IEOR 165 – Lecture 23
Affine Discrete Time Models
1 Definition
An affine discrete time model can be written as
xt+1 = Axt + But + k,
where xt ∈ Rn is a vector of states, ut ∈ Rq is a vector of inputs, and A ∈ Rn×n , B ∈ Rn×q ,
k ∈ Rn are matrices/vectors of suitably-defined size. Such models can be used for forecasting. For instance, the state xt might represent the demand of a basket of goods at time t,
and ut might be economic indicators or other determinants of demand for each item in the
basket of goods.
The benefit of these models is that identifying the models and then using them for forecasting
is a relatively simple process, consisting of the following steps:
1. Historical data (ut , xt ) for t = −n, −n + 1, . . . , 0 is used to estimate the A, B, k of the
model.
2. Different scenarios for the future inputs ut are generated.
3. The forecast is generated by recursively computing
x1 = Ax0 + Bu0 + k
x2 = Ax1 + Bu1 + k
and so on.
In some cases, we may be able to use domain knowledge to exactly specify some of the future
inputs.
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2 Parameter Identification
Identifying the A, B, k is relatively simple because we can pose the problem as a linear
regression. In particular, if we let Aj , Bj , kj denote the j-th row of A, B, k respectively, then
we can estimate the model parameters by solving the following least squares problems
 
 ′ 2
Âj
Aj j
B̂j  = arg min Y − X Bj′  ,
kj′ 2
k̂j
where we define

x′−n
u′−n 1
′
x′

 −n+1 u−n+1 1
X =  ..
.
 .

′
′
x−1
u−1 1
 j

x−n+1
xj

 −n+2 
Yj = . 
 .. 
xj0

2