Matplotlib draws a best-fit curve; it does not calculate the curve’s parameters. First choose a function that represents the relationship you want to model, then fit its parameters with a numerical method such as SciPy’s curve_fit. Finally, evaluate that function across many x-values and plot those predictions alongside your measured data.
Fit and plot a nonlinear curve
This example fits an exponential-decay model, y = a·exp(-b·x) + c. The function is only an example: select a model that makes sense for your data and question.
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
# Replace these example arrays with paired measurements.
xdata = np.asarray([0, 1, 2, 3, 4], dtype=float)
ydata = np.asarray([2.6, 1.8, 1.2, 0.9, 0.7], dtype=float)
if xdata.ndim != 1 or ydata.ndim != 1 or xdata.size != ydata.size:
raise ValueError("xdata and ydata must be aligned one-dimensional arrays")
if xdata.size == 0 or not np.isfinite(xdata).all() or not np.isfinite(ydata).all():
raise ValueError("Provide non-empty, finite measurements")
def model(x, a, b, c):
return a * np.exp(-b * x) + c
# p0 supplies starting values for the parameter search.
popt, pcov = curve_fit(model, xdata, ydata, p0=(2.0, 1.0, 0.5))
# Many x-coordinates make the plotted line appear smooth.
xfit = np.linspace(xdata.min(), xdata.max(), 300)
yfit = model(xfit, *popt)
fig, ax = plt.subplots()
ax.scatter(xdata, ydata, label="Observed data")
ax.plot(xfit, yfit, color="tab:red", label="Nonlinear least-squares fit")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.legend()
plt.show()
print("Fitted parameters (a, b, c):", popt)The example arrays are illustrative; replace them with measurements in matching positions. The model callable takes the independent variable first, followed by parameters to estimate. SciPy describes curve_fit as using nonlinear least squares to fit a function to data. Its output popt contains estimated parameters, while pcov is an approximate covariance matrix.
What each part does
Choose the model before plotting
Use a straight-line model when the relationship you want to describe is linear; use a nonlinear function when the subject calls for a different shape. The model is a mathematical assumption about how y depends on x, not a setting in Matplotlib. For a linear regression, SciPy’s linregress is an alternative; curve_fit is useful for a custom nonlinear function.
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Fit parameters with SciPy
curve_fit(model, xdata, ydata, p0=...) searches for parameters that minimize squared residuals under the model ydata = f(xdata, *params) + eps. A plausible p0 starting point can help difficult fits converge. Use bounds when the problem gives a defensible range for a parameter, rather than imposing limits merely to make a plot look better.
Draw observations and predictions separately
scatter shows the measured pairs; plot draws the fitted predictions. The fit is evaluated at 300 evenly spaced x-coordinates between the observed minimum and maximum so the line does not connect only the original measurement locations. Matplotlib’s plot draws y versus x as lines and/or markers, and scatter displays pairwise observations.
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Choose a fitting approach that matches the data
| Situation | Approach | What to consider |
|---|---|---|
| Relationship is reasonably linear | Use a linear regression method such as scipy.stats.linregress. |
Choose this because a line fits the modeling question, not because it is the easiest curve to draw. |
| A specific nonlinear function is justified | Fit that function with curve_fit. |
Provide sensible initial values; consider meaningful bounds if the parameters have known limits. |
| Measurements have known uncertainty | Pass uncertainty information through sigma to curve_fit. |
A one-dimensional sigma can represent standard deviations; a two-dimensional value can represent a covariance matrix. |
| Outliers should have less influence than in ordinary least squares | Consider scipy.optimize.least_squares with a robust loss. |
SciPy documents losses including soft_l1 and cauchy; ordinary squared residuals are not robust to outliers. |
Interpret uncertainty and diagnose unreliable fits
Understand the covariance estimate
The covariance matrix is not a guaranteed confidence interval. SciPy notes that its parameter uncertainty estimate relies on a linear approximation near the fitted optimum. When sigma represents measurement uncertainties, absolute_sigma=False (the default) scales the returned covariance to the residual variance; absolute_sigma=True treats the supplied uncertainties as absolute. Choose based on what your sigma values mean.
Check whether the model is identifiable
Redundant parameters, poor parameter scaling, an ill-conditioned covariance matrix, or a singular Jacobian can make estimates and uncertainty summaries unreliable. Simplify the model if parameters cannot be distinguished from the available data, and use parameter scaling where appropriate. A visually smooth curve does not establish that the parameters are meaningful.
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Inspect residuals and the model’s meaning
A regression curve estimates a relationship and generally does not pass through every observation; that differs from interpolation, which is designed to pass through supplied points. Inspect residuals—the differences between observations and predictions—and ask whether their pattern or size undermines the chosen model. Do not judge a fit only by smoothness or an unqualified R-squared value.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Use Matplotlib’s plotting interface
The example uses Matplotlib’s object-oriented interface: fig, ax = plt.subplots(), followed by calls on ax. It keeps plot setup and formatting organized, especially as a figure grows. The pyplot API is convenient for interactive work and simple plots; Matplotlib recommends the object-oriented Figure/Axes interface for complex plots. Label axes and include a legend so readers can distinguish measured values from model predictions. Report the chosen model and fitted coefficients when the plot is used to communicate results.
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