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I've been really rusty on math. So excuse me if I sound dumb. I am trying to find equation for the data below:

x   y
67  0.055
68  
69  
70  
71  
72  
73  
74  
75  
76  
77  
78  
79  
80  
81  
82  
83  
84  
85  0.0275
86  
87  
88  
89  
90  
91  
92  
93  
94  
95  
96  
97  
98  
99  
100 
101 
102 
103 0.01375

What would be the approach to solve this? I am trying to find the values for y.

EDIT: the x values are temp in °F and the y value @ 85 is calculated. The top and bottom values for y are double and half the value of y @ 85.

Grendizer
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  • There is no entirely universal answer, you need some kind of model. But what you are doing is interpolation of three points. One way to do it is polynomial interpolation, that can be looked up. Here you would use a quadratic. Another way would be least squares with a different model equation, like a linear equation or exponential equation. – Ian Jun 07 '16 at 15:44
  • Can you be more specific please? What kind of info do i need to derive an equation for the data? – Grendizer Jun 07 '16 at 15:46
  • It depends on where the data came from. If I had no context whatsoever and was just told "get me an interpolant of this", the first thing I would try would be polynomial interpolation. For just three data points this will probably behave pretty well. – Ian Jun 07 '16 at 15:46
  • OK, I will look into it. – Grendizer Jun 07 '16 at 15:47
  • Btw, the x values are temp in F and the y value @ 85 is calculated. The top and bottom values for y are double and half the value of y @ 85. – Grendizer Jun 07 '16 at 15:49
  • Are they double and half the value at 85 for a reason? Should that reason persist in between? If so, then exponential interpolation seems appropriate. Here the exponential interpolant would be $0.0275 \cdot 2^{-(x-85)/18}$. – Ian Jun 07 '16 at 16:00
  • Yes for both questions, Ok, i'll look into that – Grendizer Jun 07 '16 at 16:02

1 Answers1

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If there is a systematic reason that the $y$ value is cutting in half every $18$ degrees, then this relationship is exponential. (You can also check out the wikipedia article on half-life.) In this case you can model the relationship by the equation

$$y = 0.0275 \times 0.5^{(x-85)/18}$$

(Alternatively, $y = 0.055 \times 0.5^{(x-67)/18}$ or $y = 0.01375 \times 0.5^{(x-103)/18}$; they are all equivalent.)

The logic is that, since your $y$ is multiplying by $0.5$ every time $x$ increases by $18$ degrees, it must be multiplying by $0.5^{1/18}$ every time $x$ increases by a single degree, so that after $18$ degrees it has multiplied by $(0.5^{1/18})^{18} = 0.5$.

In my first equation, I picked one of the values you gave, $0.0275$, which was the value at $x=85$; then if $x$ is more than $85$, $x-85$ is the number of degrees beyond $85$, so $0.0275$ will have to be multiplied by $0.5^{1/18}$ $x-85$ times, i.e. multiplied by $(0.5^{1/18})^{x-85} = 0.5^{(x-85)/18}$. This is where the formula came from. Similar logic works if $x$ is less than $85$. And the other two equations I gave come from applying the same logic to the other two known points.

Ben Blum-Smith
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