Simple linear least squares regression

More on Simple linear least squares regression (SLLSR), also known as Simple least squares regression (SLSR) or univariate least squares regression... 

Example of a Simple Linear Least-Squares Regression... 

Example of an F-Test for Goodness of Fit for a Simple Linear Least-Squares Regression for Replicate Determinations... 

A linear fit is determined using a simple linear least-squares regression that is based on two assumptions i) the experimental uncertainty in Xj is much smaller than... 

In practice, spectral components almost invariably do overlap, and hence the simple linear least squares regression approach is not feasible. It is, however, possible to construct a linear least squares regression model for more than one component if Beer s law is modified to recognize that measured absorbance is the sum of the absorbances of all the components in the mixture. Now the model becomes a multivariate system. 

Linear least squares regression is the most common method of fitting a response that is a function of a single independent variable. Many nonlinear functions may be transformed to simple linear functions, extending the capabilities of the simplest regression algorithm. 

The next step is to apply detrending. As with SNV, each spectrum is treated independently of the others in the training set so that there is no external reference. It is a relatively simple calculation A linear least squares regression is used to fit a quadratic polynomial to the responses in the spectrum. This curve is then subtracted from the spectrum to give the result. As mentioned earlier, the quadratic curvature component attempts to correct for the effects of particle size and sample packing (Fig. 24). 

In the preceding section we showed that linear least-squares regression is limited to single components or simple mixtures where the bands are totally isolated from one another and there is no overlap in absorbance between the components. Under these conditions a separate linear least-squares regression can be completed for each of the components. This is the definition of a univariate system. 

Transformed Variables Sometimes an alternative to a simple linear model is suggested by a theoretical relationship or by examining residuals from a linear regression. In some cases, linear least-squares analysis can be used after the simple transformations shown in Table 8-3. 

Near-infrared (NIR) spectroscopy is becoming an important technique for pharmaceutical analysis. This spectroscopy is simple and easy because no sample preparation is required and samples are not destroyed. In the pharmaceutical industry, NIR spectroscopy has been used to determine several pharmaceutical properties, and a growing literature exists in this area. A variety of chemoinfometric and statistical techniques have been used to extract pharmaceutical information from raw spectroscopic data. Calibration models generated by multiple linear regression (MLR) analysis, principal component analysis, and partial least squares regression analysis have been used to evaluate various parameters. 

Earlier, it was noticed that the evaluation of the Fq by using a simple linear extrapolation of the initial Fy fluorescence rise can be significantly affected by the slope of the initial fluorescence rise (3). Therefore, it is essential to determine the validity of the Fq estimation when the slope of the initial fluorescence is different. In order to do this, we evaluated the Fq values when the initial kinetics rise were changed due to the addition of different electron acceptors. The Fig 2a, b and c represents the fluorescence induction curves of barley chloroplasts in the presence of different concentrations of the electron acceptors BQ, FeCN, and MV. The Fy values were quenched dependently to the concentrations of the electron acceptors, consequently, the quenching intensity Fy had different fluorescence initial rises and total fluorescence yields (Fig 2d) which resulted from the rate of the PSII primary electron acceptor Qa reduction (1). By using the least square regression method, we estimated an unchanged Fq value when the initial fluorescence rises were different (Fig 2d). This indicates that, when the chloroplasts were... 

Now we calculate the linear least-squares parameters A and B as before, treating all n = k.m = 12 data points individually. This case is slightly more complicated than the simple hnear regression case considered in an earlier text box since now, although there are only six different values of x, each value appears twice so we have to he sure to take this into account. 

An example of a more complex function that is linear in this sense is Yi pred = A + 5.exp(x ), but Yj p d = A + 5.exp(C.X ) can not be fitted by linear least-squares since d(S, j )/dC is not linear in C in such cases there is no simple algebraic solution for the parameters as there is for the linear case (e.g., Equations [8.21]-[8.24]), so we must use nonlinear regression techniques that involve successive iterations to the final best result. 

Linear Regression Baseline Fitting. This is a very simple approach to baseline correction in that it requires no effort to set up. In this method, a least squares regression line is fit to the responses in each spectral region selected for calibration. This line is then subtracted from the response values in the region before using the data to perform the calibration model calculations (Ref. 52). 

First, consider the simple linear model P = aT+ b. The parameter estimates are obtained using ordinary, least-squares regression. The parameter estimates with 95% confidence intervals are ... 

MSC corrects spectra according to a simple linear univariate fit to a standard spectrum a and are estimated by least squares regression using the standard spectrum. As the standard spectrum, a spectrum of a particular sample or an average spectrum is used. 

There are many different methods for selecting those descriptors of a molecule that capture the information that somehow encodes the compounds solubility. Currently, the most often used are multiple linear regression (MLR), partial least squares (PLS) or neural networks (NN). The former two methods provide a simple linear relationship between several independent descriptors and the solubility, as given in Eq. (14). This equation yields the independent contribution, hi, of each descriptor, Di, to the solubility ... 

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