Inverse Polynomial Function Research Articles

Modelling of the first lactation milk yield of Frieswal cows using different mathematical functions

A study was conducted to find the most appropriate mathematical model that describes the first lactation milk yield of Frieswal cattle. Data on 42,368 individual first lactation test day yields of 1,072 Frieswal cows calved during 2005 to 2014 in Ambala and Meerut Military dairy farms were utilized for the study. The first test day milk yield was recorded on 6thday after calving while the subsequent records were collected at seven days interval and so the average 43 test day yields were taken for fitting the lactation curve models. Five different mathematical models, viz. Exponential decline function (EDF), Parabolic exponential function (PEF), Inverse polynomial function (IPF), Gamma function (GF) and Mixed log function (MLF) were fitted.The accuracy of fitting (R2 value) the models revealed that the MLF (96.14) was more appropriate followed by IPF (95.57), GF (93.85), PEF (83.68) and EDF (69.09). The RMSE estimate of MLF was lowest (0.3483) as expected while the EDF had the highest RMSE value of 0.9858.The AIC criterion was lowest for IPF (5.7175) and highest for GF (8.0212). The BIC values of five functions ranged between –83.6262 for MLF to 3.1809 for EDF. All the DW estimates were positive and ranged between 0.3656 for EDF to 0.7106 for GF indicating positive autocorrelation between the residuals. Based on the results obtained in the present study, it may be inferred that the first lactation yield was explained accurately by the mixed log function (MLF) in Frieswal cattle. As the inverse polynomial (IPF) and gamma function (GF) also had satisfactory results, these two functions can also be used for fitting the lactation curve models in Frieswal cattle. On the other hand, exponential decline function and parabolic exponential functions least explain the first lactation curve in Frieswal cattle.

Comparative efficiency of five mathematical functions in modelling the first lactation milk yield of Kankrej cattle

Modelling of first lactation milk yield of Kankrej cattle was done using five different non-linear mathematical functions viz., exponential decline function (EDF), gamma function (GF), inverse polynomial function (IPF), mixed log function (MLF) and parabolic exponential function (PEF). We compared their efficiency in describing variations in first lactation milk yield of animals maintained under field conditions. A total of 3994 fortnightly daily milk records of 203 Kankrej cattle calved during the years between 2011 and 2017 at farmers? herds in the Banaskantha region of Gujarat State, India were utilized for the study. The PROC NLIN procedure of Statistical Analysis Software (SAS) using Newton method of iteration was applied to generate the model parameters and corresponding standard errors. Different parameters like adjusted R2 -value, root mean square error (RMSE), Durbin?Watson (DW) statistic, Akaike information criterion (AIC) and Bayesian information criterion (BIC) were used to compare the efficiency of different models. Results inferred that mixed log function was the best non-linear mathematical function for explaining variations in the first lactation milk yield of Kankrej cattle followed by gamma function and parabolic exponential function. The exponential decline function was the least efficient in fitting the lactation curve in Kankrej cattle, followed by inverse polynomial function.

Modeling and evaluation of lactation curve functions in Gir cattle

The aim of the study was to model and evaluate five different non-linear lactation curve functions for their efficiency of explaining the variations in first lactation milk yield of Gir cows maintained in the farmers herds. Information on 4,334 fortnightly test day yields of 223 cows calved during the period from 2013-2017 were used for the study. Twenty fortnightly yields starting from the day 15 of lactation were used for fitting the five different non-linear mathematical models, viz. Exponential decline function (EDF), Gamma function (GF), Inverse polynomial function (IPF), Mixed log function (MLF) and Parabolic exponential function (PEF). The curve functions (a, b and c) with standard errors and different evaluation parameters, viz. adjusted R2-value, Akaike information criterion (AIC), Bayesian information criterion (BIC), Durbin watson (DW) statistic and root mean square error (RMSE) were estimated by non-linear regression analysis using PROC NLIN procedure Newton method of SAS (SAS Institute, 2010). The adjusted R2 value of the models ranged from 69.28 (exponential decline function) to 99.36% (parabolic exponential function). All the DW estimates were positive ranging from 0.1573 for exponential decline function to 0.7707 for parabolic exponential function. The RMSE (0.1453) and AIC (4.9152) estimates were also lowest for parabolic exponential function while highest for exponential decline function. Based on the results, it may be concluded that among five functions, parabolic exponential function is the best fitted lactation curve model followed by mixed log function, gamma function, inverse polynomial function while exponential decline function was the least efficient in explaining the variations in first lactation daily yield in Gir cattle.

Comparison of lactation curve models for fortnightly test day milk yield

The study of lactation curve helps to predict lactation milk production at a certain point of time and provides useful information for taking desirable decisions on preliminary sire evaluation, early culling of poor producers, breeding and feeding management and comparison of lactation records of unequal length. The test day milk yield data on 511 crossbred cattle were obtained and utilized according to fortnightly interval upto 305th day maintained at Instructional Dairy Farm, G B Pant University of Agriculture and Technology, Pantnagar (Uttarakhand). The different statistical models predicting the shape of lactation curve considered were exponential function, parabolic exponential function, inverse polynomial function and gamma type function. For fortnightly test day milk yield, the inverse polynomial function exhibited the highest coefficient of determination (R2=99.97%) followed by parabolic exponential (89.32%), gamma type (87.26%) and exponential function (35.70). Inverse polynomial function is the best fitted function for explaining the first lactation curve by utilizing the records of fortnightly test day milk yields.

Testing the efficacy of different lactation curve models for monthly test day milk yield

The test day milk yield data on 511 crossbred cattle were obtained and utilized according to monthly interval upto 305 th day maintained at Instructional Dairy Farm, of G. B. Pant University of Agriculture and Technology, Pantnagar (Uttarakhand). The different statistical models considered in this study for predicting the shape of lactation curve were Exponential Function, Parabolic Exponential Function, Inverse Polynomial Function and Gamma Type Function. Maximum coefficient of determination (R 2 ) value was observed for Inverse Polynomial function (R 2 = 99.97%) followed by Gamma function (R 2 = 88.28%), Parabolic function (R 2 = 80.95 %), and Exponential function (R 2 = 39.10%). The root mean square error (RMSE) was least for Inverse Polynomial function (0.07 kg) followed by Gamma Type function (0.10kg ), Parabolic Exponential function (0.12 kg ) and Exponential function (0.21 kg ). The absolute mean deviation (AMD) was least for Inverse Polynomial (0.16 Kg) followed by Gamma Type function (0.24 Kg), Parabolic Exponential (0.31Kg) and Exponential function (0.48) confirming the same order of superiority.

Use of the D-Shuttle dosimeter for radiation protection of members of the public: Characterization and feasibility study

The objective of this study was to characterize the latest version of the electronic D-Shuttle dosimeter (Chiyoda Technol Corp., Tokyo, Japan) for potential use in dosimetry of members of the public. The investigations comprise: (a) reproducibility, batch homogeneity, dose linearity, dose rate and angle dependence, and radiation hardness using irradiations at the secondary standard dosimetry laboratory of the Paul Scherrer Institute (PSI); (b) estimation of the environmental dose rate and its variability; and (c) feasibility tests for guests at PSI (short-term researchers). The performed verification measurements showed overall a good reproducibility, maximum 1.4% standard deviation (sd) per dosimeter, and maximum 1.0% sd for 5 dosimeters, in agreement with the manufacturer's specifications. The deviation from linearity was <10% and the deviation with energy was <30% in the range [60 keV, 1.25 MeV], except for 164 keV photons (N-200 radiation quality), where a deviation of almost 50% was observed with respect to the Cs-137 radiation quality. The angle dependence was within specifications (±10%) for Cs-137, but up to +30% for 83 keV photons (N-100 radiation quality). The drop in response for dose rates >0.5–1 mSv/h was well described by an inverse polynomial function, and no influence of a high dose irradiation (16.37 Sv) in subsequent low dose measurements was observed. Environmental background (BG) measurements demonstrated that the dosimeters are stable, showing a mean background indication of (70 ± 30) nSv/h, where the standard deviation is compatible with Poisson statistics assuming a sensitivity of 1 count per 10 nSv. This means that hourly indications above 220 nSv/h (BG + 5 sd) can be clearly distinguished from the background. The monitoring of the guests activities showed that the high sensitivity and hourly information can be useful to investigate out-of-the-ordinary exposures, representing a valuable tool for radiation protection of members of the public.

Modeling of lactation curves for prediction of standard lactation milk yield in Marathwadi buffaloes

The present study was carried out on 538 monthly milk yield records of Marathwadi buffaloes maintained at Cattle Breeding Farm, College of Veterinary and Animal Sciences, Udgir Dist-Latur Maharashtra spread over a period of 9 years. The monthly milk yield was recorded at 6th, 36th, 66th and so on till standard lactation with 30 days interval. The goodness of fit of 3 lactation curve models, viz. parabolic exponential function, inverse polynomial function and mixed log function were used on monthly milk yields in Marathwadi buffaloes. The highest coefficient of determination for fitting of lactation curve models was with mixed log function (97.63%), followed by parabolic exponential function (95%) and inverse polynomial function (82.70%). Further, the root mean square error was found least for mixed log function (1.70), followed by parabolic exponential function (3.55) and inverse polynomial function (4.61). It is therefore inferred that mixed log function fitted best for prediction and modeling of lactation curves based upon monthly milk yields.

Effects of environmental factors on stress corrosion cracking of cold-drawn high-carbon steel wires

Premature failure of cable bolts due to stress corrosion cracking (SCC) is a phenomenon that has reported to occur in underground environments. This paper presents an experimental study to determine the impacts of environmental factors on SCC in high-carbon steel cable bolts used in ground support system. The service life of the wire strands was measured under a range of accelerated SCC conditions using three- and four-point loading jigs. Fractographic analyses of the fracture surfaces of the failed wires displayed typical features indicating that hydrogen embrittlement was involved in SCC in these tests. It was shown that the presence of hydrogen sulphide is more critical than its concentration in SCC. The pH and level of applied stress were determined to have a direct impact on the occurrence of SCC. The deflection angle of the crack path was observed to be an inverse polynomial function of the applied stress level in the wire. This demonstrated the significance of the stress level on the fracture mechanism of cable bolts. Furthermore, the use of a protective galvanised coating on cable bolts was found to be a promising countermeasure against SCC. The results of this study provide detailed insight into the environmental factors involved in SCC of high-carbon steel wires and can be further used for setting guidelines for assessment of environments which cause susceptibility to SCC.

Comparison of different lactation curve models in Sahiwal cattle up to fourth parity using monthly test day milk yields.

In present study data of 6391 monthly test day milk yield (MTDMY) records of first lactation pertaining to 691, 4048 MTDMY records of second lactation pertaining to 454, 3013 MTDMY records of third lactation pertaining to 340 and 2280 MTDMY records of fourth lactation pertaining to 249 Sahiwal cows, spread over a period of 51 years (1961- 2011), maintained at NDRI, Karnal were used to compare different lactation curve models. The five different models viz. inverse polynomial function, Wilmink’s exponential function, gamma type function, quadratic cum log function and mixed log function were fitted on MTDMY by statistical package SAS version 9. The relative efficiency of the models was determined on the basis of the higher value of coefficient of determination (R 2 -value) and lesser value of root mean square error (RMSE). The R 2 values ranged from 72.96% (0.154 kg) for Wilmink’s function in first lactation to 99.98% (0.0.043 kg) for inverse polynomial function in third lactation with RMSE values in parentheses. The R 2 value for different lactation curve models showed that the inverse polynomial function gave best fit; whereas Wilmink’s function was found to be least fit in Sahiwal cattle. Hence it can be concluded that inverse polynomial function is the model of choice for estimating the trend of milk production in Sahiwal cattle.

Curva de lactação de fêmeas mestiças Taurino x Zebu

Objetivou-se comparar equações matemáticas descritas na literatura, identificar a que melhor descreve a produção de leite e estimar alguns componentes da curva de lactação de vacas de alta variabilidade genética, resultante de cruzamento taurino (Bos taurus) x zebuíno (Bos indicus), em sistema de produção semi-extensivo em Sergipe. Foram utilizados 1496 registros de produção de leite. Os controles leiteiros foram realizados em intervalos de 30 dias no período de 2011 a 2012. O ajuste foi feito para a curva média, fazendo-se o uso de quatro modelos matemáticos: modelo de Nelder, 1966 (ND), Wood, 1967 (WD), Bianchini Sobrinho, 1984 (BS) e Wilmink, 1987 (WK). Os critérios utilizados para verificar a qualidade do ajuste para cada função foram: coeficiente de determinação ajustado (Ra2), gráfico de distribuição dos resíduos e porcentagem de desvios entre a produção observada e as estimadas. Os valores médios de Ra2 foram maiores que 0,80 para os modelos empregados, exceto para o modelo WK. Todas as curvas apresentaram padrões típicos. As funções WD e WK subestimaram a produção de leite acumulada aos 305 dias e apresentaram maiores desvios em relação à produção observada. As demais funções apresentaram comportamento semelhante no gráfico de distribuição dos resíduos, no entanto, função polinomial inversa (ND) proporcionou melhor estimativa das variáveis que compõem a curva de lactação de vacas provenientes de cruzamentos de Bos taurus x Bos indicus.

Using mathematical models for lactation curve of milk and fat production in Iranian Holstein cows

An appropriate mathematical model is required to fit into lactation curves to study the environmental and biological variables that affect milk production. This study examined the suitability of 5 mathematical models (linear and nonlinear form of Wood, Wilmink, Guo and Swalve and inverse polynomial function of Nelder) for lactation curve traits (milk, fat yield and fat content) of 305-d milk yield of Iranian Holsteins. Models were compared based on R2, RMSE, and MSPE criteria by using NLIN and REG procedures in SAS. Using GLM procedure in SAS, factors of herd, calving year, calving season, season of production, age at calving, age at test and days in milk were significant on daily milk yield and the fat content during production period (P &lt; 0.01). The study revealed that the linear form of Wood model and Wilmink function were the best for describing milk, fat yield and fat content during 305-d of lactation.

Use of Discrete Laguerre Expansions for Noniterative Identification of Nonlinear Wiener Models

A noniterative least-squares method is presented to identify a discrete-time Wiener model, which consists of a linear dynamic element followed by a nonlinear static element. The identification algorithm represents the linear dynamics as a Laguerre FIR (finite impulse response) expansion with a time-scaling factor and approximates the static nonlinearity by an inverse polynomial function with a reference point. The incorporation of an adjustable reference point allows an arbitrary input design for the identification test and renders the algorithm robust with respect to noise. Furthermore, a proper selection of the time-scaling factor and the output reference value could facilitate accurate estimation of the linear and nonlinear elements, respectively. With the two arguments specified a priori, the regression equation becomes linear-in-parameters, involving the Laguerre and polynomial coefficients to be estimated. Two error criteria concerning the internal variable are then proposed to infer their proper values. When the process is highly nonlinear, a four-segment piecewise linear function is developed to replace the polynomial function for the approximation of static nonlinearity. Another error criterion concerning the output variable is employed to find the partition points for the four segments. Simulation study with four physical examples demonstrates that the proposed identification method is valid for a variety of test conditions and nonlinearities.

PCPs with small soundness error

The soundness error of a PCP verifier is the probability (usually denoted ε) that the verifier accepts an incorrect input. We are interested in the smallest possible values of ε for which the PCP theorem holds, and in particular whether the theorem holds when ε is an inverse polynomial function of the input length. We discuss the 'sliding scale conjecture' of [BGLR93, LY94] and related questions. We then sketch some of the existing approaches and constructions of PCPs with sub-constant soundness error.

Comparação de alguns modelos matemáticos para o ajuste às curvas de lactação individuais de vacas da raça Caracu

Foram ajustadas 7239 curvas de lactação de vacas Caracu, controladas semanalmente entre os anos de 1978 a 1988, pertencentes à Fazenda Chiqueirão, Poços de Caldas, MG. As funções utilizadas foram a linear hiperbólica (FLH), a quadrática logarítmica (FQL), a gama incompleta (FGI) e a polinomial inversa (FPI). Os parâmetros foram estimados por meio de regressões não lineares, usando-se processos iterativos. A verificação da qualidade do ajuste baseou-se no coeficiente de determinação ajustado (R²A), no teste de Durbin-Watson (DW) e nas médias e desvios-padrão estimados para os parâmetros e funções dos parâmetros dos modelos. Para a curva média, os R²A foram superiores a 0,90 para todas as funções. Bons ajustes, baseados nos R²A&gt;0,80 foram obtidos, respectivamente, por 25,2%, 39,1%, 31,1% e 28,4% das lactações ajustadas pelas funções FLH, FQL, FGI e FPI. De acordo com o teste de DW, bons ajustes foram proporcionados para 29,4% das lactações ajustadas pela FLH, 54,9% pela FQL, 34,9% pela FGI e 29,6% pela FPI. Para ambos os critérios, a FQL foi superior às demais funções, indicando grande variação nas formas das curvas de lactação geradas pelos ajustes individuais. Curvas atípicas foram estimadas pelas funções, com picos ocorrendo antes do parto e algumas vezes após o término da lactação. Todas as funções apresentaram problemas quando ajustaram dados individuais.

Comparação de alguns modelos matemáticos para ajuste à curva de lactação média de um rebanho da raça Caracu

No presente estudo foi estimada a curva de lactação média de 7239 lactações de vacas da raça Caracu, com base em 299.949 controles semanais. As funções ajustadas aos dados médios da curva de lactação foram Linear Hiperbólica e Quadrática Logarítmica, como lineares, e Gama Incompleta e Polinomial Inversa, como não-lineares. Os parâmetros foram estimados por meio de processos iterativos, método de Gauss-Newton. O Coeficiente de Determinação ajustado (R²A), o Teste de Durbin-Watson (DW) e os desvios (%) entre as produções totais (PTL) observadas e estimadas foram usados como medida da qualidade do ajuste. Os resíduos estimados, apesar de pequenos, apresentaram correlação serial positiva, pois o teste DW foi significativo para as quatro funções estudadas, indicando violação da pressuposição de independência entre estes resíduos. Os R²A foram superiores a 97% e os desvios entre as PTL observadas e estimadas, mínimos, indicando que, de acordo com estes critérios, qualquer função poderia ser usada para ajustar a curva de lactação média desse rebanho. Quanto aos componentes da curva, a melhor aproximação foi obtida pela função Polinomial Inversa, para produção inicial, tempo e produção no pico. As funções Gama Incompleta e Quadrática Logarítmica apresentaram ajustes semelhantes, porém a primeira apresentou melhor definição dos parâmetros e componentes da curva de lactação.

Selection of a Mathematical Model to Generate Lactation Curves Using Daily Milk Yields of Holstein Cows

Mathematical descriptions of early stages of lactation were investigated using daily milk yields of 117 first, 78second, 57 third, and 36 fourth lactations of 120 Holstein cows fitted by 10 models. The measure of fit was the error mean squares, which were replaced by ranks to perform an analysis of variance with lactation number, model, and period as factors and with cows as replicates. The interaction of model and lactation number was significant for the fit of the entire lactation. A significant interaction of model and period was obtained for the fit of three 30-d intervals. For the entire lactation, the best fit for all four lactations occurred from the diphasic logistic function, y = d1(1 – tanh2(b1(nk – c1))) + d2(1 – tanh2(b2(n – c2))). For the first 30 d, a modified gamma function gave the best fit for the first lactation, the inverse polynomial function for the second lactation, and the quadratic log function for the third lactation. The diphasic logistic function gave the best fit for the remaining two periods and was not significantly different from the best fitting models for the first 30-d period. Hence, this function may be useful to describe the lactation curve of Holstein cows for dairy herds in which the daily milk yield of individual cows is constantly monitored with a computer.

Nitrogen Fertilization of No‐Till Winter Wheat and Rye. II. Influence on Grain Protein

AbstractNo‐till seeding into standing stubble immediately after harvest of the previous crop (‘stubbling‐in’) has been employed to successfully produce winter wheat (Triticum aestivum L.) and rye (Secale cereale L.) on the western Canadian prairies. Soil N deficiencies, which result in low grain protein concentrations, are often associated with stubbled‐in winter cereals. In the present study, the effect of N fertilization on protein concentration and protein yield of stubbled‐in winter wheat and rye was evaluated in 40 field trials conducted over a wide range of soil types and environmental conditions in Saskatchewan, Canada. Grain protein concentrations exhibited sigmoidal responses to N fertilization with minimums of 98.8 and 93.0 and maximums that varied from 130 to 231 and 107 to 177 g protein kg−1 dry grain for wheat and rye, respectively. The Gompertz equation was employed to describe the relationship between protein concentration and total available N, providing an average reduction in sums of squares, due to the model, of 96 and 97% for wheat and rye, respectively. Grain protein yield response to N fertilizer was described by an inverse polynomial function that provided an average reduction in sums of squares, due to the model, of 99% for both wheat and rye. Although grain protein concentration was significantly higher for winter wheat in most trials, winter rye demonstrated a greater N use efficiency for grain production. Higher grain yields for rye compensated for lower protein concentrations, and at equal N rates rye produced as much or more protein per hectare than wheat in all comparable trials except two.

Multitrait Estimation of Genetic Parameters of Lactation Curves in Holstein Heifers

Weekly milk yields of 1022 Holstein heifers from 61 sires were used to derive coefficients of the lactation curves using modified gamma and inverse polynomial functions. The natural logarithm of a modified gamma function was ln(yn) = ln (a) + b In (n) + en + u sin (x) + v cos (x), where a, b, c, u, and v are coefficients to be estimated; n is the day of lactation; and x is the day of year. Estimates of a, b, and c were combined to define persistency [−(b + 1) ln c], week of peak yield (b/c), and peak yield [a(b/c)be−b]. The inverse polynomial function was n/yn = A0 + A1n + A2n2, where A0, A1, and A2 are coefficients to be estimated. Variance and covariance components for the coefficients of the lactation curve were estimated by the multitrait restricted maximum likelihood method using canonical transformation. Heritability estimates were In (a) .11, b .07, c .04 u .01, v .04, A0 .28, A1 .26, A2 .21, persistency .21, week of peak .18, peak yield .23, and 308-d milk yield .41. Genetic correlations indicated that selection for faster rate of increase to peak production would result in higher 308-d milk production, higher peak yield, and greater persistency.

COMPARISON OF TWO MATHEMATICAL MODELS IN FITTING LACTATION CURVES FOR PURELINE AND CROSSLINE DAIRY COWS

Weekly milk yield of 2066 first, 1407 second, and 755 third lactation pureline and crossline cows was used to study environmental and genetic effects on the coefficients of the lactation curves derived by modified gamma and inverse polynomial functions. The inverse polynomial function provided a better fit than the modified gamma function based on comparison of R2 values. Coefficients of the two models were analyzed to evaluate environmental and genetic effects on the shape of the lactation curve. The model included station, year of calving, month of calving, age at calving, days open, line of sire, line of dam, interaction of line of sire with line of dam, and sires within line of sire. In addition, coefficients of the lactation curve were analyzed by another model which included station, year of calving, month of calving, age at calving, days open, breed additive, maternal, and heterosis effects. Effects of station, year of calving, month of calving, and days open were mostly significant (P &lt; 0.05); however the effect of age at calving was not significant on the coefficients of the lactation curves. Significant (P &lt; 0.05) line of sire and line of dam effects on the level of initial yield suggest that genetic improvement of this trait could be achieved through selection. Breed additive and maternal effects were mostly nonsignificant on the coefficients of the lactation curves indicating shape of the lactation curve could not be changed by selection. There was very little evidence of nonadditive genetic variation associated with the coefficients of lactation curves. Key words: Lactation curves, pureline, crossline, dairy cows

Crop Response to Fertilizers

The response–function approach to the effects of fertilizers on crop yield is discussed. The problem is first considered in general terms and expressions are derived for the maximum yield, the economic yield and the utilization efficiencies or recovery factors. It is described how the use of a compound fertilizer modifies the method. The theory is then applied to the inverse polynomial function, which is used to describe the response to various levels of nitrogen, phosphorous and potassium fertilizers; the response equation is modified to allow for the depressive effects of high nitrogen levels on yield. Next, a numerical example is given in which the results are applied to summer cabbage, and the most profitable fertilizer treatments are calculated. Finally, the application of the model across different soils and different crops is briefly discussed.