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An estimator for the slope with approximately median rank, having the same breakdown point as the Theil–Sen estimator, may be maintained in the data stream model (in which the sample points are processed one by one by an algorithm that does not have enough persistent storage to represent the entire data set) using an algorithm based on ε-nets.
In other words, F is proportional to the logarithm of x times the slope of the straight line of its lin–log graph, plus a constant. Specifically, a straight line on a lin–log plot containing points ( F 0 , x 0 ) and ( F 1 , x 1 ) will have the function:
Tobler's hiking function – walking speed vs. slope angle chart. Tobler's hiking function is an exponential function determining the hiking speed, taking into account the slope angle. [1] [2] [3] It was formulated by Waldo Tobler. This function was estimated from empirical data of Eduard Imhof. [4]
This is the slope of the line joining the two group means. The p-value that the slope of 4 is different from 0 is p = 0.00805. The coefficients for the linear regression specify the slope and intercept of the line that joins the two group means, as illustrated in the graph. The intercept is 2 and the slope is 4.
An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models (in particular, linear regression ), although they can also extend to non-linear models.
However, those formulas do not tell us how precise the estimates are, i.e., how much the estimators ^ and ^ vary from sample to sample for the specified sample size. Confidence intervals were devised to give a plausible set of values to the estimates one might have if one repeated the experiment a very large number of times.
The above procedure now is reversed to find the form of the function F(x) using its (assumed) known log–log plot. To find the function F, pick some fixed point (x 0, F 0), where F 0 is shorthand for F(x 0), somewhere on the straight line in the above graph, and further some other arbitrary point (x 1, F 1) on the same graph.
In the middle, the fitted straight line represents the best balance between the points above and below this line. The dotted straight lines represent the two extreme lines, considering only the variation in the slope. The inner curves represent the estimated range of values considering the variation in both slope and intercept.