Independent and dependent variables

There are two kinds of independent variables: (1) functional independent variables, and (2) physical independent variables. To avoid confusion an independent variable it is standard that a variable be of both kinds, since being of one kind does not imply being of the other kind.

A physical independent in an experiment remains the independent variable throughout the experiment. A function with a functionally independent variable that is also a physical independent variable remains a physical independent variable even if the function is changed into one with a different functional independent variable, as a non-standard case.

There are two ways of expressing an independent variable: (1) its value is fixed or controlled separately from measuring any dependent variable, or (2) its values are a pre-defined sequence of values within the experiment, but they may be imagined to continue indefinitely beyond the experiment. Once the independent variable is determined, then one or more dependent variables can be measured in relation to it.

Examples of the first way are specifying a time interval and then taking a measurement for the specified interval of time. One could also specify a distance, and then measure the elapsed time. It is important to note that if the distance is independent, it is absolute within the experiment, whereas time is relative.

The second way commonly makes time the independent variable, which is absolute within the experiment. Space in the form of distances (spaces) can also be the independent variable, which is called stance so that stance intervals are distances. In this case stance is absolute within the experiment, whereas time is relative.

If time is the independent variable, the universe of the experiment is spatio-temporal (dimensionally 3+1). If space (stance) is the independent variable, the universe of the experiment is temporo-spatial (dimensionally 1+3).

The independent variable is in the denominator of a rate. Otherwise, the rate must be inverted. For example, the spatio-temporal rate of motion is speed or velocity; the temporo-spatial rates are pace or lenticity. Addition of vector rates depends on what is common between them: either the denominator or numerator. If the denominators are in common, then regular vector addition is required. If the numerators are in common, they are added with reciprocal addition.

So in a temporo-spatial context one must use reciprocal addition for velocities since the distance is in common. Similarly, in a spatio-temporal context one must use reciprocal addition for lenticities since the time is in common. This is the reason that the harmonic mean is used to average velocities in a temporo-spatial context and lenticities in a spacio-temporal context.

If one maps the variables, then the independent variable should be the background map that the dependent variables are indicated on. For example, a map of the local geography forms the background for indicating the location of various dependent variables in the foreground. A temporo-spatial map has a time scale in the background with the chronation of various dependent event variables indicated on the foreground.