Scipy curve fit

content/scipy/concepts/scipy-optimize/terms/curve-fit/curve-fit.md

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+---
+Title: 'Curve Fit' 
+Description: 'Fits a custom function to data by adjusting its parameters to match the data as closely as possible.'
+Subjects:
+  - 'Data Science'
+  - 'Machine Learning'
+Tags:
+  - 'Python'
+  - 'Optimization'
+  - 'Mathematics'
+CatalogContent:
+  - 'learn-python'
+  - 'paths/data-science'
+---
+
+**`curve_fit()`** fits a custom function to data by adjusting its parameters to minimize the difference between the function's predictions and the actual data points, ensuring the best possible match.
+
+## Syntax
+
+```pseudo
+popt, pcov = 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)
+```
+
+- `f`: This is the function to fit to the data.
+- `xdata`: An array-like input representing the independent variable values (e.g., time, x-coordinates, etc.).
+- `ydata`: An array-like input representing the dependent variable values (e.g., measured data corresponding to `xdata`).
+- `p0`: Inital guess of the parameters.
+- `sigma`: Defines the uncertainty in `ydata`.
+- `absolute_sigma`: If `True`, `sigma` is interpreted absolutely, and the parameter covariance pcov reflects absolute values. If `False` which is the default, `sigma` is scaled to normalize residual variance. Here, pcov(absolute_sigma=False) = pcov(absolute_sigma=True) \* chisq(popt)/(M-N).
+- `check_finite`: Ensures input arrays do not contain `NaN` or `inf`. If `True`, a `ValueError` is raised when such values are found.  Defaults to `True` unless `nan_policy` is explicitly specified.
+- `bounds`: Specifies parameter bounds. Defaults to no bounds. Options include:
+  - An instance of the `Bounds` class.
+  - A 2-tuple of array-like objects or scalars: Scalars apply bounds uniformly and `np.inf` can disable bounds partially.
+- `method` - Optimization method. Choices are:
+  - `'lm'` (default for unconstrained problems): Levenberg-Marquardt.
+  - `'trf'` (default if bounds are set): Trust Region Reflective.
+  - `'dogbox'`: Dogleg.
+    `'lm'` cannot handle cases where observations < variables. Use `'trf'` or `'dogbox'` instead.
+- `jac`: Jacobian matrix computation for `jac(x, ...)`. Defaults to numerical estimation if None. Supports finite difference schemes for `'trf'` and `'dogbox'` methods.
+- `full_output`: If `True`, returns additional information such as infodict, mesg, ier.
+- `nan_policy`: Decides behavior when `NaN` values exist in input data:
+  - None (default): No special handling, behavior depends on implementation.
+  - `'raise'`: Throws an error.
+  - `'omit'`: Ignores NaN values during computation.
+- `**kwargs`: Additional keyword arguments passed to leastsq (if method = 'lm') or least_squares otherwise.
+
+It returns:
+
+- `popt`: A 1D array containing the optimal values of the parameters (`a`, `b`, `c`, etc.) that minimize the difference between the function and the data (`ydata`).
+- `pcov`: A 2D array representing the covariance matrix of the estimated parameters, which provides an estimate of the uncertainties (or standard errors) associated with the optimized parameters.
+
+## Example
+
+This code uses `curve_fit` to fit a custom exponential decay function to noisy data and estimates the parameters `a`, `b`, and `c`:
+
+```py
+import numpy as np
+from scipy.optimize import curve_fit
+
+# Define your custom function
+def func(x, a, b, c):
+  return a * np.exp(-b * x) + c
+
+# Define your data
+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
+
+# Fit for the parameters a, b, c of the function func:
+popt, pcov = curve_fit(func, xdata, ydata)
+print(popt)
+```
+
+The above will give the following output:
+
+```shell
+[2.47681145 1.32375703 0.53063146]
+```