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# Fitting a Line

Fitting a line to (x,y) data points is a common case in many areas. Examples fits are given for fitting, fitting using the uncertainties as weights, and fitting using iterative sigma clipping.

## Simple Fit

Here the (x,y) data points are fit with a line. The (x,y) data points are simulated and have a range of uncertainties to give a realistic example.

    import numpy as np
    import matplotlib.pyplot as plt
    from astropy.modeling import models, fitting

    # define a model for a line
    line_orig = models.Linear1D(slope=1.0, intercept=0.5)

    # generate x, y data non-uniformly spaced in x
    # add noise to y measurements
    npts = 30
    rng = np.random.default_rng(10)
    x = rng.uniform(0.0, 10.0, npts)
    y = line_orig(x)
    yunc = np.absolute(rng.normal(0.5, 2.5, npts))
    y += rng.normal(0.0, yunc, npts)

    # initialize a linear fitter
    fit = fitting.LinearLSQFitter()

    # initialize a linear model
    line_init = models.Linear1D()

    # fit the data with the fitter
    fitted_line = fit(line_init, x, y)

    # plot
    fig, ax = plt.subplots()
    ax.plot(x, y, 'ko', label='Data')
    ax.plot(x, line_orig(x), 'b-', label='Simulation Model')
    ax.plot(x, fitted_line(x), 'k-', label='Fitted Model')
    ax.set(xlabel='x', ylabel='y')
    ax.legend()

(`png`, `svg`, `pdf`)

## Fit using uncertainties

Fitting can be done using the uncertainties as weights. To get the standard weighting of 1/unc^2 for the case of Gaussian errors, the weights to pass to the fitting are 1/unc.

    import numpy as np
    import matplotlib.pyplot as plt
    from astropy.modeling import models, fitting

    # define a model for a line
    line_orig = models.Linear1D(slope=1.0, intercept=0.5)

    # generate x, y data non-uniformly spaced in x
    # add noise to y measurements
    npts = 30
    rng = np.random.default_rng(10)
    x = rng.uniform(0.0, 10.0, npts)
    y = line_orig(x)
    yunc = np.absolute(rng.normal(0.5, 2.5, npts))
    y += rng.normal(0.0, yunc, npts)

    # initialize a linear fitter
    fit = fitting.LinearLSQFitter()

    # initialize a linear model
    line_init = models.Linear1D()

    # fit the data with the fitter
    fitted_line = fit(line_init, x, y, weights=1.0/yunc)

    # plot
    fig, ax = plt.subplots()
    ax.errorbar(x, y, yerr=yunc, fmt='ko', label='Data')
    ax.plot(x, line_orig(x), 'b-', label='Simulation Model')
    ax.plot(x, fitted_line(x), 'k-', label='Fitted Model')
    ax.set(xlabel='x', ylabel='y')
    ax.legend()

(`png`, `svg`, `pdf`)

## Iterative fitting using sigma clipping

When fitting, there may be data that are outliers from the fit that can significantly bias the fitting. These outliers can be identified and removed from the fitting iteratively. Note that the iterative sigma clipping assumes all the data have the same uncertainties for the sigma clipping decision.

    import numpy as np
    import matplotlib.pyplot as plt
    from astropy.stats import sigma_clip
    from astropy.modeling import models, fitting

    # define a model for a line
    line_orig = models.Linear1D(slope=1.0, intercept=0.5)

    # generate x, y data non-uniformly spaced in x
    # add noise to y measurements
    npts = 30
    rng = np.random.default_rng(10)
    x = rng.uniform(0.0, 10.0, npts)
    y = line_orig(x)
    yunc = np.absolute(rng.normal(0.5, 2.5, npts))
    y += rng.normal(0.0, yunc, npts)

    # make true outliers
    y[3] = line_orig(x[3]) + 6 * yunc[3]
    y[10] = line_orig(x[10]) - 4 * yunc[10]

    # initialize a linear fitter
    fit = fitting.LinearLSQFitter()

    # initialize the outlier removal fitter
    or_fit = fitting.FittingWithOutlierRemoval(fit, sigma_clip, niter=3, sigma=3.0)

    # initialize a linear model
    line_init = models.Linear1D()

    # fit the data with the fitter
    fitted_line, mask = or_fit(line_init, x, y, weights=1.0/yunc)
    filtered_data = np.ma.masked_array(y, mask=mask)

    # plot
    fig, ax = plt.subplots()
    ax.errorbar(x, y, yerr=yunc, fmt="ko", fillstyle="none", label="Clipped Data")
    ax.plot(x, filtered_data, "ko", label="Fitted Data")
    ax.plot(x, line_orig(x), 'b-', label='Simulation Model')
    ax.plot(x, fitted_line(x), 'k-', label='Fitted Model')
    ax.set(xlabel='x', ylabel='y')
    ax.legend()

(`png`, `svg`, `pdf`)