3. 2. Forms of data

The data used in my analyses come in several forms, and in this section I briefly present the different data types. Data of fish travel times - the time taken for a fish to travel between two points - is obviously temporal in nature, with the spatial component set as the length of the river reach. Travel time data vary depending on whether fish are released as individuals or groups. In group releases, a common mark identifies the fish, and the number of fish, {n_t: t = 1,2,3,...,k}, sampled at the downstream collection site during discrete time intervals is observed. Clearly, n_t is integer valued. Each time interval is t in duration (usually 1 day), and the final time interval, k, is an interval after the last fish has been observed. Also, is the total number of fish. Group covariates - such as date of release and river flow - may be associated with the cohort.

Alternatively, a unique marking may distinguish individuals in a cohort. In this case, the data are of individual travel times, {t_i: i = 1,2,3,...,n}, where n is the total number of individuals observed. t_i is positive valued; it can be continuous or discrete. Each individual may also be characterized by covariates such as river flow, date of release, and fish length.

In radio-tracking data, an individual's position is followed through time. The data are continuous in both time and space and can be denoted as {X(t): }. The vector X can be 1, 2, or 3 dimensional and can take on both positive and negative values. It will be bounded by the system in which the fish are observed. In practice, the position of the individual is noted at successive points in time so that time is discrete, and the positional vector can be denoted as {X_t: t = 0,1,2,...,n-1}, with n the number of observations. Ideally, the time intervals are equal in duration, but this is not always the case. It is often more practical to work with displacements, Y_t = X_t - X_t-1.

Hydroacoustic instruments observe the distribution of depths of fish in the water column at a fixed location during a period of time. The data are usually discrete, {Z_i: i = 1,2,3,...,n}, with Z_i being the number of individuals observed in the ith equally spaced interval of the water column.