Least Squares Fitting - Data Analysis, CSIR-NET Physical Sciences | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

I. INTRODUCTION

Frequently we are given a set of data points (x_i, y_i ), i = 1, 2, . . . , N , through which we would like to t to a smooth function. The function could be straight line (the simplest case), a higher order polynomial, or a more complicated function. As an example, the data in Fig. 1 seems to follow a linear behavior and we may like to determine the "best" straight line through it. More generally, our tting function, y = f (x), will have some adjustable parameters and we would like to determine the "best" choice of those parameters.

The denition of "best" is not unique. However, the most useful choice, and the one nearly always taken, is the "least squares" t, in which one minimizes the sum of the squares of the difference between the observed y-value, yi , and the tting function evaluated at x_i , i.e.

   (1)

The simplest cases, and the only ones to be discussed in detail here, are where the tting

function is a linear function of the parameters. We shall call this a linear model. Examples are a straight line

y = a₀ + a₁x                                   (2)

and an m-th order polynomial

   (3)
where the parameters to be adjusted are the a_α. (Note that we are not requiring that y is a linear function of x, only of the t parameters a_α .)

An example where the tting function depends non -linearly on the parameters is

y = a₀x^a1 + a₂,                            (4)

Linear models are fairly simply because, as we shall see, the parameters are determined by linear equations, which always have a unique solution which can be found by straightforward methods. However, for tting functions which are non-linear functions of the parameters, the resulting equations are non-linear which may have many solutions or non at all, and so are much less straightforward to solve.

Sometimes a non-linear model can be transformed into a linear model by a change of variables.
For example if we want to t to

y = a₀x^a1 ,             (5)

which has a non-linear dependence on a₁, then taking logs gives

ln y = ln a₀ + a₁ ln x ,                      (6)

which is a linear function of the parameters a^'₀ = ln a₀ and a₁ . Fitting a straight line to a log-log plot is a very common procedure in science and engineering.

II. FITTING TO A STRAIGHT LINE

To see how least squares tting works, consider the simplest case of a straight line t, Eq. (2), for which we have to minimize

              (7)

with respect to a₀ and a₁. Differentiating F with respect to these parameters and setting the results to zero gives

         (8a)

            (8b)

We write this as

U₀₀ a₀ + U₀₁ a₁ = v₀,             (9a)

U₁₀ a₀ + U₁₁ a₁ = v₁,               (9b)

where

          (10)

          (11)

Equations (9) are two linear equations in two unknowns. These can be solved by eliminating one variable, which immediately gives an equation for the second one. The solution can also be determined from

           (12)

where m = 1 here, and the inverse of the 2 x 2 matrix U is given, according to standard rules, by

 (13)
where

                          (14)
and we have noted that U is symmetric so U₀₁ = U₁₀. The solution for a₀ and a₁ is therefore given by

               (15a)

               (15b)

We see that it is straightforward to determine the slope, a₁ , and the intercept, a₀ , of the t from Eqs. (10), (11), (14) and (15) using the N data points (x_i , y_i ).

III. FITTING TO A POLYNOMIAL

Frequently it may be better to t to a higher order polynomial than a straight line, as for example in Fig. 2 where the t is a parabola. Using the notation for an m-th order polynomial in Eq. (3), we need to minimize

 (16)

with respect to the M = m + 1 parameters a_α. Setting to zero the derivative of F with respect to a_α gives

  (17)

which we write as

         (18)

where U_αβ  and v_α  are dened in Eqs. (10) and (11). Eq. (18) represents M = m + 1 linear equations, one for each value of α . Their solution is given formally by Eq. (12).

Hence polynomial least squares fits, being linear in the parameters, are also quite straightforward. We just have to solve a set of linear equations, Eq. (18), to determine the fit parameters.

IV. FITTING TO DATA WITH ERROR BARS

Frequently we have an estimate of the uncertainty in the data points, the "error bar". A fit would be considered satisfactory if it goes through the points within the error bars. An example of data with error bars and a straight line t is shown in the figure below.

If some points have smaller error bars than other we would like to force the fit to be closer to those points. A suitable quantity to minimize, therefore, is

          (19)

called the "chi-squared" function, in which σ_i is the error bar for point i. Assuming a polynomial fit, we proceed exactly as before, and again find that the best parameters are given by the solution of Eq. (18), i.e.

   (20)

but where now U_αβ  and v_α are given by

           (21)

                 (22)

The solution of Eqs. (20) can be obtained from the inverse of the matrix U , as in Eq. (12).
Interestingly the matrix U ^-1 contains additional information. We would like to know the range of values of the a_α which provide a suitable fit. It turns out, that the square of the uncertainty in a_α is just the appropriate diagonal element of U ^-1 , so

                 (23)

For the case of a straight line fit, the inverse of U is given explicitly in Eq. (13). Using this information, and the values of (x_i , y_i , σ_i) for the data in the above gure, I find that the fit parameters (assuming a straight line fit) are

I had generated the data by starting with y = 1 + 2x and adding some noise with zero mean. Hence the t should be consistent with y = 1 + 2x within the error bars, and it is.

V. CHI-SQUARED DISTRIBUTION

It is all very well to get the t parameters and their error bars, but these results don't mean much unless the fit actually describes the data. Roughly speaking, this means that it goes through the data within the error bars. To see if this is the case, we look at the value X² dened by Eq. (19) with the optimal parameters. Intuitively, we expect that if the fit goes through the data within about one error bar, then X² should be about N . This indeed is correct, and, in fact, one can get much more information, including the distribution of X² , if we assume that the data points y_i have a Gaussian distribution. 

Let us rst assume that, apart from the (Gaussian) noise, the data exactly ts a polynomial, for some parameters . We rst calculate the distribution of X² where we put in the exact values for the parameters, , rather than those obtained my minimizing X² with respect to the parameters a_α. (Afterwords we will consider the effect of minimizing with respect to the a_α .) We therefore consider the distribution of

  (26)

Each of the N terms is the (square of) a Gaussian random variable with mean 0 and standard deviation unity. Denoting these variables by t_i then

           (27)

Firstly, suppose that N = 1. The single variable t has the distribution

  (28)

We actually want the distribution of t² . Let us call u = t² . To get the distribution of u, which we call P_u (u), from the distribution of t, where u is a function of t, we note that the probability that t lies between t and t + dt is the probability that u lies between u and u + du where du=dt is the derivative, i.e.

  (29)

where we noted that if the derivative is negative we must take the absolute value (since we are just equating the weight of the distribution of t over an interval dt to the weight of the distribution of u over an interval du). In this case we get

 (30)

where we also multiplied by 2 since there are two solutions for t of t² = u. Remember Eq. (30) just corresponds to the Gaussian distribution in Eq. (28) but with u = t² . It is instructive to check that the distribution in Eq. (30) is normalized.

where we made the substitution w = u/2 and used the result, discussed in class, that  .

As we have discussed, to obtain the distribution of a sum of random variables, such as in Eq. (27), it is useful to Fourier transform the distribution of the individual variables. We therefore take the Fourier transform of P_u(u),

With the substitution w = (1- 2ik)(u=2) we get

Now X² in Eq. (27) is just the sum of N random variables each of which has the distribution in Eq. (30). Hence the Fourier transform of the distribution of X² is the N -th power of the Fourier transform of the distribution of one variable, which is given by Eq. (33), so

        (34)

We will now verify that the function whose Fourier transform gives this expression is

(Note that we are determining the distribution of X² , not X, we are following convention in writing the basic variable as the square of something.) The Fourier transform of Eq. (35) is

which gives Eq. (34) as desired. We have again made the substitution w = (1 2ik)(X²=2), and we also used the denition of the τ (N/2).
Hence the X² distribution for N variables is given by Eq. (35). From this it is easy to determine the mean and standard deviation of X², if we remember the denition of the Gamma function:

              (37)
       (38)
       (39)

Hence the average value of X² is N (as we guessed intuitively above) and the standard deviation is 

In my opinion, it is often more convenient to consider the distribution of X² =N which is called the This has mean unity (independent of N ) and standard deviation   is 

Figure 4 shows the distribution of X² /N , the chi-squared per degree of freedom, for N = 2, 5 and 50. Note that for N = 2 the distribution of X² / N , is an exponential, for N > 2 it vanishes at the origin, and for N → ∞ the central limit theorem tells us (as we can verify) that it is a Gaussian of mean 1 and standard deviation 

FIG. 4: The chi-squared distribution per degree of freedom for N = 2, 5 and 50.

We have obtained the distribution of X² in Eq. (26) assuming that data ts exactly the specied polynomial apart from the Gaussian noise. However, in practice, we do not know the polynomial and we estimate it by minimizing X² in Eq. (19) with respect to the parameters a_α. It turns out that the net effect is to decrease the "number of degrees of freedom", which was N before, by the number of parameters in the t M (so for example M = m + 1 for an m-th order polynomial t).
To show this precisely requires some rather heavy math, which I will avoid, but instead indicate intuitively how the result arises.

Consider the simple case of a straight line t, y = a₀ + a₁x. Suppose we added a constant to all the y_i. Clearly X² would be unchanged once I've minimize with respect to a₀ because the modi ed value of a₀ would exactly compensate for the shift in the y values. Similarly, if I change the y_i by an amount which is proportional to the x_i then again the X² would be unchanged because the new value for a₁ would exactly compensate for the change in the data. Hence only changes in the yi which are orthogonal to δy_i = c₀, and δy_i = c₁x_i give a change in X² after minimization. There are N - 2 such linear combinations. It turns out that each gives a contribution to X² with the same distribution as each of the terms in the unoptimized case, i.e. Eq. (30).

Hence, when we minimize X² to get the best t, if the t were perfect apart from the random Gaussian noise in the data, the distribution of X² , would be the chi-squared distribution in Eq. (35), but with N replaced by ν , the "number of degrees of freedom", which is equal to the number of data points N minus the number of t parameters M, i.e. the distribution of X² is

     (40)

where

ν = N - M.                 (41)

Finally, all the discussion of t parameters and their errors, is irrelevant if the curve does t the data. It is therefore essential to also calculate a "goodness of t parameter" Q. This is dened to be the probability that one would get a value for X² greater than or equal to the observed one by chance. In otherwords it is the area under the curve of P_ν (X² ) to the right of the observed value.
If X² is less than   will be close to 1 (so the t is very likely), whereas if X² is much greater than  will be close to zero (so the t is very unlikely).
An expression for Q can be found in terms of tabulated functions, since

Where 

   (43)

is called the incomplete Gamma function. Note that