RESUMO
The problem of variable clustering is that of estimating groups of similar components of a p-dimensional vector X = (X 1, , X p ) from n independent copies of X. There exists a large number of algorithms that return data-dependent groups of variables, but their interpretation is limited to the algorithm that produced them. An alternative is model-based clustering, in which one begins by defining population level clusters relative to a model that embeds notions of similarity. Algorithms tailored to such models yield estimated clusters with a clear statistical interpretation. We take this view here and introduce the class of G-block covariance models as a background model for variable clustering. In such models, two variables in a cluster are deemed similar if they have similar associations will all other variables. This can arise, for instance, when groups of variables are noise corrupted versions of the same latent factor. We quantify the difficulty of clustering data generated from a G-block covariance model in terms of cluster proximity, measured with respect to two related, but different, cluster separation metrics. We derive minimax cluster separation thresholds, which are the metric values below which no algorithm can recover the model-defined clusters exactly, and show that they are different for the two metrics. We therefore develop two algorithms, COD and PECOK, tailored to G-block covariance models, and study their minimax-optimality with respect to each metric. Of independent interest is the fact that the analysis of the PECOK algorithm, which is based on a corrected convex relaxation of the popular K-means algorithm, provides the first statistical analysis of such algorithms for variable clustering. Additionally, we compare our methods with another popular clustering method, spectral clustering. Extensive simulation studies, as well as our data analyses, confirm the applicability of our approach.
RESUMO
The aim of this study was twofold: first, to evaluate the influence of body weight on the efficiency of dairy cows, and second, to analyze the current state of dairy cattle populations as part of the Austrian Cattle Breeding Association's Efficient Cow project. Data of Fleckvieh (FV, dual-purpose Simmental), Fleckvieh × Red Holstein (FV × RH), Holstein (HF) and Brown Swiss (BS) dairy cows (161 farms, 6098 cows) were collected at each performance recording during the year 2014. In addition to routinely recorded data (e.g., milk yield, fertility), body weight, body measurements, body condition score (BCS) and individual feed information were also collected. The following efficiency traits were considered: body weight efficiency as the ratio of energy-corrected milk (ECM) to metabolic body weight, feed efficiency (kilogram ECM per kilogram dry-matter intake) and energy efficiency expressed as the ratio of energy in milk to energy intake. The relationship of milk yield to body weight was shown to be nonlinear. Milk yield decreased in cows above the 750â¯kg body weight class for HF, BS and FV × RH with 68â¯% RH genes, but less dramatically and later for FV at 800â¯kg. This resulted in an optimum body weight for feed and energy efficiency. BS and HF had the highest efficiency in a narrower and lighter body weight range (550-700â¯kg) due to a stronger curvature of the parabolic curve. Contrary to this, the efficiency of FV did not change as much as it did in the dairy breeds with increasing body weight, meaning that FV had a similar feed and energy efficiency in a range of 500-750â¯kg. The breed differences disappeared when body weight ranged between 750 and 800â¯kg. The average body weight of the breeds studied (FV 722â¯kg, BS 649 and HF 662â¯kg) was in the optimum range. FV was located at the upper end of the decreasing segment. In conclusion, an optimum body weight range for efficiency does exist, due to the nonlinear relationship of milk yield and body weight. Specialized dairy breeds seem to respond more intensively to body weight range than dual-purpose breeds, due to the stronger curvature. Cows with medium weights within a population are the most efficient. Heavy cows ( > 750 â¯kg) produce even less milk. A further increase in dairy cows' body weights should therefore be avoided.
RESUMO
LOVE, a robust, scalable latent model-based clustering method for biological discovery, can be used across a range of datasets to generate both overlapping and non-overlapping clusters. In our formulation, a cluster comprises variables associated with the same latent factor and is determined from an allocation matrix that indexes our latent model. We prove that the allocation matrix and corresponding clusters are uniquely defined. We apply LOVE to biological datasets (gene expression, serological responses measured from HIV controllers and chronic progressors, vaccine-induced humoral immune responses) resulting in meaningful biological output. For all three datasets, the clusters generated by LOVE remain stable across tuning parameters. Finally, we compared LOVE's performance to that of 13 state-of-the-art methods using previously established benchmarks and found that LOVE outperformed these methods across datasets. Our results demonstrate that LOVE can be broadly used across large-scale biological datasets to generate accurate and meaningful overlapping and non-overlapping clusters.
RESUMO
The objective of this study was to predict cows' body weight from body size measurements and other animal data in the lactation and dry periods. During the whole year 2014, 6306 cows (on 167 commercial Austrian dairy farms) were weighed at each routine performance recording and body size measurements like heart girth (HG), belly girth (BG), and body condition score (BCS) were recorded. Data on linear traits like hip width (HW), stature, and body depth were collected three times a year. Cows belonged to the genotypes Fleckvieh (and Red Holstein crosses), Holstein, and Brown Swiss. Body measurements were tested as single predictors and in multiple regressions according to their prediction accuracy and their correlations with body weight. For validation, data sets were split randomly into independent subsets for estimation and validation. Within the prediction models with a single body measurement, heart girth influenced relationship with body weight most, with a lowest root mean square error (RMSE) of 39.0â¯kg, followed by belly girth (39.3â¯kg) and hip width (49.9â¯kg). All other body measurements and BCS resulted in a RMSE of higher than 50.0 kg. The model with heart and belly girth (Model HG BG ) reduced RMSE to 32.5â¯kg, and adding HW reduced it further to 30.4â¯kg (Model HG BG HW ). As RMSE and the coefficient of determination improved, genotype-specific regression coefficients for body measurements were introduced in addition to the pooled ones. The most accurate equations, Model HG BG and Model HG BG HW , were validated separately for the lactation and dry periods. Root mean square prediction error (RMSPE) ranged between 36.5 and 37.0â¯kg (Model HG BG HW , Model HG BG , lactation) and 39.9 and 41.3â¯kg (Model HG BG HW , Model HG BG , dry period). Accuracy of the predictions was evaluated by decomposing the mean square prediction error (MSPE) into error due to central tendency, error due to regression, and error due to disturbance. On average, 99.6â¯% of the variance between estimated and observed values was caused by disturbance, meaning that predictions were valid and without systematic estimation error. On the one hand, this indicates that the chosen traits sufficiently depicted factors influencing body weight. On the other hand, the data set was very heterogeneous and large. To ensure high prediction accuracy, it was necessary to include body girth traits for body weight estimation.