mov_med {edecob}

Moving Median over a Time Window

Description

Calculates the moving median over a time window around a time point for the all time points between the first and last time point provided.

Usage

mov_med(data, med_win = c(-42, 42), min_pts_in_win = 1)

Arguments

data

A data frame in long format containing the data for which events is to be detected. This means that each measurement corresponds to a row and the columns are (in order): source (the device or person from which the data was collected), point in time, and measurement value. If custom detection bounds are chosen, the folloing two columns must be added: lower detection bound, and upper detection bound.

The source is expected to be a string; the time point are integers; measurements, and detection bounds are expected to be numerical. The detection bounds are in absolute value in the same unit as the values and each is expected to be identical for the same source.

In case detection is wanted for a one sided change (e.g. give an event if the confidence bounds drop below a threshold) then the upper or lower detection bound can be chosen to be Inf or -Inf respectively.

med_win

A vector containing two numbers specifying the window over which the moving median is to be taken. More specifically, when given a certain time point, the numbers specify the maximum number of time units difference that a time point can have such that that data point will be considered for the moving median. Note that the first number must be smaller than the second number. The numbers may be negative.

min_pts_in_win

The minimal number of measurements required to be in the time window in order for the median to be calculated.

Details

Consider a sample X_1,\dots, X_n of size n and the reordering X_{(1)},\dots, X_{(n)} such that X_{(1)} \le X_{(2)} \le \dots \le X_{(n)}, commonly called the order statistic. Then for n even the median usually defined as

median(X_1,\dots, X_n) = X_{(k)}, \mathrm{where } \; k = n/2.

In the case where n is odd the median is defined as

median(X_1,\dots, X_n) = 1/2(X_{(k)} + X_{(k+1)}), \mathrm{where } \; k = n/2.

Let the study days at which the measurements X_1, \dots, X_n were taken be t_1, \dots, t_n. Let T a fixed positive amount of time. Then the moving median at time point t with window size T is defined as

S(t) = median({X_j | t - T/2 \le t_j \le t + T/2}).

For the initial time points where the time difference between the first data point and the time point for which we are calculating the median is less than half the width, we do not have enough data points to form a window which has the same size to both sides of the time point. In this case fewer data points are used to calculate the median and the time window is not symmetric around the time point for which we are calculating the median.

No median is calculated if the time difference between the last data point and the current time point for which we are calculating the median is less than half the width. We do not calculate the median using a smaller time window so that the values do not change upon receiving new data with time points newer than that of the old data.

Value

A data frame containing the values of the moving median, the study day to which it corresponds, the time window from which it was calculated, and the subject id corresponding to the data.