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scipy.optimize.curve\_fit(*f*, *xdata*, *ydata*, *p0\=None*, *sigma\=None*, *absolute\_sigma\=False*, *check\_finite\=None*, *bounds\=(-inf, inf)*, *method\=None*, *jac\=None*, *\**, *full\_output\=False*, *nan\_policy\=None*, *\*\*kwargs*)[\[source\]](https://github.com/scipy/scipy/blob/v1.17.0/scipy/optimize/_minpack_py.py#L590-L1074)[\#](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit "Link to this definition") Use non-linear least squares to fit a function, f, to data. Assumes `ydata = f(xdata, *params) + eps`. Parameters: **f**callable The model function, f(x, …). It must take the independent variable as the first argument and the parameters to fit as separate remaining arguments. **xdata**array\_like The independent variable where the data is measured. Should usually be an M-length sequence or an (k,M)-shaped array for functions with k predictors, and each element should be float convertible if it is an array like object. **ydata**array\_like The dependent data, a length M array - nominally `f(xdata, ...)`. **p0**array\_like, optional Initial guess for the parameters (length N). If None, then the initial values will all be 1 (if the number of parameters for the function can be determined using introspection, otherwise a ValueError is raised). **sigma**None or scalar or M-length sequence or MxM array, optional Determines the uncertainty in *ydata*. If we define residuals as `r = ydata - f(xdata, *popt)`, then the interpretation of *sigma* depends on its number of dimensions: - A scalar or 1-D *sigma* should contain values of standard deviations of errors in *ydata*. In this case, the optimized function is `chisq = sum((r / sigma) ** 2)`. - A 2-D *sigma* should contain the covariance matrix of errors in *ydata*. In this case, the optimized function is `chisq = r.T @ inv(sigma) @ r`. Added in version 0.19. None (default) is equivalent of 1-D *sigma* filled with ones. **absolute\_sigma**bool, optional If True, *sigma* is used in an absolute sense and the estimated parameter covariance *pcov* reflects these absolute values. If False (default), only the relative magnitudes of the *sigma* values matter. The returned parameter covariance matrix *pcov* is based on scaling *sigma* by a constant factor. This constant is set by demanding that the reduced *chisq* for the optimal parameters *popt* when using the *scaled* *sigma* equals unity. In other words, *sigma* is scaled to match the sample variance of the residuals after the fit. Default is False. Mathematically, `pcov(absolute_sigma=False) = pcov(absolute_sigma=True) * chisq(popt)/(M-N)` **check\_finite**bool, optional If True, check that the input arrays do not contain nans of infs, and raise a ValueError if they do. Setting this parameter to False may silently produce nonsensical results if the input arrays do contain nans. Default is True if *nan\_policy* is not specified explicitly and False otherwise. **bounds**2-tuple of array\_like or [`Bounds`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.Bounds.html#scipy.optimize.Bounds "scipy.optimize.Bounds"), optional Lower and upper bounds on parameters. Defaults to no bounds. There are two ways to specify the bounds: - Instance of [`Bounds`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.Bounds.html#scipy.optimize.Bounds "scipy.optimize.Bounds") class. - 2-tuple of array\_like: Each element of the tuple must be either an array with the length equal to the number of parameters, or a scalar (in which case the bound is taken to be the same for all parameters). Use `np.inf` with an appropriate sign to disable bounds on all or some parameters. **method**{‘lm’, ‘trf’, ‘dogbox’}, optional Method to use for optimization. See [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares") for more details. Default is ‘lm’ for unconstrained problems and ‘trf’ if *bounds* are provided. The method ‘lm’ won’t work when the number of observations is less than the number of variables, use ‘trf’ or ‘dogbox’ in this case. Added in version 0.17. **jac**callable, string or None, optional Function with signature `jac(x, ...)` which computes the Jacobian matrix of the model function with respect to parameters as a dense array\_like structure. It will be scaled according to provided *sigma*. If None (default), the Jacobian will be estimated numerically. String keywords for ‘trf’ and ‘dogbox’ methods can be used to select a finite difference scheme, see [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares"). Added in version 0.18. **full\_output**boolean, optional If True, this function returns additional information: *infodict*, *mesg*, and *ier*. Added in version 1.9. **nan\_policy**{‘raise’, ‘omit’, None}, optional Defines how to handle when input contains nan. The following options are available (default is None): - ‘raise’: throws an error - ‘omit’: performs the calculations ignoring nan values - None: no special handling of NaNs is performed (except what is done by check\_finite); the behavior when NaNs are present is implementation-dependent and may change. Note that if this value is specified explicitly (not None), *check\_finite* will be set as False. Added in version 1.11. **\*\*kwargs** Keyword arguments passed to [`leastsq`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.leastsq.html#scipy.optimize.leastsq "scipy.optimize.leastsq") for `method='lm'` or [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares") otherwise. Returns: **popt**array Optimal values for the parameters so that the sum of the squared residuals of `f(xdata, *popt) - ydata` is minimized. **pcov**2-D array The estimated approximate covariance of popt. The diagonals provide the variance of the parameter estimate. To compute one standard deviation errors on the parameters, use `perr = np.sqrt(np.diag(pcov))`. Note that the relationship between *cov* and parameter error estimates is derived based on a linear approximation to the model function around the optimum [\[1\]](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#r765c0da07a13-1). When this approximation becomes inaccurate, *cov* may not provide an accurate measure of uncertainty. How the *sigma* parameter affects the estimated covariance depends on *absolute\_sigma* argument, as described above. If the Jacobian matrix at the solution doesn’t have a full rank, then ‘lm’ method returns a matrix filled with `np.inf`, on the other hand ‘trf’ and ‘dogbox’ methods use Moore-Penrose pseudoinverse to compute the covariance matrix. Covariance matrices with large condition numbers (e.g. computed with [`numpy.linalg.cond`](https://numpy.org/doc/stable/reference/generated/numpy.linalg.cond.html#numpy.linalg.cond "(in NumPy v2.4)")) may indicate that results are unreliable. **infodict**dict (returned only if *full\_output* is True) a dictionary of optional outputs with the keys: `nfev` The number of function calls. Methods ‘trf’ and ‘dogbox’ do not count function calls for numerical Jacobian approximation, as opposed to ‘lm’ method. `fvec` The residual values evaluated at the solution, for a 1-D *sigma* this is `(f(x, *popt) - ydata)/sigma`. `fjac` A permutation of the R matrix of a QR factorization of the final approximate Jacobian matrix, stored column wise. Together with ipvt, the covariance of the estimate can be approximated. Method ‘lm’ only provides this information. `ipvt` An integer array of length N which defines a permutation matrix, p, such that fjac\*p = q\*r, where r is upper triangular with diagonal elements of nonincreasing magnitude. Column j of p is column ipvt(j) of the identity matrix. Method ‘lm’ only provides this information. `qtf` The vector (transpose(q) \* fvec). Method ‘lm’ only provides this information. Added in version 1.9. **mesg**str (returned only if *full\_output* is True) A string message giving information about the solution. Added in version 1.9. **ier**int (returned only if *full\_output* is True) An integer flag. If it is equal to 1, 2, 3 or 4, the solution was found. Otherwise, the solution was not found. In either case, the optional output variable *mesg* gives more information. Added in version 1.9. Raises: ValueError if either *ydata* or *xdata* contain NaNs, or if incompatible options are used. RuntimeError if the least-squares minimization fails. OptimizeWarning if covariance of the parameters can not be estimated. See also [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares") Minimize the sum of squares of nonlinear functions. [`scipy.stats.linregress`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.linregress.html#scipy.stats.linregress "scipy.stats.linregress") Calculate a linear least squares regression for two sets of measurements. Notes Users should ensure that inputs *xdata*, *ydata*, and the output of *f* are `float64`, or else the optimization may return incorrect results. With `method='lm'`, the algorithm uses the Levenberg-Marquardt algorithm through [`leastsq`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.leastsq.html#scipy.optimize.leastsq "scipy.optimize.leastsq"). Note that this algorithm can only deal with unconstrained problems. Box constraints can be handled by methods ‘trf’ and ‘dogbox’. Refer to the docstring of [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares") for more information. Parameters to be fitted must have similar scale. Differences of multiple orders of magnitude can lead to incorrect results. For the ‘trf’ and ‘dogbox’ methods, the *x\_scale* keyword argument can be used to scale the parameters. [`curve_fit`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit "scipy.optimize.curve_fit") is for local optimization of parameters to minimize the sum of squares of residuals. For global optimization, other choices of objective function, and other advanced features, consider using SciPy’s [Global optimization](https://docs.scipy.org/doc/scipy/tutorial/optimize.html#tutorial-optimize-global) tools or the [LMFIT](https://lmfit.github.io/lmfit-py/index.html) package. References \[[1](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#id1)\] K. Vugrin et al. Confidence region estimation techniques for nonlinear regression in groundwater flow: Three case studies. Water Resources Research, Vol. 43, W03423, [DOI:10.1029/2005WR004804](https://doi.org/10.1029/2005WR004804) Examples Try it in your browser\! ``` >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.optimize import curve_fit ``` ``` >>> def func(x, a, b, c): ... return a * np.exp(-b * x) + c ``` Define the data to be fit with some noise: ``` >>> xdata = np.linspace(0, 4, 50) >>> y = func(xdata, 2.5, 1.3, 0.5) >>> rng = np.random.default_rng() >>> y_noise = 0.2 * rng.normal(size=xdata.size) >>> ydata = y + y_noise >>> plt.plot(xdata, ydata, 'b-', label='data') ``` Fit for the parameters a, b, c of the function *func*: ``` >>> popt, pcov = curve_fit(func, xdata, ydata) >>> popt array([2.56274217, 1.37268521, 0.47427475]) >>> plt.plot(xdata, func(xdata, *popt), 'r-', ... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) ``` Constrain the optimization to the region of `0 <= a <= 3`, `0 <= b <= 1` and `0 <= c <= 0.5`: ``` >>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 1., 0.5])) >>> popt array([2.43736712, 1. , 0.34463856]) >>> plt.plot(xdata, func(xdata, *popt), 'g--', ... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) ``` ``` >>> plt.xlabel('x') >>> plt.ylabel('y') >>> plt.legend() >>> plt.show() ```  For reliable results, the model *func* should not be overparametrized; redundant parameters can cause unreliable covariance matrices and, in some cases, poorer quality fits. As a quick check of whether the model may be overparameterized, calculate the condition number of the covariance matrix: ``` >>> np.linalg.cond(pcov) 34.571092161547405 # may vary ``` The value is small, so it does not raise much concern. If, however, we were to add a fourth parameter `d` to *func* with the same effect as `a`: ``` >>> def func2(x, a, b, c, d): ... return a * d * np.exp(-b * x) + c # a and d are redundant >>> popt, pcov = curve_fit(func2, xdata, ydata) >>> np.linalg.cond(pcov) 1.13250718925596e+32 # may vary ``` Such a large value is cause for concern. The diagonal elements of the covariance matrix, which is related to uncertainty of the fit, gives more information: ``` >>> np.diag(pcov) array([1.48814742e+29, 3.78596560e-02, 5.39253738e-03, 2.76417220e+28]) # may vary ``` Note that the first and last terms are much larger than the other elements, suggesting that the optimal values of these parameters are ambiguous and that only one of these parameters is needed in the model. If the optimal parameters of *f* differ by multiple orders of magnitude, the resulting fit can be inaccurate. Sometimes, [`curve_fit`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit "scipy.optimize.curve_fit") can fail to find any results: ``` >>> ydata = func(xdata, 500000, 0.01, 15) >>> try: ... popt, pcov = curve_fit(func, xdata, ydata, method = 'trf') ... except RuntimeError as e: ... print(e) Optimal parameters not found: The maximum number of function evaluations is exceeded. ``` If parameter scale is roughly known beforehand, it can be defined in *x\_scale* argument: ``` >>> popt, pcov = curve_fit(func, xdata, ydata, method = 'trf', ... x_scale = [1000, 1, 1]) >>> popt array([5.00000000e+05, 1.00000000e-02, 1.49999999e+01]) ``` Go Back Open In Tab [previous isotonic\_regression](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.isotonic_regression.html "previous page") [next root\_scalar](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root_scalar.html "next page") On this page - [`curve_fit`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit) © Copyright 2008, The SciPy community. Created using [Sphinx](https://www.sphinx-doc.org/) 8.1.3. Built with the [PyData Sphinx Theme](https://pydata-sphinx-theme.readthedocs.io/en/stable/index.html) 0.16.1.

scipy.optimize. scipy.optimize.curve\_fit(*f*, *xdata*, *ydata*, *p0\=None*, *sigma\=None*, *absolute\_sigma\=False*, *check\_finite\=None*, *bounds\=(-inf, inf)*, *method\=None*, *jac\=None*, *\**, *full\_output\=False*, *nan\_policy\=None*, *\*\*kwargs*)[\[source\]](https://github.com/scipy/scipy/blob/v1.17.0/scipy/optimize/_minpack_py.py#L590-L1074)[\#](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit "Link to this definition") Use non-linear least squares to fit a function, f, to data. Assumes `ydata = f(xdata, *params) + eps`. Parameters: **f**callable The model function, f(x, …). It must take the independent variable as the first argument and the parameters to fit as separate remaining arguments. **xdata**array\_like The independent variable where the data is measured. Should usually be an M-length sequence or an (k,M)-shaped array for functions with k predictors, and each element should be float convertible if it is an array like object. **ydata**array\_like The dependent data, a length M array - nominally `f(xdata, ...)`. **p0**array\_like, optional Initial guess for the parameters (length N). If None, then the initial values will all be 1 (if the number of parameters for the function can be determined using introspection, otherwise a ValueError is raised). **sigma**None or scalar or M-length sequence or MxM array, optional Determines the uncertainty in *ydata*. If we define residuals as `r = ydata - f(xdata, *popt)`, then the interpretation of *sigma* depends on its number of dimensions: - A scalar or 1-D *sigma* should contain values of standard deviations of errors in *ydata*. In this case, the optimized function is `chisq = sum((r / sigma) ** 2)`. - A 2-D *sigma* should contain the covariance matrix of errors in *ydata*. In this case, the optimized function is `chisq = r.T @ inv(sigma) @ r`. Added in version 0.19. None (default) is equivalent of 1-D *sigma* filled with ones. **absolute\_sigma**bool, optional If True, *sigma* is used in an absolute sense and the estimated parameter covariance *pcov* reflects these absolute values. If False (default), only the relative magnitudes of the *sigma* values matter. The returned parameter covariance matrix *pcov* is based on scaling *sigma* by a constant factor. This constant is set by demanding that the reduced *chisq* for the optimal parameters *popt* when using the *scaled* *sigma* equals unity. In other words, *sigma* is scaled to match the sample variance of the residuals after the fit. Default is False. Mathematically, `pcov(absolute_sigma=False) = pcov(absolute_sigma=True) * chisq(popt)/(M-N)` **check\_finite**bool, optional If True, check that the input arrays do not contain nans of infs, and raise a ValueError if they do. Setting this parameter to False may silently produce nonsensical results if the input arrays do contain nans. Default is True if *nan\_policy* is not specified explicitly and False otherwise. **bounds**2-tuple of array\_like or [`Bounds`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.Bounds.html#scipy.optimize.Bounds "scipy.optimize.Bounds"), optional Lower and upper bounds on parameters. Defaults to no bounds. There are two ways to specify the bounds: - Instance of [`Bounds`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.Bounds.html#scipy.optimize.Bounds "scipy.optimize.Bounds") class. - 2-tuple of array\_like: Each element of the tuple must be either an array with the length equal to the number of parameters, or a scalar (in which case the bound is taken to be the same for all parameters). Use `np.inf` with an appropriate sign to disable bounds on all or some parameters. **method**{‘lm’, ‘trf’, ‘dogbox’}, optional Method to use for optimization. See [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares") for more details. Default is ‘lm’ for unconstrained problems and ‘trf’ if *bounds* are provided. The method ‘lm’ won’t work when the number of observations is less than the number of variables, use ‘trf’ or ‘dogbox’ in this case. Added in version 0.17. **jac**callable, string or None, optional Function with signature `jac(x, ...)` which computes the Jacobian matrix of the model function with respect to parameters as a dense array\_like structure. It will be scaled according to provided *sigma*. If None (default), the Jacobian will be estimated numerically. String keywords for ‘trf’ and ‘dogbox’ methods can be used to select a finite difference scheme, see [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares"). Added in version 0.18. **full\_output**boolean, optional If True, this function returns additional information: *infodict*, *mesg*, and *ier*. Added in version 1.9. **nan\_policy**{‘raise’, ‘omit’, None}, optional Defines how to handle when input contains nan. The following options are available (default is None): - ‘raise’: throws an error - ‘omit’: performs the calculations ignoring nan values - None: no special handling of NaNs is performed (except what is done by check\_finite); the behavior when NaNs are present is implementation-dependent and may change. Note that if this value is specified explicitly (not None), *check\_finite* will be set as False. Added in version 1.11. **\*\*kwargs** Keyword arguments passed to [`leastsq`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.leastsq.html#scipy.optimize.leastsq "scipy.optimize.leastsq") for `method='lm'` or [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares") otherwise. Returns: **popt**array Optimal values for the parameters so that the sum of the squared residuals of `f(xdata, *popt) - ydata` is minimized. **pcov**2-D array The estimated approximate covariance of popt. The diagonals provide the variance of the parameter estimate. To compute one standard deviation errors on the parameters, use `perr = np.sqrt(np.diag(pcov))`. Note that the relationship between *cov* and parameter error estimates is derived based on a linear approximation to the model function around the optimum [\[1\]](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#r765c0da07a13-1). When this approximation becomes inaccurate, *cov* may not provide an accurate measure of uncertainty. How the *sigma* parameter affects the estimated covariance depends on *absolute\_sigma* argument, as described above. If the Jacobian matrix at the solution doesn’t have a full rank, then ‘lm’ method returns a matrix filled with `np.inf`, on the other hand ‘trf’ and ‘dogbox’ methods use Moore-Penrose pseudoinverse to compute the covariance matrix. Covariance matrices with large condition numbers (e.g. computed with [`numpy.linalg.cond`](https://numpy.org/doc/stable/reference/generated/numpy.linalg.cond.html#numpy.linalg.cond "(in NumPy v2.4)")) may indicate that results are unreliable. **infodict**dict (returned only if *full\_output* is True) a dictionary of optional outputs with the keys: `nfev` The number of function calls. Methods ‘trf’ and ‘dogbox’ do not count function calls for numerical Jacobian approximation, as opposed to ‘lm’ method. `fvec` The residual values evaluated at the solution, for a 1-D *sigma* this is `(f(x, *popt) - ydata)/sigma`. `fjac` A permutation of the R matrix of a QR factorization of the final approximate Jacobian matrix, stored column wise. Together with ipvt, the covariance of the estimate can be approximated. Method ‘lm’ only provides this information. `ipvt` An integer array of length N which defines a permutation matrix, p, such that fjac\*p = q\*r, where r is upper triangular with diagonal elements of nonincreasing magnitude. Column j of p is column ipvt(j) of the identity matrix. Method ‘lm’ only provides this information. `qtf` The vector (transpose(q) \* fvec). Method ‘lm’ only provides this information. Added in version 1.9. **mesg**str (returned only if *full\_output* is True) A string message giving information about the solution. Added in version 1.9. **ier**int (returned only if *full\_output* is True) An integer flag. If it is equal to 1, 2, 3 or 4, the solution was found. Otherwise, the solution was not found. In either case, the optional output variable *mesg* gives more information. Added in version 1.9. Raises: ValueError if either *ydata* or *xdata* contain NaNs, or if incompatible options are used. RuntimeError if the least-squares minimization fails. OptimizeWarning if covariance of the parameters can not be estimated. Notes Users should ensure that inputs *xdata*, *ydata*, and the output of *f* are `float64`, or else the optimization may return incorrect results. With `method='lm'`, the algorithm uses the Levenberg-Marquardt algorithm through [`leastsq`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.leastsq.html#scipy.optimize.leastsq "scipy.optimize.leastsq"). Note that this algorithm can only deal with unconstrained problems. Box constraints can be handled by methods ‘trf’ and ‘dogbox’. Refer to the docstring of [`least_squares`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html#scipy.optimize.least_squares "scipy.optimize.least_squares") for more information. Parameters to be fitted must have similar scale. Differences of multiple orders of magnitude can lead to incorrect results. For the ‘trf’ and ‘dogbox’ methods, the *x\_scale* keyword argument can be used to scale the parameters. [`curve_fit`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit "scipy.optimize.curve_fit") is for local optimization of parameters to minimize the sum of squares of residuals. For global optimization, other choices of objective function, and other advanced features, consider using SciPy’s [Global optimization](https://docs.scipy.org/doc/scipy/tutorial/optimize.html#tutorial-optimize-global) tools or the [LMFIT](https://lmfit.github.io/lmfit-py/index.html) package. References \[[1](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#id1)\] K. Vugrin et al. Confidence region estimation techniques for nonlinear regression in groundwater flow: Three case studies. Water Resources Research, Vol. 43, W03423, [DOI:10.1029/2005WR004804](https://doi.org/10.1029/2005WR004804) Examples ``` >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.optimize import curve_fit ``` ``` >>> def func(x, a, b, c): ... return a * np.exp(-b * x) + c ``` Define the data to be fit with some noise: ``` >>> xdata = np.linspace(0, 4, 50) >>> y = func(xdata, 2.5, 1.3, 0.5) >>> rng = np.random.default_rng() >>> y_noise = 0.2 * rng.normal(size=xdata.size) >>> ydata = y + y_noise >>> plt.plot(xdata, ydata, 'b-', label='data') ``` Fit for the parameters a, b, c of the function *func*: ``` >>> popt, pcov = curve_fit(func, xdata, ydata) >>> popt array([2.56274217, 1.37268521, 0.47427475]) >>> plt.plot(xdata, func(xdata, *popt), 'r-', ... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) ``` Constrain the optimization to the region of `0 <= a <= 3`, `0 <= b <= 1` and `0 <= c <= 0.5`: ``` >>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 1., 0.5])) >>> popt array([2.43736712, 1. , 0.34463856]) >>> plt.plot(xdata, func(xdata, *popt), 'g--', ... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) ``` ``` >>> plt.xlabel('x') >>> plt.ylabel('y') >>> plt.legend() >>> plt.show() ```  For reliable results, the model *func* should not be overparametrized; redundant parameters can cause unreliable covariance matrices and, in some cases, poorer quality fits. As a quick check of whether the model may be overparameterized, calculate the condition number of the covariance matrix: ``` >>> np.linalg.cond(pcov) 34.571092161547405 # may vary ``` The value is small, so it does not raise much concern. If, however, we were to add a fourth parameter `d` to *func* with the same effect as `a`: ``` >>> def func2(x, a, b, c, d): ... return a * d * np.exp(-b * x) + c # a and d are redundant >>> popt, pcov = curve_fit(func2, xdata, ydata) >>> np.linalg.cond(pcov) 1.13250718925596e+32 # may vary ``` Such a large value is cause for concern. The diagonal elements of the covariance matrix, which is related to uncertainty of the fit, gives more information: ``` >>> np.diag(pcov) array([1.48814742e+29, 3.78596560e-02, 5.39253738e-03, 2.76417220e+28]) # may vary ``` Note that the first and last terms are much larger than the other elements, suggesting that the optimal values of these parameters are ambiguous and that only one of these parameters is needed in the model. If the optimal parameters of *f* differ by multiple orders of magnitude, the resulting fit can be inaccurate. Sometimes, [`curve_fit`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit "scipy.optimize.curve_fit") can fail to find any results: ``` >>> ydata = func(xdata, 500000, 0.01, 15) >>> try: ... popt, pcov = curve_fit(func, xdata, ydata, method = 'trf') ... except RuntimeError as e: ... print(e) Optimal parameters not found: The maximum number of function evaluations is exceeded. ``` If parameter scale is roughly known beforehand, it can be defined in *x\_scale* argument: ``` >>> popt, pcov = curve_fit(func, xdata, ydata, method = 'trf', ... x_scale = [1000, 1, 1]) >>> popt array([5.00000000e+05, 1.00000000e-02, 1.49999999e+01]) ```