## 1 Introduction  
Fluorescence correlation spectroscopy (FCS) is a technique that provides molecular diffusion dynamics and the number of molecules of fluorescently labeled target molecules [1; 2; 3; 4]. The FCS system comprise a confocal fluorescence detection system. The number of target molecules inside the confocal volume (measurement volume) fluctuates owing to the translational diffusion in the sample solution. As a result, the detected fluorescence intensity signal fluctuated. This means that the fluctuation analysis of the fluorescence intensity signal can provide information about translational diffusion, which is a concept of FCS. Because FCS measurements can be performed even in living cells with low damage, FCS is a useful tool, especially in the field of biology [5]. For example, [PERSON] et al. analyzed the mobility of a focal adhesion protein zyxin at focal adhesions of living HC11 cells by FCS. They found that the stiffness of the substrate for cell culturing did not affected to the mobility of zyxin [6]. A polarization dependent FCS (Pol-FCS), an advanced method of FCS, was used to evaluate the macromolecular crowding environment in cells [7]. Recently, FCS is used also in the field of soft matter to evaluate the structure of materials [7; 8]. In FCS, the speed of fluctuation of fluorescence intensity is analyzed by transforming a fluorescence intensity signal into an auto-correlation function (ACF). The characteristic decay time of the ACF, which is called the diffusion time in FCS, is inversely proportional to the diffusion coefficient of the target molecules. The amplitude of the ACF is inversely proportional to the number of molecules in the measurement volume. If a sample includes two species with different diameters, the ACF is a superposition of two ACFs with different diffusion times as described by Equation (A9) in Appendix A. If the two ACFs are separated by fitting analysis of the resulting ACF, the diffusion coefficients and the number of molecules can be obtained for each species. Therefore, FCS can be used to evaluate the molecular interactions. For example, [PERSON] et al. analyzed the complex formation of enhanced green fluorescent protein (EGFP) and anti-EGFP antibodies [9]. [PERSON] et al. established a phage display based on FCS. They successfully detected specific interactions between fluorescently labeled antibodies and their antigens using FCS [10]. Thus, FCS is useful for evaluating molecular interactions. However, if the sample contains two species with different brightness levels, interpreting the result is difficult. The estimated number of each species becomes incorrect because their amplitude in the ACF is weighted, corresponding to the brightness of each species [5; 11]. We can correctly obtain the number of each species only if their brightness was measured in advance. [PERSON] et al. successfully measured the dimerization of green fluorescent protein-fused human glucocorticoid receptor alpha (GFP-GR\(\alpha\)) using FCS. They assumed that there were only monomers and dimers of GFP-GR\(\alpha\), and the brightness of dimers was two times larger than that of the monomer. In addition, the brightness of the GFP-GR\(\alpha\) monomer was measured as a negative control before the sample measurement [12]. There are similar approaches for the dimer and oligomer formation analysis [13; 14; 15; 16]. [PERSON] et al. proposed the high-order image correlation spectroscopy (HICS) which is based on high-order ACF analysis [17]. They successfully evaluate the numbers of each species and the brightness ratio between multiple species without any assumption. This approach may be useful for FCS too. However, most of the available FCS instruments and hardware correlators do not provide high-order autocorrelation function so far. Introducing high-order ACF is difficult for current FCS users.