Estimation of the Form of the Best Fitting Model

1965 ◽  
Vol 19 (1) ◽  
pp. 64-69
Author(s):  
J. Vlcek

Fitting of linear models occupies a large portion of the adjustments of surveying data. In this paper, an attempt is made to present the main ideas of the statistical theory connected with the estimation of linear models, based on the method of least squares. A method of solution of normal equations is given, which lends itself to a simple geometric interpretation and has particular statistical advantages. Also, some possibilities for testing the adequacy of the fitted model are discussed.

2021 ◽  
Vol 99 (Supplement_1) ◽  
pp. 58-59
Author(s):  
Larissa L Becker ◽  
Emily E Scholtz ◽  
Joel M DeRouchey ◽  
Mike D Tokach ◽  
Jason C Woodworth ◽  
...  

Abstract A total of 2,124 barrows and gilts (PIC 1050′DNA 600, initially 48.9 kg) were used in a 32-d study to determine the optimal dietary standardized ileal digestibility (SID) Lys level in a commercial setting. Pigs were randomly allotted to 1 of 5 dietary treatments with 24 to 27 pigs/pen and 16 replications/treatment. Similar number of barrows and gilts were placed in each pen. Diets were fed over 3 phases (48.9 to 58.6, 58.6 to 70.9, and 70.9 to 80.8 kg respectively). Dietary treatments were corn-soybean meal-based and contained 10 (phase 1 and 2) or 5% (phase 3) distillers dried grains with solubles. Diets were formulated to 85, 95, 103, 110, or 120% of the current Pig Improvement Company (PIC, Hendersonville, TN) SID Lys gilt recommendations with phase 1 SID Lys levels of 0.90, 1.01, 1.09, 1.17 and 1.27%, phase 2 levels of 0.79, 0.87, 0.94, 1.03, and 1.10%, and phase 3 levels of 0.71, 0.78, 0.85, 0.92, and 0.99%, respectively. Dose response curves were evaluated using linear (LM), quadratic polynomial (QP), broken-line linear (BLL), and broken-line quadratic (BLQ) models. For each response variable, the best-fitting model was selected using the Bayesian information criterion. Overall (d 0 to 32), increasing SID Lys increased (linear, P< 0.001) BW, ADG, G:F, Lys intake/d, and Lys intake/kg of gain. Modeling margin over feed cost (MOFC), BLL and QP estimated the requirement at 105.8% and 113.7% respectively. In summary, while growth increased linearly up to 120% of the PIC current feeding level, the optimal MOFC was 106% to 114% depending on the model used.


2020 ◽  
Vol 23 (6) ◽  
pp. 330-337
Author(s):  
Olatz Mompeo ◽  
Rachel Gibson ◽  
Paraskevi Christofidou ◽  
Tim D. Spector ◽  
Cristina Menni ◽  
...  

AbstractA healthy diet is associated with the improvement or maintenance of health parameters, and several indices have been proposed to assess diet quality comprehensively. Twin studies have found that some specific foods, nutrients and food patterns have a heritable component; however, the heritability of overall dietary intake has not yet been estimated. Here, we compute heritability estimates of the nine most common dietary indices utilized in nutritional epidemiology. We analyzed 2590 female twins from TwinsUK (653 monozygotic [MZ] and 642 dizygotic [DZ] pairs) who completed a 131-item food frequency questionnaire (FFQ). Heritability estimates were computed using structural equation models (SEM) adjusting for body mass index (BMI), smoking status, Index of Multiple Deprivation (IMD), physical activity, menopausal status, energy and alcohol intake. The AE model was the best-fitting model for most of the analyzed dietary scores (seven out of nine), with heritability estimates ranging from 10.1% (95% CI [.02, .18]) for the Dietary Reference Values (DRV) to 42.7% (95% CI [.36, .49]) for the Alternative Healthy Eating Index (A-HEI). The ACE model was the best-fitting model for the Healthy Diet Indicator (HDI) and Healthy Eating Index 2010 (HEI-2010) with heritability estimates of 5.4% (95% CI [−.17, .28]) and 25.4% (95% CI [.05, .46]), respectively. Here, we find that all analyzed dietary indices have a heritable component, suggesting that there is a genetic predisposition regulating what you eat. Future studies should explore genes underlying dietary indices to further understand the genetic disposition toward diet-related health parameters.


1999 ◽  
Vol 26 (1) ◽  
pp. 177-185 ◽  
Author(s):  
BYRON F. ROBINSON ◽  
CAROLYN B. MERVIS

Expressive vocabulary data gathered during a systematic diary study of one male child's early language development are compared to data that would have resulted from longitudinal administration of the MacArthur Communicative Development Inventories spoken vocabulary checklist (CDI). Comparisons are made for (1) the number of words at monthly intervals (9;10.15 to 2;0.15), (2) proportion of words by lexical class (i.e. noun, predicate, closed class, ‘other’), (3) growth curves. The CDI underestimates the number of words in the diary study, with the underestimation increasing as vocabulary size increases. The proportion of diary study words appearing on the CDI differed as a function of lexical class. Finally, despite the differences in vocabulary size, logistic curves proved to be the best fitting model to characterize vocabulary development as measured by both the diary study and the CDI. Implications for the longitudinal use of the CDI are discussed.


2016 ◽  
Vol 91 (1-2) ◽  
pp. 161-176
Author(s):  
Maral Kichian

The natural rate of interest is an unobservable entity and its measurement presents some important empirical challenges. In this paper, we use identification-robust methods and central bank real-time staff projections to obtain estimates for the equilibrium real rate from contemporaneous and forward-looking Taylor-type interest rate rules. The methods notably account for the potential presence of endogeneity, under-identification, and errors-in-variables concerns. Our applications are conducted on Canadian data. The results reveal some important identification difficulties associated with some of our models, reinforcing the need to use identification-robust methods to estimate such policy functions. Despite these challenges, we are able to obtain fairly comparable point estimates for the real equilibrium interest rate across our different models, and in the case of the best fitting model, also remarkable estimate precision.


1977 ◽  
Vol 99 (2) ◽  
pp. 345-352 ◽  
Author(s):  
A. T. Chatas

The purpose of this paper is to indicate a method for estimating values of specified aquifer parameters from an investigation of the reservoir performance of an associated oilfield. To achieve this objective an analysis was made of the simultaneous solution of the material-balance and diffusivity equations, followed by an application of the method of least squares. Three analytical functions evolved, which in dimensionless form were numerically evaluated by computer and tabulated herein. Application of the proffered method requires the simultaneous solution of the three normal equations developed in the paper.


2021 ◽  
Author(s):  
Assel Akimbekova ◽  
Paolo Mancinelli ◽  
Massimiliano Rinaldo Barchi ◽  
Cristina Pauselli ◽  
Giorgio Minelli

<p>Abstract</p><p> </p><p>In the present study, starting from original measurement stations, we created the Bouguer anomaly map of Southern Italy with a reduction density of 2670 kg m<sup>-3</sup>. We perform a regional gravity modelling at crustal scale along the trace of the CROP-04 (on-shore) and MB6 (off-shore) deep seismic reflection profiles crossing the Southern Apennines and the Southern Tyrrhenian Sea. Along the 320 km-long modelled profile, we investigate crustal-scale sources for the observed gravity anomalies. </p><p>After a compelling review of the published Moho geometries in the area, that were retrieved from either active or passive seismic methods, we test them in the observed gravity field through forward modeling of the Bouguer gravity anomalies. The comparison between the different Moho interpretations shows that the steepness of the subducting slab, the position of the step between the western (Tyrrhenian) and the eastern (Adriatic) Moho and Moho depth represent the main features influencing the observed Bouguer anomalies at crustal scale.</p><p>Finally, we provide a best-fitting model across both onshore and offshore areas. In the proposed best-fitting model, the wide wavelength and strong regional Bouguer anomalies correlate with the geometry of the Moho discontinuity and deep tectonic structures. On the other hand, the small-amplitude oscillations of the gravity anomalies were attributed to the low-density values of the Pliocene-Quaternary deposits both on- (e.g. the Bradanic trough) and off-shore (e.g. recent deposits in the Tyrrhenian sea bottom). Gravity minima correspond to the crustal doubling underneath the Southern Apennines where the Tyrrhenian Moho (~27 km depth) overlies the deeper Adriatic Moho (~50 km depth). The positive trend of the observed anomaly toward NE is related to the shallowing of the Adriatic Moho to depths of ~28 km in the Adriatic. Similarly, towards SW, the observed anomaly follows a positive trend towards the maxima located in the Central Tyrrhenian Sea. We model this trend as representative of crustal thinning and shallowing to values of ~12 km depth of the Tyrrhenian Moho. We also model a crustal transition from geometries and density values typical of a continental crust in the Adriatic domain towards a more oceanic structure and composition in the Tyrrhenian domain. This crustal model locates the westward flexure of the Adriatic Moho, mimicking the subduction of the Adriatic lithosphere beneath the Peri-Tyrrhenian block and locates step between the western (Tyrrhenian) and the eastern (Adriatic) Moho beneath the Apennines range.</p><p>The resulted gravity forward model provide contributions to the tectonic settings understanding of the area by providing a robust crustal model ranging from the Tyrrhenian Sea to the Apulian foreland.</p><p> Finally, we believe that the proposed model can serve as a starting point for future studies investigating the upper crustal geometries in the area and addressing open questions about its relations with seismicity distribution.</p><p> </p>


2019 ◽  
Vol 97 (10) ◽  
pp. 4032-4040
Author(s):  
Carine M Vier ◽  
Steve S Dritz ◽  
Fangzhou Wu ◽  
Mike D Tokach ◽  
Joel M DeRouchey ◽  
...  

Abstract Two experiments were conducted to determine the standardized total tract digestible phosphorus (STTD P) requirement for 11- to 23-kg nursery pigs fed diets with or without phytase. A total of 1,080 and 2,140 pigs (PIC 359 × Camborough, Hendersonville, TN; initially 11.4 ± 0.29 and 11.1 ± 0.24 kg) were used in Exp. 1 and Exp. 2, respectively. There were 23 to 27 pigs per pen with 6 and 12 replicate pens per treatment in Exp. 1 and Exp. 2, respectively. After weaning, pigs were fed a common pelleted diet with 0.45% STTD P for 7 d, and a common phase 2 meal diet with 0.40% STTD P for 14 d in Exp. 1 and 18 d in Exp. 2. Pens of pigs were then allotted to dietary treatments in a randomized complete block design with body weight (BW) as the blocking factor. In Exp. 1, dietary treatments consisted of 0.26%, 0.30%, 0.33%, 0.38%, 0.43%, 0.48%, and 0.53% STTD P. Treatments were achieved with the inclusion of monocalcium phosphate at the expense of corn. In Exp. 2, diets contained 1,000 phytase units (FYT; Ronozyme Hiphos 2500, DSM Nutritional Products, Inc., Parsippany, NJ) with assumed release value 0.132% STTD P, and treatments consisted of 0.30%, 0.33%, 0.38%, 0.43%, 0.48%, 0.53%, and 0.58% STTD P. These STTD P concentrations included the expected phytase release of 0.132% STTD P. In both experiments, a similar 1.17:1 Ca:P ratio was maintained across treatments. Statistical models included linear model (LM), quadratic polynomial (QP), broken-line linear (BLL), and broken-line quadratic (BLQ). In Exp. 1, increasing STTD P increased (linear, P < 0.001) ADG, ADFI, G:F, final BW, and grams of STTD P intake per day and per kilogram of gain. There was also a marginal quadratic response for G:F (P < 0.066). In Exp. 2, ADG and G:F increased quadratically (P < 0.05), whereas ADFI increased linearly (P = 0.060) with increasing STTD P. The BLL and QP model provided similar fit to G:F in Exp. 1, estimating the requirement for maximum G:F at 0.34% and 0.42%, respectively. The BLL was the best fitting model for ADG and G:F in Exp. 2, estimating the breakpoint at 0.40% and 0.37% STTD P, respectively. The BLL and BLQ models estimated the breakpoint for ADG as a function of STTD P intake in grams per day at 2.92 and 3.02 g/d, respectively. These data provide empirical evidence that for 11- to 23-kg pigs, the NRC (2012) accurately estimates the STTD P requirement on a g/d basis. As a percentage of the diet, the STTD P requirement for diets without or with 1,000 FYT added phytase ranged from 0.34% to 0.42%.


1997 ◽  
Vol 64 (1) ◽  
pp. 63-69 ◽  
Author(s):  
A. Amici ◽  
S. Bartocci ◽  
S. Terramoccia ◽  
F. Martillotti

AbstractFive mathematical models were compared to select the most satisfactory model to describe digesta kinetics of solids and fluids in the gastrointestinal tract of buffaloes (Mediterranean bulls), cattle (Friesian bulls) and sheep (Delle Langhe rams) given food at maintenance level, according to a Latin-square arrangement for four consecutive periods of 21 days. Chromium mordanted alfalfa hay and cobalt-ethylenediamine tetraacetic acid were used as nonabsorbable markers and were administered through the rumen cannula in a single dose. Four different isonitrogenous diets (N × 6·25 = 140 g/kg dry matter) with different concentrate:forage ratios (12·5:87·5, 25:75, 37·5:62·5, 50:50) were used.Faecal chromium and cobalt concentration curves were fitted with five non-linear models: three gamma (G2, G3, G4) age-dependent one-compartment, one gamma age-dependent/age-independent two-compartment (G2G1) and one multicompartment (MC).Wilcoxon tests on residual sums of squares of the different models for solids showed that MC and G4 gave a better fit than G2G1, G2, G3 for all the data and within the species. The comparison of MC v. G4 did not show any significant difference (P > 0·05) for all the data computed together or within each species. Nevertheless, MC had a higher number of curves with lower residual sums of squares in comparison with G4 and was also able to produce estimates of digesta kinetics in the second compartment.The cobalt excretion curves for fluids, considering all the data, and only within sheep, showed G4 as the best fitting model. When G4 was compared with other models no significant differences were recorded either for cattle: G4 v. G2 (F = 0·6645), G4 v. G2G1 (P = 0·0620) and for buffalo: G4 v. G2 (P = 0·1575), G4 v.G3(P = 0·0796) and G4 v. G2G1 (P = 0·1641).It is concluded that the multicompartment model (MC) and G4 model were the best fits for solids and for fluids respectively.


Geophysics ◽  
1984 ◽  
Vol 49 (12) ◽  
pp. 2143-2158 ◽  
Author(s):  
Robert L. Parker

The electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two‐layer systems. This new representation is called the bilayer expansion for the Green’s function. It enables the forward problem of resistivity sounding to be solved very efficiently, even for complicated profiles. Also, the bilayer expansion facilitates the solution of the resistivity inverse problem: the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model. Specifically, I consider models comprising uniformly conducting layers of equal thickness; for a given finite data set a quadratic program can be used to find the best‐fitting model in this class for any specified thickness. As the thickness is reduced, models of this kind can approximate arbitrary profiles with unlimited accuracy. If there is a model that satisfies the data well, there are other models equally good or better whose variation takes place in an infinitesimally thin zone near the surface, below which there is a perfectly conducting region. This extraordinary class of solutions underscores the serious ambiguity in the interpretation of apparent resistivity data. It is evident that strong constraints from outside the electrical data set must be applied if reliable solutions are to be discovered. Previous work seems to have given a somewhat overly optimistic impression of the resolving abilities of this kind of data. I consider briefly a regularization technique designed to maximize the smoothness of models found with the bilayer inversion.


A surface of the second degree is constructed from the five components of the second rank tensor which describes the alinement of a spin-assembly undergoing magnetic resonance. The functions which characterize alinement are given a simple geometric interpretation in terms of radii vectores of this ellipsoid. The time-dependence of the different resonance functions at frequencies 0, ω and 2 ω is easily understood in terms of the rotation of the ellipsoid. In the absence of an r.f. field, and with pumping and relaxation processes only, the ellipsoid is uniaxial with its axes in the direction of the static field ( Z axis). With a weak r.f. field the shape of the ellipsoid is unchanged, but it is tilted and precesses round the Z axis at the frequency of the driving field. With stronger r.f. fields the shape of the ellipsoid changes, but at resonance one of the principal axes is always in the direction of the r.f. field and the length of this axis is independent of the field strength. At resonance also, the tilt increases to a limiting value of 1/4π with increasing r.f. field strength and the lengths of the axes in the plane perpendicular to the r.f. field tend to equality.


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