Using Pandas and statsmodels for ordinary least squares

Question

Apologies in advance for the tedious beginner question.

I'm trying to translate a least-squares problem from a manual process (using Excel for matrix transposition and multiplication) to using the Python statsmodels package. In this case, I'm performing an affine transform from a set of observed coordinates to a set of ground coordinates in eastings (E) and northings (N). I've used the following formula to form the A (design) matrix: $$ \begin{equation} f_i(a_{0}, a_{1}, a_{2}) = a_{0} + a_{1}x_{i} + a_{2}y_{i} \\ f_i(b_{0}, b_{1}, b_{2}) = b_{0} + b_{1}x_{i} + b_{2}y_{i} \end{equation} $$ Which gives me a matrix that looks like:

$$ \begin{bmatrix} 1 & E_1 & N_1 & 0 & 0 & 0 \\ 1 & E_n & N_n & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & E_1 & N_1 \\ 0 & 0 & 0 & 1 & E_n & N_n \\ \end{bmatrix} $$

and a b vector, which is all x co-ordinates, followed by all y co-ordinates:

$$ \begin{bmatrix} x_1 \\ x_n \\ y_1 \\ y_n \end{bmatrix} $$

I also construct a square covariance matrix, W, using the square of the standard errors of the eastings and northings, arranged in a diagonal. This is used as the weight matrix during the least-squares process (the standard errors are assumed to be independent)

I then calculate:

$A^TWA$, $A^TWb$, and $(A^TWA)^{-1}$, then multiply $(A^TWA)^{-1}$ by $A^TWb$ to determine a vector x, which contains values for $a_0, a_1, a_2$ and $b_0, b_1, b_2$ 5. Multiply A by x, and subtract b from the result, to determine a residuals vector, v.

I can calculate the unit variance ($\sigma0$) by obtaining the square root of $\frac{v^TWv}{observations - unknowns}$, and multiplying it by the a priori standard error of each co-ordinate, in order to assess the quality (a posteriori standard error) of the transform. I can also calculate the standard error of my x vector (the diagonal values of Cx) by multiplying $(A^TWA)^{-1}$ by $\sigma0^2$

Now that the tedious step-by-step manual explanation is out of the way, let's say I have Pandas DataFrames for A, b and W:

In [124]: A_matrix
Out[124]: 
<class 'pandas.core.frame.DataFrame'>
Int64Index: 108 entries, 0 to 107
Data columns (total 6 columns):
As             108  non-null values
Eastings_a     108  non-null values
Northings_a    108  non-null values
Bs             108  non-null values
Eastings_b     108  non-null values
Northings_b    108  non-null values
dtypes: float64(4), int64(2)

In [125]: b_vector
Out[125]:
<class 'pandas.core.frame.DataFrame'>
Int64Index: 108 entries, 0 to 107
Data columns (total 1 columns):
coordinates    108  non-null values
dtypes: float64(1)

In [162]: Weight_matrix
Out[162]: 
<class 'pandas.core.frame.DataFrame'>
Int64Index: 108 entries, 0 to 107
Columns: 108 entries, 0 to 107
dtypes: float64(108)

How do I use statsmodels.ols_regression to easily calculate my residuals and $\sigma 0$?

Answer 1

Calculating σ0 is simple: take the square root of the returned fit object's mse_resid property.

Calculating the residuals: It's just the resid property of the returned fit object.