**Abstract**:

**Abstract**
**Microsaccades, small involuntary eye movements that occur once or twice per second during attempted visual fixation, are relevant to perception, cognition, and oculomotor control and present distinctive characteristics in visual and oculomotor pathologies. Thus, the development of robust and accurate microsaccade-detection techniques is important for basic and clinical neuroscience research. Due to the diminutive size of microsaccades, however, automatic and reliable detection can be difficult. Current challenges in microsaccade detection include reliance on set, arbitrary thresholds and lack of objective validation. Here we describe a novel microsaccade-detecting method, based on unsupervised clustering techniques, that does not require an arbitrary threshold and provides a detection reliability index. We validated the new clustering method using real and simulated eye-movement data. The clustering method reduced detection errors by 62% for binocular data and 78% for monocular data, when compared to standard contemporary microsaccade-detection techniques. Further, the clustering method's reliability index was correlated with the microsaccade-detection error rate, suggesting that the reliability index may be used to determine the comparative precision of eye-tracking devices.**

*x*is the eye position (horizontal or vertical) at time

_{i}*i*,

*v*is the instantaneous eye velocity (horizontal or vertical) at time

_{i}*i*, and

*F*is the sampling rate. This operation is equivalent to smoothing the eye position with a triangular window (normalized Barlett window) of six samples and then differentiating to obtain the velocity. To maintain proper alignment between the velocity and position signals, the output must correspond with the center of the window. The length of the window may be adapted depending on the data-collection sampling rate to maintain a constant bandwidth—for example, by using a six-sample window (0, 1/6, 2/6, 2/6, 1/6, 0) for data recorded at 500 Hz and a 12-sample window (0, 1/12, 2/12, 3/12, 4/12, 5/12, 5/12, 4/12, 3/12, 2/6, 1/6, 0) for data recorded at 1000 Hz. The bandwidth of this triangular smoothing filter is approximately 100 Hz, which matches the bandwidth of fixational saccades (Findlay, 1971).

_{S}*k*-means. This consists of an iterative algorithm that assigns observations to

*k*groups or clusters to minimize the within-cluster variability, that is, the sum of distances from each observation to the center of its own cluster. Each observation is characterized by a vector of features, and the separation between observations is typically measured with euclidean distance. Here we used the implementation of the algorithm from the function kmeans of the Statistical Toolbox within the MATLAB framework (MathWorks, Inc., Natick, MA).

*x*be the vector of features for the microsaccade candidate

_{i}*i*, and

*X*be the matrix formed by all the candidates:

*C*is the covariance matrix of

*X*,

*M*the mean vector of

*X*, V the matrix formed by the eigenvectors of

*C*, and

*D*the diagonal matrix with the inverse square root of the eigenvalues of

*C*, then the new matrix

*X**corresponds with the uncorrelated components of

*X*:

*p*columns of

*X**(which account for the most variance) and apply the

*k*-means algorithm to this new set of observations (Figure 1F, G). We select

*p*= 1, 2, or 3 depending on how many columns have an eigenvalue larger than 5% of the maximum eigenvalue (to ensure that the selected uncorrelated components of

*X*explain most of the variance). To select the initial condition for the algorithm (i.e., the starting center of each cluster), we divide the data into groups of equal number of candidates by sorting them by peak velocity, and we calculate the mean features within each group. To select the value of

*K*, the number of clusters, we test multiple values (2, 3, and 4) and select the one with the smallest average silhouette (Rousseeuw, 1987; see next subsection for details). The algorithm is able to detect more than two potential clusters (i.e., microsaccades and noise) because the sources of noise are diverse; thus noisy events may separate into multiple clusters.

*a*is the average distance between observation

_{i}*x*and all the elements in its own cluster. If there are only two clusters,

_{i}*b*is the average distance between observation

_{i}*x*and the elements of the other cluster. If there are more than two clusters,

_{i}*b*is the shortest of all average distances between observation

_{i}*x*and the elements of each cluster, excluding its own.

_{i}^{2}) plus the 200 ms before and after. Such periods are probably due to partial blinks, where the pupil is never fully occluded (thus failing to be identified as a blink by the eye-tracker software; Troncoso, Macknik, Otero-Millan, et al., 2008). We note that this method to remove partial blinks is specific to EyeLink systems, and that other eye-tracking systems may require different methods to remove partial blinks.

*λ*, which, multiplied by an estimation of the level of noise in the data, determines the final value of the velocity threshold. Here we used

*λ*= 6 when performing analyses with a single

*λ*value. When comparing multiple values of

*λ*we used

*λ*= (2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 10, 12, 15, 20). To reduce the amount of potential noise (Engbert, 2006), we analyzed only binocular microsaccades (that is, microsaccades with a minimum overlap of one data sample in both eyes; Laubrock, Engbert, & Kliegl, 2005; Engbert, 2006; Rolfs, Laubrock, & Kliegl, 2006; Troncoso et al., 2008a). We also imposed a minimum intersaccadic interval of 20 ms so that dynamic overshoots observed right after microsaccades were not categorized as extra microsaccades (Møller, Laursen, Tygesen, & Sjølie, 2002; Otero-Millan et al., 2008; Troncoso, Macknik, & Martinez-Conde, 2008b).

*λ*. For most

*λ*values, there were more microsaccades detected in the original data than in the surrogate data. We selected the

*λ*that resulted in the largest difference between the number of microsaccades detected in the original and surrogate data.

*σ*= 50 ms,

*τ*= 300 ms) and random peak velocities following a log-normal distribution (mean = 40°/s,

*σ*= 13°/s) to create a sequence of microsaccades. In some of the simulations, microsaccade magnitudes (and therefore peak velocities) were constant for all microsaccades in the sequence.

*λ*. The clustering method does not have a criterion level, because it yields the optimal operating point by design. To obtain different points in the ROC curve for the clustering method, we parametrically modified the distance between each point and the center of the microsaccade cluster, thereby reducing or increasing the number of candidates detected as microsaccades. In this scenario, the total number of true negatives is arbitrary, because any point in time may be counted as a nonmicrosaccade event. Thus, we held the total number of true negatives constant and equal to five per second.

*λ*, typically 6 but sometimes 4 or 5; see Engbert & Mergenthaler, 2006; McCamy et al., 2012; Mergenthaler & Engbert, 2010) that is multiplied by the standard deviation of the eye-movement velocity to obtain the final velocity threshold (see Methods for details).

*λ*values in the E&K method, and to compare them with the results from the clustering method, we used a qualitative approach based on the shape of the resultant distributions of saccadic parameters. Nyström and Holmqvist (Nyström & Holmqvist, 2010) used this approach previously to evaluate the quality of automatic saccade detection.

*λ*values produced many false positives and bimodal distributions of microsaccade magnitudes and peak velocities, where the first mode corresponds to false positives (noise or slow eye movements) and the second mode to true microsaccades (Figure 3). In an ideal method, all detected microsaccades would be included in the second mode and none in the first mode.

*λ*values (<4) produce many microsaccades in the first mode (i.e., false positives). High

*λ*values (>7) decrease the size of the second mode, thereby reducing the number of true microsaccades detected (i.e., more false negatives). The standard value of

*λ*= 6 works well, but some false positives remain in the first mode (especially clear in Figure 3B). The clustering method results in a single (i.e., second) mode, suggesting an improved performance when compared to set

*λ*values (Figure 3A, B).

*R*= −0.9,

*p*= 0.000000003), thus suggesting that our metric is a good predictor of the reliability of the detection. A recording with a high mean silhouette is more likely to result in few detection errors.

*λ*= 6) and clustering methods. Figure 5A shows the results for all recordings, sorted by mean silhouette (higher silhouette first). The clustering method shows an improved performance in most cases (19 out of 24 recordings). The median error rate in the E&K method is 0.25 errors per second; the median error rate in the clustering method is 0.1 errors per second (we used median error rates to reduce the influence of outliers; Figure 5B). The cluster method's improvement in overall detection error rate results from a lesser number of false positives than in the E&K method (Figure 5C). False negatives were slightly more prevalent with the cluster method than with the E&K method, however (Figure 5D).

*λ*= 6 varied across recordings (see also Figure 3), suggesting the possibility that other

*λ*values might have been preferable in some instances. To address this issue, Engbert and Mergenthaler (Engbert & Mergenthaler, 2006; Mergenthaler & Engbert, 2010) developed an additional method to select the best

*λ*value for any given recording. Their method, based on surrogate data, selects the value of

*λ*that maximizes an estimation of difference between the numbers of true and false positives (see Methods). Figure 6A shows the error rates obtained when selecting the best

*λ*for each recording: Some individual error rates remain quite large, but the median error rate improves slightly. We also calculated the error rates when selecting for each recording the

*λ*value that produced the smallest error rate a posteriori. This resulted in a much lower median error rate, but still slightly higher than with the clustering method. Thus, the clustering method selected the optimum boundary between true microsaccades and noisy or other nonmicrosaccadic events. (See Figure 6B for ROC curves for the different methods.)

*λ*= 6) and clustering methods for monocular recordings. The clustering method was more robust and outperformed the E&K method in most recordings (17 of 24), reducing the median error rate from 1.1 to 0.25 errors per second.

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