ANIMAL GENETICS & BREEDING Biometrical Techniques

Transcript:ANIMAL GENETICS & BREEDING Biometrical Techniques:
ANIMAL GENETICS & BREEDING Biometrical Techniques in Animal Breeding Course No. AGB – 605 UNIT - II Lecture – 9 Introduction to Matrix Algebra Dr K G Mandal Department of Animal Genetics & Breeding Bihar Veterinary College, Patna Bihar Animal Sciences University, Patna Introduction to Matrix Algebra Science of today is becoming increasingly quantitative in nature. Scientists are being confronted with large volumes of numerical data gathered from their laboratories, field experiments and surveys. Mere collection and recording of data achieves nothing until or unless those data are analysed and interpreted. Mathematics is made great use of in describing this analysis and interpretation, and one of the most important and useful branches of mathematics for this purpose is matrix algebra. It is useful not only for simplifying description and development of many analysis methods but also in organising computer techniques to execute those methods and to present the results. As a branch of mathematics, it dates back more than a century, but its application in today’s world are widespread, particularly in statistics. Matrices are simply rectangular arrays of numbers arranged in rows and columns, matrix algebra is the algebra of those arrays. Application: Useful in the field of population dynamics to investigate the distribution of individuals according to their age and sex. Calculation of transition probabilities over a range of time intervals. continued on the next page 3. In the field of Animal Breeding. For genetic improvement of farm animals, the animals are selected by using a selection methods known as selection index. The numerical score of selection index is obtained through solving partial regression coefficients (b’s) for all the characters included in the index. The partial regression coefficients are solved through the technique of matrix algebra. SI = b1X1 + b2X2 + b3X3 + . . . . + bnXn Instead of using numbers, “ i “ is used as subscript to denote the row and “ j “ is used to denote column, where i = 1, 2, 3, and j = 1, 2, 3, 4. Thus, the element aij means the element is located at ith row and jth column. Accordingly, the element a11 is located at first row and first column of the matrix, A. Similarly, the matrix, A = [ aij ] where, i = 1, 2, 3, and j = 1, 2, 3, 4 like this. 7. Subdiagonal elements: The elements of a square