The chloro-substitution network of phenol is represented here as a Hasse diagram

H(

P) (

Figure 1) which mathematically represents a finite poset

P. An oriented edge in the Hasse diagram here represents the transition

α→

β from a chemical compound

α with

n chlorine atoms to one

β with

n+1 chlorine atoms, and we attach a real variable

x_{α}_{→ β} ranging from 0 to 1, that represents the transformation of

α into

β. When formulating the splinoid QSSPR model for a property

X, one considers cubic spline polynomials (in

x_{α}_{→β}) on the oriented edges

α→

β of the Hasse diagram

H(

P). Further each vertex

α of

H(

P) or

P is identified by a value

a_{α} and a slope

b_{α} for the spline polynomials incident at

α. The splinoid poset QSSPR model is generated based on known values of the property

X for a subset

K⊆

P of the chemical compounds. Briefly, the splinoid fit consists of the following steps: first, the cubic splines match values

a_{α} at the nodes

α ∈

K to the known property values; second, the incoming and outgoing slopes through each node match to the corresponding

b_{α} value; and third, a relevant total “curvature” of the overall spline fit is minimized (subject to the constraints of the first two conditions). With the splinoid QSSPR determined for the vertices from

K, one can predict the property values for the remaining chemical compounds that do not have an experimental value for the property

X these being the compounds that form the “unknown” set

U of vertices

α ∉

K.

A mathematical derivation [

27] leads to a closed formula predicting the values of

X for the set

U of chemical compounds. Let

**A** denote the adjacency matrix of the Hasse diagram

H(

P), and let

**S** denote the oriented adjacency matrix of

H(

P), where:

The in-degree on vertex

α ∈

P is denoted by

d_{→α}, and the out-degree on vertex

α ∈

P is denoted by

d_{α}_{→}. Then, we introduce two diagonal matrices:

Further define the matrices

**U** (the |U|×|

P| submatrix of the unity matrix

**I**, with rows indexed by the elements of

U), and

**K** (the |

K|×|

P| submatrix of the unity matrix

**I**, with rows indexed by the elements of

K), and the derived matrix:

The (column) vector of known property values is denoted by

. Then, the vector

that contains the predictions for the unknown property values

a_{α} is computed from:

For a few different reaction networks we have studied the matrix

**UMU**^{t} which appears in practice to be invertible regardless of how sparse the “known” data is in the network up to the point that very few (≤2) known data are available. The coefficients appearing in the spline polynomials do not explicitly appear in our splinoid formula for

, but they are complicit in the derivation of this formula for

. The present formula gives

in terms of the poset structure, and thence completes the splinoid QSSPR algorithm, which turns out to give a robust model in accommodating a diversity of missing values for several compounds (which may possibly even be adjacent). This is a significant advantage of the splinoid model, which uses the topology of the Hasse diagram to generate a response network for the investigated property. To achieve comparison with the results from the other poset QSSPR models, we have used the splinoid model in the leave-one-out cross-validation procedure.

**Figure 2.**
The reaction poset diagram of chlorophenols with the experimental values of the octanol/water partition coefficients (log

K_{ow}) [

36,

37].