Unit 5: Activation and Context
The
goal of this unit is to introduce the components of the activation equation
that reflect the context of a declarative memory retrieval.
5.1 Spreading Activation
The
first context issue we will consider is called spreading activation. The chunks in the slots of the chunk in the
goal buffer provide a context in which to perform a retrieval. Those chunks spread activation to the other
chunks in declarative memory based on their relations to the other chunks,
which we call their strengths of association.
This essentially results in increasing the activation of those chunks
which are related to the current context.
The
equation for the activation Ai
of a chunk i including spreading
activation is defined as:
Measures of Prior Learning,
Bi:
The base-level activation reflects the recency and frequency of practice
of the chunk as described in the previous unit.
Sources of Activation: The elements j being summed over are the chunks which are in the slots of the
goal chunk.
Weighting: Wj
is the amount of activation from source j.
Strengths of Association: Sji
is the strength of association from source j to chunk i.
e: The noise value as described in the
last unit.
The
weights, Wj, of the activation spread defaults to an even
distribution. The total amount of source activation is called W and defaults to
1 where it is usually left, though it can be set as the goal activation global
parameter (:ga in the sgp command). The Wj
values are then set to W/n where n is the number of chunks in goal slots.
The
strength of association, Sji, between two chunks
is 0 if chunk j is not in a slot of
chunk i or is not itself chunk j and is set using this equation when
chunk j is in a slot of chunk i or is itself chunk j:
S: A parameter to be estimated (set with
the maximum associative strength, :mas, parameter)
fanj: is the number of chunks in which j is the value of a slot plus one for
chunk j being associated with itself.
Here
is a diagram to help you visualize how the spreading activation works. Consider a goal chunk that has two chunks in
its slots when a retrieval is requested and that there are three chunks which
match the retrieval request for which the activation needs to be determined.
The
activations of the three chunks are then:
5.2 The Fan Effect
1. A hippie is in the park.
2. A hippie is in the church.
3. A hippie is in the bank.
4. A captain is in the park.
5. A captain is in the cave.
6. A debutante is in the bank.
7. A fireman is in the park.
8. A giant is in the beach.
9. A giant is in the dungeon.
10. A giant is in the castle.
11. A earl is in the castle.
12. A earl is in the forest.
13. A lawyer is in the store.
…
After studying these facts, they had to judge whether they saw facts such as the following:
A
hippie is in the park.
A hippie is in the cave.
A lawyer is in the store.
A lawyer is in the park.
A debutante is in the bank.
A
debutante is in the cave.
A captain is in the bank.
which contained both studied sentences (targets) and new sentences (foils).
The people and locations for the study sentences could occur in any of one, two, or three of the study sentences. That is called their fan. The following tables show the recognition latencies in seconds for targets and foils as a function of person fan and location fan:
Targets Foils
Location Person Fan Person Fan
Fan 1 2
3 Mean 1
2 3 Mean
1
1.111 1.174 1.222 1.169 1.197 1.221 1.264 1.227
2
1.167 1.198 1.222 1.196 1.250 1.356 1.291 1.299
3
1.153 1.233 1.357 1.248 1.262 1.471 1.465 1.399
Mean 1.144 1.202 1.357 1.20 1.236 1.349 1.340 1.308
The
main effects in the data are that as the fan increases the time to respond
increases and that foil sentences take longer to respond to than the
targets. We will now show how these
effects can be modeled using spreading activation.
5.3 Fan Effect Model
The fan model in the unit 5 materials contains a model for the testing phase of the experiment. The study portion of the task is not included for simplicity and the model already has chunks defined that encode all of the studied sentences and it can perform one trial of the testing phase. Here is a trace of the model performing one trial for the target sentence “The lawyer is in the store”:
> (test-sentence-model "lawyer" "store" t
'person)
Time 0.000: Vision found LOC4349
Time 0.000: Find-Person Selected
Time 0.050: Find-Person Fired
Time 0.050: Module :VISION running command
FIND-LOCATION
Time 0.050: Vision found LOC4350
Time 0.050: Attend-Person Selected
Time 0.100: Attend-Person Fired
Time 0.100: Module :VISION running command
MOVE-ATTENTION
Time 0.185: Module :VISION running command
ENCODING-COMPLETE
Time 0.185: Vision sees TEXT4344
Time 0.185: Retrieve-Person Selected
Sources of activation are:
(Retrieving-Person)
CHUNK Lawyer Activation
10.000 Latency 0.000
Time 0.235: Retrieve-Person Fired
Time 0.235: Lawyer Retrieved
Time 0.235: Encode-Person Selected
Time 0.285: Encode-Person Fired
Time 0.285: Module :VISION running command
FIND-LOCATION
Time 0.285: Vision found LOC4352
Time 0.285: Attend-Location Selected
Time 0.335: Attend-Location Fired
Time 0.335: Module :VISION running command
MOVE-ATTENTION
Time 0.420: Module :VISION running command
ENCODING-COMPLETE
Time 0.420: Vision sees TEXT4343
Time 0.420: Retrieve-Location Selected
Sources of activation are:
(Lawyer Retrieving-Location)
CHUNK Store Activation
10.000 Latency 0.000
Time 0.470: Retrieve-Location Fired
Time 0.470: Store Retrieved
Time 0.470: Encode-Location Selected
Time 0.520: Encode-Location Fired
Time 0.520: Retrieve-From-Person Selected
Sources of activation are:
(Lawyer Store)
Spreading
activation 0.453 from source Store
level 0.500 times IA 0.907
Spreading
activation 0.453 from source Lawyer
level 0.500 times IA 0.907
CHUNK P13 Activation 0.907 Latency
0.254
Time 0.570: Retrieve-From-Person Fired
Time 0.824: P13 Retrieved
Time 0.824: Yes Selected
Time 0.874: Yes Fired
Time 0.874: Module :MOTOR running command
PRESS-KEY
Time 1.024: Module :MOTOR running command
PREPARATION-COMPLETE
Time 1.074: Module :MOTOR running command
INITIATION-COMPLETE
Time 1.084: Device running command OUTPUT-KEY
<< Window "Sentence Experiment" got key #\k at
time 1084 >>
Time 1.174: Module :MOTOR running command
FINISH-MOVEMENT
Time 1.174: Checking for silent events.
Time 1.174: * Nothing to run: No productions, no events.
To run the model through one trial of the test phase you can call the function test-sentence-model. It takes four parameters. The first is a string of the person for the probe sentence. The second is a string of the location for the probe sentence. The third is whether the correct answer is true (t) or false (nil), and the last is either ‘person or ‘location to choose which of the retrieval productions is used (more on that later). The model can be run over each of the conditions to produce a data fit using the average-person-location function:
> (average-person-location)
CORRELATION: 0.864
MEAN DEVIATION: 0.049
TARGETS:
Person fan
Location
1 2 3
fan
1
1.084 (T ) 1.142 (T
) 1.190 (T )
2
1.142 (T ) 1.212 (T
) 1.271 (T )
3
1.190 (T ) 1.271 (T
) 1.339 (T )
FOILS:
1
1.230 (T ) 1.275 (T
) 1.313 (T )
2
1.275 (T ) 1.320 (T
) 1.358 (T )
3
1.313 (T ) 1.358 (T
) 1.396 (T )
Two parameters were estimated to produce that fit to the data. They are the latency factor, F, set to .63 and the maximum associative strength, S, set to 1.6. We will now look at how this model performs the task and how spreading activation leads to the effects in the data.
5.4 Model Representations
The study sentences are encoded in comprehend-sentence chunks like this:
(p13 isa
comprehend-sentence
state study
relation in
arg1 lawyer
arg2 store)
which are propositions encoding the result of past study in the form of an association among the concepts (in this case in, lawyer, and store for “The lawyer is in the store”).
There are also meaning chunks which connect the text read from the display to the concepts. For instance, relevant to the case above we have
(lawyer isa meaning
word "lawyer")
(store
isa meaning
word "store")
The base-level activations of these meaning chunks have been set to 10 to reflect the fact that they are well practiced and should not fail to be retrieved while the activations of the comprehend-sentence chunks are left at the default of 0.
5.5 Perceptual Encoding
In this section we will briefly describe the productions that perform the perceptual portion of the trial. This is similar to the steps that have been done in previous models and thus it should be fairly familiar.
The entire sentence is presented on the screen, but the model only reads the person and location words from the display to perform the task. If the model were to read all of the words in the sentence it would be difficult to be able to respond fast enough to match the experimental data, and in fact recent studies of the fan effect done using an eye tracker verify that in general participants only fixate those two words from the sentences during the testing trials. To make this easier to handle for the model the sentences are presented with the words in fixed locations on the display. To read and encode the words the model goes through a four step process.
The first production to fire issues a request to the visual-location buffer to find the person word:
(P
find-person
=goal>
ISA comprehend-sentence
arg1 nil
state test
==>
=goal>
state looking-for-person
+visual-location>
ISA visual-location
screen-x (within 105 135)
)
Although the text for the word always starts at the same location its exact position will vary based on the length of the word, and thus a within test is used to specify where that word should be found.
The next production harvests that location and requests a shift of attention to it:
(P
attend-person
=goal>
ISA comprehend-sentence
state looking-for-person
=visual-location>
ISA visual-location
=visual-state>
ISA module-state
modality free
==>
=goal>
state attending-person
+visual>
ISA visual-object
screen-pos =visual-location
)
Then the chunk in the visual buffer is harvested and a retrieval request is made to request the chunk that represents the meaning of that word:
(P
retrieve-person
=goal>
ISA comprehend-sentence
state attending-person
=visual>
ISA text
value =word
==>
=goal>
state retrieving-person
+retrieval>
ISA meaning
word =word
)
Finally, that retrieval request is harvested and the meaning chunk is placed into a slot of the goal:
(P
encode-person
=goal>
ISA comprehend-sentence
state retrieving-person
=retrieval>
ISA meaning
==>
=goal>
arg1 =retrieval
state looking-for-location
+visual-location>
ISA visual-location
screen-x (within 400 430)
)
This
production then issues the visual-location
request to find the location word and a similar set of productions attend
and encode the information for that word.
5.4 Determining the Response
After
the encoding has happened the goal chunk will look like this for the sentence
“The lawyer is in the store.”:
Goal
isa COMPREHEND-SENTENCE
relation nil
arg1 Lawyer
arg2 Store
state Get-Retrieval
and
then one of these two productions will be selected and fired to retrieve a
study sentence:
(P
retrieve-from-person
=goal>
ISA comprehend-sentence
arg1 =person
state get-retrieval
==>
=goal>
state nil
+retrieval>
ISA comprehend-sentence
arg1 =person
state study
)
(P
retrieve-from-location
=goal>
ISA comprehend-sentence
arg2 =location
state get-retrieval
==>
=goal>
state nil
+retrieval>
ISA comprehend-sentence
arg2 =location
state study
)
A
thorough model of the task would have those two productions competing and one
would randomly be selected. However, to
simplify things for demonstration the experiment code forces one or the other
to be selected for each trial. The data
is averaged over two runs of each trial – one instance of each of those
productions being selected.
One
important thing to notice is that those productions request the retrieval of a
studied chunk based only on one of the items from the probe sentence. By doing so it ensures that one of the study
sentences will be retrieved instead of a failure for the event of a foil. If retrieval failure were used by the model
to detect the foils then there would be no difference in response times for the
foil probes because the time to fail is based solely upon the retrieval
threshold, but the data clearly shows that the fan of the items affects the
time to respond to foils.
After
one of those productions fires a comprehend sentence chunk representing a study
trial will be retrieved and one of the following productions will fire to
produce a response:
(P yes
=goal>
ISA comprehend-sentence
arg1 =person
arg2 =location
state nil
=retrieval>
ISA comprehend-sentence
arg1 =person
arg2 =location
state study
==>
=goal>
state done
+manual>
ISA press-key
key "k"
)
(P
mismatch-person
=goal>
ISA comprehend-sentence
arg1 =person
arg2 =location
state nil
=retrieval>
ISA comprehend-sentence
-
arg1 =person
==>
=goal>
state done
+manual>
ISA press-key
key "d"
)
(P
mismatch-location
=goal>
ISA
comprehend-sentence
arg1 =person
arg2 =location
state nil
=retrieval>
ISA comprehend-sentence
-
arg2 =location
==>
=goal>
state done
+manual>
ISA
press-key
key "d"
)
If the retrieved sentence matches the probe then the model
responds with the true response, “k”, and if either one of the components does
not match then the model responds with “d”.
5.5 Retrieving the Critical
Study Chunk
The
perceptual encoding productions have a fixed cost of 520 ms and the time to
respond after retrieving a comprehend-sentence chunk is 260 ms. Those times are constant across all trials. The difference in the conditions will result
from the times to retrieve the studied sentences. Recall from the last unit that the time to
retrieve a chunk i is based on its activation and specified by
the equation:
Thus,
it is differences in activation of the comprehend-sentence chunks which will
result in the different times to respond to different trials.
The
goal chunk at the time of the retrieval (after either retrieve-from-person or
retrieve-from-location fires) will look like this:
Goal
isa COMPREHEND-SENTENCE
relation nil
arg1 person
arg2 location
state nil
where
person and location will be the chunks that represent the meanings for the
particular probe being presented. The
retrieval request will look like:
+retrieval>
isa
comprehend-sentence
arg1 person
state study
or
this:
+retrieval>
isa
comprehend-sentence
arg2 location
state study
depending
on which of the productions was chosen to perform the retrieval.
The
important thing to note is that because the sources of activation in the goal
are the same for either retrieval request the spreading activation will not
differ between the two cases. You might
wonder then why we would need to have both options. That will be described in the detailed
examples below.
5.5.1 A simple target trial
The
first case we will look at is the target sentence “The lawyer is in the
store”. Both the person and location in
this sentence have a fan of one – they each only occur in that one study
sentence.
Thus,
the goal chunk will look like this at the time of the critical retrieval:
Goal
isa COMPREHEND-SENTENCE
relation nil
arg1 lawyer
arg2 store
state nil
We
will now look at the retrieval that results from the retrieve-from-person
production firing. For the following traces we have enabled the activation
trace parameter (:act) and the exact match trace (:emt) parameter which show a
lot of additional information when a retrieval attempt is made (the :emt
parameter is disabled in the model as provided because it generates a lot of
additional trace material which is not really important). Here is the trace of the model when that
retrieval occurs:
Time 0.520: Retrieve-From-Person Selected
Sources of activation are:
(Lawyer Store)
Matching CHUNKs of type COMPREHEND-SENTENCE.
Matching CHUNK P1.
Value Hippie is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P2.
Value Hippie is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P3.
Value Hippie is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P4.
Value Captain is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P5.
Value Captain is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P6.
Value Debutante is
different from condition Lawyer in slot arg1: test fails.
Matching CHUNK P7.
Value Fireman is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P8.
Value Giant is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P9.
Value Giant is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P10.
Value Giant is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P11.
Value Earl is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P12.
Value Earl is different
from condition Lawyer in slot arg1: test fails.
Matching CHUNK P13.
Spreading
activation 0.453 from source Store
level 0.500 times IA 0.907
Spreading activation 0.453 from source Lawyer level 0.500 times IA 0.907
Activation 0.907 is larger than previous best 0.000: selecting P13.
Matching CHUNK Goal.
Value nil is different from
condition Study in slot state: test fails.
CHUNK P13 Activation 0.907 Latency
0.254
Time 0.570: Retrieve-From-Person Fired
With
those traces on it shows the chunks that are the sources of activation – lawyer
and store. Then it shows all of the chunks that are attempted to be matched to
the retrieval request and the activation that is spread from the source to
those chunks.
In
this case, the only chunk which matches the request is chunk p13 and it is also
the only one to receive any spreading activation. Note that this would look exactly the same if
the retrieve-from-location production had fired because it would still be the
only chunk that matched the request and the sources of activation are the same
regardless of which one fires.
This
portion of the trace shows the equations we have described above being computed
by the model:
Matching CHUNK P13.
Spreading
activation 0.453 from source Store
level 0.500 times IA 0.907
Spreading
activation 0.453 from source Lawyer
level 0.500 times IA 0.907
Activation 0.907 is larger than previous best 0.000: selecting P13.
…
CHUNK P13 Activation 0.907 Latency
0.254
Remember
that we have set the parameter F to .63, the parameter S to 1.6, and the
base-level activation for the chunks is 0 in this model.
Looking
at this trace, we see the Sji values from store to P13 and lawyer to
P13 (called the IA in the trace) are both .907.
That comes from the equation:
The
value of S was estimated to fit the data as 1.6 and the fan of both the store
and lawyer chunks is 2. They each occur
as a slot value in only the P13 chunk plus one for their reference to
themselves. Then substituting into the
equation we get:
The
Wj values are .5 because the source activation is the default value
of 1 and there are two source chunks.
Thus the activation of chunk P13 is:
Finally,
we see the time to complete the retrieval (the latency listing in the trace)
computed as:
Adding
that retrieval time to the fixed costs of .52 seconds to do the perception, .05
seconds to fire the production that requests the retrieval, and .26 seconds to
perform the response gives us a total of 1.084 seconds, which the value in the
fan 1-1 cell of the model data presented above.
Now
that we have looked at the intricate details of how the retrieval times are
determined for the simple case we will look at a few other cases.
5.5.2 A different target trial
The target sentence “The hippie
is in the bank” is a more interesting case to look at. Hippie is the person in three of the study
sentences and bank is the location in two of them. Now we will see why it takes the model longer
to respond to such a probe. Here are the
critical components from the trace when retrieve-from-person is chosen:
Time
0.520: Retrieve-From-Person Selected
Sources of activation are: (Hippie Bank)
Matching CHUNKs of type COMPREHEND-SENTENCE.
Matching CHUNK P1.
Spreading activation 0.107 from source Hippie level 0.500 times IA 0.214
Activation
0.107 is larger than previous best
0.000: selecting P1.
Matching CHUNK P2.
Spreading activation 0.107 from source Hippie level 0.500 times IA 0.214
Activation
0.107 is larger than previous best
0.107: selecting P2.
Matching CHUNK P3.
Spreading activation 0.251 from source Bank level 0.500 times IA 0.501
Spreading activation 0.107 from source Hippie level 0.500 times IA 0.214
Activation
0.358 is larger than previous best
0.107: selecting P3.
…
CHUNK P3
Activation 0.358 Latency 0.441
There are three chunks that
match the request for a comprehend-sentence chunk with an arg1 value of
hippie. Each receives the same amount of
activation being spread from hippie.
Because hippie is a member of three chunks it has a fan of 4 and thus
the S(hippie)i value is:
Chunk
P3 also contains the chunk bank in its arg2 slot and thus receives the source
spreading from it as well.
Now we
will look at the case when retrieve-from-location fires for this probe
sentence:
Time
0.520: Retrieve-From-Location Selected
Sources of activation are: (Hippie Bank)
Matching CHUNKs of type COMPREHEND-SENTENCE.
…
Matching CHUNK P3.
Spreading activation 0.251 from source Bank level 0.500 times IA 0.501
Spreading activation 0.107 from source Hippie level 0.500 times IA 0.214
Activation
0.358 is larger than previous best
0.000: selecting P3.
Matching CHUNK P6.
Spreading activation 0.251 from source Bank level 0.500 times IA 0.501
…
CHUNK P3 Activation 0.358 Latency
0.441
In this
case there are only two chunks which match the request for a
comprehend-sentence chunk with an arg2 value of bank.
Regardless
of which production fired to request the retrieval, chunk P3 had the highest
activation because it received spreading activation from both sources. Thus, even if there is more than one chunk
which matches the retrieval request issued by retrieve-from-person or
retrieve-from-location the correct study sentence will always be retrieved
because its activation will be the highest, and that activation value will be
the same in both cases.
Notice
that the activation of P3 is less than the activation that P13 had in the
previous example because the source activation being spread is less. That is because the sources here have a
higher fan, and thus a lesser Sji.
Because the activation is smaller, it takes longer to retrieve such a
fact and that gives us the difference in response time effect of fan in the
data.
5.5.3 A foil trial
Now we
will look at a foil trial. The foil
probe “The giant is in the bank” is similar to the target that we looked at in
the last section. The person has a fan
of three and the location has a fan of two.
This time however there is no matching study sentence. Here are the critical components from the
trace when retrieve-from-person is chosen:
Time
0.520: Retrieve-From-Person Selected
Sources of activation are: (Giant Bank)
Matching CHUNKs of type COMPREHEND-SENTENCE.
…
Matching CHUNK P8.
Spreading activation 0.107 from
source Giant level 0.500 times IA 0.214
Activation
0.107 is larger than previous best
0.000: selecting P8.
Matching CHUNK P9.
Spreading activation 0.107 from
source Giant level 0.500 times IA 0.214
Activation
0.107 is larger than previous best
0.107: selecting P9.
Matching CHUNK P10.
Spreading activation 0.107 from
source Giant level 0.500 times IA 0.214
Activation
0.107 is larger than previous best
0.107: selecting P10.
…
CHUNK P10 Activation 0.107 Latency
0.566
There are three chunks that
match the request for a comprehend-sentence chunk with an arg1 value of giant
and each receives the same amount of activation being spread from giant.
However, none contain an arg2 value of bank. Thus they only get activation
spread from one source and have a lesser activation value than the
corresponding target sentence probe has.
Because the activation is smaller, the retrieval time is greater. This results in the effect of foil trials
taking longer than target trials.
Before concluding this section
however let us look at the trace if retrieve-from-location were to fire for
this foil:
Time
0.520: Retrieve-From-Location Selected
Sources of activation are: (Giant Bank)
Matching CHUNKs of type COMPREHEND-SENTENCE.
…
Matching CHUNK P3.
Spreading activation 0.251 from source Bank level 0.500 times IA 0.501
Activation
0.251 is larger than previous best
0.000: selecting P3.
…
Matching CHUNK P6.
Spreading activation 0.251 from source Bank level 0.500 times IA 0.501
Activation
0.251 is larger than previous best
0.251: selecting P6.
…
CHUNK P6 Activation 0.251 Latency
0.490
In this
case there are only two chunks which match the request for a
comprehend-sentence chunk with an arg2 value of bank.. Again, the activation of the chunk retrieved
is less than the corresponding target trial, but it is not the same as when
retrieve-from-person fired. That is why
the model is run with each of those productions fired once for each probe and
the results averaged together. Otherwise
the foil data would only show the effect of fan for the item that was used to
retrieve the study chunk.
5.6 Partial Matching
Up
to now models have either always retrieved a chunk which matched the retrieval
request or resulted in a failure to retrieve anything. Now we will look at modeling errors in recall
in more detail. There are two kinds of
errors that can occur. One is an error
of commission when the wrong thing is recalled.
This will occur when the activation of the wrong chunk is greater than
the activation of the correct chunk. The
second is an error of omission when nothing is recalled. This will occur when no chunk has activation
above the retrieval threshold.
We
will continue to look at productions from the fan model for now. In particular, this production requests the
retrieval of a chunk:
(P
retrieve-from-person
=goal>
ISA comprehend-sentence
arg1 =person
state get-retrieval
==>
=goal>
state nil
+retrieval>
ISA comprehend-sentence
arg1 =person
state study
)
In this case an attempt is being made to retrieve a comprehend-sentence chunk with a particular person (bound to =person) that had been studied. If =person were the chunk giant, this retrieval request would be looking for a chunk of the form:
+retrieval>
isa comprehend-sentence
arg1 giant
state study
As was shown above, there were three chunks which matched that
request in the study set and one of those will be retrieved.
However, let us consider the case where there had been no study
sentences with the person giant but there had been a sentence with the person
titan in the location being probed with giant i.e. there was a study sentence “The
titan is in the bank” and the test sentence is now “The giant is in the
bank”. In this situation one might
expect that real participants might incorrectly classify the probe sentence as
one that was studied because of the similarity between the words giant and
titan. The model however could not make
such an error.
This is what the partial matching mechanism is designed to
address. When partial matching is
enabled (by setting the :pm parameter to t) the similarity between the chunks
in the retrieval specification and the chunks in the slots of the chunks in
declarative memory are taken into consideration. The chunk with the highest activation will
still be the one retrieved, but with partial matching enabled that chunk may
not have the exact slot values as specified in the retrieval request.
The
activation Ai of a chunk i is defined fully as:
Bi,
Wj, Sji, and e
have been discussed previously. The new
term is the partial matching component.
Specification
elements k: The matching summation is computed over the slot values of
the retrieval specification.
Match Scale, P: This reflects the amount of weighting given to the
similarity in slot k. By default this is
a constant across all slots with the value of 1 and is set with the :mp
parameter.
Match Similarities, Mki: The similarity between the value k in the
retrieval specification and the value in the corresponding slot of chunk i.
Similarity
values, the Mki’s, can be set by the modeler along with the
scale on which they are defined. The
scale range is set with a maximum similarity (set using the :ms parameter) and
a maximum difference (set using the :md parameter). By default :ms is 0 and :md is -1. The similarity between anything and itself is
automatically set to the maximum similarity and by default the similarity
between any other pair of values is the maximum difference. However, it is possible to choose some value
between maximum similarity and maximum difference for items that are deemed to
be somewhat similar. Thus, unless one sets specific similarities or changes the
range of similarities, the match similarity
will be 0 when the element in the chunk’s slot matches the retrieval
specification and –1 when it does not.
5.7 Grouped Recall
To illustrate these ideas we will use a model called grouped. This is a demonstration model of a grouped recall task which is based on a larger model of a complex recall experiment. As with the fan model, the studied items are already specified in the model, so it does not model the encoding and study of the items. In addition, the response times and error profiles of this model are not fit to any data. This demonstration model is designed to show the mechanism of partial matching and how it can lead to errors of commission and errors of omission. Because the model is not fit to any data, and the mechanism being studied does not rely on any of the perceptual or motor modules of ACT-R, they are not being used, and instead the chunks are manipulated “directly” by the Lisp code that provides the experiment. This technique of using only the cognitive system in ACT-R can be useful when modeling a task where the timing is not important or other situations where accounting for a “real world” interaction is not necessary to accomplish the objectives of the model. The experiment description text for this unit gives the details of how that is accomplished in this model.
If you check the global parameters you will see that it has a retrieval threshold of -0.5 and a value of .15 for the transient noise s parameter. Also, to simplify the demonstration the spreading activation described above is disabled by setting the goal activation to 0. This model is set up to recall a list of nine items which are encoded in groups of three elements. The list that should be recalled is (123) (456) (789). To run the model, call the run-grouped-recall function. You will get a trace like the one below and the list of responses will be returned. Because of the activation noise it will likely vary from run to run. In this run, notice that it mis-ordered the recall of the 5 and 6 and failed to recall the last item, producing (123) (465) (78).
Time
0.000: Recall-First-Group Selected
Time
0.050: Recall-First-Group Fired
Time
0.851: Group1 Retrieved
Time
0.851: Start-Recall-Of-Group Selected
Time
0.901: Start-Recall-Of-Group Fired
Time
1.868: Item1 Retrieved
Time
1.868: Harvest-First-Item Selected
Time
1.918: Harvest-First-Item Fired
Time
2.676: Item2 Retrieved
Time
2.676: Harvest-Second-Item Selected
Time
2.726: Harvest-Second-Item Fired
Time
3.648: Item3 Retrieved
Time
3.648: Harvest-Third-Item Selected
Time
3.698: Harvest-Third-Item Fired
Time
5.347: Failure Retrieved
Time
5.347: Second-Group Selected
Time
5.397: Second-Group Fired
Time
6.471: Group2 Retrieved
Time
6.471: Start-Recall-Of-Group Selected
Time
6.521: Start-Recall-Of-Group Fired
Time
7.443: Item4 Retrieved
Time
7.443: Harvest-First-Item Selected
Time
7.493: Harvest-First-Item Fired
Time
8.528: Item6 Retrieved
Time
8.528: Harvest-Second-Item Selected
Time
8.578: Harvest-Second-Item Fired
Time
9.724: Item5 Retrieved
Time
9.724: Harvest-Third-Item Selected
Time
9.774: Harvest-Third-Item Fired
Time 11.423: Failure Retrieved
Time 11.423: Third-Group Selected
Time 11.473: Third-Group Fired
Time 12.433: Group3 Retrieved
Time 12.433: Start-Recall-Of-Group Selected
Time 12.483: Start-Recall-Of-Group Fired
Time 13.338: Item7 Retrieved
Time 13.338: Harvest-First-Item Selected
Time 13.388: Harvest-First-Item Fired
Time 13.969: Item8 Retrieved
Time 13.969: Harvest-Second-Item Selected
Time 14.019: Harvest-Second-Item Fired
Time 15.668: Failure Retrieved
Time 15.668: Checking for silent events.
Time 15.668: * Nothing to run: No productions, no events.
("1"
"2" "3" "4" "6" "5” "7"
"8")
5.8 Error of
Commission
If one turns on the activation trace and the partial matching trace (set the :act and :pmt parameters to t) one will get a much more exhaustive trace explaining the computations taking place. The following is from the detailed trace of the above run relevant to an error of commission when ACT-R recalls 6 in the second position where 5 is the correct item. The critical comparison is between item5, which should be retrieved and item6, which is retrieved:
Sources of activation are: (List Group2
Recalling-Item Second)
Matching CHUNKs of type ITEM.
Adding noise 0.360
...
Adding noise -0.241
Partial matching chunk Item5 with activation
-0.241
Similarity between chunks Group2 and Group2
is 0.000
Adjusting activation by 0.000 to -0.241
Similarity between chunks Second and Second
is 0.000
Adjusting activation by 0.000 to -0.241
Similarity between chunks Unused and Retrieved
is -1.000
Matching score of chunk Item5 is -0.241.
Activation -0.241 is larger than previous best
-0.500: selecting Item5.
Adding noise 0.465
Partial matching chunk Item6 with activation 0.465
Similarity between chunks Group2 and Group2
is 0.000
Adjusting activation by 0.000 to
0.465
Similarity between chunks Third and Second is
-0.500
Adjusting activation by -0.500 to -0.035
Similarity between chunks Unused and Retrieved
is -1.000
Matching score of chunk Item6 is -0.035.
Activation -0.035 is larger than previous best
-0.241: selecting Item6.
These report the calculations of the activation equation (ignoring
associative strength because W is
set to 0).
In these examples the base-level activations, Bi, have their default value of 0, the match scale, P, has the default value of 1, and the
only noise value is the transient component.
So the calculations are really just a matter of adding up the match
similarities, Mki and
adding the transient noise. The matching
of item5 is repeated below. While the base level is 0 it starts out at
-0.241 because of transient noise. The
retrieval pattern is
+retrieval>
isa item
parent Group2
position second
- status retrieved
and
item5 is:
Item5
isa ITEM
name "5"
parent Group2
position Second
status unused
which matches both on the parent and position slots resulting in the addition 0 to the base-level activation. It also satisfies the negative test on the status slot and thus does not have to pay the penalty of a mismatch. Thus its final activation is -0.241.
Adding noise -0.241
Partial matching chunk Item5 with activation
-0.241
Similarity between chunks Group2 and Group2
is 0.000
Adjusting activation by 0.000 to -0.241
Similarity between chunks Second and Second
is 0.000
Adjusting activation by 0.000 to -0.241
Similarity between chunks Unused and Retrieved
is -1.000
Matching score of chunk Item5 is -0.241.
Activation -0.241 is larger than previous best
-0.500: selecting Item5.
This activation value is higher than the threshold of -.5, so item5 is a temporary choice but then we come to the matching of item6:
Item6
isa ITEM
name "6"
parent Group2
position Third
status unused
Adding noise 0.465
Partial matching chunk Item6 with
activation 0.465
Similarity between chunks Group2 and Group2
is 0.000
Adjusting activation by 0.000 to
0.465
Similarity between chunks Third and Second is
-0.500
Adjusting activation by -0.500 to -0.035
Similarity between chunks Unused and Retrieved
is -1.000
Matching score of chunk Item6 is -0.035.
Activation -0.035 is larger than previous best
-0.241: selecting Item6.
The similarity between second and third is -0.5. However, the initial activation starts out at 0.465 because of transient noise and so the final activation is 0.465 + -0.5 = -0.035 which is greater than the activation for item5. Thus, this wrong item is selected because of the fluctuation in activations.
The similarities between the different positions are defined in the model using the setsimilarities command:
(setsimilarities
(first second -0.5)
(second third -0.5)
(first third -1))
Similarity values are symmetric, thus it is not necessary to also specify (second first -0.5). The similarity between a chunk and itself has the value of maximum similarity by default, and thus also does not need to be specified.
5.9 Error of Omission
Here is the portion of the detailed trace relevant to the failure to recall the ninth item:
Sources of activation are: (List Group3
Recalling-Item Third)
Matching CHUNKs of type ITEM.
...
Adding noise 0.301
Partial matching chunk Item8 with
activation 0.301
Similarity between chunks Group3 and Group3
is 0.000
Adjusting activation by 0.000 to
0.301
Similarity between chunks Second and Third is
-0.500
Adjusting activation by -0.500 to -0.199
Similarity between chunks Retrieved and
Retrieved is 0.000
Adjusting activation by -1.000 to -1.199
Matching score of chunk Item8 is -1.199.
Adding noise -0.594
Partial matching chunk Item9 with activation
-0.594
Similarity between chunks Group3 and Group3
is 0.000
Adjusting activation by 0.000 to -0.594
Similarity between chunks Third and Third
is 0.000
Adjusting activation by 0.000 to -0.594
Similarity between chunks Unused and Retrieved
is -1.000
Matching score of chunk Item9 is -0.594.
CHUNK failure Activation -0.500 Latency 1.649
No chunk reached activation threshold -0.500:
matching fails.
Item9 starts out with an activation of –0.594 and because it does
not get any penalties added that is its activation, but it is below the
threshold of -0.5, so it is not retrieved.
One other
thing to note about that trace is the activation of item8. It starts out with an
activation of 0.301 and gets a penalty of -0.500 based on the similarity
between second and third. So, at that
point it has an activation of -0.199 which is above the threshold. However, it matches perfectly on the status
slot, but the test is one of negation so it also gets the maximum difference
penalty added to it. If the retrieval
specification did not request an item that had not been previously retrieved item8 would have been incorrectly
retrieved for the ninth position instead of a complete retrieval failure. The important thing about a negative test is
that it will not change the activation if the similarity between the chunks in
the test is less than the maximum similarity, but if the similarity between the
chunks is the maximum similarity it will receive an activation penalty of the
maximum difference, which, with the default values is -1.0.
5.10 Unit
Exercise: Simple Addition
The following are data obtained by Siegler and Shrager on the
relative frequencies of different responses by 4 year olds to addition
problems. It seems likely that many of the kids did not know the
answers to the larger problems so we will only focus on the addition table from
1+1 to 3+3.
0
1 2 3
4 5 6
7 8 Other (includes no response)
1+1
- .05 .86
- .02 -
.02 - -
.06
1+2
- .04 .07
.75 .04 -
.02 - -
.09
1+3
- .02 -
.10 .75 .05
.01 .03 -
.06
2+2 .02
- .04 .05
.80 .04 -
.05 - -
2+3
- - .07
.09 .25 .45
.08 .01 .01
.06
3+3 .04
- - .05
.21 .09 .48
- .02 .11
The siegler model
contains the functions to perform a version of this task with a model. As with the grouped model, there is no interface generated for the task so it
is not possible to run yourself through the experiment, but one would guess
that you would probably not make the same mistakes as 4 year olds anyway. For
the model, the goal chunk is directly modified to provide the stimuli. This simplification is done because we are
not concerned with matching latencies so the time of the encoding is not a factor and also because it is assumed
that the errors are not a result of improper encoding of the items.
The function called test-fact
will present 1 trial with the specified numbers and return the contents of the
goal chunk’s answer slot, which should be a string from the name slot of a
number chunk if the model has an answer. The numbers that are passed to
test-fact need to be the names of the numbers as strings. Thus, to run a trial of 1+2 you would call (test-fact "one" "two"). When the model is doing the task it is reset
before each trial, then the arg1 and arg2 slots of the goal chunk (named goal)
are set to the names of the numbers being presented. The model then needs to encode those names
and determine a response. The name of
that number should be placed in the answer slot of the goal. This is similar to how one would map a visual
representation to a corresponding number chunk before attempting a retrieval
and then retrieve a chunk to extract it’s response value from one of its slots.
The function called do-one-set takes no parameters and runs one trial of each problem returning the list of responses. Since the model is reset each time the order of presentation does not matter and it is not randomized.
The function run-subjects takes
one parameter which is the number of times to run do-one-set. It will then
tally all of the responses, report the fit to the data, and display the results
in a table.
The following is the output from a run of my model of the task:
CORRELATION: 0.976
MEAN DEVIATION: 0.058
0
1 2 3
4 5 6
7 8 Other
1+1 0.00
0.07 0.85 0.07
0.00 0.00 0.00
0.00 0.00 0.00
1+2 0.00
0.00 0.06 0.86
0.07 0.00 0.00
0.00 0.00 0.00
1+3 0.00
0.00 0.00 0.09
0.86 0.04 0.00
0.00 0.00 0.01
2+2 0.00
0.00 0.00 0.09
0.90 0.00 0.00
0.00 0.00 0.01
2+3 0.00
0.00 0.00 0.02
0.34 0.55 0.03
0.00 0.00 0.06
3+3 0.00
0.00 0.00 0.00
0.01 0.05 0.70
0.06 0.00 0.17
Your task is to write a model that encodes the question presented and then responds with an answer (by placing it in the answer slot of the goal) and produces a distribution of errors that is comparable to the data. This should be achievable by manipulating the similarities, the activation noise, the retrieval threshold, and the base-level activations. The values that these parameters have in the given model are most likely not at the optimal values for fitting the data. The effects that these parameters will have on the data will be described below. Source activation, W, is set to zero and can be left there to do the assignment, but if you would like to explore the effect it has feel free to set W to a non-zero value and experiment.
You are given a set of chunks that encode the numbers from 0-9,
the addition table from 0+0 to 5+5, and a goal of type problem. The chunk-type of the goal looks like this:
(chunk-type
problem arg1 arg2 answer state)
The arg1 and arg2 slots are set to contain the string names of the
numbers at the beginning of each trial automatically, so your model then needs
to encode those so that a plus-fact can (possibly) be retrieved.
The similarities between the number chunks will affect the
distribution of incorrect retrievals.
While this looks like 45 free parameters to be fit, in practice that is
just not reasonable. For a situation
like this, where the chunks represent numbers, it is better to set the
similarity between two numbers based on the numerical difference between them
using a single formula to specify all of them.
To set the similarities you need to use the setsimilarities command.
Here is an example of its use from the grouped model:
(setsimilarities
(first second -0.5)
(second third -0.5)
(first third -1))
The similarities are symmetric, so the order of the chunks in the
lists does not matter, and you only need to set the similarity for the pair
once.
The activation noise is going to impact the frequency of incorrect
retrievals. The more noise there is the
less likely it is that the correct chunk will have the highest activation. The noise is set with the :ans parameter.
The retrieval threshold is going to establish the chance for a
retrieval failure. Its setting is going
to depend on the average activation levels which will depend on how the other
parameters are set. It may be best to
set it to a very low value at the start (like -10) so that there is always a
plus-fact retrieved. Then, once the distribution looks good you can increase it
until it starts to introduce some failures to improve the fit. It is set using the :rt parameter.
You may also want to set the base-level activation of some chunks directly. Because base-level learning is not enabled, those base-level values will not change as the model runs. This will make particular chunks more likely to be retrieved than others. In the model code provided the base-levels of the number chunks are set high so that there should never be a failure to retrieve a number:
(set-general-base-levels
(zero 10) (one 10) (two 10) (three 10) (four
10) (five 10)
(six 10) (seven 10)(eight 10) (nine 10))
Depending on how you set your other parameters, you may want to adjust the base-level for the numbers even higher than that. This is another instance where it looks like there are a lot of free parameters that could be used to fit the data, but again a principled approach is advised. If you need to adjust the base-levels do it in a manner that seems justified. For example, if +0 facts are more often retrieved as errors than the rest you could set all of those to the same larger base-level. A model that sets a different base-level for each plus-fact does not really demonstrate anything.
There is one other thing that you will need to do to produce the response that has not yet been demonstrated in the tutorial. That is to retrieve a chunk directly. If you have a chunk bound to a variable in a production you can request that that chunk be retrieved like this:
(P demo
=goal>
ISA test-goal
value =val
==>
+retrieval> =val)
That will attempt to retrieve that chunk specifically. It does not attempt to retrieve any other chunk, so it will not retrieve a chunk that is similar even if it has a higher activation.
In addition to the parameters suggested, you may also want to adjust the match scale, P (set with :mp). Because the default range of similarities is -1.0 to 0.0 there is not going to be much difference in the activations due to similarities between the numbers, and as a result the predictions can be very sensitive to small changes in the parameters. Thus, if you adjust P you may find it easier to arrive at a good fit to the data. While this is not essential we suggest changing it to a value like 10 which we used in the reference solution.
Siegler, R. S., & Shrager, J. (1984). Strategy
choices in addition and subtraction: How do children know what to do? In C.
Sophian (Ed.), Origins of cognitive skills (pp. 229-293).