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Magnitude and the production pattern (or shape) of milk produced by dairy cattle are determined by the physiological process of the mammary gland. The production pattern or shape, projected surface on a plane by graphical representation and which can be regarded as a biological form, lacks its appropriate description. We developed the application of the relatively new geometric morphometrics method, which visualizes, measures, and tests differences in the form of biological shapes. We applied the landmark-based geometric morphometrics technique to quantify variation of magnitude and the shape projected on plane by graphical depiction representing the relationship between milk yield and time. We used a free software and small dataset of milk production, monthly time series data from 2007 to 2015, of two leading dairy industries: New Zealand and United States. The results of the analysis showed production patterns of cardioid shape in New Zealand and heart shape in United States. Those forms varied in siz e and shape within and between countries, and only shape within country were statistically non-significant. The landmark-based geometric morphometric is effective to quantify variation of the shape of the milk production pattern under different setting. This may not only complement the analysis of milk prediction, but also reveal profound information about the biological process represented through the shape, by allowing the control of co-variation with other variables.

Milk produced by dairy species is important worldwide in providing food, employment, and income. Driven mainly by the income factor, researchers, following the requirement of dairy industries at some countries, have developed the mathematical representation of milk production during lactation. Mathematical models are the unparalleled data analysis tool generating knowledge for the progress of the dairy industries, and the past 90 years have seen their development and evolution under different setting (for details of the different mathematical functions see the review by [

Those mathematical models can describe and predict, with minimum error, the milk yields of lactation records, commonly of 305-day lactation length and two different phases, ascending and descending. They have been also used to investigate other aspects of the cow’s lactation including evaluating its variation by the influence of factors and variables related to management, environment, and physiology [^{2}). Second, it is constructed fixed, which neglects important effects about the management, environment, and physiology of cow on shape.

The current approach to describe shape of milk production can be compared to a designer that is designing clothes without the person’s measures of body, and instead he is using the person’s weight and age. Working with those data would be of more confusion than usefulness because the subject is described as a blob, of known weight and age, but unknown shape. For the purpose of the tailor, use of body shape measures is the appropriate. As for the development of dairy cattle industry, regarding any lactation pattern as any other biological shape and being able to appropriately measure it, may not only complement to the analysis by mathematical models, but also reveal profound information about the biological process represented through the shape. The well- known shape of lactation curve is the phenotypical expression of the physiological process of milk secretion [

The appropriate analysis of shapes in this field has been hindered by difficulty and lack of a standard method [

Milk production data exhibit two intrinsical features, magnitude and shape; the former refers to the quantity expressed in units of mass (kg) or volume (L) and the later states the relationship between milk quantity and time (projected surface on a plane by graphical depiction). To measure the lactation curve shape, phenotypical expression of this physiological process, we have developed a graphical depiction of the data. Line graph has been essential in visualizing data for fitting lactation models (

graph for further analysis. The orbital graph consisted of six axes to account for the twelve months of the year. The monthly data were plotted into the graph and the order of the months, calendar basis, was rotated counterclockwise to relate the time cows produced milk to the season.

The milk production data are converted to cubic meter to use its meter-side-length that is in accordance with the shape unit (linear or areal). For instance, the quantity of milk 9 m^{3} in January can be thought, for the objective of providing a linear measure, as a rectangular solid of 9 m length, 1 m height, and 1 m width; then the 9 m length of the rectangular solid provides the linear measure that is in accordance with units of shape. Each of the 12 linear measures in a year comprises the two-dimensional (2D) shape of milk production, which can be now treated as any other biological form and a group of shapes of milk production can be quantified with the appropriate analysis.

Different types of data on milk production are generated in dairy industries. They are commonly found of complete individual and aggregated lactations, single farm and region, and whole industry. Such data can be recorded daily, weekly or monthly. For instance, the test-day-model system usually records milk quantity at month interval during the entire lactation of a cow. Such continuous records can be grouped into lactation cycles; then each lactation cycle can be viewed as an individual. The lactation curve represents an individual (a cow). This framework of representing an individual by a curve was first introduced by Kirpatrick and Lofsvold [

For instance, the milk quantity of a cow at January (time) can be viewed as a different character from the milk quantity of the same cow at February. This framework can be also applied to monthly data on milk production of a farm, region, and industry, and consideration should be given to the different factors influencing each data type.

The complete lactation of cow has been the metric unit to develop the current analyses (lactation models) of data in the field. Those lactation models are the data analysis tool and provide the theory, concept, and idea from the research findings that are then applied to production records of farm or whole industry (e.g., [

The weather conditions were typical climatically for the 9 years under study (temperature above the average and rainfall below the average); exceptions were 2008 and 2013 for New Zealand and 2011 for the United States that reported the incident of droughts. In New Zealand, the main effect of drought on milk production is through decreasing the pasture growth, which is main feed supply. In the United States, while drought also reduces pasture, corn, and soybean growths, the main effect on milk production is the heat stress on dairy cows. In addition, another effect is the lower reproduction rates, which causes problems in the subsequent production year.

The New Zealand and United States industries vary in size and structure, as shown in

Morphometrics is overall defined in the biological-shape literature as the measurement of shape variation and its covariation with other variables [

In the theory and measurement of morphometric, it is commonly distinguished

Items | New Zealand | United States |
---|---|---|

Average from 2011-2015 | ||

Cows (head) | 4,717,504 (SD = 172,407) | 9,345,000(SD = 46,135) |

Milk volume (m^{3}) | 19,751,980 (SD = 1,244,264) | (83,118,501 (SD = 2,244,089) |

Average farm size (hd.) | 398.5 (SD = 11.7) | 192 (SD = 10.8) |

Number of farm | 11,837.8 (SD = 87.4) | 47,295 (SD = 3,065) |

Lactation length (day) | 266.8 (SD = 11.7) | |

Correspond to 2014 | ||

Herd genetic | HF and Jersey Crossbred (42.6%) | Holstein |

Holstein-Friesian (37.0) | Jersey | |

Jersey (11.7) | Brown Swiss | |

Ayrshire and other (8.7) | ||

Management system | Grazing based | confinement |

Milking frequency | 2 times | 2 times |

Milking system | Mechanial | mechanical |

Feeding | Pasture and concentration | Total mixed ration |

Breedng system | Artificial insemination 73% | Artificial insemination |

Calculated with data obtained from [

between the size (magnitude) and shape of a biological structure. Shape corresponds an outline with landmarks after differences in location, scale, and orientation has been removed [

GM relies on multivariate analysis such as General Procrustes Analysis, Principal Component Analysis, ANOVA, MANOVA, Regression, and other tests. Several freeware programs to conduct analysis on GM are available such as Morpho J [

Landmark-based GM techniques begin with the collection of 2 or 3D coordinates (which are based on the x, y, and z Cartesian plane) of biologically definable landmarks [

The coordinates were zero on the x-axis and the respective milk quantity on the y-axis for the January and July landmarks (see

The calculation yielded the width and length of the rectangle which corresponded to the x and y coordinates, respectively. Similar calculation was done to obtain the coordinate values of the March, May, September, and November landmarks with the difference that the length of the rectangle represented the x-axis and the width the y-ones. The negative signs were also added to either the x- or y-axis depending on the direction on the plane.

This is the most widely used superimposition method in GM (others: full Procrustes superimposition, resistant-fit method, Bookstein registration, and sliding baseline registration [

parameters [

Removal of location differences is done by centering configurations, for which the centroid of each configuration is calculated and used as the origin of the new coordinates system. The centroid of a configuration is the mean values of the x- and y- (and z-) coordinates for all k landmarks in the configuration. Removal of size differences between configurations is done by rescaling each configuration so that they share a common centroid size of 1. Centroid size is the square root of the sum of the squared distances between each landmark and the centroid of the form [

The Procrustest shape coordinates obtained by GPA is of a high dimensionality, which makes it difficult to visualize and interpret. PCA is a method that summarizes multivariate data by building linear combinations of the original variables that are uncorrelated and maximizes the sample total amount of variance explained [

There are several options to visualized shape variation in a sample, such as thin plate spline, displace vector (called lollipop graph in MorphoJ), and wireframe. In our study, the last option was adopted because it was commonly observed in previous studies. Wireframe diagrams are lines that connect landmarks in to resemble a 2 or 3D shape.

The previous methods (PCA and visualization) are largely exploratory and others statistical analysis are employed for testing hypotheses [

Differences in the shape and size of the production pattern can occur within and between dairy industries. To test within country variations, we used one factor ANOVA for repeated measures, because the set of landmarks associated with the shape are generated from the monthly milk produce in a year by the same dairy herd. The calculation was done separately for each country using the freeware Real Statistic Resource Pack for Excel [

Followed from the depiction on orbital graph is the result of the geometric mophometrics assessment of shapes.

Given the farming system, geographical region, and the genetic merit and the lactation stage of the cows’ population, the New Zealand and United States dairy industries projected production patterns, hereafter named as, of cardioid and heart shapes,

On the other hand, the development of the heart shape of the United States may be more difficult to explain, because it is the result of production by dairy farms distributed in seven geographical regions, among which, each one presents its own environmental conditions, adopts its own management practices, and develops its own pattern of production. Overall, a general pattern can be cast. Yet production in the United States is more uniform throughout the months, the seasonal pattern can still be seen. The greater proportion of cows calving in spring ascends milk production to a peak, from March-to-May, followed by two slightly-gradual decreases, first from June-to- August and second from September-to-October, and an increase of up to the June-to- July level, from November to December (caused perhaps by another proportion of cows calving during autumn, October onward, which may be practiced by some farms). The cleft of the heart shape (February), which is the minimum point in the year, may be the result by those cows calving in spring and that are dried off during December and January.

The influence of environment on milk production was exhibited. The peak production in both countries occurred in their respective grass growing season, observed on the bigger upper half of the March-September axis for New Zealand, and the lower one of the same axis for United States (

GPA superimposed the configuration of landmarks, then suitable for the statistical analysis commonly done in GM studies. The procedure produced an average shape with the scatter of landmark positions around the overall mean.

The assumption of sphericity variation at each landmark position was questionable because the scatters of landmark positions around the overall consensus were not circular (

over the ANOVA for repeated measures. The results for size and shape variations are reported in

On the other hand, similar results to the significant of size within countries were obtained for their comparison (

We only present the summary for the pattern of individual variations, though the non-significance of shape, because for that of the cross comparison will produce an average shape from the cardioid and heart; this may not be appropriate since they correspond to two completely different forms of milk production. The pattern of variation showed by the PCs visualized the major changes in milk production over the 9 years under study; it also showed the specific time of the year that such changes occur. In the New Zealand, the shape variations were described by the first four PCs, which accounted for 93.1% of total variance in the cardioid shape. The analysis of variation of production pattern (the wireframe depiction in

Item | t-test | P | |
---|---|---|---|

Size | New Zealand’s cardioid | 1.859 | <0.001 |

United States’ heart | 1.859 | <0.001 | |

Shape | T^{2} | P (parametric) | |

New Zealand’s cardioid | 6.34E−24 | 1 | |

United States’ heart | 3.95E−24 | 1 |

Item | t-test | P | Subhead |
---|---|---|---|

Size | 1.8124 | <0.001 | |

Mahalanobis distance | P (permutation) | P (permutation) | |

Shepe | 1812.23 | <0.001 | <0.001 |

right lobe, wider left lobe, and right shifted cleft. This pattern of variation agrees well with changes in the cows’ lactations, because majority of New Zealand’s cows calve in a concentrated pattern, in late-August to early-September, the overall shape of the national milk production could be similar to that of a standard lactation curve. The PC2 and PC3 were also associated with changes of the left lobe, summer to autumn production, and the PC4 featured variation in the right lobe.

Similarly, in the United States (

The application of GM as a new data analysis tool for dairy science was demonstrated using a dataset of monthly milk production at the industry level of two leading countries. The projected surface on a plane, representing milk yield over time by orbital graph, was the basis for the landmark based GM analysis. The monthly quantities of milk were the shapes’ landmarks and their respective coordinates were estimated using Pythagorean Theorem. Given the landmark (or a quantity of milk), the set of coordinates can be exactly calculated, if mathematical calculation are free of error, which overall is straightforward and convenient compared to that of GM that depends on special equipment and software to acquire image and locate landmarks. In addition, the counterclockwise rotation of the milk production data in orbital graph also provided a visual combination of the biological process of milk production coupled to its environment (seasons), which may provide understanding and guidance to appropriately analyze the interaction of these two processes. Rather than following the calendar year, the starting month can be adjusted to that of the enterprise since most, if not all, agricultural activities obey a seasonal pattern; the New Zealand milk production, for instance, follows a seasonal operation that begins in June prior to the months of grass growing season and ends in May of the followed year. The application of orbital graph to represent time series data is new, and its extent to lactation records has to be developed, especially in the number of axis representing lactations of different lengths.

GPA superimposed the landmark configurations, which influences the results in testing and visualizing shape variations. Different superimposition methods have different pros and cons, and those of GPA are well discussed by studies applying GM See [

The individual variation of the production pattern was non-significant. This observation is reasonable about the genetic, management, and environment shared by the cows’ population throughout the years. The small differences may also provide evidence of the standardized production of milk by these two countries. Another factor to bear in mind is the type of data, national time series, which may also play a roll, since it is subject to methodological procedure in collection as well as the unequal number of days comprising each month. Differences between the production patterns were larger than within them, which may have been expected from

About 58.8% of shape variation in New Zealand was associated with the PC1 (

The application of GM and the results, however, should be seen as preliminary and exploratory. First, because it has to be considered whether the invariance procedure (translation, scaling, and rotation) done with Procrustes methods applied to the kind of data here used. For instance, the need for rotational invariance since the data might be already in a fixed orientation. The shape comparison in this study can benefit from using other GM methods such as Fourier analysis, which treat most of the landmarks as semi-landmarks. Regarding the results, because the statistical significance marked by the permutation test of DFA, though applicable to sample sizes very small [

We have developed the use of landmark-based geometric morphometric method to measure the form (magnitude and shape) projected on a plane by the graphical representation of milk production data. We demonstrated the new methodology in a small case study of milk production, monthly time series data, from New Zealand and United States dairy industries for their distinct farming system and geographical location. The monthly time series data were appropriate for implementing the methodology, which in turns demonstrated their use as morphological data. The closed curve depicted by orbital graph and regarded as the biological form of milk production was the basic for applying the landmark-based GM analysis.

The analysis revealed production patterns of cardioid shape in New Zealand and heart shape in United States. The data indicated that individual and group variations in size and shape of these patterns were significant, except for shape within country. Overall, GM method seems to be effective to quantify variation of shape and the magnitude of the milk production patterns. This may complement the analysis of milk prediction and reveal profound information of the biological process represented through the shape by controlling the different factors influencing it. The new methodological framework can be also applied to analyze the production pattern of individual and aggregated lactations and farm and region. The assumptions in applying the methodology and environmental, managerial, and physiological factors affecting each type of data may differ.

We profoundly thank to Edgardo Reyes Calderon and Wen-I Chan for their comments during the time we were conceiving this study and for their carefully review and suggestions on several version of this manuscript. We also thank to Meidiana Purnamasari and Thuline Phasha Zwane for their comments and suggestions on this manuscript as well as to Omkar Byadgi for his advice in preparing our response to the reviewers, to whom gratitude is also given for helping to craft the study.

Durón-Benítez, Á.A. and Huang, W.-C. (2016) Using Geo- metric Morphometrics to Quantify Variation of Shape and Magnitude of the Pattern of Milk Production of Dairy Cattle. Open Access Library Journal, 3: e2928. http://dx.doi.org/10.4236/oalib.1102928