Calibration

Calibration is the process of finding a relationship between two quantities that are unknown (when the measurable quantities are not given a particular value for the amount considered or found a standard for the quantity). When one of quantity is known, which is made or set with one device, another measurement is made as similar way as possible with the first device using a second device.The measurable quantities may differ in two devices which are equivalent. The device with the known or assigned correctness is called the standard. The second device is the unit under test, test instrument, or any of several other names for the device being calibrated.

The formal definition of calibration by the International Bureau of Weights and Measures is the following: "Operation that, under specified conditions, in a first step, establishes a relation between the quantity values with measurement uncertainties provided by measurement standards and corresponding indications with associated measurement uncertainties (of the calibrated instrument or secondary standard) and, in a second step, uses this information to establish a relation for obtaining a measurement result from an indication."

Calibrated probability assessment

Calibrated probability assessments are subjective probabilities assigned by individuals who have been trained to assess probabilities in a way that historically represents their uncertainty. In other words, when a calibrated person says they are "80% confident" in each of 100 predictions they made, they will get about 80% of them correct. Likewise, they will be right 90% of the time they say they are 90% certain, and so on.

Calibration training improves subjective probabilities because most people are either "overconfident" or "under-confident" (usually the former). By practicing with a series of trivia questions, it is possible for subjects to fine-tune their ability to assess probabilities. For example, a subject may be asked:

If a person has no idea whatsoever, they will say they are only 50% confident. If they are absolutely certain they are correct, they will say 100%. But most people will answer somewhere in between. If a calibrated person is asked a large number of such questions, they will get about as many correct as they expected. An uncalibrated person who is systematically overconfident may say they are 90% confident in a large number of questions where they only get 70% of them correct. On the other hand, an uncalibrated person who is systematically underconfident may say they are 50% confident in a large number of questions where they actually get 70% of them correct.

Calibrated geometry

In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibration in the sense that

Set G_x(φ) = { ξ as above : φ|_ξ = vol_ξ }. (In order for the theory to be nontrivial, we need G_x(φ) to be nonempty.) Let G(φ) be the union of G_x(φ) for x in M.

The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier (in 1966) Edmond Bonan introduced G₂-manifold and Spin(7)-manifold, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifold were simultaneously studied in 1965 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form.

Calibrated submanifolds

A p-dimensional submanifold Σ of M is said to be a calibrated submanifold with respect to φ (or simply φ-calibrated) if TΣ lies in G(φ).