---
title: "PKH1982"
author: "Victor Navarro"
output: rmarkdown::html_vignette
bibliography: references.bib
csl: apa.csl
vignette: >
  %\VignetteIndexEntry{PKH1982}
  %\VignetteEngine{knitr::rmarkdown}
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---

# The mathematics behind PKH1982

Another departure from global error term models such 
as RW1972 [@rescorla_theory_1972], the PKH1982 model 
[@pearce_predictive_1982] does not use an error term 
for learning excitatory associations (but does for inhibitory 
associations), and ties stimulus associability ($\alpha$) 
to absolute global prediction error.

*note: The implementation of this model closely 
follows the technical note from the 
[CAL-R](https://www.cal-r.org/index.php?id=PHsim) group where possible. 
Divergences are noted.*

## 1 - Generating expectations

Let $v_{k,j}$ denote the excitatory strength from 
stimulus $k$ to stimulus $j$, and $v_{k,\overline j}$ the 
inhibitory strength from stimulus $k$ to stimulus $j$ 
(effectively, a "no j" representation). On any given trial, 
the net expectation of stimulus $j$, $e_j$, is given by:

$$
\tag{Eq.1}
e_j = \sum_{k}^{K}x_k v_{k,j} - \sum_{k}^{K}x_k v_{k,\overline j}
$$

where $x_k$ denotes the presence (1) or absence (0) of stimulus 
$k$, and the set $K$ represents all stimuli in the design.

## 2 - Learning associations

Changes to the excitatory and inhibitory associations 
between stimuli are given by:

$$
\tag{Eq.2a}
\Delta v_{i,j} = \delta_jx_i \beta_{ex,j} \alpha_i \lambda_j
$$
$$
\tag{Eq.2b}
\Delta v_{i,\overline j} = x_i \beta_{in,j} \alpha_i  |\overline{\lambda_j}|
$$

where $\beta_{ex,j}$ and $\beta_{in,j}$ represent learning 
rates for excitatory and inhibitory associations, respectively, 
as determined by stimulus $j$, $\alpha_i$ is the associability of 
stimulus $i$, respectively, and $\lambda_j$ and $\overline {\lambda_j}$ 
are the excitatory asymptote and the overexpectation of stimulus $j$, 
respectively.

Importantly, $\delta_j$ in Eq.2a is a parameter that is equal to 
1 if the expectation of stimulus $j$, is lower than its excitatory 
asymptote (i.e., $e_j < \lambda_j$), but 0 if not. This implies 
that the model stops strengthening $v_{i,j}$ if the expectation of 
$j$ is higher than its excitatory asymptote.

As mentioned in the introductory note, the PKH1982 model 
does not learn excitatory associations via correction error. 
However, the model **does** learn inhibitory associations via 
correction error, as the overexpectation term above, $\overline 
{\lambda_j}$ is equal to $min(\lambda_j - e_j, 0)$, where $min$ 
is the minimum function. This implies $\overline {\lambda_j}$ 
only takes non-zero values when the expectation of $j$ is higher 
than its intensity on the trial ($\lambda_j$).

## 3 - Learning to attend

The associability parameter $\alpha_i$ changes completely 
from trial to trial as a function of learning (note the lack 
of $\Delta$ below), with the change being equal to the difference 
of the absolute global error, via:

$$
\tag{Eq.3}
\Delta \alpha_{i} = x_i \sum_{j}^{K}\gamma_j(|\lambda_j - e_j|)
$$
where $\gamma_j$ denotes the contribution of the 
prediction error based on the jth stimulus. In this regard, 
it is important to note that @pearce_predictive_1982 did not extend 
their model to account for the predictive power of within-compound 
associations, yet the implementation of the model in this package does. 
This can sometimes result in unexpected behaviour, 
and as such, Eq. 3 above includes the extra parameter 
$\gamma_j$ (defaulting to 1/K) that denotes whether the expectation 
of stimulus $j$ contributes to attentional learning. As such, the 
user can set these parameters manually to reflect the contribution 
of the different experimental stimuli. For example, in a simple 
"AB>(US)" design, setting $\gamma_{US}$ = 1 and $\gamma_{A} = 
\gamma_{B} = 0$ leads to the behavior of the original model.

The PKH1982 model improves upon the @pearce_model_1980 
model by adding an extra parameter that controls the rate 
at which associability changes. If we qualify the changes 
in associability determined by Eq.3 via $\Delta\alpha_{i}^{n}$ 
(meaning they happened after trial $n$), then we can quantify the 
total associability of stimulus $i$ after trial $n$ via:

$$
\tag{Eq.4}
\alpha_{i}^{n} = 
\begin{cases}
  (1-\theta_i) \alpha_{i}^{n-1} + \theta_i\Delta\alpha_{j}^n &\text{, if } x_i = 1\\
  \alpha_{i}^{n} & \text{, otherwise}
\end{cases}
$$
where $\theta_i$ is a parameter determining both the 
rate at which associability decays (via $1-\theta_i$), and the 
rate at which increments in attention occur. Note that changes 
in associability only apply to stimuli presented on the trial 
(i.e., $x_i = 1$); attention to absent stimuli remains unchanged.

## 4 - Generating responses

There is no specification of response-generating mechanisms in 
PKH1982. However, the simplest response function that can be 
adopted is the identity function on stimulus expectations. If so, 
the responses reflecting the nature of $j$, $r_j$, are given by:

$$
\tag{Eq.5}
r_j = e_j
$$

### References
