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PHENOMENAL DATA MINING: FROM
DATA TO PHENOMENA
Computer Science Department
Stanford, CA 94305
2000 Feb 23, 5:07 p.m.
Phenomenal data mining ﬁnds relations between the data and the
phenomena that give rise to data rather than just relations among the
For example, suppose supermarket cash register data does not
identify cash customers. Nevertheless, there really are customers, and
these customers are characterized by sex, age, ethnicity, tastes, income
distribution, and sensitivity to price changes. A data mining program
might be able to identify which baskets of purchases are likely to have
been made by the same customers. In this example, the receipts are
the data, and the customers are phenomena not directly represented
in the data. Once the “baskets” of purchases are grouped by customer,
the way is open to infer further phenomena about the customers, e.g.
their sex, age, etc.
This article concerns what can be inferred by programs about phe-
nomena from data and what facts are relevant to doing this.1 We work
mainly with the supermarket example, but the idea is general.
1In a sense, all data mining is phenomenal; it’s just that the phenomenal part is usually
done by hand. We want the computer to do the phenomenal part also.
In order to infer phenomena from data, facts about their relations
must be supplied. Sometimes these facts can be implicit in the pro-
grams that look for the phenomena, but more generality is achieved
if the facts are represented as sentences of logic in a knowledge base
used by the programs.
The result of phenomenal data-mining might include an extended
database with additional ﬁelds on existing relations and new relations.
Thus the relations describing supermarket baskets might be extended
with a customer ﬁeld, and new relations about customers and their
properties might be introduced.
Science and common sense both tell us that the facts about the world are not
directly observable but can be inferred from observations about the eﬀects
of actions. What people infer about the world is not just relations among
observations but relations among entities that are much more stable than
observations. For example, 3-dimensional objects are more stable than the
image on a person’s retina, the information directly obtained from feeling
2 Likewise the fact
an object or on an image scanned into a computer.
that a customer has children is more stable than the fact that a particular
basket includes Roll-ups. The fact that a customer has diabetes is more
stable than a particular pattern of food purchases that may allow inferring
that he has diabetes. The phenomenal facts, partly because they are more
stable than observations, are more predictive of future behavior than simple
The extreme positivist philosophical view that science concerns relations
among observations still inﬂuences the design of learning programs, and
that’s what data miners are. However, science never worked that way, nei-
ther do babies and neither should data mining programs. All obtain and use
representations of the objects and use observations only as a means to that
2Even very young babies have a lot of innate knowledge of the world. My on-line article
The Well-Designed Child, http://www-formal.stanford.edu/jmc/child.html concerns
what innate knowledge children probably do have about the world and what knowledge
robots should be given. Elizabeth Spelke, (Spelke 1994), investigates innate knowledge in
Data mining involves computer programs that infer relations among dif-
ferent kinds of data in large databases. The goal has been to infer useful
relations that might not have been noticed or at least could not have been
conﬁrmed among this data. We use the standard example of a supermarket
chain with a database of all the cash register receipts for some long time
period—many gigabytes of data. The company wants to mine this database
for information useful for improving its business.
Data-mining can be made to do more than just ﬁnd relations among data.
Data amounts to observations of the world, and it is possible to infer relations
among entities in the world from the data. Such relations are likely to be as
useful to know about as are relations among the entities directly represented
in the data.
In the supermarket chain example, there are people, groups
of people, their homes with pantries, refrigerators and freezers and facts
about what they cook and what they eat. It should even be possible to infer
the existence of entities in the world, such as previously unidentiﬁed groups
of people with distinct eating and purchasing habits. Another example is
to identify bellwether groups; what they buy today, many more will buy
Moreover, the information will usually admit a more compact description
in terms of the underlying phenomena than in terms of the data.
Although all common sense level knowledge of the world is potentially
relevant to data mining, formalizing common sense has proved to be a dif-
ﬁcult AI problem, and progress has been slow. Nevertheless, we can expect
that certain phenomena will be related to the information in databases in a
straightforward enough way so that information about them can be found
by data miners.
2 Phenomena in the World
What phenomena in the world should a data mining program have built into
it, be told or be able to discover for itself?
At ﬁrst, knowledge of the general phenomena will be built into the data
miners (data mining programs), and the programs will infer speciﬁc values.
Later data miners should use the information expressed in a logical form.
This will permit them to use databases of common sense facts about the
world. Very ambitious data mining projects might hope to make programs
that will come up with entirely new phenomena.
Here are some phenomena and facts relevant to the supermarket domain
together with logical expressions for some of these facts. We give just two
example formulas, and these are not part of a worked out scheme for con-
structing a knowledge base.
people There are the shoppers themselves and also family members. The
data may not identify them directly, but learning about them is the
point of data mining.
Shopper(x) → F
amily(x) ⊂ P eople.
ownership and purchases People buy things and then own them and keep
them somewhere. Maybe the facts about where people keep things are
not relevant for most data mining. The distinction between durable
goods and consumables is important. Here’s a situation calculus ex-
pression of two facts.
Durable(x) ∧ Has(person, x, s)
→ Has(person, x, Result(U
ses(person, x), s))
¬Durable(x) ∧ Has(person, x, s)
→ ¬Has(person, x, Result(U
ses(person, x), s))
possessions Freezers, refrigerators, cars and microwave ovens are items that
some customers will have and others won’t. Having them aﬀects be-
events The observed events are purchases in the stores for which we have
Unobserved are the trips to the store and the cooking and eating and
the inspections of the larder. Maybe these can usefully be discrimi-
nated, but maybe they should be lumped into consumption. Other
unobserved events include purchases from the competitors. When a
person purchases a freezer, his status changes to that of a freezer owner
and that fact will persist. The event of acquiring a freezer is more com-
mon than that of giving up the possession of a freezer.
preferences People have preferences among states of aﬀairs—or more specif-
ically among objects.
distributions of properties over people The customers have age, sex,
income and ethnic distributions.
customers appear and disappear There are causes for the appearance
and disappearance of customers, and supermarket chains will be inter-
ested in ﬁnding them out. These include moving in or out of the area,
change in family circumstances, advertising campaigns by the chain or
its competitors, changes in the store or its hours of operation, satisfac-
tion or dissatisfaction with goods, prices or service.
The present state of AI is not up to formulating a full common sense
database, but full common sense knowledge is not necessary. We can expect
to do a lot with very limited knowledge. A sophisticated data mining system
might be able to use the following facts in its formulation of hypotheses. An
ambitious logic-based system might use logical expressions of the facts. Less
ambitiously, programmers would use them in designing data mining systems.
functions and logical assertions are sometimes needed to express the
frigerators, some into freezers and some are eaten right away. When a
food object is eaten it is removed from where it was stored.
11. There are bounds on the rate at which people eat. What they don’t
get from one store they get from another.
12. A person has an age which increases with time. Very young people are
13. There are lots of people an lots of stores. The data miner will have
information about only some of them.
14. Customers who buy substantial quantities of frozen or freezable goods
15. Owners of microwave ovens can be identiﬁed.
16. Consistent purchase of the most expensive items indicates prosperity.
It can be asked whether consistent purchase of expensive items is all
the data miner wants to know anyway. I don’t know about that.
17. Suppose a customer comes rarely and always buys frozen spinach in
bags and a few other items. Inference: the store where he buys most
of his food doesn’t sell frozen spinach in bags.
The point is that all the above are a priori facts that may be used to
infer phenomena. We suppose that only some phenomena need be taken into
account. For this phenomenal mining we ignore birth and death, physical
motion, and shape. Mass is taken into account only in connection with
quantities purchased and rates of consumption.
It is clear that a very large number of facts are relevant to getting infor-
mation out of databases of customer purchases. These include general facts
of common sense and speciﬁc facts about consumer properties, consumer
goods and consumer behavior. I see no alternative to a big project like CyC,
as described by Lenat and Guha (Lenat and Guha 1990), for putting facts
into a knowledge base by hand. However even a small knowledge base may
be useful and adequate for experiments. Perhaps substantial parts of the
existing CyC knowledge base may provide part of the information needed for
supermarket phenomenal data mining.
3 Grouping supermarket purchases by cus-
We propose programs to determine from the cash register receipts which
baskets were purchased by the same customer. The putative customers can
then be given identiﬁers. Programs can infer more facts about customer
characteristics and behavior with facts about purchases of an identiﬁed cus-
tomer over time than could be inferred from mere statistics about the baskets
This example of phenomenal data mining is straightforward in that it is
reasonably clear what a successful result would be and how it might be used.
We hope to make it plausible that enough information is present in the data
to usefully distinguish customers. However, experiment is needed to verify
that feasible algorithms exist.
Demographic information about customers is known to be useful, e.g.
their ages, occupations, sexes and incomes. When this information is sup-
in mail order situations where credit is granted, it is extensively
used. However, in our supermarket chain example, that information is not
in the database of transactions. Let us consider inferring it; it might then be
used in any of the presently conventional ways.
There are several approaches to associating baskets purchased by the
3.1 Minimizing anomaly in assignments of baskets to
One approach involves minimizing total anomaly in the assignment of baskets
Deﬁnition: A partial assignment α groups some of the baskets of pur-
chases according to whether they were purchased by the same customer.
Each group also includes an identiﬁer c for the customer and a classiﬁcation
3It has been suggested that grouping baskets by customer is an example of clustering as
treated in learning theory. This is incorrect, although there are some similarities. Consider
two large identical baskets purchased ten minutes apart. Clustering would assign them
to the same category, but these baskets would almost certainly have been purchased by
diﬀerent customers. Identical baskets purchased far enough apart would have an increased
probability of having been purchased by the same customer, but it wouldn’t be certain.
Still, the literature on clustering might tell us something useful for the present problem.
class(c) of the customer. The set of baskets associated with the putative
customer c will be denoted by
Deﬁnition: A complete assignment groups all of the purchase baskets.
If there are N baskets in the database, there are something like 22N
complete assignments—less because the customers may be permuted.
Deﬁnition: Associated with each assignment will be a numerical total
anomaly measuring how anomalous the assignment is. The program’s goal is
to ﬁnd an assignment (or maybe many assignments) that minimize the total
The total anomaly
anom(α) of an assignment α is the sum of two main
anom1(α) is itself a sum
anom1(α) = (cid:88)
where the variable c ranges over the set of customers to which the baskets
anom2(α) concerns global properties of the set of assignments.
Deﬁnition: Associated with an assignment α and a customer c is a char-
char(c, α) of the putative customer. The characterization may
include qualitative characeristics like sex or owning a freezer, quantitative
characeristics like age or income group and other customer characteristics
like a certain purchase signature. The anomaly
anom11(c) associated with a
customer c depends on the characterization
char(c, α). Thus buying chew-
ing tobacco or baby food is more anomalous for some customers than others.
A program that generates assignments will generate characterizations as it
groups the baskets by customer. The characterization itself will contribute
to the anomaly if it is an unusual characterization.
Deﬁnition: A signature is a set of choices among alternate brands or
sizes of certain commodities. The commodities most useful for signatures
are those for which variety is not normally considered desirable. While a
person may want variety in food he is unlikely to want variety per se in dish-
washing soap, toilet paper or size of dog food. Signatures are included in the
characterization of a customer.
The part of the anomaly
anom11(c) associated with the putative customer
c is computed relative to the characterization. Thus if c is characterized as
single young female, a purchase of chewing tobacco should have a higher
anomaly score than for a male.
One way of looking at minimizing anomaly of assignments is that we
want to explain as much of the purchasing behavior as possible by allowable
characterizations of the customers.
We regard the notions of minimizing anomaly in the space of assignments
as a guiding theoretical idea. Programs may ﬁnd complete assignments, but
they are unlikely to do it by comparing a large number of alternative complete
assignments. Instead they are likely to do hill climbing in the space of partial
Here are some kinds of terms that may be associated with the customer
part of the anomaly function.
fairly regular rate. If baby food is purchased, it also is consumed at a
regular rate, although it can be stored. Some customers will be very
irregular, but an assignment shouldn’t make most of them irregular.
contribute to the anomaly.
all those baskets.
purchasing but not directly about the eating or the state of the larder.
We can attribute a larder function of time to a customer as part of the
ascription and use some measure of its irregularity as a component of
Here are some ideas about programs for ﬁnding assignments.
both customers’ ascribed larder functions.
baskets, the program would try to reduce the number by combining
How can it be inferred that several cash purchases involved the same
customer? We only need to be correct often enough so that the statistics
come out right. Each customer has his own pattern of purchases. Here are
sig(c, α)is a purchase pattern unique to the customer c. Consider items where variety is not normally desired, e.g. dishwasher
soap. There are several brands, but a customer will normally stick
with one for quite a long time. If there are 5 brands and 50 such kinds
of items, there are enough possible signatures to distinguish far more
customers than a store or even a chain is likely to have. Of course,
a customer is unlikely to purchase a complete signature package each
time he goes to the store, so partial signatures will have to be used.
varied in a unique way.
refrigerator or freezer.
indicative than whether peanut butter is bought at all on a particular
occasion, since the customer may not have run out yet.
that there is enough information to identify the customers over some
20 shopping trips. The information theory numbers can be analyzed,
but experiment is still required to determine feasibility.
each buy a six pack of the same brand of beer and nothing else. Which
one made which purchase will be impossible to tell, but it won’t matter
which purchase is assigned to which customer.
4 The Customer as a Stochastic Process
The methods discussed in section 3 group purchases by customer. However,
the speciﬁc purchases made by the customer are of interest only in so far as
they enable prediction of his future behavior and how he might respond to
things the store might do, e.g. advertisements, sales, changes in products
oﬀered, changes in prices.
In general, we might regard the customer as a stochastic process, i.e.
what he will buy (and whether he will come to the store at all), depends
probabilistically on the state of his larder, and the actions of the store.
A regular customer may visit the store once per week for 5 years, i.e.
make 250 visits. Some may make as many as 1,000 visits. Nevertheless,
there often won’t be enough information to make a very sophisticated model
of a customer. Therefore, simpliﬁed models are worth considering.
The simplest model is that customer c has probability
p(c, i) of buying
item i. The matrix ||
p(c, i)|| is likely to be approximable by a matrix of much
lower rank, i.e. the customers form a space of lower dimension. If this is true,
customers can be approximately characterized by a much smaller number of
parameters than are needed for a complete probability distribution. This in
turn means that accurate information about the customers can be obtained
with smaller samples that would otherwise be required. If the assumption of
independence of the members of the signature is valid, it still takes quite a
lot of information to characterize the customer.
The next more elaborate model might take into account the state of the
customer’s larder. He won’t buy more of certain items until he has consumed
what he previously bought. If we regard the customer’s state as given by the
contents of his larder, we can regard his purchases as determined by a Markov
The model might be further elaborated to take into account his probable
response to sales, etc. Economists would be tempted to try to ascribe a
demand curve, most likely just two numbers—the demand at a base price
and an elasticity.
We will not pursue these elaborations further in this article, but it seems
likely that the most useful information to supermarket companies doing data
mining will involve the probabilities of response of diﬀerent kinds of cus-
tomers to diﬀerent stimuli.
5 Mail Order Bookstores
Consider a store selling books by mail. The customers are identiﬁed, so we
don’t have that problem.
However, we can suppose that many characteristics of customers are not
identiﬁed such as age, income, occupation. Literary tastes can be identiﬁed
in so far as they correspond to a tendency to buy books in predeﬁned clas-
siﬁcations. However, the data may allow inferring new classiﬁcations with
respect to which the customers behave more consistently than with respect
to the traditional classiﬁcations.
Some classes of customers are leaders in that their preferences today can
be used to predict the market for a book later. Identifying such customers
and classes of customers may be useful.
The above phenomena—age, etc.—are not in the data per se, but they are
rather close to it. Their identiﬁcation should not be as ambitious a project as
identifying the customers of a supermarket. Almost all of the computations
will involve the individual customers. The leadership phenomenon involves
more but still has a rather simple character.
6 Proposed Experiments
Grouping supermarket purchases by customer as proposed in Section 3 can
be tested with the aid of a supermarket database that does contain customer
identiﬁcation. We discard the customer identiﬁcation, run our grouping al-
gorithm and compare the results with the genuine customer data.
My present opinion is that grouping baskets by customers is likely to be
a diﬃcult but feasible task. As will be seen, it will involve taking advantage
special features of the behavior of supermarket customers. In this respect, it
may resemble cryptanalysis which often takes advantage of special features
of the behavior of senders of messages. Moreover, the results cannot be per-
fect in terms of identifying the purchasers, but the uncertainties about who
bought what may not aﬀect the interesting statistics of customer behavior.
Here are some ideas about how to proceed.
baskets to customers. Maybe the computational resources will be ad-
equate to deal with hundreds to thousands of possible assignments.
Each of these assignments will have an anomaly computed on the basis
of what has been assigned so far.
milk and skim milk every time, because of the needs of diﬀerent family
down the number of open choices.
7 The Logic of Phenomenal Data Mining
The ways in which mathematical logic has been used in database
theory and database systems are likely to require extension for
phenomenal data mining..
Database theory and database system commonly use mathematical logic
to represent facts. However, subsets of logic are adequate for most present
database systems. For example, the databases can often be considered as
collections of ground literals, i.e. predicate symbols applied to constants.
More general sentences are used as rules and given a special status.
One example that immediately arises in the supermarket problem is the
fact that the customers who bought particular baskets are unknown, and it
is not known a priori whether two given baskets were bought by the same
customer. In Prolog and similar systems, the unique names hypothesis, i.e.
that distinct constant expressions represent distinct objects, is usually built
into the system.
buyer(b1) = buyer (b2),
which asserts that baskets b1 and b2 were purchased by the same customer.
Unlike the common situation in database systems, the truth of this assertion
is not in the database. Neither is a unique identiﬁer for b1 available.
The set of customers is unknown, although it is known to exist. Facts
about it may be known or conjectured.
We follow Quine in taking the ontology of a system to be given by the col-
lection of domains over which variables range. In the supermarket example,
the ontology includes
products These can be represented by their product codes.
purchased items The particular instances of items purchased as part of a
baskets The collection of items purchased on a particular occasion.
customers The set of customers is unknown but is known to exist
characteristics of products durable, edible, assumes a dog
characteristics of people female, married, has baby, has toddler, Hispanic
There’s lots more in the ontology.
at the approximate rank of the matrix Pij.
as a continuous variable.
We can tell this from a situation in which everyone buys meat but less,
because certain other purchase patterns are associated with not buying
knowing this fact give more than just the correlation?
customers that buy it rarely buy another, and these customers are only
those with young girls in the family, and those customers have almost
all bought one. Under these hypotheses, which identifying customers
might verify, it is reasonable to conclude that the fad for hula hoops
has reached its peak, and that if a lot more are ordered, the store is
likely to be stuck with them.
we determine how far the customers live from the store? The informa-
tion might be useful in anticipating how much business might be lost to
a newly opened competitor. No immediate idea occurred to me when I
thought of the question. However, it is rash to conclude that it can’t be
done. Someone cleverer than I, or who knows more about customers of
supermarkets, might ﬁgure a way. One just shouldn’t jump to negative
not show up as a direct correlation in the data unless item 531 were
bought in quantities that signiﬁcantly aﬀected sales of some other
10. If a customer buys a certain product but doesn’t buy a necessary com-
plementary product, we can infer that he buys the complementary
product from someone else.
The only experimental work with phenomenal data mining is reported by
Lyons and Tseytin (Lyons and Tseytin 1998).
I am indebted to Rakesh Agrawal of the IBM Almaden Research Laboratory
for introducing me to data mining in conversation and through his papers. I
also had useful discussions with Ted Selker, Joe Halpern, Jeﬀ Ullman, Nimrod
Megiddo, Rajeev Motwani, Brad Efron, Christos Papadimitriou, Ron Fagin,
Tom Costello, and Gregory Tseytin.
This work was partly supported by ARPA (ONR) grant N00014-94-1-
0775 and partly by IBM while I was visiting faculty during the summer of
1996 and since.
Lenat, D. B., and R. V. Guha. 1990. Building Large Knowledge-Based
Systems: Representation and Inference in the CYC Project. Addison-
Lyons, D., and G. S. Tseytin. 1998. Phenomenal data mining and link
analysis. In D. Jensen and C. Henry Goldberg (Eds.), Artiﬁcial Intelligence
and Link Analysis, 1998 Fall Symposium, 68–75. AAAI, AAAI Press.
Spelke, E. 1994. Initial knowlege: six suggestions. Cognition 50:431–445.