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Comparison of two computing algorithms for solving mixed
model equations for multiple trait random regression test day
models
*
J. Jamrozik, L.R. Schaeffer
Centre for Genetic Improvement of Livestock, Department of Animal and Poultry Science, University of Guelph, Guelph, ON, Canada N1G 2 W1
Received 29 July 1999; received in revised form 24 January 2000; accepted 9 February 2000
Abstract
Two computing algorithms for solving mixed model equations for a multiple lactation, multiple trait random regression test day model were compared. The model for each trait (yields of milk, fat, and protein, and somatic cell scores in the first three lactations) included fixed contemporary groups, fixed regressions within levels of time–region–age–season parity subclasses at calving and two sets of random regressions: animal genetic and permanent environmental effects, giving a total of twelve traits and 36 equations for each animal genetic effect and each animal permanent environmental effect. Algorithm A utilized the iteration on data with blocking strategy (with contemporary group and animal blocks) in a Gauss–Seidel iteration scheme. Block sizes for animal genetic and permanent environmental effects were of order 36. Algorithm B utilized an alternative blocking strategy for animal effects with separate blocks for each lactation of order 12. This allowed for significant reduction in memory requirements, less time per iteration, but slightly slower convergence compared to Algorithm A. The algorithms were compared in an application of the test day model to the national Canadian Jersey test day data set. Memory and disk space requirements for the two algorithms as well as extensions of the model were discussed.
2000 Elsevier Science B.V. All rights reserved.
Keywords: Test day models; Genetic evaluation; Computing
1. Introduction the complexity of the model, the size and idiosyncrasies of the data, the computing environ-The implementation of computing algorithms to ment, and possible future enhancements to the solve BLUP mixed model equations (MME) for model, data, or computing environment. Data sets genetic evaluation requires careful consideration of tend to grow in numbers of animals and records over time; models tend to become more complex; and computing tends to become faster and to have more
*Corresponding author. Tel.:11-519-8244-120; fax:1
1-519-available memory, disk space, and multiple
pro-7670-573.
E-mail address: [email protected] (L.R. Schaeffer). cessors. Generalized computing software is often
sufficient for many research tasks and saves the user mink’s function (Wilmink, 1987) with three parame-from software development time and parame-from possible ters per trait giving 72 equations per cow with TD programming errors. However, efficient, routine records and 36 equations per animal without data. genetic evaluations from specially developed soft- For the Holstein breed, CTDM required processing ware can save time for delivery of results and may over 21 million TD records on 1.3 million cows in 2 be necessary when general software can not accom- million contemporary groups and 2.2 million animals modate the model or data size. in total. The total number of equations was more
Iteration on data (Schaeffer and Kennedy, 1986) than 135 million.
has been used widely as a method of solving MME. Several requests were received from Europe by The MME are not constructed explicitly, but data Canadian Dairy Network (CDN) to acquire the files are read each round of iteration (or stored in computer programs used in the CTDM, but the cost memory), and diagonal elements, right hand sides, of the programs was a major obstacle. The decision and solutions need to be stored in memory. Iteration was made, therefore, to publish the gory computa-on data allows for a variety of techniques to be tional details in this journal so that others may write applied, such as Gauss–Seidel, Jacobi, or combina- their own programs if they want. Also, the details tions of both, sparse matrix techniques (Misztal, given here can serve as the beginning of the history 1999), transformations to simplify multiple trait on computing algorithms for random regression problems (Ducrocq and Besbes, 1993), or parallel models. First attempts, such as given here, are processor algorithms for solving large sparse equa- usually replaced with better algorithms over time. tion systems (Madsen and Larsen, 1998). Thus, the objectives of this paper were to present the The scale of the equations to be solved has computing details used in the CTDM, to present an dramatically increased with the introduction of test outline for an alternative computing procedure that day (TD) models (Ptak and Schaeffer, 1993) for uses less memory and disk space, and to compare the genetic evaluation of dairy cattle. Reents et al. computing requirements of the two algorithms when (1995) presented an efficient iteration on data algo- applied to data of the Canadian Jersey dairy breed. rithm for a multiple lactation TD model that has been
applied in Canada and Germany. Random regression
(RR) TD models (Jamrozik et al., 1997) added 2. Material and methods
another level of complexity, mainly through an
enormous increase in the size of the MME. Savings 2.1. Model in computing requirements for TD models have been
suggested by using transformations (Van der Werf et The CTDM has been described by Schaeffer et al. al., 1998). Canonical transformations for the random (2000a) with a discussion of the experiences in using regression model does not lead to a series of a TD model for routine genetic evaluation. In matrix univariate analyses, but to a multiple trait model of notation, the multiple lactation, multiple trait, ran-reduced rank in which only the variables with dom regression TD model could be written as significant eigenvalues are evaluated. Then the
multi-y5Hc1Xb1Wp1Za1e,
ple trait model with missing traits of Ducrocq and
where Besbes (1993) is applied. Parallel computing
tech-niques have also been attempted (Stranden, 1999).
The introduction of a multiple lactation, multiple y is the vector of observations on T traits in L trait, random regression test day (TD) model in lactations ordered traits within lactations, Canada in February 1999 was possible through the c is the vector of fixed contemporary group design and development of specialized software. effects defined as herd-test date-parity sub-Milk, fat, and protein yields plus somatic cell scores classes,
p is the vector of random regression coeffi- where cients for animal permanent environmental
(PE) effects, A is the additive genetic relationship matrix,
a is the vector of random regression coeffi- P0 are covariance matrices of order R*T *L for cients for animal additive genetic effects, and the PE and genetic regression coefficients,
e is the vector of random residual effects, G0 respectively, and
H is an incidence matrix that relates contem- Rij is the covariance matrix for cow i on a given porary groups to observations, and test day j.
X,W,Z are matrices of covariates involving number
of days in milk associated with a cow on a The rank of R can vary from 1 to T depending onij
given test date and corresponding to the the traits that are missing on a cow. The values in Rij
observations. depend on the lactation number and number of days in milk within that lactation.
The corresponding mixed model equations Genetic groups for unknown parents are included
(MME) for this model are in the vector a, for simplicity of notation, and in the
definition of A, the matrix of additive genetic 21 21 21 21
ˆ H9R H H9R X H9R W H9R Z c
relationships. In the CTDM, the fixed regressions 21 21 21 21 ˆ
X9R H X9R X X9R W X9R Z b
and the random genetic and PE regressions were 21 21 21 21 21
ˆ W9R H W9R X W9R W1P W9R Z p
1
212
modelled after Wilmink’s function (Wilmink, 1987),
21 21 21 21 21
ˆ Z9R H Z9R X Z9R W Z9R Z1G a
but in general, these regressions do not need to be
21
H9R y
modelled after the same function. For example, the
21
X9R y
fixed regressions could be modelled as classification
5 21 .
variables for every 10 days in milk because the
1 2
W9R y21
shapes of the lactation curves would be allowed to Z9R y
take whatever form was appropriate rather than being
Let forced to fit a particular function. Let R be the
number of regression coefficients in the function
NPE equal the number of animals with TD used for the lactation curve, i.e. for Wilmink’s
records, function, R53.
NAN equal the total number of animals, The expectations and covariance matrices are
NCG equal the number of contemporary groups, and
y Hc1Xb
p 0 NFR equal the number of subclasses of fixed
E 5
a 0 regressions.
1 2 1 2
e 0Then the total number of equations in MME for and
where but because NFR is usually much smaller than NCG, NPE, or NAN, and because NCG does not involve
P5I^P ,
0 any regression functions, then NEQ can be roughly
21
108 000 000. If all calculations are performed as ing the elements of A . Recall that the relationship double precision (i.e. eight bytes per variable), then matrix inverse can be decomposed as
storing the solutions, diagonal elements, and right 21 22
A 5T9D T,
hand sides to MME would require a minimum of 2.6
gigabytes (GB) of memory. If the computer on where T is a triangular matrix with ones on the which the calculations are to be performed has only diagonals and at most two non-zero elements per row 2 GB of memory, then not all of these elements can in columns corresponding to the parents of an animal
22
be stored in memory at one time or computations with value equal to 20.5, and D is a diagonal
must be done in single precision, or a combination of matrix. In a non-inbred population the diagonal both. In the United States NAN would be greater elements are 2 if both parents are known, (4 / 3) if than 10 million for their Holstein population. only one parent is known, and 1 if both parents are unknown. With an inbred population then there are many more possible values for these diagonal ele-2.2. Algorithm A
ments which can be computed using the methods of Meuwissen and Luo (1992). The variables in the A multiple trait model provides an obvious
block-PEDIGREE file are ing structure of MME by traits. With the TD model,
blocks can be defined on different levels of
generali-Animal number (ID), ty. Two different blocks will be used through the
Its sire ID, description of Algorithm A, which is the algorithm
Its dam ID, and currently used with the CTDM. Record blocks (RB) 22
Value from D . are determined by the residual covariance matrix, R ,ij
on a given test day for an animal. These blocks are
The ID numbers of animals were consecutive from of order T or 4 for the CTDM. RB matches the way
1 to NAN, i.e. youngest to oldest sequence. Genetic in which data are stored, i.e. four traits within an
groups were assigned for all missing parents and animal on a given test day. The contemporary group
were numbered from NAN11 onwards. Obviously,
effects have diagonal blocks of order T as well as T
there must be another file that links the consecutive elements in the right hand sides (RHS). That is,
ID number to an animal’s registration number, name,
21
H9R H is a block diagonal matrix with blocks of
and ownership, but such a file is not needed in the order T.
computation process. The entire PEDIGREE file is The other type of blocking is called an animal
stored in memory during the iteration process, and block, (AB), which is defined by G and P of order0 0 requires approximately 16*NAN bytes of memory.
21
R*T *L or 36 in the CTDM. That is, W9R W and
The TD records data file on all cows contains the
21
Z9R Z are block diagonal matrices with blocks of
following information: order R*T *L. Data associated with an AB or RB are
processed at the same time, and all equations
pertain-Animal ID, (matches the ID in PEDIGREE file) ing to a block are solved simultaneously. Data
Cow ID, (numbering for PE effects) processing in blocks is simple and speeds
conver-Contemporary group number gence, but to implement a blocking strategy requires
Fixed regression subclass number specific preparation of data files.
Days in milk (DIM) Parity number 2.2.1. Data files Missing traits code
Two types of data files are required; the pedigree Accuracy of TD yields code
file and the data file with TD yields. The pedigree Yields for milk, fat, protein and somatic cell score. file, (PEDIGREE), contains one record per
milk yields present, but other traits may be missing. work vector large enough to store the RHS for The missing trait codes specify the correct R to beij animal additive genetic effects, and this vector must used in conjunction with parity number and DIM. be double precision because the RHS of the MME TD yields are estimates of 24 h yield and if for some animals can become quite large in mag-estimated from two supervised weighings receives an nitude. Elements of RHS for CG and PE effects are accuracy of 100. If 24 h yield is estimated from an created sequentially while processing the CG file and evening or morning weighing only, then accuracy is COW file, respectively. Because the fixed regression 89%. If 24 h yields are estimated from one weighing effects are relatively small in number of levels, the in herds that are milking three times a day, then solutions, diagonal blocks, and RHS for fixed regres-accuracy would be lower around 80%. These num- sions are stored in memory. The PEDIGREE file is bers are provided by the milk recording organiza- also stored in memory, as mentioned previously. tions (Schaeffer et al., 2000b). Each record in the Inverses of the residual covariance matrices (by yield file requires 2014*T bytes of storage. With parity number, four DIM intervals, missing trait
over 21 million records in this file for Canadian code, and accuracy of TD yields) are stored in Holsteins, storage of the information in memory is memory as half-stored T3T matrices. Inverses for
impossible, and therefore the file must be re-read G and P are created prior to iteration.0 0
during each iteration. In fact, two copies of the data The iteration process proceeds as follows: file are needed: one sorted by contemporary group 1. The CG file is read sequentially.
numbers (CG file) and one sorted by cow ID (COW (a) All records within a CG are stored in memory
21
file), and each file needs to be read once during every and the appropriate Rij is selected for each record round of iteration. Reading these files in an efficient based on DIM, parity number, missing trait combina-manner is facilitated by special input / output (I / O) tion, and accuracy of the TD information. Let the routines (in the C language) and writing the data in model for the jth TD record in the ith CG and kth an unformatted manner. fixed regression subclass be
y 5c 1X b 1W p1Z a1e .
ijk i ij k ij ij ij
2.2.2. Iteration scheme
Prior to iteration the diagonal blocks for
contem-(b) Adjust the observations for the current solu-porary groups, animal PE, and animal genetic effects
tions for fixed regressions, animal PE, and animal need to be created, inverted, and stored on disk as
additive genetic effects, and accumulate into the three separate data files written in standard
RHS for that CG (call it CGRHS),
FORTRAN 77 as unformatted. Animal genetic and
21
PE diagonal blocks for cows with TD records are CGRHS5
O
R ( y 2X b 2W p2Z a).ij ijk ij k ij ij
functions of DIM on which the cow’s records were j
made. Because there is a very large number of
Because Wilmink’s function is used for both fixed possible combinations of DIM, missing trait codes,
and random regressions in the CTDM, the values of and accuracy codes, these diagonal blocks have to be
the covariates that appear in X , W , and Z are theij ij ij
created and stored explicitly. For animals without
same. This is not essential, but it does make pro-TD records (i.e. ancestors), the diagonal block for
gramming a little easier. the genetic effect is
(c) Adjust the observations in each TD record for
ii 21 animal PE and animal additive genetic effects and
a G0 ,
accumulate into the RHS for the kth fixed regression
ii 21
subclass (call them FRHS ), where a is the diagonal element of A for animal i, k
which can be created as needed or stored using an 2
1
9
FRHS 5FRHS 1X R ( y 2W p2Z a).
implicit representation as shown by Tier and Graser k k ij ij ijk ij ij
(1991).
21 21
9
HINV5(H R H ) , p 5WINV*PERHS.
i i i i
and obtain a new solution for that CG,
(e) Adjust the animal genetic RHS for the new
c 5HINV*CGRHS. animal PE solution, i
21
9
ARHS 5ARHS 2Z R W p .
i i ij ij ij i
(e) Go through the records for that CG again and adjust the RHS of the fixed regressions for the new
4. To get new animal genetic solutions, the CG solution,
PEDIGREE file (in memory) must be processed.
21
9
FRHS 5FRHS 2X R c .
k k ij ij i Remember that animals are sorted from youngest to
oldest, and that this ordering is critical. Let i Continue until all CG have been processed.
represent the ith animal, s represents the sire of 2. Compute new solutions for the fixed
regres-animal i, and d represents the dam of regres-animal i, and sions. The block diagonal inverses are already stored km 21
let a represent elements of A between animals k in memory,
and m.
21 21 (a) Adjust the animal genetic RHS for its sire and
*
model for the jth TD record on the ith cow be
denoted as (b) If the animal has TD records, read in the inverted diagonal block for animal i as
y 5H c1X b1W p 1Z a 1e ,
with Var(e )5R . For simplicity, the same
ij ij
subscript, i, has been used to denote PE and animal or if the animal has no TD records, then genetic effects, but remember the PE effects are
ii 21 21
referenced by the cow ID in the data file and animal ZINV5(a G ) .
0
genetic effects are referenced by the animal ID.
(a) Read and store in memory all TD records for a Calculate a new animal genetic solution vector as given cow.
a 5ZINV*ARHS .
(b) Adjust the observations for fixed regressions, i i
CG, and animal genetic effects and accumulate in the
RHS for the PE effects (i.e. a 36 by 1 vector). (c) Adjust the sire and dam genetic RHS for the new animal genetic solution and the solution for its
21
(c) Adjust the observations for CG and fixed s s 0 i d
regressions and accumulate into the RHS for animal
21 di ds
genetic effects, which is the large work vector in ARHS 5ARHS 2G (a a 1a a ).
d d 0 i s
memory for all animals,
5. Solve for new genetic group solutions as
21
(d) Read in the diagonal block inverse for the
animal PE effect, The iteration process is continued until satisfactory
21 21 21 convergence is obtained. The CTDM applies up to
9
WINV5(W R W1P ) ,
i i i 0
2.3. Algorithm B would only be of length 78, but three could be necessary for each animal. In terms of disk space this Inspection of the MME for the CTDM reveals the would be only 35% of that required for the larger following structures as a result of partitioning ac- blocks. If an animal does not have any second or cording to lactation number. That is, third lactation TD records, then their inverted diag-onal blocks for those lactations are not written to
21 21
W9R W1P 5 22 21
disk because those blocks would be equal to (P )0
21 11 12 13
these matrices. Thus, the actual savings in disk space
P P W R W3 3 31P
would be greater than 65%. Cows, however, need to
21 21
with a similar structure for Z9R Z1G , and be coded in the program to know which ones do not
have records in second or third lactations.
21
9
W R1 1 Z1 0 0 Memory storage is still required for solutions to all
21
21 9
0 W R Z 0
W9R Z5 .
2 2 2 effects in the model, but now the RHS for animal
1
212
genetic effects only needs to be large enough for all
9
0 0 W R3 3 Z3
animals for one lactation, i.e. NAN*R*T rather than Note that there are no data connections between NAN*R*T *L. The iteration process proceeds as lactations, but only connections via the non-zero follows:
covariances of PE and genetic effects between 1. The CG file is read sequentially and calcula-lactations. These structures suggest blocking PE and tions are performed exactly the same as in Algorithm genetic effects on a lactation by lactation basis, A. RHS for fixed regressions are handled in the same rather than all three lactations simultaneously. Let manner.
11 12 13 2. New solutions for fixed regressions are
calcu-P0 P0 P0
lated as in Algorithm A.
21 22 23
3. The COW file is processed. Remember that this
1
31 32 332
P0 P0 P0 file is now sorted by cow ID within parity number.
Let the model for the jth TD record on the ith cow in and
(a) Read and store in memory all TD records for a
21
where each partition is of order R*T. cow and determine the appropriate Rijm.
(b) Adjust the observations for contemporary 2.3.1. Data files groups, fixed regressions, and animal genetic effects The PEDIGREE file is needed as before with no and accumulate in the RHS for the PE effects (i.e. changes. The TD records data files are also the same within lactation RHS is a 12 by 1 vector),
as before except that the COW file must be sorted by 9 21
PERHS5
O
W R ( y 2H c2X bijm ijm ijm ijm ijm
cow ID within parity number. The CG file, sorted by
j
contemporary group numbers, remains the same.
2Z a ). ijm im
2.3.2. Iteration scheme
(c) Further adjustment to PERHS is needed for the Blocks are now defined by animals within
lacta-PE effects in the other lactations which are correlated tions. The order of these blocks is R*T rather than
to the PE effects in lactation m for cow i,
R*T *L. For the CTDM with R53, T54, and L53,
,m
the block size of a half-stored matrix per animal was PERHS5PERHS2
O
P p .0 i,
(d) Adjust the observations for CG and fixed (d) Adjust the sire and dam genetic RHS for the regressions and accumulate over j into the RHS for new animal genetic solution and for the mate’s animal genetic effects for cow i. genetic solution as
21 L
(e) If a cow has TD records in lactation m, then
L
read in the inverted diagonal block for cow i PE m, di ds
ARHS 5ARHS 2
O
G (a a 1a a ).dm dm 0 i, s,
effects in lactation m, ,51
21 mm 21
Compute the new animal PE solution for lactation m 2.4. Comparison of algorithms as
Algorithms A and B were applied to the national
p 5WINV*PERHS. im
Canadian Jersey dairy data set. Data were 543 769 TD records from the first three lactations of 35 502 (f) Adjust the animal genetic RHS for animal i
cows (5NPE) that calved after January 1, 1988. The
and lactation m for the new PE solution,
total number of animals in the evaluation was 69 946
21
9
ARHS 5ARHS 2Z R W p . (5NAN). Contemporary groups, formed on the basis
im im ijm ijm ijm im
of herd-test date-parity subclasses (with second and third parities combined) numbered 71 038 (5NCG).
4. The animal genetic solutions for lactation m are
Seventeen phantom parent groups were formed for obtained by processing the PEDIGREE file (in
unknown sires and dams based on sex of parent and memory).
year of birth of offspring. The number of fixed (a) Adjust the animal’s RHS for the sire and dam
regression subclasses, formed on the basis of region– solutions in all lactations as
parity-age at calving-season of calving, was 38
L
m, is id (5NFR). The model for each trait was the same and
ARHS 5ARHS 2
O
G (a a 1a a ).im im 0 s, d,
,51 was described in detail by Schaeffer et al. (2000a).
Wilmink’s function was utilized so that R53.
(b) Further adjustment to ARHSim is needed for
The MME comprised a total of 4 081 096 equa-the genetic effects in equa-the oequa-ther lactations which are
tions. Starting solutions for all effects were zero for correlated to the genetic effects in lactation m for
both algorithms prior to iteration. Algorithms were cow i.
compared on the basis of total computing time per
ii ,m
ARHS 5ARHS 2
O
a G a . iteration, convergence properties, and memory andim im 0 i,
,±m disk storage requirements. Convergence was attained
when the sum of squares of differences in animal (c) If the animal has TD records in lactation m
genetic solutions between iterations divided by the then read in the inverted diagonal block for animal i,
sum of squares of animal genetic solutions in the
21 ii mm 21
9
ZINV5(Z R Z 1a G ) ,
im im im 0 latest iteration all times 100 was less than 0.00001.
This criterion is unitless compared to a comparison otherwise,
of the squared differences between actual and
re-ii mm 21
ZINV5(a G ) . generated right hand sides which would require
0
additional storage to re-generate the right hand sides Calculate a new animal genetic solution vector as
for comparisons.
a 5ZINV*ARHS . Algorithms were written and implemented in
Table 2
standardFORTRAN 77. Programs were run on an
HP-Expected storage requirement for Algorithm B as applied to
UX 9000 / 800 workstation. All solution and RHS
Canadian Holstein data set for different numbers of covariates as
vectors were declared as single precision except for random regressions the RHS work vector that was allocated for NAN
Number of Memory space Disk storage
animal genetic effects, which was declared as double
covariates (MB) (MB)
precision, and was critical to achieving convergence
3 715 2433
in the Holstein breed.
4 954 4242
5 1192 6552
6 1430 9360
3. Results 7 1669 12 666
A summary of the comparison statistics for the Table 3
two algorithms is given in Table 1. Algorithm B Number of iterations to reach different levels of the convergence criterion for two algorithms applied to Canadian Jersey data set
required just 90% of the time per iteration than
Algorithm A. Algorithm B required 80% less disk Value of convergence Algorithm Algorithm
storage space than Algorithm A. Therefore, there criterion A B
were many cows with TD records in only lactation 1 1.0e0 6 6 which would not have had any inverted diagonal 1.0e-1 14 13
1.0e-2 36 35
blocks written to disk for lactations 2 and 3. The
1.0e-3 97 95
expected savings in disk storage space was 66%. The
1.0e-4 231 264
memory storage space for the RHS vector for animal
1.0e-5 439 656
genetic effects was as expected 66% smaller in Algorithm B.
In the Canadian Holstein population, Algorithm A, rates are also key factors in deciding on expansion of with Wilmink’s function (i.e. three parameters), the CTDM. The number of iterations for different required approximately 1140 megabytes (MB) of values of the convergence criteria is given in Table memory for RHS and solution vectors for animal 3. Algorithm A required 439 iterations to reach the genetic and PE effects, and 6930 MB of disk storage desired criterion value at 1.46 min per iteration for a space for animal genetic and PE block diagonal total of 641 min. Algorithm B required 656 iterations matrices. Algorithm B required 715 MB of memory at 1.31 min per iteration for a total of 859 min, or and 2433 MB of disk storage, respectively, for the 34% longer. Both algorithms performed similarly same problem (Table 2). The numbers in Table 2 during the first 200 iterations.
indicate that a function with five parameters could be accommodated by Algorithm B within the same
space as Algorithm A with Wilmink’s function. 4. Discussion
Computing time per iteration and convergence
A blocking strategy of cows within parities achieved significant savings in disk storage and
Table 1
memory requirements while giving slightly faster
Computing requirements for two algorithms applied to the
Cana-times per iteration than a strategy using larger
dian test day model on the Jersey data set
blocks. Such savings from Algorithm B allows
Item Algorithm A Algorithm B
consideration of possible expansions of the CTDM.
Time per iteration (min) 1.46 1.31
Wilmink’s function could be replaced by a function
Disk storage for animal PE 94 22
with more than three parameters thereby giving a
diagonal blocks (MB)
better approximation to the genetic and PE variances
Memory space for animal 20 7
RHS (MB) and covariances through the lactations and
conse-Number of iterations to 439 656 quently giving more accurate estimated breeding reach convergence
Another alternative would be to keep Wilmink’s or a preparation program to split the data would be function, but to increase the number of lactations. required, so that each processor is working equally The Canadian dairy industry would like to see one or hard. Thus, there are computing times and communi-two more lactations added to the analyses, especially cation times that need to be balanced. Computing for cows. Breeders believe that EBV on cows based time should reduce inversely to the number of only on first or second lactations are not indicative of processors that are used, but communication time later lactation production or longevity. Because cows between processors may increase. Because animal tend to average around three lactations per lifetime, genetic effects must be processed from youngest to adding more lactations to the CTDM may only oldest, splitting the work among processors may not benefit a few animals and dairy producers. Probably be possible or efficient. Work in this area is proceed-a more beneficiproceed-al use of the spproceed-ace sproceed-aved by Algo- ing in Finland.
rithm B would be to increase the number of parame- Algorithms A and B have not made use of ters for the random regressions to five and increase transformations of the data (Van der Werf et al., the accuracy of EBV for all producers. 1998). The transformations of Gengler et al. (1999) The iteration scheme was not a true Gauss–Seidel parameterize the random regression model in terms approach because during the calculation of animal of a hierarchical model. There has been no com-genetic solutions and the processing of the PEDI- parison of Algorithm A to any other proposed GREE file, in particular, the sire and dam right hand computing algorithms. With different accuracies sides are adjusted for the new solution of their possible for TD records due to the testing scheme progeny which includes an adjustment for the mate. chosen by the dairy producer and different residual At some point either the sire’s solution will be matrices depending on parity and days in milk, updated before the dam’s or vice versa, and there- transformations would have to be very complicated fore, the latest ‘mate’ solution would never be used or else make lots of approximations or assumptions. during this adjustment. For all other factors in the A straight-forward multiple lactation, multiple trait model, however, iteration is by Gauss–Seidel. Be- approach seemed to be the most logical and simple, cause animals are processed from youngest to oldest, as a first computing algorithm. Also, changes may be this slight departure from true Gauss–Seidel does not needed in the future to the model or data files, and seem to cause any problems in convergence. There is therefore, model changes to Algorithms A or B a way of creating a special pedigree file so that true should be easy to implement. Once the model is Gauss–Seidel is maintained, but this would require a finalized (for at least the next 5 year), then alter-PEDIGREE file that was three times as large. The native algorithms may be explored. If the existing extra space was not warranted. algorithm is sufficient for routine usage, then perhaps Second order Jacobi iteration would require stor- faster alternatives are not necessary. Every computer ing two vectors of animal solutions which would use center has to make their own decisions in this twice as much memory space. Convergence of respect.
Jacobi schemes are often slower than Gauss–Seidel The algorithms described in this paper have been and may need special tricks to attain convergence. specifically for genetic evaluation and have assumed The newly proposed preconditioned conjugate gra- that parameters in G , P , and R0 0 ij were known. If dient method (Lidauer and Stranden, 1999) may be these matrices have to be estimated, then Algorithm worth considering in the future. The applicability to A using the bigger blocks for animal PE and genetic multiple trait, random regression models has not effects is better for this purpose in the Gibbs been examined. Parallel processing of TD data has sampling approach (Jamrozik et al., 1998).
Lidauer, M., Stranden, I., 1999. Fast and flexible program for
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Ptak, E., Schaeffer, L.R., 1993. Use of test day yields for genetic Acknowledgements evaluation of dairy sires and cows. Livest. Prod. Sci. 34,
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Data were provided by the Canadian Dairy Net- Reents, R., Dekkers, J.C.M., Schaeffer, L.R., 1995. Genetic evaluation for somatic cell score with a test day model for
work, Guelph, Ontario, Canada. Financial support for
multiple lactations. J. Dairy Sci. 78, 2858–2870.
this research from the Ontario Ministry of
Agricul-Schaeffer, L.R., Kennedy, B.W., 1986. Computing strategies for
ture, Food, and Rural Affairs, the Cattle Breeding solving mixed model equations. J. Dairy Sci. 69, 575–579. Research Council of Canada, and the Natural Sci- Schaeffer, L.R., Jamrozik, J., Kistemaker, G.J., Van Doormaal,
ences and Engineering Research Council are grate- B.J., 2000a. Experience with a test day model. J. Dairy Sci. 83, 1135–1144.
fully acknowledged.
Schaeffer, L.R., Jamrozik, J., Van Dorp, R., Kelton, D.F., Lazenby, D.W., 2000b. Estimating daily yields of cows from different milking schemes. Livest. Prod. Sci. 65, 219–228.
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