Exploratory Calibration of a Retail Location Model Using Search by Golden Section

1971 ◽  
Vol 3 (4) ◽  
pp. 411-432 ◽  
Author(s):  
M Batty
Keyword(s):  
Goodness Of Fit ◽  
Spatial Interaction ◽  
Golden Section ◽  
Fibonacci Numbers ◽  
Location Model ◽  
Fundamental Result ◽  
Calibration Methods ◽  
Retail Location ◽  
Two Parameters ◽  
Parameter Values

This paper presents some experiments in calibrating a retail location model designed for the Kristiansand region in southern Norway. The form and behaviour of the model under extremes of its parameter values is first investigated, and then some calibration methods proposed by Hyman are tested. The need for a more general method is evident and a search procedure based on the Fibonacci numbers is outlined. A generalisation of this method, based on the golden section number, is derived and this method is used to explore the sensitivity of the model's goodness-of-fit to changes in parameter values. A fundamental result of these explorations shows that there is no unique fit for retail models with two parameters, and this result is likely to hold for other models of spatial interaction.

The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction

1972 ◽  
Vol 4 (2) ◽  
pp. 205-233 ◽  
Author(s):  
M Batty ◽  
S Mackie
Keyword(s):  
Maximum Likelihood ◽  
Spatial Interaction ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Calibration Methods ◽  
The Family ◽  
Newton Raphson ◽  
Best Parameter ◽  
Parameter Values ◽  
Raphson Method

This paper presents a methodology for deriving best statistics for the calibration of spatial interaction models, and several procedures for finding best parameter values are described. The family of spatial interaction models due to Wilson is first outlined, and then some existing calibration methods are briefly reviewed. A procedure for deriving best statistics based on the principle of maximum-likelihood is then developed from the work of Hyman and Evans, and the methodology is illustrated using the example of a retail gravity model. Five methods for solving the maximum-likelihood equations are outlined: procedures based on a simple first-order iterative process, the Newton—Raphson method for several variables, multivariate Fibonacci search, search using the Simplex method, and search based on quadratic convergence, are all tested and compared. It appears that the Newton—Raphson method is the most efficient, and this is further tested in the calibration of disaggregated residential location models.

scholarly journals On Insolvability of the 4-th Hilbert Problem for Hyperbolic Geometries

2019 ◽  
pp. 1-21
Author(s):  
Alexey Stakhov ◽  
Samuil Aranson
Keyword(s):  
Diophantine Equations ◽  
Hilbert Problem ◽  
Golden Section ◽  
Fibonacci Numbers ◽  
Fundamental Result ◽  
Minkowski Geometry ◽  
Russian Mathematician ◽  
Mathematical Result ◽  
Non Euclidean Geometry ◽  
Elliptic Geometry

The article proves the insolvability of the 4-th Hilbert Problem for hyperbolic geometries. It has been hypothesized that this fundamental mathematical result (the insolvability of the 4-th Hilbert Problem) holds for other types of non-Euclidean geometry (geometry of Riemann (elliptic geometry), non-Archimedean geometry, and Minkowski geometry). The ancient Golden Section, described in Euclid’s Elements (Proposition II.11) and the following from it Mathematics of Harmony, as a new direction in geometry, are the main mathematical apparatus for this fundamental result. By the way, this solution is reminiscent of the insolvability of the 10-th Hilbert Problem for Diophantine equations in integers. This outstanding mathematical result was obtained by the talented Russian mathematician Yuri Matiyasevich in 1970, by using Fibonacci numbers, introduced in 1202 by the famous Italian mathematician Leonardo from Pisa (by the nickname Fibonacci), and the new theorems in Fibonacci numbers theory, proved by the outstanding Russian mathematician Nikolay Vorobyev and described by him in the third edition of his book “Fibonacci numbers”.

Prediction of Haemoglobin Level by Some Probability Distribution

2020 ◽  
Vol 9 (1) ◽  
pp. 84-88
Author(s):  
Govinda Prasad Dhungana ◽  
Laxmi Prasad Sapkota
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Goodness Of Fit ◽  
Continuous Variable ◽  
Hemoglobin Level ◽  
Chi Square ◽  
Chi Square Test ◽  
Log Normal ◽  
Two Parameters ◽  
Best Fit

 Hemoglobin level is a continuous variable. So, it follows some theoretical probability distribution Normal, Log-normal, Gamma and Weibull distribution having two parameters. There is low variation in observed and expected frequency of Normal distribution in bar diagram. Similarly, calculated value of chi-square test (goodness of fit) is observed which is lower in Normal distribution. Furthermore, plot of PDFof Normal distribution covers larger area of histogram than all of other distribution. Hence Normal distribution is the best fit to predict the hemoglobin level in future.

Superconductors with spin–orbit interactions

2016 ◽  
Vol 30 (25) ◽  
pp. 1650183 ◽  
Author(s):  
Yu. N. Ovchinnikov
Keyword(s):  
Magnetic Field ◽  
Superconducting Films ◽  
Critical Magnetic Field ◽  
Spin Orbit ◽  
Zero Temperature ◽  
Interband Pairing ◽  
Two Parameters ◽  
Spin Orbit Interactions ◽  
Parameter Values ◽  
Thin Superconducting Films

The effect of spin-orbit (SO) interaction on the formation of the critical states in thin superconducting films in magnetic field oriented along the film is investigated. Hereby, the case of interband pairing is considered. It was found that eight branches exist in the plane of two parameters [Formula: see text] determined by the value of magnetic field and SO interaction. Six modes leads to inhomogeneous states with different values of the impulse [Formula: see text]. Each state is doubly degenerate over direction of impulse [Formula: see text]. The parameter values at critical point are found for all eight branches in explicit form for zero temperature. The optimal two branches are estimated, corresponding to largest critical magnetic field value for given SO interaction.

A Retail Location Model with Overlapping Market Areas: Hotelling's Problem Revisited

Urban Studies ◽  
1977 ◽  
Vol 14 (2) ◽  
pp. 203-205 ◽  
Author(s):  
H.C.W.L. Williams ◽  
M.L. Senior
Keyword(s):  
Location Model ◽  
Retail Location ◽  
Market Areas

scholarly journals A Cumulative Rainfall Function for Subhourly Design Storm in Mediterranean Urban Areas

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Marco Carbone ◽  
Michele Turco ◽  
Giuseppe Brunetti ◽  
Patrizia Piro
Keyword(s):  
Urban Areas ◽  
Goodness Of Fit ◽  
Nonlinear Least Squares ◽  
Least Square ◽  
Rainfall Time Series ◽  
Storm Events ◽  
Design Storm ◽  
Double Exponential ◽  
Two Parameters ◽  
Precipitation Records

Design storms are very useful in many hydrological and hydraulic practices and are obtained from statistical analysis of precipitation records. However considering design storms, which are often quite unlike the natural rainstorms, may result in designing oversized or undersized drainage facilities. For these reasons, in this study, a two-parameter double exponential function is proposed to parameterize historical storm events. The proposed function has been assessed against the storms selected from 5-year rainfall time series with a 1-minute resolution, measured by three meteorological stations located in Calabria, Italy. In particular, a nonlinear least square optimization has been used to identify parameters. In previous studies, several evaluation methods to measure the goodness of fit have been used with excellent performances. One parameter is related to the centroid of the rain distribution; the second one is related to high values of the standard deviation of the kurtosis for the selected events. Finally, considering the similarity between the proposed function and the Gumbel function, the two parameters have been computed with the method of moments; in this case, the correlation values were lower than those computed with nonlinear least squares optimization but sufficiently accurate for designing purposes.

scholarly journals Evidence That Supertriangles Exist in Nature from the Vertical Projections of Koelreuteria paniculata Fruit

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 23
Author(s):  
Yuping Li ◽  
Brady K. Quinn ◽  
Johan Gielis ◽  
Yirong Li ◽  
Peijian Shi
Keyword(s):  
Root Mean Square ◽  
Goodness Of Fit ◽  
Mean Square ◽  
Sea Stars ◽  
Different Types ◽  
Koelreuteria Paniculata ◽  
Two Parameters ◽  
Mean Square Errors

Many natural radial symmetrical shapes (e.g., sea stars) follow the Gielis equation (GE) or its twin equation (TGE). A supertriangle (three triangles arranged around a central polygon) represents such a shape, but no study has tested whether natural shapes can be represented as/are supertriangles or whether the GE or TGE can describe their shape. We collected 100 pieces of Koelreuteria paniculata fruit, which have a supertriangular shape, extracted the boundary coordinates for their vertical projections, and then fitted them with the GE and TGE. The adjusted root mean square errors (RMSEadj) of the two equations were always less than 0.08, and >70% were less than 0.05. For 57/100 fruit projections, the GE had a lower RMSEadj than the TGE, although overall differences in the goodness of fit were non-significant. However, the TGE produces more symmetrical shapes than the GE as the two parameters controlling the extent of symmetry in it are approximately equal. This work demonstrates that natural supertriangles exist, validates the use of the GE and TGE to model their shapes, and suggests that different complex radially symmetrical shapes can be generated by the same equation, implying that different types of biological symmetry may result from the same biophysical mechanisms.

Research on Flow Capturing Location of Commercial Complex Based on Double Design Variable Constraint

2014 ◽  
Vol 1065-1069 ◽  
pp. 2525-2529
Author(s):  
Bo Yuan ◽  
Zhi Xia Zhang ◽  
Jian Zheng
Keyword(s):  
Construction Projects ◽  
Heuristic Algorithms ◽  
Spatial Interaction ◽  
Interaction Model ◽  
Commercial Real Estate ◽  
Location Model ◽  
Location Selection ◽  
Design Variables ◽  
Development Costs ◽  
Average Running Time

Due to the complicated development process, larger operation difficulty, higher development costs of the commercial real estate, the location selection problem is critical to project success or failure. Based on the location variables and design variables as decision variables, the design variables are expanded as double factors variables, combining the spatial interaction model combined with the greedy algorithm, competitive environment of commercial complex intercepting location model is built. Combined with a practical example, the validity of the model for commercial complex location is verified, the average running time is compared between the heuristic algorithms and the enumeration method, the results show that: commercial complex intercepting location model can quickly and effectively locate the commercial facilities, and provide a theoretical basis for locating commercial complex construction projects.

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