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The majority of all commonly used machine learning methods can not be applied directly to data sets with missing values. However, most such methods only depend on the relative differences between samples instead of their particular values, and thus one useful approach is to directly estimate the pairwise distances between all samples in the data set. This is accomplished by fitting a Gaussian mixture model to the data, and using it to derive estimates for the distances. Experimental simulations confirm that the proposed method provides accurate estimates compared to alternative methods for estimating distances.

The possibility of missing or incomplete data is often ignored when describing statistical or machine learning methods, but as it is a common problem in practice, it is relevant to consider. A popular strategy is to fill in the missing values by imputation as a pre-processing step, but for many methods this is not necessary, and can yield sub-optimal results. Instead, appropriately estimating pairwise distances in a data set directly enables the use of any machine learning methods using nearest neighbours or otherwise based on distances between samples. In this paper, it is shown how directly estimating distances tends to result in more accurate results than calculating distances from an imputed data set, and an algorithm to calculate the estimated distances is presented. The theoretical framework operates under the assumption of a multivariate normal distribution, but the algorithm is shown to be robust to violations of this assumption. The focus is on numerical data with a considerable proportion of missing values, and simulated experiments are provided to show accurate performance on several data sets.

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BackgroundSome of the temporal variations and clock-like rhythms that govern several different health-related behaviors can be traced in near real-time with the help of search engine data. This is especially useful when studying phenomena where little or no traditional data exist. One specific area where traditional data are incomplete is the study of diurnal mood variations, or daily changes in individuals’ overall mood state in relation to depression-like symptoms.ObjectiveThe objective of this exploratory study was to analyze diurnal variations for interest in depression on the Web to discover hourly patterns of depression interest and help seeking.MethodsHourly query volume data for 6 depression-related queries in Finland were downloaded from Google Trends in March 2017. A continuous wavelet transform (CWT) was applied to the hourly data to focus on the diurnal variation. Longer term trends and noise were also eliminated from the data to extract the diurnal variation for each query term. An analysis of variance was conducted to determine the statistical differences between the distributions of each hour. Data were also trichotomized and analyzed in 3 time blocks to make comparisons between different time periods during the day.ResultsSearch volumes for all depression-related query terms showed a unimodal regular pattern during the 24 hours of the day. All queries feature clear peaks during the nighttime hours around 11 PM to 4 AM and troughs between 5 AM and 10 PM. In the means of the CWT-reconstructed data, the differences in nighttime and daytime interest are evident, with a difference of 37.3 percentage points (pp) for the term “Depression,” 33.5 pp for “Masennustesti,” 30.6 pp for “Masennus,” 12.8 pp for “Depression test,” 12.0 pp for “Masennus testi,” and 11.8 pp for “Masennus oireet.” The trichotomization showed peaks in the first time block (00.00 AM-7.59 AM) for all 6 terms. The search volumes then decreased significantly during the second time block (8.00 AM-3.59 PM) for the terms “Masennus oireet” (P<.001), “Masennus” (P=.001), “Depression” (P=.005), and “Depression test” (P=.004). Higher search volumes for the terms “Masennus” (P=.14), “Masennustesti” (P=.07), and “Depression test” (P=.10) were present between the second and third time blocks.ConclusionsHelp seeking for depression has clear diurnal patterns, with significant rise in depression-related query volumes toward the evening and night. Thus, search engine query data support the notion of the evening-worse pattern in diurnal mood variation. Information on the timely nature of depression-related interest on an hourly level could improve the chances for early intervention, which is beneficial for positive health outcomes.