numpy.ma.polyfit#

ma.polyfit(x,y,deg,rcond=None,full=False,w=None,cov=False)[source]#

Least squares polynomial fit.

Note

This forms part of the old polynomial API. Since version 1.4, thenew polynomial API defined innumpy.polynomial is preferred.A summary of the differences can be found in thetransition guide.

Fit a polynomialp(x)=p[0]*x**deg+...+p[deg] of degreedegto points(x, y). Returns a vector of coefficientsp that minimisesthe squared error in the orderdeg,deg-1, …0.

ThePolynomial.fit classmethod is recommended for new code as it is more stable numerically. Seethe documentation of the method for more information.

Parameters:

xarray_like, shape (M,)

x-coordinates of the M sample points(x[i],y[i]).

yarray_like, shape (M,) or (M, K)

y-coordinates of the sample points. Several data sets of samplepoints sharing the same x-coordinates can be fitted at once bypassing in a 2D-array that contains one dataset per column.

degint

Degree of the fitting polynomial

rcondfloat, optional

Relative condition number of the fit. Singular values smaller thanthis relative to the largest singular value will be ignored. Thedefault value is len(x)*eps, where eps is the relative precision ofthe float type, about 2e-16 in most cases.

fullbool, optional

Switch determining nature of return value. When it is False (thedefault) just the coefficients are returned, when True diagnosticinformation from the singular value decomposition is also returned.

warray_like, shape (M,), optional

Weights. If not None, the weightw[i] applies to the unsquaredresidualy[i]-y_hat[i] atx[i]. Ideally the weights arechosen so that the errors of the productsw[i]*y[i] all have thesame variance. When using inverse-variance weighting, usew[i]=1/sigma(y[i]). The default value is None.

covbool or str, optional

If given and notFalse, return not just the estimate but also itscovariance matrix. By default, the covariance are scaled bychi2/dof, where dof = M - (deg + 1), i.e., the weights are presumedto be unreliable except in a relative sense and everything is scaledsuch that the reduced chi2 is unity. This scaling is omitted ifcov='unscaled', as is relevant for the case that the weights arew = 1/sigma, with sigma known to be a reliable estimate of theuncertainty.

Returns:

pndarray, shape (deg + 1,) or (deg + 1, K)

Polynomial coefficients, highest power first. Ify was 2-D, thecoefficients fork-th data set are inp[:,k].

residuals, rank, singular_values, rcond

These values are only returned iffull==True

• residuals – sum of squared residuals of the least squares fit

• rank – the effective rank of the scaled Vandermonde

  coefficient matrix

• singular_values – singular values of the scaled Vandermonde

  coefficient matrix

• rcond – value ofrcond.

For more details, seenumpy.linalg.lstsq.

Vndarray, shape (deg + 1, deg + 1) or (deg + 1, deg + 1, K)

Present only iffull==False andcov==True. The covariancematrix of the polynomial coefficient estimates. The diagonal ofthis matrix are the variance estimates for each coefficient. If yis a 2-D array, then the covariance matrix for thek-th data setare inV[:,:,k]

Warns:

RankWarning

The rank of the coefficient matrix in the least-squares fit isdeficient. The warning is only raised iffull==False.

The warnings can be turned off by

>>>importwarnings>>>warnings.simplefilter('ignore',np.exceptions.RankWarning)

See also

polyval

Compute polynomial values.

linalg.lstsq

Computes a least-squares fit.

scipy.interpolate.UnivariateSpline

Computes spline fits.

Notes

Any masked values in x is propagated in y, and vice-versa.

The solution minimizes the squared error

\[E = \sum_{j=0}^k |p(x_j) - y_j|^2\]

in the equations:

x[0]**n*p[0]+...+x[0]*p[n-1]+p[n]=y[0]x[1]**n*p[0]+...+x[1]*p[n-1]+p[n]=y[1]...x[k]**n*p[0]+...+x[k]*p[n-1]+p[n]=y[k]

The coefficient matrix of the coefficientsp is a Vandermonde matrix.

polyfit issues aRankWarning when the least-squares fit isbadly conditioned. This implies that the best fit is not well-defined dueto numerical error. The results may be improved by lowering the polynomialdegree or by replacingx byx -x.mean(). Thercond parametercan also be set to a value smaller than its default, but the resultingfit may be spurious: including contributions from the small singularvalues can add numerical noise to the result.

Note that fitting polynomial coefficients is inherently badly conditionedwhen the degree of the polynomial is large or the interval of sample pointsis badly centered. The quality of the fit should always be checked in thesecases. When polynomial fits are not satisfactory, splines may be a goodalternative.

References

[1]

Wikipedia, “Curve fitting”,https://en.wikipedia.org/wiki/Curve_fitting

[2]

Wikipedia, “Polynomial interpolation”,https://en.wikipedia.org/wiki/Polynomial_interpolation

Examples

Try it in your browser!

>>>importnumpyasnp>>>importwarnings>>>x=np.array([0.0,1.0,2.0,3.0,4.0,5.0])>>>y=np.array([0.0,0.8,0.9,0.1,-0.8,-1.0])>>>z=np.polyfit(x,y,3)>>>zarray([ 0.08703704, -0.81349206,  1.69312169, -0.03968254]) # may vary

It is convenient to usepoly1d objects for dealing with polynomials:

>>>p=np.poly1d(z)>>>p(0.5)0.6143849206349179 # may vary>>>p(3.5)-0.34732142857143039 # may vary>>>p(10)22.579365079365115 # may vary

High-order polynomials may oscillate wildly:

>>>withwarnings.catch_warnings():...warnings.simplefilter('ignore',np.exceptions.RankWarning)...p30=np.poly1d(np.polyfit(x,y,30))...>>>p30(4)-0.80000000000000204 # may vary>>>p30(5)-0.99999999999999445 # may vary>>>p30(4.5)-0.10547061179440398 # may vary

Illustration:

>>>importmatplotlib.pyplotasplt>>>xp=np.linspace(-2,6,100)>>>_=plt.plot(x,y,'.',xp,p(xp),'-',xp,p30(xp),'--')>>>plt.ylim(-2,2)(-2, 2)>>>plt.show()

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