  The signatures of polynomials were originally introduced by Faug\`{e}re for
the efficient computation of Gr\"obner bases [Fau02], and redefined by
Arri-Perry [AP11] as the standard monomials modulo the module of syzygies.
Since it is difficult to determine signatures, Vaccon-Yokoyama [VY17]
introduced an alternative object called guessed signatures. In this paper, we
consider a module $\mathrm{Gobs}(F)$ for a tuple of polynomials $F$ to analyse
computation of Gr\"obner bases via theory of signatures. This is the residue
module
$\mathrm{ini}_{\prec}(\mathrm{Syz}(\mathrm{LM}(F)))/\mathrm{ini}_{\prec}(\mathrm{Syz}(F))$
defined by the initial modules of the syzygy modules with respect to the
Schreyer order. We first show that $F$ is a Gr\"obner basis if and only if
$\mathrm{Gobs}(F)$ is the zero module. Then we show that any homogeneous
Gr\"obner basis with respect to a graded term order satisfying a common
condition must contain the remainder of a reduction of an S-polynomial. We give
computational examples of transitions of minimal free resolutions of
$\mathrm{Gobs}(F)$ in a signature based algorithm. Finally, we show a
connection between the module $\mathrm{Gobs}(F)$ and Gr\"obner degenerations.
