Related post: Using Log-Log Plots to Determine Whether Size Matters Curve Fitting with Nonlinear Regression You’re absolutely correct that the biased and unbiased models can have similar R-squared and S values because those statistics don’t evaluate bias. You can have high values of R-squared (or, equivalently, low values of S) and still have a biased model. And you can have low R-squared (high S) with unbiased models. So, those statistics don’t relate to bias. For our data, the increases in Output flatten out as the Input increases. There appears to be an asymptote near 20. Let’s try curve fitting with a reciprocal term. In the data set, I created a column for 1/Input (InvInput). I fit a model with a linear reciprocal term (top) and another with a quadratic reciprocal term (bottom). If you are dealing with count data, you might look into zero inflated models. I discuss those a bit in my post about choosing the correct type of regression analysis. You’ll find that in the count data section at the end. Coope, I.D. (1993). "Circle fitting by linear and nonlinear least squares". Journal of Optimization Theory and Applications. 76 (2): 381–388. doi: 10.1007/BF00939613. hdl: 10092/11104. S2CID 59583785.

There are several reasons given to get an approximate fit when it is possible to simply increase the degree of the polynomial equation and get an exact match.: Using log transformations is a powerful method to fit curves. There are too many possibilities to cover them all. Choosing between a double-log and a semi-log model depends on your data and subject area. If you use this approach, you’ll need to do some investigation. Basically, after running your example, you will obtain the best parameters (a the slope and b the intercept) for your linear function to fit your example data. Chernov, N.; Ma, H. (2011), "Least squares fitting of quadratic curves and surfaces", in Yoshida, Sota R. (ed.), Computer Vision, Nova Science Publishers, pp.285–302, ISBN 9781612093994Please note that during particularly busy periods, it may take a little longer to receive your delivery and our carrier may attempt to deliver to you on a Saturday. You’re right, the names of the analyses (linear and nonlinear regression) really gives the wrong impression about when you should use each one! A log transformation allows linear models to fit curves that are otherwise possible only with nonlinear regression. For the model that uses the reciprocal, I had to actually create the Linear vs Quadratic Reciprocal Model comparison graph by hand because the software couldn’t do that for reciprocal variables. However, once I created the graph, I can use it to describe the relationship because it’s all in natural units at that point. Liu, Yang; Wang, Wenping (2008), "A Revisit to Least Squares Orthogonal Distance Fitting of Parametric Curves and Surfaces", in Chen, F.; Juttler, B. (eds.), Advances in Geometric Modeling and Processing, Lecture Notes in Computer Science, vol.4975, pp.384–397, CiteSeerX 10.1.1.306.6085, doi: 10.1007/978-3-540-79246-8_29, ISBN 978-3-540-79245-1

The standard error of the regression for the nonlinear model (0.179746) is almost as low the S for the reciprocal model (0.134828). The difference between them is so small that you can use either. However, with the linear model, you also obtain p-values for the independent variables (not shown) and R-squared. Super-strong: Durable, high-quality PVC construction can support up to 110 kg and will help you feel safe and supported as you exercise. Other types of curves, such as conic sections (circular, elliptical, parabolic, and hyperbolic arcs) or trigonometric functions (such as sine and cosine), may also be used, in certain cases. For example, trajectories of objects under the influence of gravity follow a parabolic path, when air resistance is ignored. Hence, matching trajectory data points to a parabolic curve would make sense. Tides follow sinusoidal patterns, hence tidal data points should be matched to a sine wave, or the sum of two sine waves of different periods, if the effects of the Moon and Sun are both considered. If a function of the form y = f ( x ) {\displaystyle y=f(x)} cannot be postulated, one can still try to fit a plane curve. Anything of a nature that for hygiene or associated health and safety - this includes the Outdoor Spas, Mattresses and Divan SetsHere’s one final caution. You’d like a great fit, but you don’t want to overfit your regression model. An overfitmodel is too complex, it begins to model the random error, and it falsely inflates the R-squared. Adjusted R-squared and predicted R-squared are tools that can help you avoid this problem. If you first visually inspect a scatterplot of the data you would pass to curve_fit(), you would see (as in the answer of @Nikaido) that the data appears to lie on a straight line. Here is a graphical Python fitter similar to that provided by @Nikaido: We have two models at the top that are equally good at producing accurate and unbiased predictions. These two models are the linear model that uses the quadratic reciprocal term and the nonlinear model.