Web Appendix for Linear Modeling of Steady-State Behavioral Dynamics

This appendix supplies further and more detailed data supplementing the paper "Linear Modeling of Steady-State Behavioral Dynamics".

A Tutorial on Linear Modeling of Steady-State Behavioral Dynamics is also available.

Procedure

Results

In the following dot plots the trial duration is represented across the X axis. The heavy blue function depicts the mean rate in each portion of the trial. The red function depicts the predicted mean rate generated by linear analysis. The green bars depict the portion of the interval containing the VI-20 schedule. Each dot in the frame represents the rate for an 8 second bin for one trial, for that position in the trial. By noting the relative density of dots the modal behavior for each portion of the trial can easily be discerned. Additionally the dispersion of behavior is also clearly depicted.

Because only a quantal number of pecks could occur in each 8 second bin thus producing quantal rates, and because a single quantal columns of data one dot wide would represent the responding across each 8 second bin; the dot representing the response rate for each bin was dithered on the x-axis (± 1/2 bin) and on the y-axis (± 1/2 peck per 8 second bin). This portrayal therefore assumes that responding is spread equally across the 8 second bin and that response rates are continuous. The width of the dithering on the y-axis can be seen by noting the vertical dithering of the dots representing zero pecks in an 8 second bin. They are plotted ± 1/2 peck per 8 second bin on either side of the x-axis.

The following figures portray the response rate every 8 seconds throughout the trial for 100 sessions.

The following figures portray the mean response rate in each portion of the trial (blue function) (the same information as was provided by the blue function in the earlier set of frames). The standard error of the mean (as calculated by dividing the standard deviation of the data points in each 8 second bin by the square root of the sample size) is depicted by the whiskers. The standard error of the mean in each bin is a function of the vertical spread of the dots in that bin as is shown in the preceding set of frames. The predicted rate generated by linear analysis is given by the red function. This function is identical to the red function in the earlier set of frames. The dotted functions depict the obtained reinforcement rate in each 8 second bin throughout the trial. The green bars depict the portion of the interval containing the VI-20 schedule. The quality of the fit as indexed by reduced chi square (Chi Sq r) as well as the root mean squared residual (RMSR) is provided in the upper right of each frame.

The following figures are animations of how the response rate across the trial changed from trial to trial. By running the animation the trial by trial behavior dynamics can be observed. The common behavior change across the trial across the phase is labeled synchronous dynamics. The change in the behavior change across the early exposure to the procedure is labeled "transient dynamics". The unsystematic changes in the behavior from trial to trial are labeled "asynchronous dynamics". The sliding average animations portray a 5 day sliding window of the same data and portrays the changes in the measure most typically used to portray static data (i.e. the mean of the last 5 days for a phase). It removes a substantial amount of the asynchronous dynamics.

As an alternate visualization example the following 3 dimensional color projection plots show the distribution of response rates across the trial obtained during the last 20 days of Phase 5 for bird 558.

The X axis shows the time in trial where the beginning of the trial is on the right. The Y axis shows the response rate, and the Z axis and color index show the probability of rate occurance.

The second figure reduces the Z axis range to facilitate viewing the smaller changes in rate without being swamped by the large peak at a 0 rate.

All predictions were bases on the Phase 3 data. Two predictions were generated for each bird for each of phases 4,5, and 6. The predictions were based on either one sample (20 sessions) or on 5, 20 session samples who's transfer functions had been averaged (note that the data was resampled in the analysis program and the resultant predictions were 256 elements long.

The data table contains:

1. The input and output for each bird under; the last 20 sessions, and the last 5 20 session samples under phase 3 (the 5th sample of the 5 sample set is identical to the single sample).
2. The input and output data files for each bird for each of phases 4, 5, and 6
3. A prediction for each bird in each phase based on the single sample and the MatLab program used to carry out the analysis.
4. A prediction for each bird in each phase based on the 5 sample data and the MatLab program used to carry out the analysis.

			Bird 554	Bird 558	Bird 568	Bird 589	Bird 593
Phase 3	1 Sample	Obtained	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance
	5 Samples	Obtained	Input1 Input2 Input3 Input4 Input5 Output1 Output2 Output3 Output4 Output5	Input1 Input2 Input3 Input4 Input5 Output1 Output2 Output3 Output4 Output5	Input1 Input2 Input3 Input4 Input5 Output1 Output2 Output3 Output4 Output5	Input1 Input2 Input3 Input4 Input5 Output1 Output2 Output3 Output4 Output5	Input1 Input2 Input3 Input4 Input5 Output1 Output2 Output3 Output4 Output5
Phase 4		Obtained	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance
	1 Sample	Predict	Prediction	Prediction	Prediction	Prediction	Prediction
		Program	P3 to P4	P3 to P4	P3 to P4	P3 to P4	P3 to P4
	5 Samples	Predict	Prediction	Prediction	Prediction	Prediction	Prediction
		Program	P3 to P4	P3 to P4	P3 to P4	P3 to P4	P3 to P4
Phase 5		Obtained	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance
	1 Sample	Predict	Prediction	Prediction	Prediction	Prediction	Prediction
		Program	P3 to P5	P3 to P5	P3 to P5	P3 to P5	P3 to P5
	5 Samples	Predict	Prediction	Prediction	Prediction	Prediction	Prediction
		Program	P3 to P5	P3 to P5	P3 to P5	P3 to P5	P3 to P5
Phase 6		Obtained	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance	Input Variance Output Variance
	1 Sample	Predict	Prediction	Prediction	Prediction	Prediction	Prediction
		Program	P3 to P6	P3 to P6	P3 to P6	P3 to P6	P3 to P6
	5 Samples	Predict	Prediction	Prediction	Prediction	Prediction	Prediction
		Program	P3 to P6	P3 to P6	P3 to P6	P3 to P6	P3 to P6

Date Last Modified : October 6, 2003