* ~*k*~ − *t* ~*k*−1~)/Δ*t* is an integer for *k* = 1,..., *T*. For unequally spaced observations, the KS estimates of the hidden regulators can be obtained in an similar manner to the equidistant time point case, where the system matrix *F* is replaced with the appropriate *F* ^*j*~*k*~^ in both the filtering and smoothing recursion. However, the covariance matrix *τ* ~*j*~*k*~~Σ~*δ*~ of the error terms *δ* ~*t*~*k*~~ will depend on the spacing *j* ~*k*~ between observations. Icaza and Jones \[[@B28]\] show how to apply the Kalman filtering and smoothing procedure to the case of multivariate observations that are unequally spaced in time. The terms in the conditional likelihood function that depend on the system matrix *F* will no longer be quadratic in *F* (as is the case in model ([1](#EEq1){ref-type="disp-formula"})). They can be represented as a polynomial expression in *F*. The solution to the matrix equation

$$\frac{\partial}{\partial F}L_{F,G,\Sigma_{\delta},\Sigma_{\epsilon},\mu,\Sigma}\left( Y_{0},\ldots,T_{T}\, \mid \, Z_{1},\ldots,Z_{T} \right) = 0$$ may not exist in closed form but can be obtained numerically for instance through a gradient ascent.

2.6. Partitioning Method for Large Data Sets {#sec2.6}
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Many gene expression (time series) data sets have tens of thousands of observations. For example, the Arabidopsis ATH1 Affymetrix microarray represents more than 24000 genes. Since the numerical expense of estimating the parameters of a state space model increases quadratically in the number of observed genes, parameter estimation for the complete set of (genes) observations quickly becomes computationally challenging. To address this challenge without restricting the gene space, or limiting the KM-algorithm, one can randomly partition the data into several smaller subsets of approximately equal size. The KM-algorithm can then be implemented, with different initial starting values, repeatedly for each subset. When sufficient computing capacity is