Pardon me if I am mistaken, but my trigonometry (taken 34 years ago) tells me  that x is hypotenuse multiplied by costheta,not plain costheta.The axis of abscissa here is measured in terms of angle, theta.

I believe that for you to make the conversion, you need to determine, based on your given coordinates (your adjacent and opposite sides), the value of your theta and the hypotenuse.and make the succeeding transformations.

First thing I wonder if this is repeated measures data. I think some sharks are observed, so there should be traces for (a few?) individual sharks. This should be considered by a possible model.

For the model, one plus three things seem to be relevant:

1) the trace is approximately sinusoidal and circadian.

1.1) the average depth (obviousely different between "their" data and "your" data)

1.2) the amplitude, possibly in relation to the depth

1.3) the phase

as another aspect one might question the frequency, if this does not need to be a cirardian thing.

So one may set up a (non-linear, repeated-measures) model fitting the depth, the (relative) amplitude and the phase depending on the data source ("theirs" versus "yours"). Then one can investigate where the differences are, for instance if there is a phase-shift and how large it is etc etc.

After trying to look at it again today I'm starting to think I need to use circular regression instead? It is a repeated measures design and the mean average depth for each minute observed will be on the y axis. Will using circular regression result in a straight line graph? I'm hoping it's something I can create using excel or minitab as these are available. If not  I can try and find different software :)

You may find these of value

http://www.teacherschoice.com.au/maths_library/coordinates/polar_-_rectangular_conversion.htm

http://www.mathsisfun.com/polar-cartesian-coordinates.html

http://www.analyzemath.com/polarcoordinates/polar_rectangular.html

http://www.mathwords.com/p/polar_rectangular_conversion_formulas.htm

Zoe, what is theta?

Thanks again for the feed back!

I think I've worked out what I needed to do and why the journal article used cos2theta. I could be wrong as I've never been taught this but....

As theta isn't used as a set value and often just represents a parameter, I'm guessing the theta is just showing thats what the cosine represents or X (not 100% which).

The secondary data looks at day and night vertical use and the pattern is similar to a cosine wave repeated hence cos2theta used to transform time to then do a linear regression graph.

For the data I've collected it only looks at 9 hours, therefore I've ended up using a shorter wave- cos0.3theta... and it seems to have worked!

The only thing I've been wondering about is whether or not I should use 0.375 or 0.3. 

0.375 is 9/24 hours which works but the curve doesn't fit as well... or I can use 0.3 where the curve fits better and results in a higher r value. Am I looking for a curve which fits well with the data to then transform time or should I use 0.375?

I'd really appreciate any opinions as to if it sounds right what I've done ?

(Or if I've completely missed the point?) 

Zoe, the transform is simply θ = 2∙π∙x/24. This transformation obviously makes the depth data linear so that linear regression can be applied to it.

I found the article and digitized Figure 6A. I did linear regression on the transformed digitized data. I got results very close to the ones presented in the figure caption.

You can find a Python script and an Excel work book in the attachments.

Can anyone explain why such data is forced into a linear shape when there are straight-forward applicable tools to analyze wave data directly?

Jochen, what do you mean by "wave data"?

It's certainly easier to fit a straight line to data (linear problem) than to fit a sine/cosine function (non-linear problem).

The paper, by the way, is provided below.

http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0060331

With "wave data" I meant data that can be fitted by some sine function, sinusodial data.

It is 2015. We have computers. Any cheap PC from today can do more flops than the entire NASA computer centre could do 1969 during the Apollo missions. So I think that "ease of calculation" is not the problem. There are 2 or 3 clicks in a software sufficient to fit a linear or a non-linear model, and robustness is rarely an issue (and if so, then one could still think of using other tools).

The main point should be a well-framed scientific question. Comparing slopes of straight-line models of some heavily transformed data seems to be an unneccessary complication. It becomes difficult to judge the relevance of the findings and conclusions. I don't say that this won't be possible - but it clearly is very indirect and very complicated, unneccessarily!

If one is interested if there are changes or correlations in amplitude, frequency, or phase, then there are tools available that directly address these questions.

Jochen, it's not the speed of calculation but the complexity of the problem. If you want to fit a cosine model to data, the model would be something like

y(x) = A∙cos(B∙x+C)+D.

That is, four parameters instead of two for the linear regression. This is a much more difficult problem to solve. For example, the choice of initial guess becomes important when solving the problem with non-linear least squares. See the Python script.

I do not agree that cosine model is complex.

Your staright-line model makes two implicit assumptions that are only made explicit by the cosine model: the frequency and the phase.

You use a frequency of 2/day implicitely. This could also be done with the cosine model substituting B by an appropriate constant.

You use a phase of 0, what accidentally fits to the data. This could again be done with the cosine model by omitting the parameter C.

I attached a modified version of your xls-sheet to show that the straigh-line model would have to be "adjusted" for data that is (accidently) different! Set C to 0 to get the correct solution for the "straight-line model". Try different parameters (cosine corves) to see that the transformation works only for specific values of B and C.

I also added the least-squares solution. And to get these values was NOT more difficult than performing the transformation and do a linear regression.

@Jochen

First of all, it is not "my" straight line model. Secondly, all the parameters (A, B, C and D) are optimized by the Python script. It produces the following result:

Second attempt: [ 5.07987235 0.44548775 1.16078858 38.12494774]

which is equal to the solution in your Excel work book.

I think you miss the point. The fact that one can fit a cosine model does not make it, in itself, the right choice. I have not read the paper, but the authors might actually have some reason for doing the analysis the way they chose to do.

@Zoe

θ is most likely hour transformed into an angle in radians (0:00 = 0 rad, 12:00 = π rad, 24:00 = 2∙π rad). I have not read the paper but you should really understand why the authors chose to use 2∙θ. It may not be meaningful to use 0.3 or 0.375. It probably makes the statistics different.

If the paper does not provide the reasoning for this (it definitely should), you can always contact the authors and ask them directly.

Thanks Kimmo,

It makes a lot more sense now! I'll probably email the authors then as I couldn't find why they'd used it in their paper...which is partly why I was struggling. 

Really appreciate all the comments 😊 

Kimmo,

I know it's not "your model", I only needed to give it a short name that we both know what I was referring to.

You said you think I miss the point.

Well, that may be, but then I need some help to get it.

The fact that one can fit a cosine model does not make it, in itself, the right choice.

Agreed.

I have not read the paper, but the authors might actually have some reason for doing the analysis the way they chose to do.

I neither read the paper. But I see that the straight line model estimated two parameters (slope and intercept) that are simply the parameters A (=amplitude) and D (=average depth) of the cosine model. The other two parameters required for the cosine model are also required for the straight line model, only being taken implicitely. So what the authors get from this analysis is nothing else but A and D, requiring to implicitely choose "good" values for B and C. So I do not understand what else the straight-line model can provide what the cosine model can not.

I suspect that the straight line model is a work-around, being in fact suboptimal. Even if *only* A and D are of interest, it seems circumstatial that this is estimated as a slose and an intercept when is can be directly fitted as average depth (D) and amplitude (A).

Can you give me a possible scientific advantage of the "straight-line analysis"? This might be helpful for me to get the point.

My point is simply: why make things complicated? Especially, since Zoe wanted to make a comparison to results already published. So, she should do a similar analysis.