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Search for the Missing Dimensions: Building a Feature-Space Representation for a Natural-Science Category Domain

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Abstract

An important goal in cognitive and mathematical psychology is to scale up the application of computational models of human classification learning to real-world, naturalistic domains. Application of the models, however, requires the derivation of the complex, multidimensional “feature spaces” in which the to-be-classified objects are embedded and to which the formal models make reference. In recent work, using rock classification in the geologic sciences as an example target domain, we used multidimensional scaling (MDS) of similarity-judgment data as an approach to deriving the feature space (Nosofsky et al., Behavior Research Methods 50:530–556, 2018c). However, subsequent work involving the modeling of independent sets of classification-learning data led us to the hypothesis that the MDS solution had many “missing dimensions” that were crucial to categorization performance (Nosofsky et al., Psychonomic Bulletin & Review 26:48–76, 2019). In the present work, we conduct a “search for the missing dimensions” in an effort to develop a more comprehensive feature-space representation for the rock stimuli. By supplementing the original MDS solution with the missing dimensions, we achieve dramatically improved accounts of varied sets of classification-learning data in this domain. We outline future steps for continuing and expanding the work to meet the goal of achieving meaningful computational modeling of human classification in naturalistic object domains.

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Notes

  1. A direct link to the interactive plots is https://craasand.shinyapps.io/Rocks_Data_Explorer/. Click on the plotting tool option, choose the 360-Rocks data set, and choose which dimensions should serve as the x-axis and y-axis of the plot. Finally, click on “Create Plot.” The interactive website provides the user with a variety of other options for exploring the high-dimensional scaling solution for the rock images.

  2. As will be seen, we use a maximum-likelihood criterion for fitting the GCM to matrices of classification-confusion data. In many instances, the cells of the classification-confusion matrices have zero or near-zero entries. The guessing parameter provides a convenient mechanism for avoiding singularities in the model-fitting procedures in these situations.

  3. We use the term “in-sample” predictions here and elsewhere to emphasize that the model is being fitted to the data through the estimation of its free parameters. Later in our article, we also test predictions based on cross-validation methods. We should acknowledge as well that in these initial stages of the project, our focus is on data that are aggregated across individual subjects. As we argue in more depth in our “General Discussion,” the analysis of the averaged data appears to yield meaningful results. The goal of characterizing similarity-judgment and classification performance at the level of individual subjects is a crucial one for future work (e.g. Lee and Pope 2003; Okada and Lee 2016; Shen and Palmeri 2016), but would introduce monumental complications at these initial stages of the project.

  4. Use of the AIC criterion (Akaike 1974) and corrected-AIC criterion (Hurvich and Tsai 1989), which include alternative penalty terms than the BIC, yielded the same model-selection results in all cases. Although future work is needed that makes use of more sophisticated model-selection methods, we will argue that the BIC selections turned out to be sensible ones with readily apparent psychological interpretations.

  5. The supplemental “green” dimension is partially redundant with the red-green hue dimension that is already part of the MDS solution derived from analysis of the similarity-judgment data. It is possible that similar improvements in fit could have been achieved by allowing for greater attention-weighting of the red-green hue dimension from the standard MDS solution.

  6. Our motivation for use of the Equation-5 scaling function was to introduce a flexible non-linear function for relating psychological scale values to the rated dimension values, and where the form of the non-linear relation might vary with position on the scale. An anonymous reviewer suggested use of a more conventional function such as the generalized logistic function (Richards 1959). We were not familiar with the generalized logistic in initiating this research, and make no strong claims regarding the precise function that is best suited for transforming the supplemental dimension ratings. We re-fitted the version of the GCM that made use of all five supplemental dimensions to the Table-3 data, except using the generalized logistic rather than our proposed Equation-5 scaling function. As it turned out, despite making use of an additional free parameter, the generalized logistic provided slightly worse absolute (log-likelihood) fits to the data.

  7. A related concern is that the MDS solution derived from the similarity-judgment data made use of only eight dimensions. To review, we chose the eight-dimensional solution based on a combination of fit to the similarity-judgment data and interpretability of the resulting dimensions. Nevertheless, to address the concern, we also fitted the GCM to the present classification data while allowing the model to make reference to a twelve-dimensional similarity-scaling solution. The resulting fit was worse than the one achieved by the eight-dimensional version of the model that made allowance for only a single supplemental dimension. In a nutshell, it does not appear to be the case that the various missing dimensions discovered here would have emerged in higher-dimensional scaling solutions derived from the similarity-judgment data.

  8. In additional analyses, we also fitted a version of the GCM in which the scaling-related parameters (u, v, p, q, and the Rm’s) for the supplemental dimensions were also allowed to vary. Although allowing these parameters to vary led to still further improvements in BIC fit, the improvements were very small in magnitude relative to the pairwise comparisons reported in Table 5.

References

  • Akaike, H. (1974). A new look at the statistical model identification. IEEE Transcations on Automatic Control, 19, 716–723.

    Google Scholar 

  • Anderson, J. R. (1991). The adaptive nature of human categorization. Psychological Review, 98(3), 409–429.

    Google Scholar 

  • Ashby, F. G. (Ed.). (1992). Multidimensional models of perception and cognition. Hillsdale: LEA.

    Google Scholar 

  • Ashby, F. G., & Maddox, W. T. (1993). Relations between prototype, exemplar, and decision bound models of categorization. Journal of Mathematical Psychology, 37(3), 372–400.

    Google Scholar 

  • Austerweil, J. L., & Griffiths, T. L. (2011). A rational model of the effects of distributional information on feature learning. Cognitive Psychology, 63(4), 173–209.

    PubMed  Google Scholar 

  • Austerweil, J. L., & Griffiths, T. L. (2013). A nonparametric Bayesian framework for constructing flexible feature representations. Psychological Review, 120(4), 817–851.

    PubMed  Google Scholar 

  • Battleday, R. M., Peterson, J. C., & Griffiths, T. L. (2017). Modeling human categorization of natural images using deep feature representations. arXiv preprint arXiv:1711.04855.

  • Guest, O., & Love, B. C. (2017). What the success of brain imaging implies about the neural code. eLife, 6, e21397.

    PubMed  PubMed Central  Google Scholar 

  • Honke, G., & Kurtz, K. J. (2019). Similarity is as similarity does? A critical inquiry into the effect of thematic association on similarity. Cognition, 186, 116–138.

    Google Scholar 

  • Hout, M. C., Goldinger, S. D., & Ferguson, R. W. (2013). The versatility of SpAM: a fast, efficient, spatial method of data collection for multidimensional scaling. Journal of Experimental Psychology: General, 142(1), 256–281.

    Google Scholar 

  • Hurvich, C. M., & Tsai, C. L. (1989). Regression and time series model selection in small samples. Biometrika, 76(2), 297–307.

    Google Scholar 

  • Jamieson, R. K., Avery, J. E., Johns, B. T., & Jones, M. N. (2018). An instance theory of semantic memory. Computational Brain & Behavior, 1, 119–136.

    Google Scholar 

  • Jones, M. N., Kintsch, W., & Mewhort, D. J. K. (2006). High-dimensional semantic space accounts of priming. Journal of Memory and Language, 55, 534–552.

    Google Scholar 

  • Khajah, M. M., Lindsey, R. V., & Mozer, M. C. (2014). Maximizing students’ retention via spaced review: practical guidance from computational models of memory. Topics in Cognitive Science, 6(1), 157–169.

    PubMed  Google Scholar 

  • Kruschke, J. K. (1992). ALCOVE: an exemplar-based connectionist model of category learning. Psychological Review, 99(1), 22–44.

    PubMed  Google Scholar 

  • Kruskal, J. B., & Wish, M. (1978). Multidimensional scaling (quantitative applications in the social sciences). Beverly Hills: Sage.

    Google Scholar 

  • Lake, B. M., Zaremba, W., Fergus, R., & Gureckis, T. M. (2015). Deep neural networks predict category typicality ratings for images. In CogSci.

  • LeCun, Y., Bengio, Y., & Hinton, G. (2015). Deep learning. Nature, 521(7553), 436–444.

    Google Scholar 

  • Lee, M. D. (2001). Determining the dimensionality of multidimensional scaling representations for cognitive modeling. Journal of Mathematical Psychology, 45(1), 149–166.

    PubMed  Google Scholar 

  • Lee, M. D., & Navarro, D. J. (2002). Extending the ALCOVE model of category learning to featural stimulus domains. Psychonomic Bulletin & Review, 9(1), 43–58.

    Google Scholar 

  • Lee, M. D., & Pope, K. J. (2003). Avoiding the dangers of averaging across participants when using multidimensional scaling. Journal of Mathematical Psychology, 47(1), 32–46.

    Google Scholar 

  • Louviere, J. J., Hensher, D. A., & Swait, J. D. (2000). Stated choice methods: analysis and applications. Cambridge University Press.

  • Love, B. C., Medin, D. L., & Gureckis, T. M. (2004). SUSTAIN: a network model of category learning. Psychological Review, 111(2), 309–332.

    PubMed  Google Scholar 

  • Marshak, S. (2015). Earth: Portrait of a planet (5th ed.). New York: W. W. Norton & Company.

    Google Scholar 

  • Mathy, F., & Feldman, J. (2009). A rule-based presentation order facilitates category learning. Psychonomic Bulletin & Review, 16(6), 1050–1057.

    Google Scholar 

  • McKinley, S. C., & Nosofsky, R. M. (1995). Investigations of exemplar and decision bound models in large, ill-defined category structures. Journal of Experimental Psychology: Human Perception and Performance, 21(1), 128–148.

    PubMed  Google Scholar 

  • Meagher, B. J., Cataldo, K., Douglas, B. J., McDaniel, M. A., & Nosofsky, R. M. (2018). Training of rock classifications: the use of computer images versus physical-rock samples. Journal of Geoscience Education, 66(3), 221–230.

    Google Scholar 

  • Nosofsky, R. M. (1986). Attention, similarity, and the identification–categorization relationship. Journal of Experimental Psychology: General, 115(1), 39–57.

    Google Scholar 

  • Nosofsky, R. M. (1992). Similarity scaling and cognitive process models. Annual Review of Psychology, 43(1), 25–53.

    Google Scholar 

  • Nosofsky, R. M. (2011). The generalized context model: An exemplar model of classification. In M. Pothos & A. J. Wills (Eds.), Formal approaches in categorization (pp. 18–39). Cambridge: Cambridge University Press.

    Google Scholar 

  • Nosofsky, R. M., & McDaniel, M. A. (2019). Recommendations from cognitive psychology for enhancing the teaching of natural-science categories. Policy Insights from the Behavioral and Brain Sciences: FABBS, 6, 21–28.

    Google Scholar 

  • Nosofsky, R. M., & Palmeri, T. J. (1997). An exemplar-based random walk model of speeded classification. Psychological Review, 104(2), 266–300.

    PubMed  Google Scholar 

  • Nosofsky, R. M., & Zaki, S. R. (2002). Exemplar and prototype models revisited: response strategies, selective attention, and stimulus generalization. Journal of Experimental Psychology: Learning, Memory, and Cognition, 28(5), 924.

    PubMed  Google Scholar 

  • Nosofsky, R. M., & Zaki, S. R. (2003). A hybrid-similarity exemplar model for predicting distinctiveness effects in perceptual old-new recognition. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29(6), 1194.

    PubMed  Google Scholar 

  • Nosofsky, R. M., Sanders, C. A., Gerdom, A., Douglas, B. J., & McDaniel, M. A. (2017). On learning natural-science categories that violate the family-resemblance principle. Psychological Science, 28(1), 104–114.

    PubMed  Google Scholar 

  • Nosofsky, R. M., Sanders, C. A., & McDaniel, M. A. (2018a). A formal psychological model of classification applied to natural-science category learning. Current Directions in Psychological Science, 27, 129–135.

    Google Scholar 

  • Nosofsky, R. M., Sanders, C. A., & McDaniel, M. A. (2018b). Tests of an exemplar-memory model of classification learning in a high-dimensional natural-science category domain. Journal of Experimental Psychology: General, 147, 328–353.

    Google Scholar 

  • Nosofsky, R. M., Sanders, C. A., Meagher, B. J., & Douglas, B. J. (2018c). Toward the development of a feature-space representation for a complex, natural-category domain. Behavior Research Methods, 50, 530–556.

    PubMed  Google Scholar 

  • Nosofsky, R. M., Sanders, C. A., Zhu, X., & McDaniel, M. A. (2019). Model-guided search for optimal training exemplars in a natural-science category domain: a work in progress. Psychonomic Bulletin & Review, 26, 48–76.

    Google Scholar 

  • Okada, K., & Lee, M. D. (2016). A Bayesian approach to modeling group and individual differences in multidimensional scaling. Journal of Mathematical Psychology, 70, 35–44.

    Google Scholar 

  • Patil, K. R., Zhu, X., Kopec, L., & Love, B. C. (2014). Optimal teaching for limited-capacity human learners. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, & K. Q. Weinberger (Eds.), Advances in neural information processing systems (NIPS), 2014. Neural Information Processing Systems Foundation.

  • Peterson, J. C., Abbott, J. T., & Griffiths, T. L. (2016). Adapting deep network features to capture psychological representations. arXiv preprint arXiv:1608.02164.

  • Pothos, E. M., & Wills, A. J. (Eds.). (2011). Formal approaches in categorization. Cambridge: Cambridge University Press.

    Google Scholar 

  • Richards, F. J. (1959). A flexible growth function for empirical use. Journal of Experimental Botany, 10, 290–300.

    Google Scholar 

  • Rosch, E., & Mervis, C. B. (1975). Family resemblances: studies in the internal structure of categories. Cognitive Psychology, 7(4), 573–605.

    Google Scholar 

  • Rumelhart, D. E., & Todd, P. M. (1992). Learning and connectionist representations. In D. Meyers & S. Kornblum (Eds.), Attention and performance. Cambridge: MIT Press.

    Google Scholar 

  • Sanborn, A. N., Griffiths, T. L., & Navarro, D. J. (2010). Rational approximations to rational models: alternative algorithms for category learning. Psychological Review, 117(4), 1144–1167.

    PubMed  Google Scholar 

  • Sanders, C. A. (2018). Using deep learning to automatically extract psychological representations of complex natural stimuli. Unpublished Ph.D. Dissertation, Indiana University.

  • Sanders, C. A., & Nosofsky, R. M. (2018). Using deep learning representations of complex natural stimuli as input to psychological models of classification. Proceedings of the 2018 conference of the cognitive science society, Madison, WI.

  • Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2), 461–464.

    Google Scholar 

  • Schyns, P. G., Goldstone, R. L., & Thibaut, J. P. (1998). The development of features in object concepts. Behavioral and Brain Sciences, 21(1), 1–17.

    PubMed  Google Scholar 

  • Shen, J., & Palmeri, T. J. (2016). Modelling individual difference in visual categorization. Visual Cognition, 24(3), 260–283.

    PubMed  PubMed Central  Google Scholar 

  • Shepard, R. N. (1957). Stimulus and response generalization: a stochastic model relating generalization to distance in psychological space. Psychometrika, 22(4), 325–345.

    Google Scholar 

  • Shepard, R. N. (1980). Multidimensional scaling, tree-fitting, and clustering. Science, 210(4468), 390–398.

    PubMed  Google Scholar 

  • Shepard, R. (1987). Toward a universal law of generalization for psychological science. Science, 237(4820), 1317–1323.

    PubMed  Google Scholar 

  • Shepard, R. N., & Arabie, P. (1979). Additive clustering: representation of similarities as combinations of discrete overlapping properties. Psychological Review, 86(2), 87–123.

    Google Scholar 

  • Steyvers, M., & Busey, T. (2000). Predicting similarity ratings to faces using physical descriptions. In M. Wenger & J. T. Townsend (Eds.), Computational, geometric, and process perspectives on facial cognition: Contexts and challenges (pp. 115–146). Hillsdale: Erlbaum.

    Google Scholar 

  • Storms, G., De Boeck, P., & Ruts, W. (2000). Prototype and exemplar-based information in natural language categories. Journal of Memory and Language, 42(1), 51–73.

    Google Scholar 

  • Tarbuck, E. J., & Lutgens, F. K. (2014). Earth science (14th ed.). Boston: Pearson.

    Google Scholar 

  • Tenenbaum, J. B. (1996). Learning the structure of similarity. Advances in Neural Information Processing Systems, 3–9.

  • Tversky, A. (1977). Features of similarity. Psychological Review, 84(4), 327–352.

    Google Scholar 

  • Vanpaemel, W., & Storms, G. (2008). In search of abstraction: the varying abstraction model of categorization. Psychonomic Bulletin & Review, 15(4), 732–749.

    Google Scholar 

  • Verguts, T., Ameel, E., & Storms, G. (2004). Measures of similarity in models of categorization. Memory & Cognition, 32(3), 379–389.

    Google Scholar 

  • Verheyen, S., Ameel, E., & Storms, G. (2007). Determining the dimensionality in spatial representations of semantic concepts. Behavior Research Methods, 39(3), 427–438.

    PubMed  Google Scholar 

  • Voorspoels, W., Vanpaemel, W., & Storms, G. (2008). Exemplars and prototypes in natural language concepts: a typicality-based evaluation. Psychonomic Bulletin & Review, 15(3), 630–637.

    Google Scholar 

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Funding

This work was supported by the National Science Foundation Grant 1534014 (EHR Core Research) to Robert Nosofsky, Bruce Douglas, and Mark McDaniel.

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Authors and Affiliations

  1. Psychological and Brain Sciences, Indiana University Bloomington, 1101 E. Tenth Street, Bloomington, IN, 47405, USA

    Robert M. Nosofsky & Brian J. Meagher

  2. Vanderbilt University, Nashville, TN, USA

    Craig A. Sanders

  3. Earth and Atmospheric Sciences, 1101 E. Tenth Street, Bloomington, IN, 47405, USA

    Bruce J. Douglas

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  1. Robert M. Nosofsky
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Correspondence to Robert M. Nosofsky.

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Appendix

Appendix

Procedure for Collecting Ratings on Supplemental Dimensions

The procedure for collecting the ratings on the supplemental dimensions followed closely the one already described in detail by Nosofsky et al. 2018c In brief, in all conditions, on each trial, a single picture from the 360-picture set was presented at the center of the computer screen and subjects provided a rating for the rock that was appropriate to the condition in which they were being tested. In all conditions, the order in which the pictures were presented was randomized for each individual subject. Detailed instructions (available from the first author upon request) were provided in defining the dimensions, and example pictures to illustrate the definitions were also shown. The subjects were all undergraduate or graduate students from Indiana University who reported little or no previous experience with geologic rock identification. The number of subjects who participated in each dimension-rating condition was a follows: porphyritic—21, holes—19, greenness—13, pegmatitic—20, conchoidal fracture—15.

With the exception of the “holes” dimension, subjects provided ratings for the rock pictures on a 1–9 scale. For example, in the “porphyritic texture” condition, subjects were instructed to provide a rating of 1 for the rocks that showed the least evidence of porphyritic texture, a rating of 9 for the rocks that showed the greatest evidence, and a rating of 5 for rocks that showed medium evidence. In an attempt to promote the use of consistent scale values across subjects, anchor pictures were displayed along with scale values on the computer screen throughout each rating session. One anchor picture corresponded to the lowest rating, a second anchor picture corresponded to the highest rating, and a third anchor corresponded to a rock that we judged to be roughly average on the rated dimension. The anchors and scale values were displayed at the bottom of the screen throughout the rating session (see Nosofsky et al. 2018c, Fig. 1, for an example screen shot). Note that the extreme anchor pictures were displayed midway between the 1–2 and 8–9 scale values to provide subjects with some flexibility in assigning their ratings.

The data for the “holes” dimension had been collected in the previous study reported by Nosofsky et al. (2018c). Those data were not based on continuous ratings; rather, they consisted of the proportion of subjects who judged that each individual rock contained holes. To place the “holes” data on roughly the same scale as the other supplementary dimensions, the proportions were multiplied by 10. Because the model assigned a freely estimated dimension weight and reference-value parameter to each dimension, this multiplication was done for scaling convenience and has no effect on the resulting model fits.

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Nosofsky, R.M., Sanders, C.A., Meagher, B.J. et al. Search for the Missing Dimensions: Building a Feature-Space Representation for a Natural-Science Category Domain. Comput Brain Behav 3, 13–33 (2020). https://doi.org/10.1007/s42113-019-00033-2

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