Supplementary Information for:
Singh R, Palmer N, Gifford D, Berger B and BarJoseph Z. Active Sampling for Sampling in TimeSeries Experiments: With Application to Gene Expression Experiments Proceedings of the 22nd Int'l Conference on Machine Learning, 2005. To Appear.
To obtain a continuous time formulation, we use cubic Bsplines, which are sets of piecewise defined cubic polynomials, to represent temporal gene expression profiles. Splines in general, and cubic Bsplines in particular, are mathematically convenient for data approximation are often used to produce smooth lowdegree polynomial curves, while avoiding the problems of overfitting, numerical instability and oscillations that arise if single highdegree polynomials are used.
Bsplines are described as a linear combination of a set of basis polynomials. By knowing the value of these splines at a set of control points, one can generate the entire set of polynomials from these basis functions. We assume that the expression of gene at time can be represented by a spline curve and additional noise using the following equation:
Here is the expression profile for gene , is a vector of spline control points for gene and is a matrix of spline basis functions evaluated at the sampling points of the experiment. is a vector of the noise terms, which is assumed to be normally distributed with mean . Because the data is expected to be noisy, and may contain missing values, determining the parameters of the above equation ( and ) for each gene separately may lead to overfitting. Instead, when estimating these splines from expression data, we constrain the control point values of genes in the same cluster (coexpressed genes) to covary, thus using other coexpressed genes to overcome noise and missing values for a single gene. The parameters of this model are determined using an EM algorithm. In the E step we determine cluster membership for each gene, while the other parameters of the model are maximized with respect to cluster assignment in the M step. See [#!BGGJS03!#] for complete details.
Given the spline coefficient matrix, , in the above model, for a data set containing timepoints when control points are used, the smoothing matrix is defined as:
(2) 
where is a matrix. Thus rows of define the ``coordinates'' of the timepoints in terms of the basis splines.
In previous work [BarJoseph et al. 2002], we showed that this method provides a superior fit for time series expression data when compared to all other previous methods.


The hardness of a dataset is controlled by varying the frequency of sinusoids and their positions across the clusters: higher frequency implies that more timepoints will be needed to characterize the curve in that region. Also, introducing flat regions makes the dataset harder in the opposite way: if we require that sampling strategy use fewer timepoints in total, then it will then have to find out a way to identify flat regions and sample at a low rate there.
Plots for some other generated datasets are shown here:
The adaptive cost strategy is quite intuitive: if the addition of new data
doesn't change , decrease the cost a bit; otherwise, set it back to the
default value (= 1). More precisely,
We have used the periodogram^{*} to rank order all genes in the alpha and cdc28 experiments. Periodogram uses Fourier analysis to determine which genes cycle during the experiment. For both experiments we used the cell cycle duration supplied in the Spellman paper (64 minutes for alpha and 85 minutes for cdc28). FV and FC28 consisted of the top 500 genes according to the ranked lists of alpha and cdc28 respectfully. In order to generate a consensus set from both datasets we summed the rank of the gene in both lists (so that a gene that was ranked 50 in alpha and 34 in cdc28 received a score of 84). We then selected the 500 genes with the lowest score and used these as our consensus set which we denoted in the paper as FCOMB.
*: See Ref 21 in paper: Wicher, Fokianos and Strimmer, Bioinformatics 20:520, 2004.