3. SOB model for recognition
3.1 The extension of SOB and C-SOB
3.1.1 Measuring energy of probe
3.1.1 Measuring energy of probe
SOB model
SOB model assumes that the memory traces are self-associated items. The energy of incoming item is the difference between matrix of memory traces and the outer product of incoming item, as seen in Equation 3.1. The larger the difference, the larger the energy is. The consistency between probe and memory traces could be measured in the same way. The more consist between probe and memory traces, the smaller the difference is, and results in smaller energy of probe. The consistency of probe is defined as the energy of probe in SOB mode.
E =−(ϵ/2)∑i∑
jwijxixj, i̸= j. (3.1) Since that the negative probes are never presented while learning, the source of consistency should only comes from the prior, and results in a large inconsistency and energy. Positive probes, in the other hand, are encoded in memory trace while learning, should have smaller energy. Also, the increasing of set size decreases the consistency of probe and results in larger energy, as shown in Figure 3.1.
3 4 5 6
Figure 3.1: The energy of positive probe and negative probe among different set sizes in SOB.
However, the consistency of positive probe comes from both prior knowledge
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strength, as a result of energy-gated encoding, shows the primacy gradient that the strength decreases along with serial position. The consistency of positive probe shows only primacy gradient as Figure 3.2.
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Serial position
Energy
Figure 3.2: The energy position curve among different serial position.
C-SOB model
Unlike a straightforward way to measuring the energy of probe in SOB, C-SOB model requires more works. In C-SOB model, the memory traces of learned items are stored in the associations between the context and content layers. The system energy for an input item can be viewed as the resonance of the memory traces to the current item. Presumably, the resonance to an item is larger if it is more similar to the previously learned items in the memory traces. Due to that the energy value is always negative as seen in Equation 3.2, the larger the resonance, the smaller the energy is. Applying this logic to the recognition of an item, it is straightforward to expect that a probe (i.e., the item presented in the recognition stage) would induce a smaller energy if it is similar to the items in the memory traces and vice versa. In other words, this probe is familiar to the system. Thus, the familiarity to a probe can be defined as the energy of system induced by it.
Ei =−viTCi−1pi. (3.2)
In C-SOB model, the energy is the product of the content vector (vi) of an
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item i, the context vector (pi) for the item i, and the memory trace matrix (Ci−1).
Parallel to this computation, the energy in the recognition task is the product of the content vector of a probe (vp), the context vector for the probe (pp), and the memory trace matrix (Ci−1). This turns Equation 3.2 into Equation 3.3.
Ep =−vptCpp. (3.3)
The context vector for the probe pp is created using the same algorithm in C-SOB model, with the correlation to the context vector of the learned item pi, which is defined as
cos(pi, pp) = tptnc−i, (3.4) where 0 ≤ tc ≤ 1 and 0 ≤ tp ≤ 1 respectively denote the correlation between two adjacent context vectors and the correlation between the context vector of the last item and that of the probe, with set size = n. The energy of positive probes and negative probes among different set size are shown in Figure 3.3 According to Equation 3.4, the closer the item to the probe, the higher the correlation between them is. During the encoding stage, the energy decreases as encoding proceeds.
However, this is not necessary the case for recognizing the probe. For the positive probe, the energy actually varies along the position of the probe in the learning list. Figure 3.4 shows that the energy is the largest for the middle position and the smallest for the first and the last position when set size is greater than 3. As mentioned before, the larger the energy, the less familiar the system is with the probe. Therefore, this reversed U curve indicates that the very first and very last items are most likely to be recognized and the middle items are relatively hard to be recognized; namely the primacy and recency effect in recognition memory. The mathematical proof is provided as follow.
First, the energy for a probe is computed by Equation 3.3. With the set size
= n, given the memory trace Cn is Cn =
∑n i=1
ηivipTi , (3.5)
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Setsize
Energy
Positive probe Negative probe
Figure 3.3: The energy of positive probe and negative probe among different set sizes in C-SOB.
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Serial Position
Energy
Setsize 3 Setsize 4 Setsize 5 Setsize 6
Figure 3.4: The energy position curve among different set size and serial position.
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Equation 3.3 will then become
Ep =−vpT(
Because the length of pi and pp are equal to 16, given that the product of two vector lengths equals the inner product between these two vectors divided by the cosine value between them
cos(a, b) = a· b
√a· a√
b· b (3.7)
and Equation 3.4, the energy of probe could be reform as Ep =−
The similarity on context between the probe and the learning item i is tnc−i, which decays along the distance to the last item from the learning item. Although vTp · vi
is the same for the probe being on any position, ηi decays along the position of the learning item, with the largest for the first item, as shown in Figure 2.3. Together with the similarity on context between item and probe, the energy would be the most negative for the probe being on the earlier or the latter position in the learning list and would be the least negative for the probe being on the middle position. This is the reason for a reserved U profile of the energy for a positive probe. One thing to note is that although tp does not change the shape of the energy curve, it might be sensitive to the consistency of the context for the stimulus between the encoding and retrieval stages, larger for the consistent case and smaller for the inconsistent case. For the simulations in this study, tp is set to be constant.
The energy Ep for the negative probe must be larger than that for the positive probe, as exactly one product of vpT · vi is larger in the case for the positive probe than in the case for the negative probe.