For instance, when discussing the relationship between "distance from sea level" to "temperature": you can't get a to height of 1 million miles (because the limit of the relationship is within atmosphere), nor can you get a temperature of -500F if you just keep going further and further up. And yet these two are strongly negatively related (increase in height, decrease in temperature, within limits). http://www.engineeringtoolbox.com/air-altitude-temperature-d_461.html there for a chart
So what I can't wrap my head around, is where you are coming from when you say that a causation is not a superset of a correlation, or that a correlation is not a subset of a causation (unless it's a purely definition based objection; saying that the terms subset and superset come from math, and therefore are inappropriate terms to use here). You are saying that causation does not have intervals. Is it my imagination or are you assuming that you could get values that don't exist in the real world?
Cause this part, I don't get. Why are you limiting it to just one value? If x=-3 then -2(-3)+1=7. That's another point along the graph. However, real life correlations are not graphs first that then get put into real form; they are points of real world observational or experimentally collected data that get plotted on a graph. So you *first* start with plot points of: "temperature of 59F" and "at Sea Level", "51.9" at "2000 feet above sea level", "44.7" at "4000 feet", 37.6 at "6000 feet", etc. You then graph those points, and then you ... what's the correct term here... approximating an average? Is that the right term? It's been a while. Where you draw a line that averages out the plot points at the various intervals along the x and y axis (which, there's a couple of ways to do it, depending on what you want to illustrate). Ie, the lines in the middle of the scattergram here http://www.simplypsychology.org/correlation.html
So, we have plot points, a scattergram, a graph. Now, how do we establish causation from there? Do we remove plot points? Do we kill the graph? Do we alter the slope? What do we lose when we go from correlation to causation, where both are in the pragmatic sense of having collected real world data?
The interval is (-∞, ∞). I already told you this. This means that it includes from a range from negative infinity to positive infinity without including negativity infinity or positive infinity. If we look at point and we are given a y, the x will go to the other coordinate. Secondly that psychology link is kind of wrong. A continuous function is created for a scatter plot via a regression where you use a least squares method to come up with it and estimate how approximate it is if it is not an exact fit. The problem is you are assuming that continuous functions lead to extrapolations not supported by real world data in such a way you end up with speculation or conjecture; however, you can create a regression from a scatter plot in such a way as to interpolate within the range of your data in conjunction with using a least squares method to say what R^2 is - how much you were off. In other words, for a discrete function, you can derive a continuous one and fill in data within your range and it be as accurate as the the description of the discrete point. You get values that don't exists in the real world if you extrapolate, but interpolation constrains you to a range where you pretty much predict what something should be from within. How do you alter a slope if you don't have one? All you have is a dot! You have to first come up with a regression which gives you the slope and solves your problem. There is nothing you are adding or removing. The only thing you'll lose is accuracy, really, but there is nothing in that which denotes causation. You're asking how do you get something that is never notated in the scatter of data you have to begin with. There is no way you can mathematically derive causation, for you can only interpret something to be a function of something else.
In the case of there existing a limit in say height, you don't use a linear function; you can use a quadratic function with a negative second degree coefficient, or you can frame it as linear inequality with a conjunction. For example, if you throw a ball up, it won't keep going up and up; rather, it will go up and then come down as a parabola, or you put y in between two values in such a way to make it bounded - you can also do this with an x, too. We can shift this to 1<-2x+1<3, for example. Quadratic equations always give you an absolute value. Absolute values are plus or minus, so they give you a maximum and a minimum. Pretty much, when you take the square root of both sides, you get an absolute value in that -1^2 and 1^2 both give you 1, so it can be either one of those. Things increase until they hit the vertex and then they decrease. I've just been sticking to linear functions for simplicity.
I've just been sticking to linear equations because they are simplistic, but not all correlations are linear. For example, if you were looking at exponential population growth in terms of a correlation, you would use a power function. Part of coming up with a good regression is figuring out what function likely the one you want to use. You can also use a piecewise function where you end up with some finite sub-domains even though you have an infinite one in a direction(when you take a big function and break it down into smaller ones you end up with sub-functions where there are sub-domains which are finite from which you can derive a finite cardinality):
Pretty much, your restricted boundary determines which parts of the function you apply. This results in a function that behaves differently based on your x value. For example, your x may satisfy 2 out of three of the functions in the piecewise where a different x may satisfy all three parts. Not only does your restricted boundary make it not go on forever, the pieces of the function are specifically applied versus the whole function being applied at once. If you look at that graph, you will see one has a part that has an infinite part where there is a segment of that which is actually bounded. Since you seem to be talking about the real world all of the time, you pretty much can use this in the real world in terms of circuits with switches, or any other disturbed system. For example, when you close a loop that is being used to charge something in such a way as to let it discharge, your switch is a piecewise defined function. You can also use it when discussing friction and entropy, too, but I honestly do not feel like going into it that in depth(again I'm in an engineering program where I have to learn about switches, circuits, and all the neat little things and functions that go to them).
Algorithmically, you can write something like so:
We can write the parameter as an identity function of 0<=x<=10 which gives us [0,10] as our interval. The iteration is just to increase a counter variable, though. it follows the same function. num++ is really num=num+1 which is incrementing it by 1, so it is really f(x)=x in that it is 1/1x which is an increasing slope of 1. This means that the loop will run at 0 and up until and including 10 and then stop. x is the value that is returned which gives you an identity which will be between 0 and 10. An identity function is f(x)=x so f(1)=1, for example. In y intercept form y=mx+b where m is your slope, x the input, and b the intercept, so, for this, you get an implied 1 for the slope, so the slope, or rate of change, is 1. In other words, if you graph it, it will keep increasing by one where the inequality makes it start at 0, and include it, and stop at 10, and include it.
You can see where it stops. f(x)=x is really 1/1x where y=1 and x=1, so for every 1 unit of y, you get 1 unit of x. You get a function that increments by 1 over x and y which starts at 0 and ends at 10. Your conditional statement that tells the loop to terminate(which you would have in an if switch statement) is set up via an inequality.
The link you posted is inaccurate. It is called a regression
. What determines if something is increasing or decreasing is your slope. The strength of the correlation to the function is taken from the R^2
value where the first difference of the inputs and outputs tell you if it is linear or not. You would use a least squares method to calculate a regression and get your R^2 which is how much it is off. Also, experiments, generally, do not establish cause and effect. There are plenty of studies that show there is a correlation between things; however, they can't tell you what is causing it. You typically have to encapsulate that into some sort of theory where you then deduce a causal thing. They show one thing being a function of another; however, they don't show, in themselves, what causes it. You're misinterpreting being a function of something as being the effect of something where that is really just your codomain.
You get that in Biology, a lot. Take a look at these videos. They are more accurate. How to calculate linear regression using least square method How to Calculate R Squared Using Regression Analysis
You don't have a science degree, per your own admission, so you can't get a teacher's certification along with your other licenses to be a science teacher because you have no science degree.
OH, and since we're on the topic of our degrees and authority derived from those degrees: You have a Bachelors in Biology, not mathematics nor statistics. If you really wanted to push the idea that I have to accept what you say at face value, in the discipline of your degree, then that would be in topics of Biology, not mathematics nor statistics.
I already told you I am finishing up an Engineering degree that is mathematically rigorous. In other words, my academic fields cover Biology and Mathematics and Information Technology and Computer Science and Engineering.
In roughly a year and a half, I'll have four degrees along with certifications and licenses in a wide range of things covering all of those disciplines. And, guess what? I'm getting more, so in three and a half years, I'll have 5 degrees. I have this weird complex where I just develop tangent skills so as to be a very well-rounded person. I already have two. I also said I could teach, I did not say I did, but I did not say I did not, which means for all you know, I could have already gotten what was required or maybe not. One thingthat is in my program is Discrete Mathematics where you pretty much have to take some course in mathematical modeling if you are getting a science degree of any type. I attend one of the most rigorous schools around here,in terms of mathematics and engineering, though. By the way, your entire argument is grounded in neurobiology and biochemistry which happens to fall under one of my disciplines. You know; Biology. I'm just not addressing everything wrong you said about that, because I've done that already in other threads.
You'll still end up with what I told you, though. You'll just end up with an interval that tells you about the segment of the function. That is your set. I kind of find it absurd that I have to provide literally pages of mathematical proof for something simple that everyone is pretty much taught all because you don't want to believe what I've said because I'm me and you don't trust me. It is like asking for a drawn out proof of a sequence just because you distrust a person so much you won't believe them when they say 1+1=2.