Help with finding an equation for a set of data

[harlayhar]

I need some help finding the equation to a graph. The graph data is pictured below.

The graph is for a piece of fishing equipment that is used to get your fishing line out and away from your boat and down to your desired depth.

Here is the data that we have using actual tests. It's not exact which I why I want to find an equation for this;

[PLAIN]http://www.slidediver.com/img/dptchart1.jpg

[PLAIN]http://www.slidediver.com/img/dptchart2.jpg

Here is an explanation of the data:

The depth (in green) is the target depth that we wish to achieve.

Ring: Each data set was done with a ring attached that increases the surface area of the diver. There are two sizes that we call #2 and #3

Speed: This is how fast the boat is moving. We did sample data for 2.8 MPH and 2.3 MPH. I am hoping that we can compute data for speeds down to 1 MPH.

12" lead is a constant so ignore that.

Angle Settings (blue): This is a dial that sets the angle of the dive. This can be set from 0-6 in increments of 1/2. It makes the diver move away from the boat.

Length (red): This is how much line needs to be sent out to achieve the desired depth.

In actual use here is how it should work:

I am trolling at 2.3 MPH, I have a #2 ring on my diver and I'm setting my angle at 3. I see fish on my graph at 25 feet down. How much line do I need to send out to get down to 25 feet? (In this case it would be 92ft)

Here is a picture of the diver:

 

[Stephen Tashi]

You did a good job of explaining your problem, but I think you would get the best advice if you posted that data as some sort of spreadsheet file or at least as a text file. I think there are many forum members who are very talented with working spreadsheet data. (I'm not one of them!)

From a statistical point of view, it would be interesting to know if your data is an average of several measurements and, if so, how much variability is present in the raw data.

 

[harlayhar]

Thanks Stephen, I'll work on putting the data in a spreadsheet.

 

[harlayhar]

Spreadsheet attached.

 

[hotvette]

The approach you take depends on what you want to do. I assume the task is interpolation (i.e. find a value between two other values). If that's true, a number of approaches could be satisfactory. Any book on numerical methods should discuss a variety of interpolation schemes.

If you really are looking for an equation of the form y = f(x) for each condition, then polynomials are probably your best bet since the data look reasonably linear. A 6th order polynomial will fit the 7 data points (of each condition) exactly but the result probably won't be very satisfactory since the curve will wiggle too much at the extremes. A least squares fitting using a 3rd or 4th order polynomial could give adequate results.

If you are looking for a single equation to model ALL conditions where the input variables are (1) desired depth, (2) ring # 0-3, and (3) trolling speed, that is a much, much more difficult problem.

Even more difficult would be extrapolation where you want to find values outside the data range provided. The most rigorous approach would be to use the basic physics of the problem (fluid dynamics, drag, etc.), which isn't trivial.

 

[harlayhar]

Thanks for the reply. The data would never fall outside of those values that we have because at those values the product begins to lose effectiveness.

What I'm really trying to do is find is a formula that at least gets you close to were you need to be. After all, if you are seeing your targets at a depth of 20', in reality they will be anywhere between 18-22 ft.

 

[hotvette]

Attached is an example of fitting a 5th order polynomial ( i.e. y = ax⁵ + bx⁴ + cx³ + dx² + ex + f ) through the 7 data points associated with no ring, setting = 0, speed = 2.0 knots. The table lists the error produced by the function at each data point. Given the shape of the curve you might get equally satisfactory results by just connecting straight line segments from point to point and forgetting about any kind of curve fitting.

 

[harlayhar]

I'm trying to put different numbers in there and I'm just not getting it. I haven't worked with any thing this complex for nearly 30 years. I guess it's true that if you don't use it you lose it. Three strokes haven't helped much either.

 

[hotvette]

I'm trying to put different numbers in there

Put different numbers in where? If you can better describe where you are having trouble perhaps we can help explain.

 

[harlayhar]

Actually after looking at it again I don't think that this will get me to where I am trying to go. If I can solve that for the value of x, I don't think that it will work for different values of speed and diver setting because I can't see how different values for those variables play into that equation?

 

[Stephen Tashi]

It's possible to fit functions of several variables to data. We do need to be clear about what the variables are. As I understand the problem, the variables and their ranges include:

Number of rings: 0,2,3 ( would you ever use 1 ?)
Trolling speed: 2 to 2.5 knots
Setting: 0,1,2,3,4,5,6

I not sure whether the following vary:
lead length: 12 inches
driver number: Is there a driver number besides #1 ?

I gather what we want to compute from these inputs is depth.

When you say you want to use different numbers, how many of the above things do you want to change?

Are we also trying compute "line out in feet" and "reel level winds" from the inputs or are these also inputs?

 

[harlayhar]

Yes, the three things that vary are speed, diver angle (0-6), and ring (0,2,3).

Given any set of those three variables I want to be able to "tell" the function how many feet down I want to put the diver and have it give me back how many feet of line need to be put out to achieve that depth.

Thanks for any help an this.

 

[Stephen Tashi]

You didn't explain what numbers you will need to change. For example, are you trying to condense the book by Holt and Romanack into one equation so it can be put on a calculator?

 

[harlayhar]

I wasn't aware of that book. That is way more in depth that I want to go. I just want to be able to put people "close" to where they need to be with this one product.

So we just assume that things like lead length, bait weight/drag, line type, etc are not part of the equation for now.

I'm only changing four variables;

S - speed in mph from say 1.5 to 2.8 mph
A - the angle of the dive from 1-6
R - ring size 0,2,3
D - target depth in feet

Then given those four factors I want to be able to compute how many feet of line need to be let out to get down to the target depth.

All of the data on the current charts was gathered from actual field testing. Now I want be be able to find out if I can get the length of line that needs to be let out to say get down to a depth of 25' using angle setting of 3 with a #2 ring at 1.6 mph.

Is there enough information in the current charts to be able to compute that?

 

[Stephen Tashi]

Now I want be be able to find out if I can get the length of line that needs to be let out to say get down to a depth of 25' using angle setting of 3 with a #2 ring at 1.6 mph.

Is there enough information in the current charts to be able to compute that?

The answer to that depends on how we fit the equation.

We can talk about two different ways to fit equations to data. In one way, you pick an general formula for an equation that has some unknown constants in it. You pick the formula just because you like it for some reason. Then you try to find the values of the constants that make it come close to your data. There is no reason that this type of equation will work for values that aren't part of your data. In fact, people are advised not have faith in "extrapolating", which means to apply the equation to values larger or smaller than were measured. For example, the equation might not work for the speed 1.6 since you have no data for speeds that small. Of course, if you intend to do more testing then you can guess at an equation and change it later if it doesn't fit the new data.

A second way of fitting an equation to data is to use a physical model with unknown parameters. For example, as I visualize the situation (I'm not a fisherman!) there are certain known physical properties about the situation ( such as the density, elasticity and length of line) and certain unknown ones such as the exact weight of what will be on the end of the line, it's drag coefficient in water etc. You find the correct equation using physics. It has the unknown values in it represented by letters. You solve for these unknowns by determing what values best fit the equation to the data. An equation fit to the data in this manner is more likely to work for values outside the range that you have measured since it is based on a physical theory.

To do things the second way, a physicist would have to understand exactly the exact geometry of the problem and what the tension adjustment and rings do. (I don't understand this yet and I don't know if I'm enough of a physicist to figure out the equation. What is your deadline for getting this work done?)

 

[harlayhar]

We are pulling the boat out of the water in October, so I would like to be able to go out and test this before that happens. Given that, extrapolating the data ONLY for speed is what I would want to try, all the other variables would stay within the ranges that we have already done.

A successful test would be something like this;

The equation tells us that to get down 20ft with no ring, angle of 2, moving at 1.5 mph we need to put out 28ft of line. We then set that up in 30ft of water and begin to move into 20ft of water. If in 25ft of water we hit the bottom then the equation did not work. However if anywhere between 22ft & 18ft of water we hit bottom, then it works.

Does that make sense?

This should also give you and idea of why I am trying to find an equation for this. To manually find all these values from 1.0 mph to 2.7 mph would take us;

16 speeds * 6 angles * 3 ring settings * 8 depths = 2,304 trials

and I haven't even begun to change the weight yet!

 

[Stephen Tashi]

I thing you should at least incorporate the extremes of the values in your tests. That would involve 2 speeds * 2 angles * 2 ring settings * 2 depths.

(There is actually a field of math that studies how one should do an experiment when there are too many combinations of values of variables to test them all. This field of study is called "The Design Of Experiments". I don't know much about it. Any expert on the subject is welcome to jump in.)

Do you have a diagram that shows the entire picture of how your apparatus is used - one that shows everything from the boat to the end of the line in the water? I don't understand how the "ring" and "angle" settings affect the geometry of the problem.

 

[harlayhar]

Unfortunately I don't have such a picture, but I'll explain how those things work the best that I can.

Rings: With the addition of either the #2 or #3 ring the surface area of the diver is increased. As you can see in chart #1, to achieve a depth of 30ft you need to release less line with the increase in surface area (ring #2 & #3). It basically increases the performance of the diver, the more surface area it has the easier it is to control it.

Angle: The main purpose of adjusting the weight position (i.e. angle) is to allow to cover more water area. Say for instance you put three lines off the right side of your boat without an adjustable angle. You would end up dragging three divers directly behind your boat resulting in a tangled mess. An angle setting of 6R will tilt the diver in the water sending it out about 90 degrees off the right side. So three divers at different angle settings might look something like this;

http://www.u-charters.com/img/no_angle.jpg

http://www.u-charters.com/img/3_angles.jpg

Hope this helps.

 

[Stephen Tashi]

Those diagrams were helpful. Let me make sure that I understand a very elementary point. Other products on the web that I see which are "divers" for trolling have a streamlined look and I think they are mean to be pulled under the water. Is the appartus you are dealing with like this?

If so, let me ask you about a possible simplification of the problem. Suppose that the diver is moving along under the water at a known speed and depth and we have a given length of line attached between it and the lure or bait. Would that be enough to determine the depth at which the lure is riding? ( For example, I can see that if the wake of the boat somehow affected things, we would need to know how far the diver was from the boat. But do things like that really matter if you know the depth and speed of the diver?)

The above thought suggests that the problem could be subdivided into two parts. Part 1: Get data on how the variables ( speed of diver, depth of diver, length of line from diver to lure) determine the depth at which the lure rides. Part 2) Get data on how (speed of boad, ring setting, angle setting) affect the depth at which the diver rides.

I don't know how hard it would be for you to measure the depth at which the diver is traveling. (Also I don't know if the variable "length of line" in your data refers to the line from the boat to the diver or the line from the diver to the lure.

 

[vector22]

HI
I'll look at it but if I think I can solve your prob then I want to get paid for it thanks

I dont' think a polynomial is the way to go here.

 

[harlayhar]

Those diagrams were helpful. Let me make sure that I understand a very elementary point. Other products on the web that I see which are "divers" for trolling have a streamlined look and I think they are mean to be pulled under the water. Is the appartus you are dealing with like this?

Yes, that's what this is.

If so, let me ask you about a possible simplification of the problem. Suppose that the diver is moving along under the water at a known speed and depth and we have a given length of line attached between it and the lure or bait. Would that be enough to determine the depth at which the lure is riding? ( For example, I can see that if the wake of the boat somehow affected things, we would need to know how far the diver was from the boat. But do things like that really matter if you know the depth and speed of the diver?)

The only thing that really matters is how deep the diver is, whatever the diver is pulling behind it will be close enough to the same depth as the diver.

The above thought suggests that the problem could be subdivided into two parts. Part 1: Get data on how the variables ( speed of diver, depth of diver, length of line from diver to lure) determine the depth at which the lure rides. Part 2) Get data on how (speed of boad, ring setting, angle setting) affect the depth at which the diver rides.

Part two is all that matters.

 

[Stephen Tashi]

When the diver is being towed by a line, does the line stay roughly straight as it stretches between the top of the water and the diver or does it have a noticeable curve to it? I don't know how you are measuring depth in your tests. If the line was straight you could compute depth from the line length and the "angle of depression" that the line makes with the surface of the water.

 

[harlayhar]

I'm going have to say that it would be mostly straight, not having ever been under the water to look at it. If it's being pulled forward at a constant rate it seems that it would be roughly a straight line from the diver to the point of tow.

 

[Stephen Tashi]

Assuming the line is roughly straight, I have some thoughts about how you could measure depth in your tests. (Of course, maybe you already do it this way.)

If you measure the angle $\theta$ in the attached diagram and know the distance H and line length L, you can compute the depth D by trigonometry.


$$L \sin{\theta} = H + D$$

This formula can be solved for D given the other variables.

If you can measure angle $\alpha$ with a "clinometer" then you can compute $\theta = 90$ deg $- \alpha$.

From watching web videos it appears that the trolling line is usually held on the top of a rod that is in the water or nearly in the water. To measure the angles you would have to suspend the line from a greater height H. I think data from this configuration would still be relevant to a line suspended near the water level as far as predicting depth from speed, ring, angle and line length. You'd just have to subtract the length of line that is above the water if the goal is to have "line length" mean line below the water.

I don't know whether this is a practical method. Do you think it would be?


-----

On an unrelated subject, I notice some of the cells in the spreadsheet are defined as averages of other cells, so the spreadsheet data contains some interpolations as well as test data. A person fitting a curve would want to omit the interpolated data.

 

[harlayhar]

I don't know how this would help because in order to get the correct angle I would already have to have the diver at the desired depth. Isn't this just adding one more unknown to the equation?

 

[Stephen Tashi]

I don't know how this would help because in order to get the correct angle I would already have to have the diver at the desired depth. Isn't this just adding one more unknown to the equation?

I realize that the chart you want to give to fisherman should have a nice orderly series of depths like 15, 20, 25 etc and give the required line length for them. But for a person fitting curve to data, it doesn't matter if the depths are nice and orderly. They could be slightly scattered like 10.3, 14.8, 21.7 etc. So, from the viewpoint of curve fitting and doing tests fast, my thought is that you could collect depth data in an "untidy" fashion. For example, you could set a line length of 40 ft. Then you could run the boat at various speeds (1.0, 1.5, 2.0, 2.5, etc.), rings and diver angle settings. You could measure the angle of the line in each of these conditions. The you could set the line at 45 ft and vary the things again. Back on shore, you could calculate the depth numbers. This would give data was orderly in terms of line lengths, speeds, rings, diver angles but not orderly in terms of depths. However you could use the unevenly spaced depth data to interpolate what happens at an orderly series of depths like 10,15,20... (After all, your chart already had interpolation in it.)

Another testing method: I don't know if you can let out more line while to boat is moving. If you can then you could program a calculator to compute depth as a function of speed and angle of the line. If you wanted to run at 2.0 knots and depth 15 ft, you could make a guess at the line length. Measure the angle to see if you got 15 ft. If you were to shallow, you could let out more line and measure the angle again, compute the depth again. You could get the depth to be 15 by a little trial and error. (And if you record the data as you work your way to the correct depth, that would be useful to someone fitting a curve to it.

You haven't said how you measure or set the depth, so I can't visualize how time consuming your current testing method is. I'm just doing some armchair speculation that the methods above might be quicker.

 

[Stephen Tashi]

An edit to the previous post. I meant to say "you could program a calculator to compute depth as a function of the length of the line and angle of the line" instead of "you could program a calculator to compute depth as a function of speed and angle of the line".

 

[vector22]

I think that your invention has limited use. With all the depth sounding equipment available why not make a remote control diver that maintains a set depth regardless of the towing speed. It would be like an underwater sub that you tow. That would eliminate all the headache of figureing how much line to let out.