Hi Ankit, Yes, you can estimate the standard error / accuracy of a single measurement even the true value is unknown by applying cross-validation and/or bootstrap methodods on your (single) dataset.

How can the standard error be inferred from a signle data point by bootstrapping. Please refer / explain.

Michal , I am sorry, may be i did not get the question correctly. The standard error can never be inferred from a signle data point. I thought he wanted to estimate the standard error/ accuracy of a single measurement from a sample.

My understanding of his question is that he has a series of n measurements (y_i,x_i) and wants to estimate the error for the integral. If the measurement errors are i.i.d. random variables then as far as I'm aware numerically integrating bootstrap resampled data can provide an estimate of the standard error as Mahmoud suggested.

I'm not sure how valid the i.i.d. assumption would be for spectroscopic data, but the i.i.d. assumption is unnecessarily restrictive. The estimates should still be valid for errors that are not identically distributed, perhaps with some loss in asymptotic efficiency. Correctly applying bootstrap methods to data with dependent errors is significantly more complicated.

My understanding of your problem and the methods may be incomplete, hopefully someone will correct me if I've made errant statements regarding the methods. A more clear description of your data and your problem may be helpful.

Shane, I agree with you that a more clear description of the data and the problem is necessary. However, if the case was the way you described and he has a series of n measurements (y_i,x_i) and wants to estimate the error for the integral then he still can apply cross-validation using random portions of successive measurements of length k less than n. Alternatively, he can use the randomization test or compute the error terms and bootstrap them only if they are i.i.d. random variables.

Thank you for the reply.

I have a single set of data as

(pixel number, intensity value) for 1024 pixels for a sample.

I calculate area for a sharp peak (so I use a small range of pixels), and use Integration function in Igor Pro.

Typical plot looks something like this.

The error in the peak location is the full width at half maximum FWHM. Alternately, you could fit a normal distribution to the data. Absolute error in location depends upon calibration. You say you have no reference. Do you mean you have no reference for identifying the peak? Do you mean you have no reference for calibration? You must have a reference for calibration.

There is reference for both the calibration and the peak location. And they well correspond. I want to find the error in the peak area calculation.

@Ankit

The error in the peak area depends upon how the area is determined. How the area is determined depends upon the type of spectrometry.

Some methods use counts. If counts, the variance is the sum of the squares of the total counts and the baseline counts (plus a small contribution from the determination of the baseline).

Some methods use mV plotted against time. Area is determined by peak fitting. The general area error is similar to count error; the variance derives from the peak difference from the baseline. Additional error is from the error in peak fitting and baseline determination.

There are other methods depending upon the type spectrometry and readout.

Some automated systems determine the area by a system algorithm. Information should be available in the instrument documentation.

What type spectrometry are you doing? How are the results presented? What is the basis of the presentation (counts, mV, uA)?

if its only the total area under the curve (which is sensitive to the wings of the feature)

then simply the square root of the number of pixels under the curve is a good estimator (you can convert that later into flux units of necessary)

see this practicum

http://www.physics.wisc.edu/~craigm/idl/mpfittut.html

@Joseph,

I am doing Raman measurement, where I measure intensity for a particular Raman shift. I get that now I can use variance. Since the system is not automated I get raw data. After some reading from other sources I get that variance is alright. Thank you.