find the values of polynomial to use polyfit

By using the census population data, Show that the relationship results between t and the log of the population by setting . To find and , you use the code of polyfit,

where this data is fitted in the form of

So, let's define population as p

Since ,

And log of p is

load census
t = (cdate - 1790)/10;
lnp = @(c) log(c(2)) + c(3)*t;
polyfit( lnp, t)

I write this code to find c(2) and c(3). But this code does not work. I get error.

Answer 1

[Cris LaPierre]

With polyfit, you supply it the X and Y data and it returns the equation for you. Well, the coefficients of the polynomial of the order you specify. The error message you are getting is becaue you are trying to pass in an anonymous function for X.

[Zeynep Toprak]

yes, I take a look at this website. But I cannot find a way to correct error. How can I correct according to given question? Please help me. If you want, I can re-state what I want in that question. Many thanks and I will be happy if you help me.

[Zeynep Toprak]

by the way, thanks for editting. I dont know how to write here properly.

[Cris LaPierre]

I've fixed your LaTex expressions.

Using polyfit with 2 coefficients (polyfit(x,y,1)) will fit the equation of a straight line (y=m*x+b) to your data. See if you can make your equation for log(p) look like m*x+b.

Then figure out what x and y for your equation are, and put those into polyfit. What polyfit will return are the values of the coefficients (m and b, or whatever variable those map to in your equation).

[Cris LaPierre]

It's based on the live editor, if that helps.

[Zeynep Toprak]

Thanks for your great explanation. Here what are my variables x and y? y is lnp and x is t. right? But in this case again I face with different size error.

[Cris LaPierre]

The problem is lnp is an anonymous function. You don't need it. Instead, make it the y vector of numbers you need using the data you loaded from census.

[Zeynep Toprak]

please only help me to create y. I cannot do it really.

[Cris LaPierre]

Sure you can. Your instructions and derivation tell you exactly how to do it.

Maybe approach this from a different angle. You are solving for the coefficients and . That means you don't know them. So what do you know? Well, at this stage, the only information you have is the information you loaded from census. What variables did you load? Where do they fit into your solution?

[Zeynep Toprak]

Hmm.. for that, I create a matrix to fit them into the solution like A = [ ones(size(t)) t ].

In fact, my function is like that f = @(t) [ ones(size(t)) t ] * c where c = [ d c_3]' with d = log(c2) and from here, I can find it by c_3 = exp(d).

Is it right? If yes, how can I continue ? by the way, many thanks for your great teaching and supports:)

[Zeynep Toprak]

I get this :(

[Cris LaPierre]

Stop trying to use an anonymous function. That is not how polyfit works.

What variables do you get when you run the load census command?

Also, I would recommend treating c1,c2 and c3 as separate variables instead of the vector c you are trying to use.

[Zeynep Toprak]

I get cdate and pop when I load census. And I directly put them into the polyfit, I get this result;

Are these three values c=[c1, c2, c3] ? Right?

[Cris LaPierre]

Closer, but not yet.

First, use 1 instead of 2 (this number represents the highest polynomial power. In m*x+b, it's x^1). You should be getting two coefficients out, m and b. Read the doc to figure out which is m and which is b.

Second, look at your derivations. You did something to the original equation to make it so that you could do a linear fit to the data. Right now, you are not fitting the transformed data, so the estimates of c2 and c3 are wrong.

[Zeynep Toprak]

you explain really in a good way. But I don't understand and know how to apply for them. I am not computer student. I just try to follow some lectures on the youtube and try to do some questions by myself. Please can you show their applications.

[Zeynep Toprak]

Okay, I use 1. And I get the coefficients.

[Zeynep Toprak]

b = 1.0e+03* (-2.2120), which is error estimate. And,

ln m = 1.0e+03* 0.0012. Right?

[Zeynep Toprak]

But I dont understand again what I should do in second?

[Cris LaPierre]

Closer. Now go back to the derivation you shared in the beginning. Reread it carefully. Look at the final equation. Compare what is shown there to what you are passing into polyfit. See any discrepancies?

It might help to write the equation using the variables to spot the differences.

[Cris LaPierre]

Yes, your m and b are correct for the data you used.

However, they are not yet c2 and c3.

[Zeynep Toprak]

I dont use population variable directly, since I take its log, I should use log. Now, is it okay?

[Cris LaPierre]

There you go. Yes, you need to fit to the natural log of pop (In MATLAB, log(pop)).

Now look at that final equation again (log(p)=...). When you fit to log(p), what will your coefficients be? Hint - they are written in your derived equation.

[Cris LaPierre]

Also, note that you fit to cdate and not t. You can use either, as the fit is the same. However, it will affect the value of b since your axis will be scaled differently.

[Zeynep Toprak]

since this gives log(c2), in order purely get c2, I will take its exponential, i.e.

[Cris LaPierre]

You can capture the coefficients in a variable:

p = polyfit(cdate,log(pop),1)

p = 1×2    
    0.0208  -35.4457

Is c2 the m or b in the m*x+b equation?

[Zeynep Toprak]

b = log(c2) = - 35.4457, thus c2 = exp(-35.4457)

and m = c3 = 0.0208.

[Cris LaPierre]

I knew you could do it! Nice job.

Once you have the coefficients captured in p, you can use polyval to compare the fit to the original.

load census
p = polyfit(cdate,log(pop),1)

% generate fit line
yfit=polyval(p,cdate);

% visualize
plot(cdate,log(pop),'-ro',cdate,yfit,'--')
legend({'ln(pop)','fit'},'Location',"best")

[Zeynep Toprak]

wow, perfect thank you so much for your great effort!! You're the best! Take care! :)

[Zeynep Toprak]

Hello professor again, yesterday, you explained this question perfectly. these are not assigments. since classes are closed down, on these leisure times, I try to improve my matlab skills by myself by following videos on youtube and solving some exercises. i have one more question. i solved it mostly. but at the end i need to solve two nonlinear equation simutanously. bu i could not here. Or can i do question with fminsearch ? My question is here if you 're going to help me one more, i will be glad. :)