numpy.polyfit(full=True)额外结果含义及拟合优度解读

Great question! Let's break down how these two values tie into your linear fit's goodness and reliability—they’re directly linked to the least-squares math polyfit uses under the hood.

Quick Context Recap #

When you set full=True, np.polyfit relies on singular value decomposition (SVD) to solve the least-squares problem. The extra return values are byproducts of this process; here’s how the two you’re curious about fit in:

1. singular_values #

These are the singular values from the SVD of your design matrix (for a degree-1 fit, that’s the matrix where each row is [1, x_i] for your data points x_i).

• Singular values measure the "information strength" of each independent dimension in your input data. Larger values mean the dimension carries meaningful signal for predicting y_data; smaller values mean the dimension is either nearly redundant (e.g., your x-values are clustered so tightly that rows of the design matrix are almost identical) or dominated by noise.
• For fit goodness:
  • If all singular values are relatively large and close in magnitude, your data is well-distributed, and your fit coefficients are stable (small changes in data won’t drastically alter your line).
  • If there’s a massive gap between the largest and smallest values (e.g., one is 1000 and another is 0.001), your design matrix is nearly singular. Even if residual error is small, the fit might be unreliable—tiny tweaks to your data could swing coefficients drastically.

2. rcond #

This is the condition threshold used during SVD to decide which singular values are "significant" enough to keep.

• Here’s the logic: any singular value smaller than rcond * max(singular_values) is treated as zero when calculating fit coefficients. The rank value returned (2 in your example) is the count of singular values that passed this threshold.
• For fit goodness:
  • The returned rcond (1.11e-15 in your case) is the threshold that was applied. Since your smallest singular value (0.551) is way larger than rcond * max(singular_value) (~1.44e-15), all dimensions were deemed significant—your fit used the full rank of the design matrix, which is exactly what we want for a linear fit (we expect rank 2 for a line with slope and intercept).
  • If rcond had excluded some singular values (i.e., rank was less than your polynomial degree + 1), that would signal your data lacks enough variation to support the polynomial degree you’re fitting. For example, fitting a quadratic to identical x-values would result in a rank of 1, as rcond filters out redundant singular values.

Applying This to Your Example #

Looking at your output:

• singular_values = [1.302, 0.551]: Both values are reasonably large, with a ratio of ~2.36—this means your x-data has enough spread, and the design matrix is well-conditioned. Your fit is stable.
• rcond = 1.11e-15: The extremely small threshold ensures even tiny meaningful singular values are kept. Since all your values clear this bar, the full rank is used, which is ideal for a linear fit.

内容的提问来源于stack exchange，提问作者M Waz