最小二乘拟合功率谱分析：振幅偏差及其消除，应用于光镊和原子力显微镜杠杆
Power spectrum analysis with least-squares fitting: Amplitude bias and its elimination, with application to optical tweezers and atomic force microscope cantilevers
摘要：
光镊和原子力显微镜杠杆的标定通常通过拟合布朗运动的实验功率谱来实现。我们在这里证明，如果这是用典型的加权最小二乘法完成的，结果是拟合扩散系数值的相对大小在-2/n和+1/n之间的偏差。这里n是功率谱的平均数，因此典型的校准包含10-20%的偏差。偏置的符号和大小都取决于所应用的加权方案。因此，基于扩散系数的长度尺度校准也是如此。特征频率的拟合值不受此偏差的影响。对于AFM，如果有独立的长度刻度校准，则力测量不受影响。光镊就没有这种幸运了，因为弹性常数是特征频率和扩散系数的比值。我们给出了分析结果的重量依赖的偏差为广泛的一类系统，其动态是由一个线性（积分）微分方程加性噪声，白色或有色。例子是光镊与流体动力学自相互作用和混叠，校准Ornstein-Uhlenbeck模型在金融，模型细胞迁移在生物学等，因为偏见采取的形式是一个简单的乘法因子的拟合幅度（例如扩散系数），它是直接删除，用户将需要最小的修改，他或她最喜欢的最小二乘拟合程序。使用合成数据演示和说明结果，因此我们可以将拟合与已知的真实值进行比较。我们还适合一些常见的功率谱一劳永逸的意义上，我们给他们的参数值和相关的误差条作为实验功率谱值的显式函数。
英文摘要：
Optical tweezers and AFM cantilevers are often calibrated by fitting their experimental powerspectra of Brownian motion. We demonstrate here that if this is done with typical weighted least-squares methods the result is a bias of relative size between -2/n and +1/n on the value of the fitted diffusion coefficient. Here n is the number of power-spectra averaged over, so typical calibrations contain 10-20% bias. Both the sign and the size of the bias depends on the weighting scheme applied. Hence, so do length-scale calibrations based on the diffusion coefficient. The fitted value for the characteristic frequency is not affected by this bias. For the AFM then, force measurements are not affected provided an independent length-scale calibration is available. For optical-tweezers there is no such luck, since the spring constant is found as the ratio of the characteristic frequency and the diffusion coefficient. We give analytical results for the weight-dependent bias for the wide class of systems whose dynamics is described by a linear (integro-)differential equation with additive noise, white or colored. Examples are optical tweezers with hydrodynamic self-interaction and aliasing, calibration of Ornstein-Uhlenbeck models in finance, models for cell-migration in biology, etc. Because the bias takes the form of a simple multiplicative factor on the fitted amplitude (e.g. the diffusion coefficient) it is straightforward to remove, and the user will need minimal modifications to his or her favorite least-square fitting programs. Results are demonstrated and illustrated using synthetic data, so we can compare fits with known true values. We also fit some commonly occurring power spectra once-and-for-all in the sense that we give their parameter values and associated error-bars as explicit functions of experimental power-spectral values.
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