Ordinal Representation Research Articles

Uncovering the interplay between drawings, mental representations, and arithmetic problem-solving strategies in children and adults.

There is an ongoing debate in the scientific community regarding the nature and role of the mental representations involved in solving arithmetic word problems. In this study, we took a closer look at the interplay between mental representations, drawing production, and strategy choice. We used dual-strategy isomorphic word problems sharing the same mathematical structure, but differing in the entities they mentioned in their problem statement. Due to the non-mathematical knowledge attached to these entities, some problems were believed to lead to a specific (cardinal) encoding compatible with one solving strategy, whereas other problems were thought to foster a different (ordinal) encoding compatible with the other solving strategy. We asked 59 children and 52 adults to solve 12 of those arithmetic word problems and to make a diagram of each problem. We hypothesized that the diagrams of both groups would display prototypical features indicating either a cardinal representation or an ordinal representation, depending on the entities mentioned in the problem statement. Joint analysis of the drawing task and the problem-solving task showed that the cardinal and ordinal features of the diagrams are linked with the hypothesized semantic properties of the problems and, crucially, with the choice of one solving strategy over another. We showed that regardless of their experience, participants' strategy use depends on their problem representation, which is influenced by the non-mathematical information in the problem statement, as revealed in their diagrams. We discuss the relevance of drawing tasks for investigating mental representations and fostering mathematical development in school.

Domain Invariant and Consistent Ordinal Representation Learning for Remaining Useful Life Prediction of Bearings

Domain Invariant and Consistent Ordinal Representation Learning for Remaining Useful Life Prediction of Bearings

Probabilistic reinforcement precludes transitive inference: A preliminary study.

In the basic verbal task from Piaget, when a relation of the form if A > B and B > C is given, a logical inference A > C is expected. This process is called transitive inference (TI). The adapted version for animals involves the presentation of a simultaneous discrimination between stimuli pairs. In this way, when A+B-, B+C-, C+D-, D+E- is trained, a B>D preference is expected, assuming that if A>B>C>D>E, then B>D. This effect has been widely reported using several procedures and different species. In the current experiment TI was evaluated employing probabilistic reinforcement. Thus, for the positive stimuli a .7 probability was administered and for the negative stimuli a .3 probability was administered. Under this arrangement the relation A>B>C>D>E is still allowed, but TI becomes more difficult. Five pigeons (Columba Livia) were exposed to the mentioned arrangement. Only one pigeon reached the criterion in C+D- discrimination, whereas the remaining did not. Only the one who successfully solved C+D- was capable of learning TI, whereas the others were not. Additionally, it was found that correct response ratios did not predict BD performance. Consequently, probabilistic reinforcement disrupted TI, but some positional ordering was retained in the test. The results suggest that TI might be affected by associative strength but also by the positional ordering of the stimuli. The discussion addresses the two main accounts of TI: the associative account and the ordinal representation account.

An Upper Palaeolithic Proto-writing System and Phenological Calendar

In at least 400 European caves such as Lascaux, Chauvet and Altamira, Upper PalaeolithicHomo sapiensgroups drew, painted and engraved non-figurative signs from at least ~42,000bpand figurative images (notably animals) from at least 37,000bp. Since their discovery ~150 years ago, the purpose or meaning of European Upper Palaeolithic non-figurative signs has eluded researchers. Despite this, specialists assume that they were notational in some way. Using a database of images spanning the European Upper Palaeolithic, we suggest how three of the most frequently occurring signs—the line<|>, the dot <•>, and the <Y>—functioned as units of communication. We demonstrate that when found in close association with images of animals the line <|> and dot <•> constitute numbers denoting months, and form constituent parts of a local phenological/meteorological calendar beginning in spring and recording time from this point in lunar months. We also demonstrate that the <Y> sign, one of the most frequently occurring signs in Palaeolithic non-figurative art, has the meaning <To Give Birth>. Thepositionof the <Y> within a sequence of marks denotes month of parturition, an ordinal representation of number in contrast to the cardinal representation used in tallies. Our data indicate that the purpose of this system of associating animals with calendar information was to record and convey seasonal behavioural information about specific prey taxa in the geographical regions of concern. We suggest a specific way in which the pairing of numbers with animal subjects constituted a complete unit of meaning—a notational system combined with its subject—that provides us with a specific insight into what one set of notational marks means. It gives us our first specific reading of European Upper Palaeolithic communication, the first known writing in the history ofHomo sapiens.

Thinking about order: a review of common processing of magnitude and learned orders in animals.

Rich behavioral and neurobiological evidence suggests cognitive and neural overlap in how quantitatively comparable dimensions such as quantity, time, and space are processed in humans and animals. While magnitude domains such as physical magnitude, time, and space represent information that can be quantitatively compared (4 "is half of" 8), they also represent information that can be organized ordinally (1→2→3→4). Recent evidence suggests that the common representations seen across physical magnitude, time, and space domains in humans may be due to their common ordinal features rather than their common quantitative features, as these common representations appear to extend beyond magnitude domains to include learned orders. In this review, we bring together separate lines of research on multiple ordinal domains including magnitude-based and learned orders in animals to explore the extent to which there is support for a common cognitive process underlying ordinal processing. Animals show similarities in performance patterns across natural quantitatively comparable ordered domains (physical magnitude, time, space, dominance) and learned orders (acquired through transitive inference or simultaneous chaining). Additionally, they show transfer and interference across tasks within and between ordinal domains that support the theory of a common ordinal representation across domains. This review provides some support for the development of a unified theory of ordinality and suggests areas for future research to better characterize the extent to which there are commonalities in cognitive processing of ordinal information generally.

Travelling salesman problem: Greedy single point crossover in ordinal representation

Genetic Algorithm (GA) is a metaheuristic algorithm which has stages; coding, selection, crossover, and mutation. The crossover operator plays an important role in producing offspring to find solutions. Single Point Crossover (SPC) and Two Point Crossover (TPC) are classic crossover operators that are easy to apply to ordinal representation coding rather than path representation coding. The greedy algorithm is applied to the SPC operator to solve the Traveling Salesman Problem (TSP) to improve the performance of the genetic algorithm in the ordinal representation coding scheme. The results show that the proposed Greedy Single Point Crossover (GSPC) has higher performance in finding the global optimal solution for all population sizes and cities used, but it costs computation time.

What we count dictates how we count: A tale of two encodings

We argue that what we count has a crucial impact on how we count, to the extent that even adults may have difficulty using elementary mathematical notions in concrete situations. Specifically, we investigate how the use of certain types of quantities (durations, heights, number of floors) may emphasize the ordinality of the numbers featured in a problem, whereas other quantities (collections, weights, prices) may emphasize the cardinality of the depicted numerical situations. We suggest that this distinction leads to the construction of one of two possible encodings, either a cardinal or an ordinal representation. This difference should, in turn, constrain the way we approach problems, influencing our mathematical reasoning in multiple activities. This hypothesis is tested in six experiments (N = 916), using different versions of multiple-strategy arithmetic word problems. We show that the distinction between cardinal and ordinal quantities predicts problem sorting (Experiment 1), perception of similarity between problems (Experiment 2), direct problem comparison (Experiment 3), choice of a solving algorithm (Experiment 4), problem solvability estimation (Experiment 5) and solution validity assessment (Experiment 6). The results provide converging clues shedding light into the fundamental importance of the cardinal versus ordinal distinction on adults' reasoning about numerical situations. Overall, we report multiple evidence that general, non-mathematical knowledge associated with the use of different quantities shapes adults' encoding, recoding and solving of mathematical word problems. The implications regarding mathematical cognition and theories of arithmetic problem solving are discussed.

Ordinal labels in machine learning: a user-centered approach to improve data validity in medical settings

BackgroundDespite the vagueness and uncertainty that is intrinsic in any medical act, interpretation and decision (including acts of data reporting and representation of relevant medical conditions), still little research has focused on how to explicitly take this uncertainty into account. In this paper, we focus on the representation of a general and wide-spread medical terminology, which is grounded on a traditional and well-established convention, to represent severity of health conditions (for instance, pain, visible signs), ranging from Absent to Extreme. Specifically, we will study how both potential patients and doctors perceive the different levels of the terminology in both quantitative and qualitative terms, and if the embedded user knowledge could improve the representation of ordinal values in the construction of machine learning models.MethodsTo this aim, we conducted a questionnaire-based research study involving a relatively large sample of 1,152 potential patients and 31 clinicians to represent numerically the perceived meaning of standard and widely-applied labels to describe health conditions. Using these collected values, we then present and discuss different possible fuzzy-set based representations that address the vagueness of medical interpretation by taking into account the perceptions of domain experts. We also apply the findings of this user study to evaluate the impact of different encodings on the predictive performance of common machine learning models in regard to a real-world medical prognostic task.ResultsWe found significant differences in the perception of pain levels between the two user groups. We also show that the proposed encodings can improve the performances of specific classes of models, and discuss when this is the case.ConclusionsIn perspective, our hope is that the proposed techniques for ordinal scale representation and ordinal encoding may be useful to the research community, and also that our methodology will be applied to other widely used ordinal scales for improving validity of datasets and bettering the results of machine learning tasks.

Young Children's Representation of Locations in a Series: A Front-Back Representation or an Ordinal Representation?

Previous research has found that 3–5-year-old children could encode and retrieve a target location in a two-location series. In a paradigm of running two symmetrical railcars on a circular track, the study suggested that children used front-back array to help coding. That is, children at this age code the railcar running in the front of another as “the location in the front” and the railcar running in the back of another as “the location in the back.” However, the children’s success could be attributed to an alternative interpretation; using an ordinal representation to encode the location in front as the first with the other as the second. The current study used a four-location series to examine the children’s mental representation. Three- to five-year-old children participated in a hide-and-seek game to remember a target location out of four locations that moved in a series. The results showed salient individual differences in children’s representation, and their performance improved as the representation progressed. An ordinal representation supported the precise encoding of each location, while a vague front-back representation and a clearer front-middle-back representation led to different performance.

Schema and Motor-Memory Consolidation

Recent research has demonstrated that memory-consolidation processes can be accelerated if newly learned information is consistent with preexisting knowledge. Until now, investigations of this fast integration of new information into memory have focused on the declarative and perceptual systems. We employed a unique manipulation of a motor-sequence-learning paradigm to examine the effect of experimentally acquired memory on the learning of new motor information. Results demonstrate that new information is rapidly integrated into memory when practice occurs in a framework that is compatible with the previously acquired memory. This framework consists of the ordinal representation of the motor sequence. This enhanced integration cannot be explained by differences in the explicit awareness of the sequence and is observed only if the previously acquired motor memory was consolidated overnight. Results are consistent with the schema model of memory consolidation and offer insights into how previous motor experience can accelerate learning and consolidation processes.

Deep Semantic-Preserving Ordinal Hashing for Cross-Modal Similarity Search.

Cross-modal hashing has attracted increasing research attention due to its efficiency for large-scale multimedia retrieval. With simultaneous feature representation and hash function learning, deep cross-modal hashing (DCMH) methods have shown superior performance. However, most existing methods on DCMH adopt binary quantization functions (e.g., [Formula: see text]) to generate hash codes, which limit the retrieval performance since binary quantization functions are sensitive to the variations of numeric values. Toward this end, we propose a novel end-to-end ranking-based hashing framework, in this paper, termed as deep semantic-preserving ordinal hashing (DSPOH), to learn hash functions with deep neural networks by exploring the ranking structure of feature dimensions. In DSPOH, the ordinal representation, which encodes the relative rank ordering of feature dimensions, is explored to generate hash codes. Such ordinal embedding benefits from the numeric stability of rank correlation measures. To make the hash codes discriminative, the ordinal representation is expected to well predict the class labels so that the ranking-based hash function learning is optimally compatible with the label predicting. Meanwhile, the intermodality similarity is preserved to guarantee that the hash codes of different modalities are consistent. Importantly, DSPOH can be effectively integrated with different types of network architectures, which demonstrates the flexibility and scalability of our proposed hashing framework. Extensive experiments on three widely used multimodal data sets show that DSPOH outperforms state of the art for cross-modal retrieval tasks.

Black people don't exercise in my neighborhood: Perceived racial composition and leisure-time physical activity among middle class blacks and whites

Using a sample of middle class blacks and whites living in urban and suburban areas, this article focuses on how perceptions of the racial composition of neighborhoods influence leisure-time physical activity. Using an ordinal representation of an underlying continuous indication of the perceived percentage of blacks and whites within an egocentric neighborhood, the results show that black men are significantly less likely to be physically active in neighborhoods perceived as predominately white. Alternatively, they are more likely to be physically active in neighborhoods perceived as racially diverse and predominately black. Conversely, for black women, white women, and white men, physical activity increases as the perception of one's neighborhood becomes increasingly white. Black women are significantly less likely to engage in physical activity in neighborhoods perceived as predominately black and urban. Drawing upon the intersectionality framework, I discuss how perceptions of criminalization and safety lead to different levels of leisure-time physical activity for middle class black women and men relative to their white middle class counterparts.

Infants’ detection of increasing numerical order comes before detection of decreasing number

Ordinality is a fundamental aspect of numerical cognition. However, preverbal infants’ ability to represent numerical order is poorly understood. In the present study we extended the evidence provided by Macchi Cassia, Picozzi, Girelli, and de Hevia (2012), showing that 4-month-old infants detect ordinal relationships within size-based sequences, to numerical sequences. In three experiments, we showed that at 4months of age infants fail to represent increasing and decreasing numerical order when numerosities differ by a 1:2 ratio (Experiment 1), but they succeed when numerosities differ by a 1:3 ratio (Experiments 2 and 3). Critically, infants showed the same behavioral signature (i.e., asymmetry) described by Macchi Cassia et al. for discrimination of ordinal changes in area: they succeed at detecting increasing but not decreasing order (Experiments 2 and 3). These results support the idea of a common (or at least parallel) development of ordinal representation for the two quantitative dimensions of size and number. Moreover, the finding that the asymmetry signature, previously reported for size-based sequences, extends to numerosity, points to the existence of a common constraint in ordinal magnitude processing in the first months of life. The present findings are discussed in the context of possible evolutionary and developmental sources of the ordinal asymmetry, as well as their implication for other related cognitive abilities.

Spatial Representation of Ordinal Information.

Right hand responds faster than left hand when shown larger numbers and vice-versa when shown smaller numbers (the SNARC effect). Accumulating evidence suggests that the SNARC effect may not be exclusive for numbers and can be extended to other ordinal sequences (e.g., months or letters in the alphabet) as well. In this study, we tested the SNARC effect with a non-numerically ordered sequence: the Chinese notations for the color spectrum (Red, Orange, Yellow, Green, Blue, Indigo, and Violet). Chinese color word sequence reserves relatively weak ordinal information, because each element color in the sequence normally appears in non-sequential contexts, making it ideal to test the spatial organization of sequential information that was stored in the long-term memory. This study found a reliable SNARC-like effect for Chinese color words (deciding whether the presented color word was before or after the reference color word “green”), suggesting that, without access to any quantitative information or exposure to any previous training, ordinal representation can still activate a sense of space. The results support that weak ordinal information without quantitative magnitude encoded in the long-term memory can activate spatial representation in a comparison task.

Genomic prediction of unordered categorical traits: an application to subpopulation assignment in German Warmblood horses.

BackgroundCategorical traits without ordinal representation of classes do not qualify for threshold models. Alternatively, the multinomial problem can be assessed by a sequence of independent binary contrasts using schemes such as one-vs-all or one-vs-one. Class probabilities can be arrived at by normalization or pair-wise coupling strategies. We assessed the predictive ability of whole-genome regression models and support vector machines for the classification of horses into four German Warmblood breeds.ResultsPrediction accuracies of leave-one-out cross-validation were high and ranged from 0.75 to 0.97 depending on the binary classifier and breeds incorporated in the training. An analysis of the population structure using eigenvectors of the genomic relationship matrix revealed clustering of individuals beyond the given breed labels. Admixture between two breeds became apparent which had substantial impact on the prediction accuracies between those two breeds and also influenced the contrasts between other breeds.Conclusions Genomic prediction of unordered categorical traits was successfully applied to subpopulation assignment of German Warmblood horses. The applied methodology is a straightforward extension of existing binary threshold models for genomic prediction.

Transitive inference of social dominance by human infants.

It is surprising that there are inconsistent findings of transitive inference (TI) in young infants given that non-linguistic species succeed on TI tests. To conclusively test for TI in infants, we developed a task within the social domain, with which infants are known to show sophistication. We familiarized 10- to 13-month-olds (M=11.53months) to a video of two dominance interactions between three puppets (bear>elephant; hippo>bear) consistent with a dominance hierarchy (hippo>bear>elephant; where '>' denotes greater dominance). Infants then viewed interactions between the two puppets that had not interacted during familiarization. These interactions were either congruent (hippo>elephant) or incongruent (elephant>hippo) with the inferred hierarchy. Consistent with TI, infants looked longer to incongruent than congruent displays. Control conditions ruled out the possibility that infants' expectations were based on stable behaviors specific to individual puppets rather than their inferred transitive dominance relations. We suggest that TI may be supported by phylogenetically ancient mechanisms of ordinal representation and visuospatial processing that come online early in human development.

Quantifying sudden changes in dynamical systems using symbolic networks

We characterize the evolution of a dynamical system by combining two well-known complex systems’ tools, namely, symbolic ordinal analysis and networks. From the ordinal representation of a time series we construct a network in which every node weight represents the probability of an ordinal pattern (OP) to appear in the symbolic sequence and each edge's weight represents the probability of transitions between two consecutive OPs. Several network-based diagnostics are then proposed to characterize the dynamics of different systems: logistic, tent, and circle maps. We show that these diagnostics are able to capture changes produced in the dynamics as a control parameter is varied. We also apply our new measures to empirical data from semiconductor lasers and show that they are able to anticipate the polarization switchings, thus providing early warning signals of abrupt transitions.

Common Neural Substrates for Ordinal Representation in Short-Term Memory, Numerical and Alphabetical Cognition

The representation and maintenance of serial order information is one of the main functions of verbal short-term memory (STM) but its neural correlates remain poorly understood. We show here that the neural substrates allowing for coding of order information in STM are shared with those supporting ordinal processing in the numerical and alphabetical domains. We designed an fMRI experiment determining the neural substrates sensitive to ordinal distance effects in numerical judgment, alphabetical judgment and serial order STM tasks. Null conjunction analyses for parametric ordinal distance effects showed a common involvement of the horizontal segment of the left intraparietal sulcus over the three tasks; in addition, right intraparietal sulcus involvement was also observed for ordinal distance effects in the STM and numerical judgment tasks. These findings demonstrate that shared neural correlates in the intraparietal cortex support processing of order information in verbal STM, number and alphabetical domains, and suggest the existence of domain general, potentially ordinal, comparison processes supported by the left intraparietal sulcus.

Inferring interdependencies in climate networks constructed at inter-annual, intra-season and longer time scales

We study global climate networks constructed by means of ordinal time series analysis. Climate interdependencies among the nodes are quantified by the mutual information, computed from time series of monthly-averaged surface air temperature anomalies, and from their symbolic ordinal representation (OP). This analysis allows identifying topological changes in the network when varying the time-interval of the ordinal pattern. We consider intra-season time-intervals (e.g., the patterns are formed by anomalies in consecutive months) and inter-annual time-intervals (e.g., the patterns are formed by anomalies in consecutive years). We discuss how the network density and topology change with these time scales, and provide evidence of correlations between geographically distant regions that occur at specific time scales. In particular, we find that an increase in the ordinal pattern spacing (i.e., an increase in the timescale of the ordinal analysis), results in climate networks with increased connectivity on the equatorial Pacific area. On the contrary, the number of significant links decreases when the ordinal analysis is done with a shorter timescale (by comparing consecutive months), and interpret this effect as due to more stochasticity in the time-series in the short timescale. As the equatorial Pacific is known to be dominated by El Nino-Southern Oscillation (ENSO) on scales longer than several months, our methodology allows constructing climate networks where the effect of ENSO goes from mild (monthly OP) to intense (yearly OP), independently of the length of the ordinal pattern and of the thresholding method employed.

Hughlings Jackson's neurological ideas

Hughlings Jackson's neurological ideas are scientifically valid and practically useful. He began by emphasizing the focal lesion as the key to analysing patients' symptoms. He proclaimed that 'Epilepsy is the name for occasional, sudden, excessive, rapid, and local discharge of grey matter.' He eliminated any need for a direct appeal to metaphysical agents by asserting that the nervous system is an exclusively sensorimotor machine constrained by the newly discovered conservation laws. In constructing his neurophysiology he accepted the phrenological assumption that the nervous system is composed of a number of physiologically discrete organs, each with a single function accessible to the diagnostician. By observing the march of epileptic seizures he developed the idea of somatotopic representation. He claimed that the nervous system is an evolutionary hierarchy of three levels connected by the process of weighted ordinal representation. His assertion of the Doctrine of Concomitance further separated the concerns, and the institutions, of the neurophysiologist from that of the psychiatrist. He came to reject the idea of the unconscious because he could not observe unequivocally unconscious behaviour at the bedside. Each of these ideas emerged from contemporaneous scientific streams, but Hughlings Jackson was the one to incorporate them into practical medicine. These neurological ideas gave physicians the methods, tools, principles and structures with which to establish a new science of clinical neurology.