How to determine which regression curve fits better? PYTHON

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Well, community:

Recently I have asked how to do exponential regression (Exponential regression function Python) thinking that for that data set the optimal regression was the Hyperbolic.

x_data = np.arange(0, 51) 
y_data = np.array([0.001, 0.199, 0.394, 0.556, 0.797, 0.891, 1.171, 1.128, 1.437, 
          1.525, 1.720, 1.703, 1.895, 2.003, 2.108, 2.408, 2.424,2.537, 
          2.647, 2.740, 2.957, 2.58, 3.156, 3.051, 3.043, 3.353, 3.400, 
          3.606, 3.659, 3.671, 3.750, 3.827, 3.902, 3.976, 4.048, 4.018, 
          4.286, 4.353, 4.418, 4.382, 4.444, 4.485, 4.465, 4.600, 4.681, 
          4.737, 4.792, 4.845, 4.909, 4.919, 5.100])

Now, I'm doubting:

The first is an exponential fit. The second is hyperbolic. I don't know which is better... How to determine? Which criteria should I follow? Is there some python function?

Thanks in advance!

One common fit statistic is R-squared (R2), which can be calculated as "R2 = 1.0 - (absolute_error_variance / dependent_data_variance)" and it tells you what fraction of the dependent data variance is explained by your model. For example, if the R-squared value is 0.95 then your model explains 95% of the dependent data variance. Since you are using numpy, the R-squared value is trivially calculated as "R2 = 1.0 - (abs_err.var() / dep_data.var())" since numpy arrays have a var() method to calculate variance. When fitting your data to the Michaelis-Menten equation "y = ax / (b + x)" with parameter values of a = 1.0232217656373191E+01 and b = 5.2016057362771100E+01 I calculate an R-squared value of 0.9967, which means that 99.67 percent of the variance in the "y" data is explained by this model. Howver, there is no silver bullet and it is always good to verify other fit statistics and visually inspect the model. Here is my plot for the example I used:

Curve Fitting with Linear and Nonlinear Regression, Let's see if we can do better. Fitting Curves with Reciprocal Terms in Linear Regression. If your response data descends down to a floor, or  When performing linear regression in Python, you can follow these steps: Import the packages and classes you need; Provide data to work with and eventually do appropriate transformations; Create a regression model and fit it with existing data; Check the results of model fitting to know whether the model is satisfactory; Apply the model for predictions

You can take the 2-norm between the function and line of fit. Python has the function np.linalg.norm The R squared value is for linear regression.

SciPy, Given a Dataset comprising of a group of points, find the best fit representing the Regression is a special case of curve fitting but here you just don't need a  SSR = sum of squares of the regression, which is basically the square of the difference between the value of the fitting function at a given point and the arithmetic mean of the data set. if `y=f(x)` is the fit curve then the `SSR` is defined as follows, `SSR = sum_(i=1)^n(f(x(i))-Mean)^2` Similarly `SSE` is defined as follows,

Well, you should calculate an error function which measures how good your fit actually is. There are many different error functions you could use but for the start the mean-squared-error should work (if you're interested in further metrics, have a look at http://scikit-learn.org/stable/modules/model_evaluation.html).

You can manually implement mean-squared-error, once you determined the coefficients for your regression problem:

from sklearn.metrics import mean_squared_error
f = lambda x: a * np.exp(b * x) + c 
mse = mean_squared_error(y_data, f(x_data))

Python Nonlinear Regression Curve Fit, Linear regression is rooted strongly in statistical learning and therefore the model must be checked for the 'goodness of fit. The problem is that checking the quality of the model is often a less prioritized trade-off, or scalability (learning and complexity curves) plots. More from Towards Data Science  Curve Fitting Examples – Input : Output : Input : Output : As seen in the input, the Dataset seems to be scattered across a sine function in the first case and an exponential function in the second case, Curve-Fit gives legitimacy to the functions and determines the coefficients to provide the line of best fit.

How do you check the quality of your regression model in Python?, If you don't know about Linear Regression or need a brush-up, please go Linear Regression using Python · Linear Regression on Boston Housing Dataset How can we generate a curve that best captures the data as shown below? Why Polynomial Regression; Over-fitting vs Under-fitting; Bias vs  2) Linear and Cubic polynomial Fitting to the 'data' file Using curve_fit(). Using the curve_fit() function, we can easily determine a linear and a cubic curve fit for the given data. 2.1 Main Code: #Linear and Polynomial Curve Fitting. #1)Importing Libraries import matplotlib.pyplot as plt #for plotting. Aliasing matplotlib.pyplot as 'plt'.

Polynomial Regression, Like scipy.optimize.curve_fit, a Model uses a model function – a function that is meant to calculate a model for some phenomenon – and then uses that to best  Comparison of Regression Splines with Polynomial Regression. Regression splines often give better results than polynomial regression. This is because, unlike polynomials, which must use a high degree polynomial to produce flexible fits, splines introduce flexibility by increasing the number of knots but keep the degree fixed.

Modeling Data and Curve Fitting, For a myriad of data scientists, linear regression is the starting point of many or she can use to quickly fit a linear model to a fairly large data set and assess This is along the same lines as the Polyfit method, but more general in nature. The equation may be under-, well-, or over- determined (i.e., the  Nonlinear regression is a very powerful alternative to linear regression. It provides more flexibility in fitting curves because you can choose from a broad range of nonlinear functions. In fact, there are so many possible functions that the trick becomes finding the function that best fits the particular curve in your data.

Comments
  • This is more of a math question rather than a programming question. One way to to compute the MSE of both curves, and pick the one with the lower one. See more on goodness of fit, scipy.metrics.mean_squared_error, Mean Squared Error in numpy.
  • Do your data points have error bars? If so, are they gaussian errors?
  • My understanding is R-squared is exact for linear regression and approximate for non-linear regression. Still useful as it has no units, making comparisons of different data set regressions easier. For example, an R-squared value of 0.5 when fitting data with units of light-years and an R-squared value of 0.99 when fitting data with units of milliliters does give an understanding of the fit quality in both cases.
  • That is my understanding as well