Uses and Limitation of Continuous-Time Models to Examine Dyadic Interactions

Abstract

In this chapter we present an application of the exact discrete model, first proposed by Bergstrom, to model daily interactions among romantic couples. The theoretical model is based on work by Felmlee and Greenberg (J Math Soc 23(3):155–180, 1999), which specifies that change in affect results from the combination of a weighted difference between long-term expectations and daily ratings as well as daily ratings between partners in the dyad. To verify the correct specification, we used simulated models using the LSDE SAS/IML package developed by Singer.

Appendices

Appendix 1: McDonald-Swaminathan Matrix Differentiation

The specification of an LSDE model in the SAS/IML^® framework requires the specification of matrix derivatives. Matrix differentiation is specified for each of the matrices that make up the state-space specification and is evaluated with respect to the parameters that make up each element. The implementation of matrix differentiation used by the LSDE package is based on the work of McDonald and Swaminathan (1973) which outlines how matrix differentiation can be performed. Given a matrix Y that is n × m in size, with elements that represent some function of the elements of another matrix X, which is p × q in size, the differentiation of Y with respect to X in the McDonald-Swaminathan rules would result in a matrix of partial derivatives $\frac {\partial Y}{\partial X}$ that is n(m) × p(q) in size.

As an example, imagine Y is a 2 × 2 matrix and is to be differentiated base on another matrix X that is also 2 × 2. The result of differentiation is a 4 × 4 matrix of partial derivatives. Specifically, each of the rows of Y are arranged into a single row vector that is 1 × n(m) in size,

(6.15)

The same process is performed for the matrix X and transposed to produce a column vector of elements that is p(q) × 1 in size,

(6.16)

The differentiation is arranged as the outer product of these vectors, resulting in a p(q) × n(m) matrix of partial derivatives,

$$\displaystyle \begin{aligned} \frac{\partial Y}{\partial X} = \left[ \begin{array}{cccc} \frac{\partial y_{11}}{\partial x_{11}}& \frac{\partial y_{12}}{\partial x_{11}}& \frac{\partial y_{21}}{\partial x_{11}}& \frac{\partial y_{22}}{\partial x_{11}}\\ \frac{\partial y_{11}}{\partial x_{12}}& \frac{\partial y_{12}}{\partial x_{12}}& \frac{\partial y_{21}}{\partial x_{12}}& \frac{\partial y_{22}}{\partial x_{12}}\\ \frac{\partial y_{11}}{\partial x_{21}}& \frac{\partial y_{12}}{\partial x_{21}}& \frac{\partial y_{21}}{\partial x_{21}}& \frac{\partial y_{22}}{\partial x_{21}}\\ \frac{\partial y_{11}}{\partial x_{22}}& \frac{\partial y_{12}}{\partial x_{22}}& \frac{\partial y_{21}}{\partial x_{22}}& \frac{\partial y_{22}}{\partial x_{22}} \end{array} \right]. \end{aligned} $$

(6.17)

An example drawn directly from the original paper is presented below. These equations are a reproduction of equations 3–5 in McDonald and Swaminathan (1973). Assume we are given a matrix Y ,

$$\displaystyle \begin{aligned} Y = \left[ \begin{array}{cc} e^{x_{11}x_{12}}&sin(x_{11}+ x_{12})\\ log(x_{11}+x_{12}+x_{21})& x_{11}x_{12}x_{21}x_{22} \end{array} \right] \end{aligned} $$

(6.18)

with elements that are a function of another matrix X,

$$\displaystyle \begin{aligned} X = \left[ \begin{array}{cc} x_{11}&x_{12}\\ x_{21}&x_{22} \end{array} \right]. \end{aligned} $$

(6.19)

The derivative of Y with respect to the elements of X would result in

$$\displaystyle \begin{aligned} \frac{\partial Y}{\partial X} = \left[ \begin{array}{cccc} x_{12} e^{x_{11}x_{12}}& cos(x_{11}+x_{22})& \frac{1}{x_{11}+x_{12}+x_{21}}& x_{12}x_{21}x_{22}\\ x_{11} e^{x_{11}x_{12}}& 0& \frac{1}{x_{11}+x_{12}+x_{21}}& x_{11}x_{21}x_{22}\\ 0& 0& \frac{1}{x_{11}+x_{12}+x_{21}}& x_{11}x_{12}x_{22}\\ 0& cos(x_{11}+x_{22})& 0& x_{11}x_{12}x_{21} \end{array} \right]. \end{aligned} $$

(6.20)

Appendix 2: Matrix Differentiation of the Felmlee–Greenberg Model

Next we illustrate how the steps outlined above are performed using the two matrices A and B, from our model from Eq. (6.11). These matrices represent the deterministic portion of the state equation for Felmlee–Greenberg model Felmlee and Greenberg (1999). Below we differentiate these matrices with respect to a parameter vector Θ.

To begin, we reparameterize the model and place the f ^∗ and m ^∗ terms in the parameter vector; thus Θ = (a ₁, b ₁, a ₂, b ₂, f ^∗, m ^∗). This allows us to express the A and B matrices as

$$\displaystyle \begin{aligned} A = \begin{bmatrix} -(\theta_{1}+\theta_{2}) & \theta_{2}\\ \theta_{4} & -(\theta_{3}+\theta_{4}) \end{bmatrix}, B = \begin{bmatrix} \theta_{1}\theta_{5}\\ \theta_{3}\theta_{6} \end{bmatrix}. \end{aligned}$$

The derivative of the matrix A with respect to the parameter vector Θ is

$$\displaystyle \begin{aligned} \frac{\partial A}{\partial \varTheta} = \begin{bmatrix} -1&0&0&0\\ -1&1&0&0\\ 0&0&0&-1\\ 0&0&1&-1\\ 0&0&0&0\\ 0&0&0&0 \end{bmatrix}. \end{aligned}$$

For the matrix B, the result is

$$\displaystyle \begin{aligned} \frac{\partial B}{\partial \varTheta} = \begin{bmatrix} \theta_5&0\\ 0&0\\ 0&\theta_6\\ 0&0\\ \theta_1&0\\ 0&\theta_3 \end{bmatrix}. \end{aligned}$$

This example is only a portion of what is required in the LSDE syntax for this model. Please see Appendix 3 for a complete listing of the LSDE syntax required to fit the model for positive affect.