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Prior studies have attempted to characterize the constructive matching and response elimination strategies with more traditional dwell time variables. The previous high-water mark was set by Vigneau et al. Goodness-of-fit R 2 and leave-one-out cross-validated R cv 2 for predicting Raven scores from eye movement data. The top line reports the performance of the novel method based on successor representations and principal component analysis PCA. It is compared to some prominent dwell time variables from the literature Vigneau et al.

Apparently, as Vigneau et al. This begs the question of how well the scanpath SR would perform on new data. We conducted leave-one-out cross-validation to test the generalization performance of our method. We partitioned the data into a training set of 34 participants and a test set of 1 participant. We ran our two-tier algorithm on the training set. Finally, we calculated the model's prediction of the test Raven score by multiplying the test SR matrix by the weight matrix estimated from the training set.

We repeated this process 35 times, testing on the data from each participant in turn. This produced 35 predicted scores, each one based on a model that had no access to the data that were subsequently used to test it. This is a much better estimate of generalization performance than the goodness-of-fit R 2 on the training set Haykin, The latter is inflated because it reflects not only the genuine regularities in the population, which will generalize to new cases, but also the idiosyncrasies of the training sample, which will not.

Note that this still is very respectable cross-validated performance, which sets a new benchmark for Raven score prediction. This suggests that the SR algorithm can extract reliable regularities from the data much better than traditional dwell time methods. The SR advantage comes from the sequential information in scanpaths and from the data-smoothing properties of the temporal-difference learning algorithm.

The success of the scanpath SR in cross-validated prediction is also a direct result of the stability of the principal components across folds. The same two components—systematicity and toggle—were chosen on all 35 cross-validation folds and were qualitatively indistinguishable from iteration to iteration. Although it is difficult to quantify the component overlap across folds because the two components sometimes switched places, the weight matrices derived from them can be combined linearly.

The average weight matrix is shown in Figure 4a and is virtually identical to the weight matrix from the global model trained on all data Figure 2c.


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This suggests that the components were not driven by outliers and reflect genuine dimensions of individual differences in scanpath patterns across the majority of observers. Figure 4. Leave-one-out cross-validation results.

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The average weight matrix a across 35 leave-one-out fits is virtually identical to the weight matrix produced by the fit to all data at once Figure 2c. Each Raven score was predicted by a separate model that had no access to the data for the respective individual. Finally, we compared the new scanpath SR method to first- and second-order transition probability matrices Table 1. We began by calculating the first-order transition matrix for each sequence. Adding variables to the regression model improved the fit only marginally e.

Introduction

This suggests that first-order transition matrices are too myopic to support robust prediction of Raven scores. It also demonstrates that the excellent performance of the SR method cannot be attributed to the PCA-based dimensionality reduction algorithm. Second-order transition probabilities are conditionalized on the two preceding fixations in the sequence.

Given that the median clipped sequence length was only 88, the second-order estimates were extremely variable even after averaging across the 28 trials.


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Still, it was interesting to check whether the PCA algorithm could identify individual differences among the participants. Hierarchical linear regression with the second-order projections yielded good fits to the Raven scores Table 1. While quite respectable and much better than the R cv 2 achievable with traditional measures, this falls far short of the SR-based prediction.

Moreover, unlike the SR-based components Figure 2 , the second-order components were extremely hard to visualize and interpret. The transition-based results suggest two conclusions. First, a single-step event horizon cannot capture the statistical regularities in our data. A temporally extended analysis seems necessary. This explains why the second-order model performed better than the first-order one. The second conclusion is that the probability estimates need to be smoothed. There are not enough data to populate the matrices by simple counting, particularly in the second-order case. This scarcity of data rather than computational constraints appears to be the limiting factor in scanpath analysis in general.

The SR learning algorithm Equation 1 updates a whole column of the matrix after each transition, thereby smoothing the estimates. Stated differently, each cell in the SR matrix aggregates a whole class of observations. For example, cell 1, 1 would be updated after observing any of the following subsequences: , , …, 1R1; , , …. This reuses the data and reduces the variance of the estimates. This smoothing effect contributed to the stability of the SR components during leave-one-out cross-validation. By contrast, the first-order probability estimates were apparently noisier, and the PCA solution was unstable even though it involved matrices of the same shape estimated from the same data.

Our novel method of eye movement analysis, the scanpath successor representation SR , produced new results in terms of both successful score prediction and insight into individual differences in problem solving strategies on Raven's Advanced Progressive Matrices.

With this method, we were able to extract the underlying structure from complex patterns of sequential eye movements during geometric problem solving. These regularities allowed us to predict APM scores with unprecedented accuracy. More importantly, the principal component analysis of the successor representations produced components that were readily interpretable and consistent with earlier strategy findings. The two components of the scanpath SRs that correlated strongly with the scores mapped clearly onto the two main processing strategies for multiple-choice matrix completion problems.

The anti-toggle component Figure 2b replicated earlier reports of negative correlations between toggling and Raven scores Bethell-Fox et al. This qualitative agreement with established results validates the new SR-based method. Quantitatively, however, it goes a step further because it could predict a larger proportion of the variance compared to traditional measures such as the number of toggles or toggle rate.

This suggests that the SR-based analysis provides a more sensitive measure of toggling and thus can better identify individuals who follow the response elimination strategy. This article did not address the question of whether response elimination is adopted at the beginning of a problem or only as a fallback strategy on difficult items.

Chapter 0. Recap of 2018 and Looking Forward

The systematicity component Figure 2a is a novel finding and arguably provides the most detailed picture of Raven performance and strategic processing to date. This component demonstrates the importance of processing the problem matrix row by row.

This lends new support to the theory that successful Raven solvers use a constructive matching strategy and explicates some important aspects of this strategy. We chose Raven's APM as the test bed for the novel scanpath SR method because decades of painstaking research have identified the two strategies most relevant for this domain e.

Thus, we knew what to expect and could validate the method against these established findings. Still, the method revealed previously unknown details about the constructive matching strategy. More importantly, armed with this powerful tool, we could have discovered these two strategies even if we had never read the Raven literature, simply by interpreting the component matrices in Figure 2.

Note that these matrices were calculated in an entirely automated manner and reflect regularities in the data rather than the prior knowledge of the authors. Thus, the scanpath SR method promises to be a great tool for exploratory data analysis, with the potential for rapid discoveries in other domains.

The power of the scanpath SR stems from the fact that it extends the event horizon of sequential eye movements to extract temporally extended patterns. It will very likely prove useful in any complex task environment that has distinct areas of interest statically or dynamically defined. The successor representation was introduced to the reinforcement learning literature by Dayan and was developed by White More recently, Gershman, Moore, Todd, Norman, and Sederberg under revision identified a formal connection between the SR and an influential model of episodic and semantic memory, the Temporal Context Model e.

We use a version of the successor representation that differs slightly from the standard definition Dayan, ; White, The difference is that, when visiting a state i , our version does not include this same visit in the total temporally discounted number of visits to i. To revert to the standard formulation of the SR learning algorithm, the term I j in Equation 1 must be replaced by I i. This indicates that the learning rate should be inversely related to the length of the data sequence.

This in turn suggests a potential improvement of our eye-tracking analysis application.

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It would be interesting to explore parameterizations that reduce the effective learning rate for longer sequences. The clipping of sequences longer than fixations described in the Methods section is a crude way of regularizing the sequence length. Our present results indicate that, even with a fixed learning rate, the learning algorithm can accommodate substantial variability in length. As mentioned earlier, this is a major advantage over string-editing methods for comparing scanpaths.

Varying the learning rate as a function of sequence length will provide additional robustness and reduce the variance of the estimates. This is a promising topic for future research. Another promising possibility is to improve the feature selection algorithm. Independent Component Analysis ICA; Stone, may be better suited for eye-tracking applications than PCA because it relaxes the orthogonality constraint on the components.

The SR matrices that correspond to psychologically relevant strategies are not necessarily orthogonal.

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The authors thank James Todd and Vladimir Sloutsky for valuable suggestions on the manuscript. Commercial relationships: none. Corresponding author: Alexander A. Email: apetrov alexpetrov. The posttest score was 1. Anders G.