Bayesian regression models outperform partial least squares methods for predicting milk components and technological properties using infrared spectral data

Bayesian functional regression as an alternative statistical analysis of high-throughput phenotyping data of modern agriculture

BACKGROUND

Modern agriculture uses hyperspectral cameras with hundreds of reflectance data at discrete narrow bands measured in several environments. Recently, Montesinos-López et al. (Plant Methods 13(4):1-23, 2017a. 10.1186/s13007-016-0154-2; Plant Methods 13(62):1-29, 2017b. 10.1186/s13007-017-0212-4) proposed using functional regression analysis (as functional data analyses) to help reduce the dimensionality of the bands and thus decrease the computational cost. The purpose of this paper is to discuss the advantages and disadvantages that functional regression analysis offers when analyzing hyperspectral image data. We provide a brief review of functional regression analysis and examples that illustrate the methodology. We highlight critical elements of model specification: (i) type and number of basis functions, (ii) the degree of the polynomial, and (iii) the methods used to estimate regression coefficients. We also show how functional data analyses can be integrated into Bayesian models. Finally, we include an in-depth discussion of the challenges and opportunities presented by functional regression analysis.

RESULTS

We used seven model-methods, one with the conventional model (M1), three methods using the B-splines model (M2, M4, and M6) and three methods using the Fourier basis model (M3, M5, and M7). The data set we used comprises 976 wheat lines under irrigated environments with 250 wavelengths. Under a Bayesian Ridge Regression (BRR), we compared the prediction accuracy of the model-methods proposed under different numbers of basis functions, and compared the implementation time (in seconds) of the seven proposed model-methods for different numbers of basis. Our results as well as previously analyzed data (Montesinos-López et al. 2017a, 2017b) support that around 23 basis functions are enough. Concerning the degree of the polynomial in the context of B-splines, degree 3 approximates most of the curves very well. Two satisfactory types of basis are the Fourier basis for period curves and the B-splines model for non-periodic curves. Under nine different basis, the seven method-models showed similar prediction accuracy. Regarding implementation time, results show that the lower the number of basis, the lower the implementation time required. Methods M2, M3, M6 and M7 were around 3.4 times faster than methods M1, M4 and M5.

CONCLUSIONS

In this study, we promote the use of functional regression modeling for analyzing high-throughput phenotypic data and indicate the advantages and disadvantages of its implementation. In addition, many key elements that are needed to understand and implement this statistical technique appropriately are provided using a real data set. We provide details for implementing Bayesian functional regression using the developed genomic functional regression (GFR) package. In summary, we believe this paper is a good guide for breeders and scientists interested in using functional regression models for implementing prediction models when their data are curves.

Milk yield, quality, and coagulation properties of 6 breeds of goats: Environmental and individual variability

Goat milk and cheese production is continuously increasing and milk composition and coagulation properties (MCP) are useful tools to predict cheesemaking aptitude. The present study was planned to investigate the extension of lactodynamographic analysis up to 60 min in goat milk, to measure the farm and individual factors, and to investigate differences among 6 goat breeds. Daily milk yield (dMY) was recorded and milk samples collected from 1,272 goats reared in 35 farms. Goats were of 6 different breeds: Saanen and Camosciata delle Alpi for the Alpine type, and Murciano-Granadina, Maltese, Sarda, and Sarda Primitiva for the Mediterranean type. Milk composition (fat, protein, lactose, pH; somatic cell score; logarithmic bacterial count) and MCP [rennet coagulation time (RCT, min), curd-firming time (k₂₀, min), curd firmness at 30, 45, and 60 min after rennet addition (a₃₀, a₄₅, and a₆₀, mm)] were recorded, and daily fat and protein yield (dFPY g/d) was calculated as the sum of fat and protein concentration multiplied by the dMY. Data were analyzed using different statistical models to measure the effects of farm, parity_, stage of lactation and breed; lastly, the direct and the indirect effect of breed were quantified by comparing the variance of breed from models with or without the inclusion of linear regression of fat, protein, lactose, pH, bacterial, somatic cell counts, and dMY. Orthogonal contrasts were performed to compare least squares means. Almost all traits exhibited high variability, with coefficients of variation between 32 (for RCT) and 63% (for a₃₀). The proportion of variance regarding dMY, dFPY, and milk composition due to the farm was moderate, whereas for MCP it was low, except for a₆₀, at 69%. Parity affected both yield and quality traits of milk, with least squares means of dMY and dFPY showing an increase and RCT and curd firmness traits a decrease from the first to the last parity class. All milk quality traits, excluding fat, were affected by the stage of lactation; RCT and k₂₀ decreased rapidly and a₃₀ was higher from the first to the last part of lactation. Alpine breeds showed the highest dMY and dFPY but Mediterranean the best percentage of protein, fat, and lactose and a shorter k₂₀ and a greater a₃₀. Among the Mediterranean goats, Murciano-Granadina goats had the highest milk yield, fat, and protein contents, whereas Maltese, Sarda, and Sarda Primitiva were characterized by much more favorable technological properties in terms of k₂₀, a₃₀, and a₄₅. In conclusion, as both the farm and individual factors highly influenced milk composition and MCP traits, improvements of these traits should be based both on modifying management and individual goat factors. As expected, several differences were attributable to the breed effect, with the best milk production for the Alpines and milk quality and coagulation for the Mediterranean goats.

Comparison of Bayesian regression models and partial least squares regression for the development of infrared prediction equations

The objective of this study was to compare the prediction accuracy of 92 infrared prediction equations obtained by different statistical approaches. The predicted traits included fatty acid composition (n = 1,040); detailed protein composition (n = 1,137); lactoferrin (n = 558); pH and coagulation properties (n = 1,296); curd yield and composition obtained by a micro-cheese making procedure (n = 1,177); and Ca, P, Mg, and K contents (n = 689). The statistical methods used to develop the prediction equations were partial least squares regression (PLSR), Bayesian ridge regression, Bayes A, Bayes B, Bayes C, and Bayesian least absolute shrinkage and selection operator. Model performances were assessed, for each trait and model, in training and validation sets over 10 replicates. In validation sets, Bayesian regression models performed significantly better than PLSR for the prediction of 33 out of 92 traits, especially fatty acids, whereas they yielded a significantly lower prediction accuracy than PLSR in the prediction of 8 traits: the percentage of C18:1n-7 trans-9 in fat; the content of unglycosylated κ-casein and its percentage in protein; the content of α-lactalbumin; the percentage of α_S2-casein in protein; and the contents of Ca, P, and Mg. Even though Bayesian methods produced a significant enhancement of model accuracy in many traits compared with PLSR, most variations in the coefficient of determination in validation sets were smaller than 1 percentage point. Over traits, the highest predictive ability was obtained by Bayes C even though most of the significant differences in accuracy between Bayesian regression models were negligible.

Predicting grain yield using canopy hyperspectral reflectance in wheat breeding data

BACKGROUND

Modern agriculture uses hyperspectral cameras to obtain hundreds of reflectance data measured at discrete narrow bands to cover the whole visible light spectrum and part of the infrared and ultraviolet light spectra, depending on the camera. This information is used to construct vegetation indices (VI) (e.g., green normalized difference vegetation index or GNDVI, simple ratio or SRa, etc.) which are used for the prediction of primary traits (e.g., biomass). However, these indices only use some bands and are cultivar-specific; therefore they lose considerable information and are not robust for all cultivars.

RESULTS

This study proposes models that use all available bands as predictors to increase prediction accuracy; we compared these approaches with eight conventional vegetation indexes (VIs) constructed using only some bands. The data set we used comes from CIMMYT's global wheat program and comprises 1170 genotypes evaluated for grain yield (ton/ha) in five environments (Drought, Irrigated, EarlyHeat, Melgas and Reduced Irrigated); the reflectance data were measured in 250 discrete narrow bands ranging between 392 and 851 nm. The proposed models for the simultaneous analysis of all the bands were ordinal least square (OLS), Bayes B, principal components with Bayes B, functional B-spline, functional Fourier and functional partial least square. The results of these models were compared with the OLS performed using as predictors each of the eight VIs individually and combined.

CONCLUSIONS

We found that using all bands simultaneously increased prediction accuracy more than using VI alone. The Splines and Fourier models had the best prediction accuracy for each of the nine time-points under study. Combining image data collected at different time-points led to a small increase in prediction accuracy relative to models that use data from a single time-point. Also, using bands with heritabilities larger than 0.5 only in Drought as predictor variables showed improvements in prediction accuracy.

Predicting grain yield using canopy hyperspectral reflectance in wheat breeding data

BACKGROUND

Modern agriculture uses hyperspectral cameras to obtain hundreds of reflectance data measured at discrete narrow bands to cover the whole visible light spectrum and part of the infrared and ultraviolet light spectra, depending on the camera. This information is used to construct vegetation indices (VI) (e.g., green normalized difference vegetation index or GNDVI, simple ratio or SRa, etc.) which are used for the prediction of primary traits (e.g., biomass). However, these indices only use some bands and are cultivar-specific; therefore they lose considerable information and are not robust for all cultivars.

RESULTS

This study proposes models that use all available bands as predictors to increase prediction accuracy; we compared these approaches with eight conventional vegetation indexes (VIs) constructed using only some bands. The data set we used comes from CIMMYT's global wheat program and comprises 1170 genotypes evaluated for grain yield (ton/ha) in five environments (Drought, Irrigated, EarlyHeat, Melgas and Reduced Irrigated); the reflectance data were measured in 250 discrete narrow bands ranging between 392 and 851 nm. The proposed models for the simultaneous analysis of all the bands were ordinal least square (OLS), Bayes B, principal components with Bayes B, functional B-spline, functional Fourier and functional partial least square. The results of these models were compared with the OLS performed using as predictors each of the eight VIs individually and combined.

CONCLUSIONS

We found that using all bands simultaneously increased prediction accuracy more than using VI alone. The Splines and Fourier models had the best prediction accuracy for each of the nine time-points under study. Combining image data collected at different time-points led to a small increase in prediction accuracy relative to models that use data from a single time-point. Also, using bands with heritabilities larger than 0.5 only in Drought as predictor variables showed improvements in prediction accuracy.

Breed of cow and herd productivity affect milk composition and modeling of coagulation, curd firming, and syneresis

Milk coagulation properties (MCP) have been widely investigated in the past using milk collected from different cattle breeds and herds. However, to our knowledge, no previous studies have assessed MCP in individual milk samples from several multi-breed herds characterized by either high or low milk productivity, thereby allowing the effects of herd and cow breed to be evaluated independently. Multi-breed herds (n=41) were classified into 2 categories based on milk productivity (high vs. low), defined according to the average milk net energy yielded daily by lactating cows. Milk samples were taken from 1,508 cows of 6 different breeds: 3 specialized dairy (Holstein-Friesian, Brown Swiss, Jersey) and 3 dual-purpose (Simmental, Rendena, Alpine Grey) breeds, and analyzed in duplicate (3,016 tests) using 2 lactodynamographs to obtain 240 curd firming (CF) measurements over 60min (1 every 15 s) for each duplicate. The 5 traditional single-point MCP (RCT, k₂₀, a₃₀, a₄₅, and a₆₀) were yielded directly by the instrument from the available CF measures. All 240 CF measures of each replicate were also used to estimate 4 individual equation parameters: RCT estimated according to curd firm change over time modeling (RCT_eq), asymptotic potential curd firmness (CF_P), curd firming instant rate constant (k_CF), and syneresis instant rate constant (k_SR) and 2 derived traits: maximum curd firmness achieved within 45min (CF_max) and time at achievement of CF_max (t_max) by curvilinear regression using a nonlinear procedure. Results showed that the effect of herd-date on traditional and modeled MCP was modest, ranging from 6.1% of total variance for k₂₀ to 10.7% for RCT, whereas individual animal variance was the highest, ranging from 32.0% for t_max to 82.5% for RCT_eq. The repeatability of MCP was high (>80%) for all traits except those associated with the last part of the lactodynamographic curve (i.e., a₆₀, k_SR, k_CF, and t_max: 57 to 71%). Reproducibility, taking into account the effect of instrument, was equal to or slightly lower than repeatability. Milk samples collected in farms characterized by high productivity exhibited delayed coagulation (RCT_eq: 18.6 vs. 16.3min) but greater potential curd firmness (CF_P: 76.8 vs. 71.9mm) compared with milk samples collected from low-productivity herds. Parity and days in milk influenced almost all MCP. Large differences in all MCP traits were observed among breeds, both between specialized and dual-purpose breeds and within these 2 groups of breeds, even after adjusting for milk quality and yield. Milk quality and MCP of samples from Jersey cows, and coagulation time of samples from Rendena cows were better than in milk from Holstein-Friesian cows, and intermediate results were found with the other breeds of Alpine origin. The results of this study, taking into account the intrinsic limitation of this technique, show that the effects of breed on traditional and modeled MCP are much greater than the effects of herd productivity class, parity, and DIM. Moreover, the variance in individual animals is much greater than the variance in individual herds within herd productivity class. It seems that improvement in MCP depends more on genetics (e.g., breed, selection) than on environmental and management factors.

Cheesemaking in highland pastures: Milk technological properties, cream, cheese and ricotta yields, milk nutrients recovery, and products composition

Summer transhumance of dairy cows to high Alpine pastures is still practiced in many mountainous areas. It is important for many permanent dairy farms because the use of highland pastures increases milk production and high-priced typical local dairy products often boost farm income. As traditional cheese- and ricotta-making procedures in Alpine pastures are central to this dairy system, the objective of this study was to characterize the quality and efficiency of products and their relationships with the quality and availability of grass during the grazing season. The milk from 148 cows from 12 permanent farms reared on a temporary farm located in Alpine pastures was processed every 2wk during the summer (7 cheesemakings from late June to early September). During each processing, 11 dairy products (4 types of milk, 2 by-products, 3 fresh products, and 2 ripened cheeses) were sampled and analyzed. In addition, 8 samples of fresh forage from the pasture used by the cows were collected and analyzed. At the beginning of the pasture season the cows were at 233±90d in milk, 2.4±1.7 parities, and produced 23.6±5.7kg/d of milk. The milk yield decreased with the move from permanent to temporary farms and during the entire summer transhumance, but partly recovered after the cows returned to the permanent farms. Similar trends were observed for the daily yields of fat, protein, casein, lactose, and energy, as we found no large variations in the quality of the milk, with the exception of the first period of Alpine pasture. The somatic cell counts of milk increased during transhumance, but this resulted from a concentration of cells in a lower quantity of milk rather than an increase in the total number of cells ejected daily from the udder. We noted a quadratic trend in availability of forage (fresh and dry matter weight per hectare), with a maximum in late July. The quality of forage also varied during the summer with a worsening of chemical composition. The evening milk (before and after natural creaming), the whole morning milk, and the mixed vat milk had different chemical compositions, traditional coagulation properties, and curd-firming modeling parameters. These variations over the pasture season were similar to the residual variations with respect to chemical composition, and much lower with respect to coagulation and curd-firming traits. Much larger variations were noted in cream, cheese, and ricotta yields, as well as in nutrient recoveries in curd during the pasture season. The protein content of forage was correlated with some of the coagulation and curd-firming traits, the ether extract of forage was positively correlated with milk fat content and cheese yields, and fiber fractions of forage were unfavorably correlated with some of the chemical and technological traits. Traditional cheese- and ricotta-making procedures showed average cream, cheese, and ricotta yields of 6.3, 14.2, and 4.9%, respectively, and an overall recovery of almost 100% of milk fat, 88% of milk protein, and 60% of total milk solids.

Genetic and environmental relationships of detailed milk fatty acids profile determined by gas chromatography in Brown Swiss cows

The aim of this study was to characterize the profile of 47 fatty acids, including conjugated linoleic acid (CLA), 13 fatty acid groups, and 5 Δ(9)-desaturation indices in milk samples from Brown Swiss cows. The genetic variation was assessed and the statistical relevance of the genetic background for each trait was evaluated using the Bayes factor test. The additive genetic, herd-date, and residual relationships were also estimated among all single fatty acids and groups of fatty acids. Individual milk samples were collected from 1,158 Italian Brown Swiss cows and a detailed analysis of fat percentages and milk fatty acid compositions was performed by gas chromatography. Bayesian animal models were used for (co)variance components estimation. Exploitable genetic variation was observed for most of the de novo synthesized fatty acids and saturated fatty acids, except for C4:0 and C6:0, whereas long-chain fatty acids and unsaturated fatty acids (including CLA) were mainly influenced by herd-date effects. Herd-date effect explained large portions of the total phenotypic variance for C18:2 cis-9,cis-12 (0.668), C18:3 cis-9,cis-12,cis-15 (0.631), and the biohydrogenation and elongation products of these fatty acids. The desaturation ratios showed higher heritability estimates than the individual fatty acids, except for CLA desaturation index (0.098). Among the medium-chain fatty acids, C12:0 had greater heritability than C14:0 (0.243 vs. 0.097, respectively). Both C14:0 and C16:0 showed negative additive genetic correlations with the main monounsaturated and polyunsaturated fatty acids of milk fat, suggesting that their synthesis in the mammary gland may be influenced by the presence of unsaturated fatty acids. No correlation was observed between C4:0 and the other short-chain fatty acids (except for C6:0), confirming the independence of C4:0 from de novo mammary fatty acid synthesis. Among the genetic correlations dealing with potentially beneficial fatty acids, C18:0 was positively correlated with vaccenic and rumenic acids and negatively with linoleic acid. Finally, fatty acids C6:0 through C14:0 showed relevant correlations due to unknown environmental effects, suggesting the potential existence of genetic variances in micro-environmental sensitivity. This study allowed us to acquire new knowledge about the genetic and the environmental relationships among fatty acids. Likewise, the existence of genetic variation for most of de novo synthetized fatty acids and saturated fatty acids was also observed. Overall, these results provide useful information to combine feeding with genetic selection strategies for obtaining a desirable milk fatty acids profile, depending on the origin of fatty acids in milk.

Proton transfer reaction time-of-flight mass spectrometry: A high-throughput and innovative method to study the influence of dairy system and cow characteristics on the volatile compound fingerprint of cheeses

The aim of this work was to study the effect of dairy system and individual cow-related factors on the volatile fingerprint of a large number of individual model cheeses analyzed by proton transfer reaction time-of-flight mass spectrometry (PTR-ToF-MS). A total of 1,075 model cheeses were produced using milk samples collected from individual Brown Swiss cows reared in 72 herds located in mountainous areas of Trento province (Italy). The herds belonged to 5 main dairy systems ranging from traditional to modern and the cows presented different daily milk yields (24.6±7.9kg × d(-1)), stages of lactation (199±138 d in milk), and parities (2.7±1.8). The PTR-ToF-MS revealed 619 peaks, of which the 240 most intense were analyzed, and 61 of these were tentatively attributed to relevant volatile organic compounds on the basis of their fragmentation patterns and data from the literature. Principal component analysis was used to convert the multiple responses characterizing the PTR-ToF-MS spectra into 5 synthetic variables representing 62% of the total information. These principal components were related to groups of volatile compounds tentatively attributed to different peaks and used to investigate the relationship of the volatile compound profile obtained by PTR-ToF-MS to animal and farm characteristics. Lactation stage is related to 4 principal components which brought together 52.9% of the total variance and 57.9% of the area of analyzed peaks. In particular, 2 principal components were positively related to peaks tentatively attributed to aldehydes and ketones and negatively related to alcohols, esters, and acids, which displayed a linear increase during lactation. The second principal component was affected by dairy system; it was higher in the modern system in which cows received total mixed rations. The third principal component was positively related to daily milk production. In summary, we report the first application of this innovative, high-throughput technique to study the effects of dairy system and individual animal factors on volatile organic compounds of model cheeses. Individual cheesemaking procedures together with this spectrometric technique open new avenues for genetic selection of dairy species with respect to both milk and cheese quality.