Estimating slope from noisy data

*To*: mathgroup at smc.vnet.net*Subject*: [mg89597] Estimating slope from noisy data*From*: andreas.kohlmajer at gmx.de*Date*: Sat, 14 Jun 2008 05:29:47 -0400 (EDT)

Hi! I have difficulties to estimate the correct slope from noisy data. This is the code to generate the noisy data: Needs["LinearRegression`"]; slope = 1.0; sigma = 0.5; xrange = 1.0; SeedRandom[123]; (* initialize random generator *) rnd = {#, #*slope + RandomReal[NormalDistribution[0, sigma]]} &; (* generate 2000 data points *) data = Table[ rnd[RandomReal[NormalDistribution[0, xrange/3.0]]], {2000}]; subset = Take[data, 8]; ListPlot[subset, PlotRange -> {{-3, 3}, {-3, 3}}, PlotStyle -> PointSize[.025]] fit = Regress[subset, x, x, IncludeConstant -> False, RegressionReport -> {SummaryReport, ParameterCITable}] The correct slope is exactly 1. As the data is quite noisy, the CI of the slope is very big. The estimated slope is far to big (1.947). If I use more data points, the estimation gets better; I could also use a wider x-range, to get a better estimate for the slope. However, I'm quite limited in the x-range, so using a wider x-range is no option for me. I could check the RSquared for significance (If[Abs[r*Sqrt[n - 2]/ Sqrt[1 - r^2]] >= Quantile[StudentTDistribution[n - 2], 1 - 0.05], r, 0] (* significance of 95% *)). I this case, it is significant. Is there any other way to get a good estimate for the slope, without using too many data points? (Keywords: fit, regression, slope, noisy, rsquared, limited data)

**Follow-Ups**:**Re: Estimating slope from noisy data***From:*mante <claude.mante@univmed.fr>