What's in a set of points? (straight line fitting)

Mathematical modeling has seen many changes over the years. These changes range from the types of situations being modeled to the types of tools used for the modeling. An extremely powerful modeling tool for many situations is the vertex-edge graph (hereafter simply called a graph). In this type of graph, a set of points, called vertices (or nodes), represent objects, people, or other ideas. The nodes can be connected with (not necessarily straight) line segments, called edges (or arcs), to show a relationship between the nodes. Graphs are used to model everything from transportation networks to groups of friends.

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THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195913600091 ◽

2013 ◽

Vol 23 (04n05) ◽

pp. 357-395 ◽

Cited By ~ 1

Author(s):

THERESE BIEDL ◽

MARTIN VATSHELLE

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Line Drawing ◽

Np Hard ◽

Bounded Treewidth ◽

Straight Line ◽

Triangulated Graphs ◽

Point Set ◽

Fixed Embedding ◽

Set Of Points

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

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An Improved Straight-Line Fitting Method for Analyzing Pumping Test Recovery Data

Ground Water ◽

10.1111/j.1745-6584.2005.00094.x ◽

2005 ◽

Vol 43 (6) ◽

pp. 939-942 ◽

Cited By ~ 9

Author(s):

Li Zheng ◽

Jian-Qing Guo ◽

Yuping Lei

Keyword(s):

Pumping Test ◽

Straight Line ◽

Fitting Method ◽

Line Fitting

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Radioimmunoassay of canine growth hormone

Acta Endocrinologica ◽

10.1530/acta.0.0980514 ◽

1981 ◽

Vol 98 (4) ◽

pp. 514-520 ◽

Cited By ~ 53

Author(s):

J. E. Eigenmann ◽

R. Y. Eigenmann

Keyword(s):

Growth Hormone ◽

Rhesus Monkeys ◽

Low Dose ◽

Cross Reactivity ◽

Gel Chromatography ◽

Assay Sensitivity ◽

Standard Curve ◽

Straight Line ◽

Plasma Gh ◽

Line Fitting

Abstract. A sensitive radioimmunoassay (RIA) for canine growth hormone (GH) was developed. Antibodies were elicited in rhesus monkeys. One antiserum exhibited a working titer at a dilution of 1:500000. Radioiodination was performed enzymatically employing lactoperoxidase. Logit-log transformation and least squares fitting resulted in straight line fitting of the standard curve between 0.39 and 50 ng/ml. Formation of largemolecular [12SI]GH during storage caused diminished assay sensitivity. Therefore [125I]GH was re-purified by gel chromatography. Using this procedure, high and reproducible assay sensitivity was obtained. Tracer preparations were used for as long as 3 months after iodination. Diluted plasma from normal and acromegalic dogs resulted in a dose-response curve parallel to the standard curve. Canine prolactin exhibited a cross-reactivity of 2%. The within-assay coefficient of variation (CV) was 3.8 and the between-assay CV was 7.2%. Mean plasma GH concentration in normal dogs was 1.92 ± 0.14 ng/ml (mean ± sem). GH levels in acromegalic dogs were appreciably higher. Insulin-induced hypoglycaemia, arginine and ornithine administration resulted in inconsistent and sluggish GH increment. A better response was obtained by injecting a low dose of clonidine. Clonidine administration to hypopituitary dogs resulted in absent or poor GH increment.

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Linear regression and correlation

10.1093/oso/9780198785699.003.0005 ◽

2017 ◽

Author(s):

Andrew Gelman ◽

Deborah Nolan

Keyword(s):

Standard Deviation ◽

Linear Regression ◽

Regression Line ◽

World Population ◽

Regression To The Mean ◽

Straight Line ◽

The Mean ◽

Line Fitting ◽

Single Predictor ◽

Over Time

This chapter addresses the descriptive treatment of linear regression with a single predictor: straight-line fitting, interpretation of the regression line and standard deviation, the confusing phenomenon of “regression to the mean,” correlation, and conducting regressions on the computer. These concepts are illustrated with student discussions and activities. Many examples are of the sort commonly found in statistics textbooks, but the focus here is on how to work the examples into student-participation activities rather than simply examples to be read or shown on the blackboard. Topics include the following relationships: height and income, height and hand span, world population over time, and exam scores.

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Weighted Total Least Squares (WTLS) Solutions for Straight Line Fitting to 3D Point Data

Mathematics ◽

10.3390/math8091450 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1450

Author(s):

Georgios Malissiovas ◽

Frank Neitzel ◽

Sven Weisbrich ◽

Svetozar Petrovic

Keyword(s):

Least Squares ◽

Iterative Solution ◽

Total Least Squares ◽

Dispersion Matrix ◽

Random Errors ◽

Direct Solution ◽

Weighted Total Least Squares ◽

Straight Line ◽

Point Data ◽

Line Fitting

In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates xi, yi, zi as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is performed at any step. Using a numerical example it is demonstrated how the newly developed WTLS approaches can be applied for 3D straight line fitting considering different weighting cases. The solutions are compared with results from the literature and with those obtained from an iteratively linearized GH model.

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