports. As with fully recognized NCAA sports, transfer windows linked to championship events open on the day after selections are made for the generally recognized championship events in emerging sports. Student-athletes whose athletic aid is reduced, canceled, or not renewed by their school, as well as those affected by a university's elimination of a sports team, may enter the transfer portal at any time without penalty. A slightly different exception applies to those undergoing a head coaching change; student-athletes so affected in sports other than Division I football can enter the portal within 30 days of the change, starting on the day after the coach's departure is announced. The coaching change window also applied to Division I football before October 2025. Less than a month after transfer windows were adopted, the Division I Council adopted a change that affected only graduate transfers. Student-athletes who are set to graduate with remaining athletic eligibility, and plan to continue competition as postgraduate students, were exempt from transfer windows. They could enter the portal at any time during the academic year, and were not subject to the standard deadlines of May 1 for fall and winter sports and July 1 for spring sports. In April 2024, graduate transfers became subject to the same deadlines as all other transfer students. This change did not affect windows for student-athletes affected by a head coaching change, a loss of athletic aid, or the discontinuation of a team. Because the Ivy League allows neither redshirting nor athletic participation by graduate students, athletes at its member schools who are set to complete four years of attendance but still have remaining athletic eligibility may enter the portal at any time during their fourth academic year of attendance. In October 2024, the Division I Council reduced transfer windows in football and basketball to a total of 30 days. For FBS and FCS football, the fall window opened for 20 days, starting on the Monday after FBS conference championship games. Participants in postseason play had a 5-day window that opened on the day after each team's final game. A 10-day spring window opened in mid-April. In men's and women's basketball, a single 30-day window opens on the day after the second round of each Division I tournament concludes. The existing exceptions regarding head coaching changes, a loss of athletic aid, or the discontinuation of a team remained in place. Almost exactly a year later, Division I adopted more significant changes to the football transfer portal for both FBS and FCS. The previous two windows were abolished and replaced by a single window that opens from January 2–16. Participants in the College Football Playoff National Championship—the only game in FBS or FCS played after the closure of the new window—receive a 5-day window that opens on the day after that game. The window for players undergoing a head coaching change was also reduced. A new window of 15 days opens five calendar days after the hiring or public announcement of a new head coach. Should a school fail to hire or publicly announce a new head coach within 30 days after the previous coach's departure, the window will open on the 31st day after departure, provided that the 31st day is no earlier than January 3. This particular window, also open for 15 days, may open at any time before June 30. No change was announced to the exceptions for those affected by a loss of athletic aid or the discontinuation of a team. == Impact on high school recruiting == Effective July 1, 2025, the NCAA Division I Board of Directors implemented new DI roster limits following the court-approved House settlement. Additionally, according to the NCAA, "NCAA rules for Division I programs will no longer include sport-specific scholarship limits." As a result, many top Division I programs, especially those in power conferences, are relying heavily on the transfer portal to bring in conference- and national-level student-athletes. This shift in recruiting focus has already been exemplified across Division I men's and women's track and field especially, beginning in the recruitment cycle for 2025 college entries. Track and field coaches formerly managing rosters of 120-plus (60-plus men and 60-plus women) are now limited to 45 per side for a total of 90 roster spots across men's and women's track and field, meaning they are recruiting fewer student-athletes out of high school and more immediately impactful scholarship-worthy student-athletes via the transfer portal. Roster limits for track and field teams are even more stringent in the Southeastern Conference (SEC): 35 men and 35 women. For high school track and field athletes seeking opportunities with top DI programs, they no longer need to display potential to be point-scorers, but demonstrate the ability to contribute immediately, often by competing at a level aligned with conference scoring standards.
Dynamic time warping
In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications. In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules: Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match) The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match) The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i {\displaystyle j>i} are indices from the first sequence, then there must not be two indices l > k {\displaystyle l>k} in the other sequence, such that index i {\displaystyle i} is matched with index l {\displaystyle l} and index j {\displaystyle j} is matched with index k {\displaystyle k} , and vice versa We can plot each match between the sequences 1 : M {\displaystyle 1:M} and 1 : N {\displaystyle 1:N} as a path in a M × N {\displaystyle M\times N} matrix from ( 1 , 1 ) {\displaystyle (1,1)} to ( M , N ) {\displaystyle (M,N)} , such that each step is one of ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) {\displaystyle (0,1),(1,0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoy number. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section. This is conceptually very similar to the Needleman–Wunsch algorithm. == Implementation == This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, d ( x , y ) {\displaystyle d(x,y)} is a distance between the symbols, e.g., d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} . int DTWDistance(s: array [1..n], t: array [1..m]) { DTW := array [0..n, 0..m] for i := 0 to n for j := 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := 1 to m cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } where DTW[i, j] is the distance between s[1:i] and t[1:j] with the best alignment. We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i − j | {\displaystyle |i-j|} is no larger than w, a window parameter. We can easily modify the above algorithm to add a locality constraint (differences marked). However, the above given modification works only if | n − m | {\displaystyle |n-m|} is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that | n − m | ≤ w {\displaystyle |n-m|\leq w} (see the line marked with () in the code). int DTWDistance(s: array [1..n], t: array [1..m], w: int) { DTW := array [0..n, 0..m] w := max(w, abs(n-m)) // adapt window size () for i := 0 to n for j:= 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) DTW[i, j] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } == Warping properties == The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements. == Complexity == The time complexity of the DTW algorithm is O ( N M ) {\displaystyle O(NM)} , where N {\displaystyle N} and M {\displaystyle M} are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in O ( N 2 / log log N ) {\displaystyle O({N^{2}}/\log \log N)} time and space for two input sequences of length N {\displaystyle N} . This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form O ( N 2 − ϵ ) {\displaystyle O(N^{2-\epsilon })} for some ϵ > 0 {\displaystyle \epsilon >0} cannot exist unless the Strong exponential time hypothesis fails. While the dynamic programming algorithm for DTW requires O ( N M ) {\displaystyle O(NM)} space in a naive implementation, the space consumption can be reduced to O ( min ( N , M ) ) {\displaystyle O(\min(N,M))} using Hirschberg's algorithm. == Fast computation == Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW. A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective. In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time. == Average sequence == Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences. NLAAF is an exact method to average two sequences using DTW. For more than two sequences, the problem is related to that of multiple alignment and requires heuristics. DBA is currently a reference method to average a set of sequences consistently with DTW. COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process. == Supervised learning == A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure. == Amerced Dynamic Time Warping == Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks. == Alternative approaches == In functional data analysis, time series are regarde
Memetic algorithm
In computer science and operations research, a memetic algorithm (MA) is an extension of an evolutionary algorithm (EA) that aims to accelerate the evolutionary search for the optimum. An EA is a metaheuristic that reproduces the basic principles of biological evolution as a computer algorithm in order to solve challenging optimization or planning tasks, at least approximately. An MA uses one or more suitable heuristics or local search techniques to improve the quality of solutions generated by the EA and to speed up the search. The effects on the reliability of finding the global optimum depend on both the use case and the design of the MA. Memetic algorithms represent one of the recent growing areas of research in evolutionary computation. The term MA is now widely used as a synergy of evolutionary or any population-based approach with separate individual learning or local improvement procedures for problem search. Quite often, MAs are also referred to in the literature as Baldwinian evolutionary algorithms, Lamarckian EAs, cultural algorithms, or genetic local search. == Introduction == Inspired by both Darwinian principles of natural evolution and Dawkins' notion of a meme, the term memetic algorithm (MA) was introduced by Pablo Moscato in his technical report in 1989 where he viewed MA as being close to a form of population-based hybrid genetic algorithm (GA) coupled with an individual learning procedure capable of performing local refinements. The metaphorical parallels, on the one hand, to Darwinian evolution and, on the other hand, between memes and domain specific (local search) heuristics are captured within memetic algorithms thus rendering a methodology that balances well between generality and problem specificity. This two-stage nature makes them a special case of dual-phase evolution. The basic idea behind an MA is to combine the advantages of a global search performed by an EA (or another global search method) with the local refinement provided by one or more local search techniques, while avoiding their drawbacks. The main disadvantage of EAs is that, when searching in the vicinity of an optimum, they perform poorly in determining the exact position of that optimum. The downside of local search methods lies simply in the locality of their search relative to the chosen starting point. The combination of these two classes of methods aims to merge global and local search so that the advantages of both approaches can be leveraged. The idea of this approach can be illustrated by the search for the highest mountain in the Alps. A local search method would climb one of the mountains near the starting point, ignoring Mont Blanc as long as the starting point is not in its vicinity. An EA, on the other hand, will likely only find Mont Blanc after examining many other mountains, valleys, and hills, and then it will have difficulty identifying the summit cross. From the perspective of an MA’s global search procedure, however, only the summits of hills and mountains are seen, and its search is limited to finding the best summit. The open question is whether the additional effort required for the local search is worthwhile. This depends not only on the design of the MA but also on the specific application and the local search methods used. In the context of complex optimization, many different instantiations of memetic algorithms have been reported across a wide range of application domains, in general, converging to high-quality solutions more efficiently than their conventional evolutionary counterparts. In general, using the ideas of memetics within a computational framework is called memetic computing or memetic computation (MC). With MC, the traits of universal Darwinism are more appropriately captured. Viewed in this perspective, MA is a more constrained notion of MC. More specifically, MA covers one area of MC, in particular dealing with areas of evolutionary algorithms that marry other deterministic refinement techniques for solving optimization problems. MC extends the notion of memes to cover conceptual entities of knowledge-enhanced procedures or representations. == Theoretical Background == The no-free-lunch theorems of optimization and search state that all optimization strategies are equally effective with respect to the set of all optimization problems. Conversely, this means that one can expect the following: The more efficiently an algorithm solves a problem or class of problems, the less general it is and the more problem-specific knowledge it builds on. This insight leads directly to the recommendation to complement generally applicable metaheuristics with application-specific methods or heuristics, which fits well with the concept of MAs. == The development of MAs == === 1st generation === Pablo Moscato characterized an MA as follows: "Memetic algorithms are a marriage between a population-based global search and the heuristic local search made by each of the individuals. ... The mechanisms to do local search can be to reach a local optimum or to improve (regarding the objective cost function) up to a predetermined level." And he emphasizes "I am not constraining an MA to a genetic representation.". This original definition of MA although encompasses characteristics of cultural evolution (in the form of local refinement) in the search cycle, it may not qualify as a true evolving system according to universal Darwinism, since all the core principles of inheritance/memetic transmission, variation, and selection are missing. This suggests why the term MA stirred up criticisms and controversies among researchers when first introduced. The following pseudo code would correspond to this general definition of an MA: Pseudo code Procedure Memetic Algorithm Initialize: Generate an initial population, evaluate the individuals and assign a quality value to them; while Stopping conditions are not satisfied do Evolve a new population using stochastic search operators. Evaluate all individuals in the population and assign a quality value to them. Select the subset of individuals, Ω i l {\displaystyle \Omega _{il}} , that should undergo the individual improvement procedure. for each individual in Ω i l {\displaystyle \Omega _{il}} do Perform individual learning using meme(s) with frequency or probability of f i l {\displaystyle f_{il}} , with an intensity of t i l {\displaystyle t_{il}} . Proceed with Lamarckian or Baldwinian learning. end for end while Lamarckian learning in this context means to update the chromosome according to the improved solution found by the individual learning step, while Baldwinian learning leaves the chromosome unchanged and uses only the improved fitness. This pseudo code leaves open which steps are based on the fitness of the individuals and which are not. In question are the evolving of the new population and the selection of Ω i l {\displaystyle \Omega _{il}} . Since most MA implementations are based on EAs, the pseudo code of a corresponding representative of the first generation is also given here, following Krasnogor: Pseudo code Procedure Memetic Algorithm Based on an EA Initialization: t = 0 {\displaystyle t=0} ; // Initialization of the generation counter Randomly generate an initial population P ( t ) {\displaystyle P(t)} ; Compute the fitness f ( p ) ∀ p ∈ P ( t ) {\displaystyle f(p)\ \ \forall p\in P(t)} ; while Stopping conditions are not satisfied do Selection: Accordingly to f ( p ) {\displaystyle f(p)} choose a subset of P ( t ) {\displaystyle P(t)} and store it in M ( t ) {\displaystyle M(t)} ; Offspring: Recombine and mutate individuals p ∈ M ( t ) {\displaystyle p\in M(t)} and store them in M ′ ( t ) {\displaystyle M'(t)} ; Learning: Improve p ′ {\displaystyle p'} by local search or heuristic ∀ p ′ ∈ M ′ ( t ) {\displaystyle \forall p'\in M'(t)} ; Evaluation: Compute the fitness f ( p ′ ) ∀ p ′ ∈ M ′ ( t ) {\displaystyle f(p')\ \ \forall p'\in M'(t)} ; if Lamarckian learning then Update chromosome of p ′ {\displaystyle p'} according to improvement ∀ p ′ ∈ M ′ ( t ) {\displaystyle \forall p'\in M'(t)} ; fi New generation: Generate P ( t + 1 ) {\displaystyle P(t+1)} by selecting some individuals from P ( t ) {\displaystyle P(t)} and M ′ ( t ) {\displaystyle M'(t)} ; t = t + 1 {\displaystyle t=t+1} ; // Increment the generation counter end while Return best individual p ∈ P ( t − 1 ) {\displaystyle p\in P(t-1)} as result; There are some alternatives for this MA scheme. For example: All or some of the initial individuals may be improved by the meme(s). The parents may be locally improved instead of the offspring. Instead of all offspring, only a randomly selected or fitness-dependent fraction may undergo local improvement. The latter requires the evaluation of the offspring in M ′ ( t ) {\displaystyle M'(t)} prior to the Learning step. === 2nd generation === Multi-meme, hyper-heuristic and meta-Lamarckian MA are referred to as second generation MA exhibiting the principles of me
Diffusion model
In machine learning, diffusion models, also known as diffusion-based generative models or score-based generative models, are a class of latent variable generative models. A diffusion model consists of two major components: the forward diffusion process, and the reverse sampling process. The goal of diffusion models is to learn a diffusion process for a given dataset, such that the process can generate new elements that are distributed similarly as the original dataset. A diffusion model models data as generated by a diffusion process, whereby a new datum performs a random walk with drift through the space of all possible data. A trained diffusion model can be sampled in many ways, with different efficiency and quality. There are various equivalent formalisms, including Markov chains, denoising diffusion probabilistic models, noise conditioned score networks, and stochastic differential equations. They are typically trained using variational inference. The model responsible for denoising is typically called its "backbone". The backbone may be of any kind, but they are typically U-nets or transformers. As of 2024, diffusion models are mainly used for computer vision tasks, including image denoising, inpainting, super-resolution, image generation, and video generation. These typically involve training a neural network to sequentially denoise images blurred with Gaussian noise. The model is trained to reverse the process of adding noise to an image. After training to convergence, it can be used for image generation by starting with an image composed of random noise, and applying the network iteratively to denoise the image. Diffusion-based image generators have seen widespread commercial interest, such as Stable Diffusion and DALL-E. These models typically combine diffusion models with other models, such as text-encoders and cross-attention modules to allow text-conditioned generation. Other than computer vision, diffusion models have also found applications in natural language processing such as text generation and summarization, sound generation, and reinforcement learning. == Denoising diffusion model == === Non-equilibrium thermodynamics === Diffusion models were introduced in 2015 as a method to train a model that can sample from a highly complex probability distribution. They used techniques from non-equilibrium thermodynamics, especially diffusion. Consider, for example, how one might model the distribution of all naturally occurring photos. Each image is a point in the space of all images, and the distribution of naturally occurring photos is a "cloud" in space, which, by repeatedly adding noise to the images, diffuses out to the rest of the image space, until the cloud becomes all but indistinguishable from a Gaussian distribution N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . A model that can approximately undo the diffusion can then be used to sample from the original distribution. This is studied in "non-equilibrium" thermodynamics, as the starting distribution is not in equilibrium, unlike the final distribution. The equilibrium distribution is the Gaussian distribution N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} , with pdf ρ ( x ) ∝ e − 1 2 ‖ x ‖ 2 {\displaystyle \rho (x)\propto e^{-{\frac {1}{2}}\|x\|^{2}}} . This is just the Maxwell–Boltzmann distribution of particles in a potential well V ( x ) = 1 2 ‖ x ‖ 2 {\displaystyle V(x)={\frac {1}{2}}\|x\|^{2}} at temperature 1. The initial distribution, being very much out of equilibrium, would diffuse towards the equilibrium distribution, making biased random steps that are a sum of pure randomness (like a Brownian walker) and gradient descent down the potential well. The randomness is necessary: if the particles were to undergo only gradient descent, then they will all fall to the origin, collapsing the distribution. === Denoising Diffusion Probabilistic Model (DDPM) === The 2020 paper proposed the Denoising Diffusion Probabilistic Model (DDPM), which improves upon the previous method by variational inference. ==== Forward diffusion ==== To present the model, some notation is required. β 1 , . . . , β T ∈ ( 0 , 1 ) {\displaystyle \beta _{1},...,\beta _{T}\in (0,1)} are fixed constants. α t := 1 − β t {\displaystyle \alpha _{t}:=1-\beta _{t}} α ¯ t := α 1 ⋯ α t {\displaystyle {\bar {\alpha }}_{t}:=\alpha _{1}\cdots \alpha _{t}} σ t := 1 − α ¯ t {\displaystyle \sigma _{t}:={\sqrt {1-{\bar {\alpha }}_{t}}}} σ ~ t := σ t − 1 σ t β t {\displaystyle {\tilde {\sigma }}_{t}:={\frac {\sigma _{t-1}}{\sigma _{t}}}{\sqrt {\beta _{t}}}} μ ~ t ( x t , x 0 ) := α t ( 1 − α ¯ t − 1 ) x t + α ¯ t − 1 ( 1 − α t ) x 0 σ t 2 {\displaystyle {\tilde {\mu }}_{t}(x_{t},x_{0}):={\frac {{\sqrt {\alpha _{t}}}(1-{\bar {\alpha }}_{t-1})x_{t}+{\sqrt {{\bar {\alpha }}_{t-1}}}(1-\alpha _{t})x_{0}}{\sigma _{t}^{2}}}} N ( μ , Σ ) {\displaystyle {\mathcal {N}}(\mu ,\Sigma )} is the normal distribution with mean μ {\displaystyle \mu } and variance Σ {\displaystyle \Sigma } , and N ( x | μ , Σ ) {\displaystyle {\mathcal {N}}(x|\mu ,\Sigma )} is the probability density at x {\displaystyle x} . A vertical bar denotes conditioning. A forward diffusion process starts at some starting point x 0 ∼ q {\displaystyle x_{0}\sim q} , where q {\displaystyle q} is the probability distribution to be learned, then repeatedly adds noise to it by x t = 1 − β t x t − 1 + β t z t {\displaystyle x_{t}={\sqrt {1-\beta _{t}}}x_{t-1}+{\sqrt {\beta _{t}}}z_{t}} where z 1 , . . . , z T {\displaystyle z_{1},...,z_{T}} are IID (Independent and identically distributed random variables) samples from N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . The coefficients 1 − β t {\displaystyle {\sqrt {1-\beta _{t}}}} and β t {\displaystyle {\sqrt {\beta _{t}}}} ensure that Var ( X t ) = I {\displaystyle {\mbox{Var}}(X_{t})=I} assuming that Var ( X 0 ) = I {\displaystyle {\mbox{Var}}(X_{0})=I} . The values of β t {\displaystyle \beta _{t}} are chosen such that for any starting distribution of x 0 {\displaystyle x_{0}} , if it has finite second moment, then lim t → ∞ x t | x 0 {\displaystyle \lim _{t\to \infty }x_{t}|x_{0}} converges to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . The entire diffusion process then satisfies q ( x 0 : T ) = q ( x 0 ) q ( x 1 | x 0 ) ⋯ q ( x T | x T − 1 ) = q ( x 0 ) N ( x 1 | α 1 x 0 , β 1 I ) ⋯ N ( x T | α T x T − 1 , β T I ) {\displaystyle q(x_{0:T})=q(x_{0})q(x_{1}|x_{0})\cdots q(x_{T}|x_{T-1})=q(x_{0}){\mathcal {N}}(x_{1}|{\sqrt {\alpha _{1}}}x_{0},\beta _{1}I)\cdots {\mathcal {N}}(x_{T}|{\sqrt {\alpha _{T}}}x_{T-1},\beta _{T}I)} or ln q ( x 0 : T ) = ln q ( x 0 ) − ∑ t = 1 T 1 2 β t ‖ x t − 1 − β t x t − 1 ‖ 2 + C {\displaystyle \ln q(x_{0:T})=\ln q(x_{0})-\sum _{t=1}^{T}{\frac {1}{2\beta _{t}}}\|x_{t}-{\sqrt {1-\beta _{t}}}x_{t-1}\|^{2}+C} where C {\displaystyle C} is a normalization constant and often omitted. In particular, we note that x 1 : T | x 0 {\displaystyle x_{1:T}|x_{0}} is a Gaussian process, which affords us considerable freedom in reparameterization. For example, by standard manipulation with Gaussian process, x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} x t − 1 | x t , x 0 ∼ N ( μ ~ t ( x t , x 0 ) , σ ~ t 2 I ) {\displaystyle x_{t-1}|x_{t},x_{0}\sim {\mathcal {N}}({\tilde {\mu }}_{t}(x_{t},x_{0}),{\tilde {\sigma }}_{t}^{2}I)} In particular, notice that for large t {\displaystyle t} , the variable x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} converges to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . That is, after a long enough diffusion process, we end up with some x T {\displaystyle x_{T}} that is very close to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} , with all traces of the original x 0 ∼ q {\displaystyle x_{0}\sim q} gone. For example, since x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} we can sample x t | x 0 {\displaystyle x_{t}|x_{0}} directly "in one step", instead of going through all the intermediate steps x 1 , x 2 , . . . , x t − 1 {\displaystyle x_{1},x_{2},...,x_{t-1}} . ==== Backward diffusion ==== The key idea of DDPM is to use a neural network parametrized by θ {\displaystyle \theta } . The network takes in two arguments x t , t {\displaystyle x_{t},t} , and outputs a vector μ θ ( x t , t ) {\displaystyle \mu _{\theta }(x_{t},t)} and a matrix Σ θ ( x t , t ) {\displaystyle \Sigma _{\theta }(x_{t},t)} , such that each step in the forward diffusion process can be approximately undone by x t − 1 ∼ N ( μ θ ( x t , t ) , Σ θ ( x t , t ) ) {\displaystyle x_{t-1}\sim {\mathcal {N}}(\mu _{\theta }(x_{t},t),\Sigma _{\theta }(x_{t},t))} . This then gives us a backward diffusion process p θ {\displaystyle p_{\theta }} defined by p θ ( x T ) = N ( x T | 0 , I ) {\displaystyle p_{\theta }(x
Dave's Redistricting
Dave's Redistricting App (DRA) is an online web app originally created by Dave Bradlee that allows anyone to simulate redistricting a U.S. state's congressional and legislative districts. == Purpose == According to Bradlee, the software was designed to "put power in people's hands," and so that they "can see how the process works, so it's a little less mysterious than it was 10 years ago." Bradlee has noticed that many citizens are taking this process seriously and using his app to create legitimate redistricting maps that could be put in place. Some websites have called Bradlee the pioneer and cause of the rise of do-it-yourself redistricting. States such as Montana in 2021 allowed the general population to use it to submit redistricting proposals following the 2020 United States Census. Dave's Redistricting has frequently been mentioned as a resource that can be used to combat gerrymandering, given that the public has free access to it. Political science firms such as FiveThirtyEight have used the website to draw examples of gerrymandered districts, including on their famous Atlas of Redistricting. Dave Bradlee built the first generation of DRA. DRA 2020 is built by a small team of volunteers—Dave Bradlee, Terry Crowley, Alec Ramsay, and David Rinn—all with a shared passion for technology & democracy and all Microsoft veterans. Their mission is to empower civic organizations and citizen activists to advocate for fair congressional and legislative districts and increased transparency in the redistricting process. == Functions == Users can redraw the congressional and state legislative districts for all 50 states, the District of Columbia, and Puerto Rico using a variety of census and election datasets including Cook PVI. Maps can be optimized for different criteria. DRA 2020 added several major features to the first generation app: Sharing & collaborative editing of maps, like Google Docs Multiple statewide elections for all 50 states including the ability to import your own data Comprehensive analytics for evaluating and comparing maps Custom overlays, and Block-level editing DRA remains free to use. == Versions == 2.2: This uses Bing Maps, an outdated software that projects the districts of a single state onto a map of the United States. 2.5: After Bing Maps announced that it would no longer be updating for the foreseen future, the U.S. Map feature was removed. DRA 2020: At the end of 2018, a beta version of 2020 was released. This version that did not require Microsoft Silverlight and could be used in any web browser. DRA 2020 has been under continuous development since and is built using React (JavaScript library), Mapbox, OpenStreetMap, TypeScript, Node.js, Amazon Web Services, as well as many open source components, tools, and icons.
Mathematics of neural networks in machine learning
An artificial neural network (ANN) or neural network combines biological principles with advanced statistics to solve problems in domains such as pattern recognition and game-play. ANNs adopt the basic model of neuron analogues connected to each other in a variety of ways. == Structure == === Neuron === A neuron with label j {\displaystyle j} receiving an input p j ( t ) {\displaystyle p_{j}(t)} from predecessor neurons consists of the following components: an activation a j ( t ) {\displaystyle a_{j}(t)} , the neuron's state, depending on a discrete time parameter, an optional threshold θ j {\displaystyle \theta _{j}} , which stays fixed unless changed by learning, an activation function f {\displaystyle f} that computes the new activation at a given time t + 1 {\displaystyle t+1} from a j ( t ) {\displaystyle a_{j}(t)} , θ j {\displaystyle \theta _{j}} and the net input p j ( t ) {\displaystyle p_{j}(t)} giving rise to the relation a j ( t + 1 ) = f ( a j ( t ) , p j ( t ) , θ j ) , {\displaystyle a_{j}(t+1)=f(a_{j}(t),p_{j}(t),\theta _{j}),} and an output function f out {\displaystyle f_{\text{out}}} computing the output from the activation o j ( t ) = f out ( a j ( t ) ) . {\displaystyle o_{j}(t)=f_{\text{out}}(a_{j}(t)).} Often the output function is simply the identity function. An input neuron has no predecessor but serves as input interface for the whole network. Similarly an output neuron has no successor and thus serves as output interface of the whole network. === Propagation function === The propagation function computes the input p j ( t ) {\displaystyle p_{j}(t)} to the neuron j {\displaystyle j} from the outputs o i ( t ) {\displaystyle o_{i}(t)} and typically has the form p j ( t ) = ∑ i o i ( t ) w i j . {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}.} === Bias === A bias term can be added, changing the form to the following: p j ( t ) = ∑ i o i ( t ) w i j + w 0 j , {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}+w_{0j},} where w 0 j {\displaystyle w_{0j}} is a bias. == Neural networks as functions == Neural network models can be viewed as defining a function that takes an input (observation) and produces an output (decision) f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} or a distribution over X {\displaystyle \textstyle X} or both X {\displaystyle \textstyle X} and Y {\displaystyle \textstyle Y} . Sometimes models are intimately associated with a particular learning rule. A common use of the phrase "ANN model" is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons, number of layers or their connectivity). Mathematically, a neuron's network function f ( x ) {\displaystyle \textstyle f(x)} is defined as a composition of other functions g i ( x ) {\displaystyle \textstyle g_{i}(x)} , that can further be decomposed into other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between functions. A widely used type of composition is the nonlinear weighted sum, where f ( x ) = K ( ∑ i w i g i ( x ) ) {\displaystyle \textstyle f(x)=K\left(\sum _{i}w_{i}g_{i}(x)\right)} , where K {\displaystyle \textstyle K} (commonly referred to as the activation function) is some predefined function, such as the hyperbolic tangent, sigmoid function, softmax function, or rectifier function. The important characteristic of the activation function is that it provides a smooth transition as input values change, i.e. a small change in input produces a small change in output. The following refers to a collection of functions g i {\displaystyle \textstyle g_{i}} as a vector g = ( g 1 , g 2 , … , g n ) {\displaystyle \textstyle g=(g_{1},g_{2},\ldots ,g_{n})} . This figure depicts such a decomposition of f {\displaystyle \textstyle f} , with dependencies between variables indicated by arrows. These can be interpreted in two ways. The first view is the functional view: the input x {\displaystyle \textstyle x} is transformed into a 3-dimensional vector h {\displaystyle \textstyle h} , which is then transformed into a 2-dimensional vector g {\displaystyle \textstyle g} , which is finally transformed into f {\displaystyle \textstyle f} . This view is most commonly encountered in the context of optimization. The second view is the probabilistic view: the random variable F = f ( G ) {\displaystyle \textstyle F=f(G)} depends upon the random variable G = g ( H ) {\displaystyle \textstyle G=g(H)} , which depends upon H = h ( X ) {\displaystyle \textstyle H=h(X)} , which depends upon the random variable X {\displaystyle \textstyle X} . This view is most commonly encountered in the context of graphical models. The two views are largely equivalent. In either case, for this particular architecture, the components of individual layers are independent of each other (e.g., the components of g {\displaystyle \textstyle g} are independent of each other given their input h {\displaystyle \textstyle h} ). This naturally enables a degree of parallelism in the implementation. Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where f {\displaystyle \textstyle f} is shown as dependent upon itself. However, an implied temporal dependence is not shown. == Backpropagation == Backpropagation training algorithms fall into three categories: steepest descent (with variable learning rate and momentum, resilient backpropagation); quasi-Newton (Broyden–Fletcher–Goldfarb–Shanno, one step secant); Levenberg–Marquardt and conjugate gradient (Fletcher–Reeves update, Polak–Ribiére update, Powell–Beale restart, scaled conjugate gradient). === Algorithm === Let N {\displaystyle N} be a network with e {\displaystyle e} connections, m {\displaystyle m} inputs and n {\displaystyle n} outputs. Below, x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } denote vectors in R m {\displaystyle \mathbb {R} ^{m}} , y 1 , y 2 , … {\displaystyle y_{1},y_{2},\dots } vectors in R n {\displaystyle \mathbb {R} ^{n}} , and w 0 , w 1 , w 2 , … {\displaystyle w_{0},w_{1},w_{2},\ldots } vectors in R e {\displaystyle \mathbb {R} ^{e}} . These are called inputs, outputs and weights, respectively. The network corresponds to a function y = f N ( w , x ) {\displaystyle y=f_{N}(w,x)} which, given a weight w {\displaystyle w} , maps an input x {\displaystyle x} to an output y {\displaystyle y} . In supervised learning, a sequence of training examples ( x 1 , y 1 ) , … , ( x p , y p ) {\displaystyle (x_{1},y_{1}),\dots ,(x_{p},y_{p})} produces a sequence of weights w 0 , w 1 , … , w p {\displaystyle w_{0},w_{1},\dots ,w_{p}} starting from some initial weight w 0 {\displaystyle w_{0}} , usually chosen at random. These weights are computed in turn: first compute w i {\displaystyle w_{i}} using only ( x i , y i , w i − 1 ) {\displaystyle (x_{i},y_{i},w_{i-1})} for i = 1 , … , p {\displaystyle i=1,\dots ,p} . The output of the algorithm is then w p {\displaystyle w_{p}} , giving a new function x ↦ f N ( w p , x ) {\displaystyle x\mapsto f_{N}(w_{p},x)} . The computation is the same in each step, hence only the case i = 1 {\displaystyle i=1} is described. w 1 {\displaystyle w_{1}} is calculated from ( x 1 , y 1 , w 0 ) {\displaystyle (x_{1},y_{1},w_{0})} by considering a variable weight w {\displaystyle w} and applying gradient descent to the function w ↦ E ( f N ( w , x 1 ) , y 1 ) {\displaystyle w\mapsto E(f_{N}(w,x_{1}),y_{1})} to find a local minimum, starting at w = w 0 {\displaystyle w=w_{0}} . This makes w 1 {\displaystyle w_{1}} the minimizing weight found by gradient descent. == Learning pseudocode == To implement the algorithm above, explicit formulas are required for the gradient of the function w ↦ E ( f N ( w , x ) , y ) {\displaystyle w\mapsto E(f_{N}(w,x),y)} where the function is E ( y , y ′ ) = | y − y ′ | 2 {\displaystyle E(y,y')=|y-y'|^{2}} . The learning algorithm can be divided into two phases: propagation and weight update. === Propagation === Propagation involves the following steps: Propagation forward through the network to generate the output value(s) Calculation of the cost (error term) Propagation of the output activations back through the network using the training pattern target to generate the deltas (the difference between the targeted and actual output values) of all output and hidden neurons. === Weight update === For each weight: Multiply the weight's output delta and input activation to find the gradient of the weight. Subtract the ratio (percentage) of the weight's gradient from the weight. The learning rate is the ratio (percentage) that influences the speed and quality of learning. The greater the ratio, the faster the neuron trains, but the lower the ratio, the more accurat