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My SQL server contains 2 tables containing a similar set of fields for a mailing (physical) address. NB these tables are populated before the data gets to my database (can't change that). The set of fields in the tables are similar though not identical - most exist in both tables, some only in one, some the other. The goal is to determine with "high confidence" whether or not two mailing addresses match.
Example fields:
Street Number
Predirection
Street Name
Street Suffix
Postdirection (one table and not the other)
Unit name (one table) v Address 2 (other table) --adds complexity
Zip code (length varies in each table 5 v 5+ digits)
Legal description
Ideally I'd like to a simple way to call a "function" which returns either a boolean or a confidence level of match (0.0 - 1.0). This call can be made in SQL or Python within my solution; free/open source highly preferred by client.
Among options such as SOUNDEX, DIFFERENCE, Levenshtein distance (all SQL) and usaddress, dedupe (Python) none stand out as a good-fit solution.
Ideally I'd like to a simple way to call a "function" which returns
either a boolean or a confidence level of match (0.0 - 1.0).
A similarity metric is what you're looking for. You can use Distance Metrics to calculate similarity. The Levenshtein Distance, Damerau-Levenshtein Distance and Hamming Distance are examples of Distance Metrics.
Given the shortest of the two: M the shorter of the two, N the longest, and your distance metric (D) you can measure string Similarity using (M-D)/N. You can also use the Longest Common subsequence or Longest Common Substring (LCS) to measure similarity by dividing LCS/N.
If you can use CLRs I HIGHLY recommend mdq.similarity which you can get from here. It will give a similarity metric using these algorithms:
The Damarau-Levenshtein distance (the documentation only says, "Levenshtein" but they are mistaken)
The Jaccard Similarity coefficient algorithm.
a form of the Jaro-Winkler distance algorithm.
4 a longest common subsequence algorithm (which grows by one when transpositions are involved)
If performance is important (these metrics can be quite slow depending on what you're feeding them) then I would get familiar with my Bernie function. It's designed to help measure similarity using any of the aforementioned algorithms much, much faster. Bernie is 100% open source and can be easily re-created in any language (Python, C#, etc.) Ditto my N-Grams function.
You can easily create your own metric using NGrams8K.
For pure T-SQL versions of Levenshtein or the Longest Common Subsequence you can check Phil Factor's blog. (Note these cannot compete with the CLR I mentioned).
I'll stop for now. The best advice can be given after we better understand what is making the strings different (note my question under your comment).
Firstly, sorry for the vague title and if this question has been asked before, but I was not entirely sure how to phrase it.
I am looking for general design principles for finding pairs of 'similar' objects from two different data sources.
Lets for simplicity say that we have two databases, A and B, both containing large volumes of objects, each with time-stamp and geo-location, along with some other data that we don't care about here.
Now I want to perform a search along these lines:
Within as certain time-frame and location dictated as search tiem, find pairs of objects from A and B respectively, ordered by some similarity score. Here for example some scalar 'time/space distance' function, distance(a,b), that calculates the distance in time and space between the objects.
I am expecting to get a (potentially ginormous) set of results where the first result is a pair of data points which has the minimum 'distance'.
I realize that the full search space is cardinality(A) x cardinality(B).
Are there any general guidelines on how to do this in a reasonable efficient way? I assume that I would need to replicate the two databases into a common repository like Hadoop? But then what? I am not sure how to perform such a query in Hadoop either.
What is this this type of query called?
To me, this is some kind of "fuzzy inner join" that I struggle wrapping my head around how to construct, let along efficiently at scale.
SQL joins don't have to be based on equality. You can use ">", "<", "BETWEEN".
You can even do something like this:
select a.val aval, b.val bval, a.val - b.val diff
from A join B on abs(a.val - b.val) < 100
What you need is a way to divide your objects into buckets in advance, without comparing them (or at least making a linear, rather than square, number of comparisons). That way, at query time, you will only be comparing a small number of items.
There is no "one-size-fits-all" way to bucket your items. In your case the bucketing can be based on time, geolocation, or both. Time-based bucketing is very natural, and can also scales elastically (increase or decrease the bucket size). Geo-clustering buckets can be based on distance from a particular point in space (if the space is abstract), or on some finite division of the space (for example, if you divide the entire Earth's world map into tiles, which can also scale nicely if done right).
A good question to ask is "if my data starts growing rapidly, can I handle it by just adding servers?" If not, you might need to rethink the design.
I need a way of storing sets of arbitrary size for fast query later on.
I'll be needing to query the resulting data structure for subsets or sets that are already stored.
===
Later edit: To clarify, an accepted answer to this question would be a link to a study that proposes a solution to this problem. I'm not expecting for people to develop the algorithm themselves.
I've been looking over the tuple clustering algorithm found here, but it's not exactly what I want since from what I understand it 'clusters' the tuples into more simple, discrete/aproximate forms and loses the original tuples.
Now, an even simpler example:
[alpha, beta, gamma, delta] [alpha, epsilon, delta] [gamma, niu, omega] [omega, beta]
Query:
[alpha, delta]
Result:
[alpha, beta, gama, delta] [alpha, epsilon, delta]
So the set elements are just that, unique, unrelated elements. Forget about types and values. The elements can be tested among them for equality and that's it. I'm looking for an established algorithm (which probably has a name and a scientific paper on it) more than just creating one now, on the spot.
==
Original examples:
For example, say the database contains these sets
[A1, B1, C1, D1], [A2, B2, C1], [A3, D3], [A1, D3, C1]
If I use [A1, C1] as a query, these two sets should be returned as a result:
[A1, B1, C1, D1], [A1, D3, C1]
Example 2:
Database:
[Gasoline amount: 5L, Distance to Berlin: 240km, car paint: red]
[Distance to Berlin: 240km, car paint: blue, number of car seats: 2]
[number of car seats: 2, Gasoline amount: 2L]
Query:
[Distance to berlin: 240km]
Result
[Gasoline amount: 5L, Distance to Berlin: 240km, car paint: red]
[Distance to Berlin: 240km, car paint: blue, number of car seats: 2]
There can be an unlimited number of 'fields' such as Gasoline amount. A solution would probably involve the database grouping and linking sets having common states (such as Gasoline amount: 240) in such a way that the query is as efficient as possible.
What algorithms are there for such needs?
I am hoping there is already an established solution to this problem instead of just trying to find my own on the spot, which might not be as efficient as one tested and improved upon by other people over time.
Clarifications:
If it helps answer the question, I'm intending on using them for storing states:
Simple example:
[Has milk, Doesn't have eggs, Has Sugar]
I'm thinking such a requirement might require graphs or multidimensional arrays, but I'm not sure
Conclusion
I've implemented the two algorithms proposed in the answers, that is Set-Trie and Inverted Index and did some rudimentary profiling on them. Illustrated below is the duration of a query for a given set for each algorithm. Both algorithms worked on the same randomly generated data set consisting of sets of integers. The algorithms seem equivalent (or almost) performance wise:
I'm confident that I can now contribute to the solution. One possible quite efficient way is a:
Trie invented by Frankling Mark Liang
Such a special tree is used for example in spell checking or autocompletion and that actually comes close to your desired behavior, especially allowing to search for subsets quite conveniently.
The difference in your case is that you're not interested in the order of your attributes/features. For your case a Set-Trie was invented by Iztok Savnik.
What is a Set-Tree? A tree where each node except the root contains a single attribute value (number) and a marker (bool) if at this node there is a data entry. Each subtree contains only attributes whose values are larger than the attribute value of the parent node. The root of the Set-Tree is empty. The search key is the path from the root to a certain node of the tree. The search result is the set of paths from the root to all nodes containing a marker that you reach when you go down the tree and up the search key simultaneously (see below).
But first a drawing by me:
The attributes are {1,2,3,4,5} which can be anything really but we just enumerate them and therefore naturally obtain an order. The data is {{1,2,4}, {1,3}, {1,4}, {2,3,5}, {2,4}} which in the picture is the set of paths from the root to any circle. The circles are the markers for the data in the picture.
Please note that the right subtree from root does not contain attribute 1 at all. That's the clue.
Searching including subsets Say you want to search for attributes 4 and 1. First you order them, the search key is {1,4}. Now startin from root you go simultaneously up the search key and down the tree. This means you take the first attribute in the key (1) and go through all child nodes whose attribute is smaller or equal to 1. There is only one, namely 1. Inside you take the next attribute in the key (4) and visit all child nodes whose attribute value is smaller than 4, that are all. You continue until there is nothing left to do and collect all circles (data entries) that have the attribute value exactly 4 (or the last attribute in the key). These are {1,2,4} and {1,4} but not {1,3} (no 4) or {2,4} (no 1).
Insertion Is very easy. Go down the tree and store a data entry at the appropriate position. For example data entry {2.5} would be stored as child of {2}.
Add attributes dynamically Is naturally ready, you could immediately insert {1,4,6}. It would come below {1,4} of course.
I hope you understand what I want to say about Set-Tries. In the paper by Iztok Savnik it's explained in much more detail. They probably are very efficient.
I don't know if you still want to store the data in a database. I think this would complicate things further and I don't know what is the best to do then.
How about having an inverse index built of hashes?
Suppose you have your values int A, char B, bool C of different types. With std::hash (or any other hash function) you can create numeric hash values size_t Ah, Bh, Ch.
Then you define a map that maps an index to a vector of pointers to the tuples
std::map<size_t,std::vector<TupleStruct*> > mymap;
or, if you can use global indices, just
std::map<size_t,std::vector<size_t> > mymap;
For retrieval by queries X and Y, you need to
get hash value of the queries Xh and Yh
get the corresponding "sets" out of mymap
intersect the sets mymap[Xh] and mymap[Yh]
If I understand your needs correctly, you need a multi-state storing data structure, with retrievals on combinations of these states.
If the states are binary (as in your examples: Has milk/doesn't have milk, has sugar/doesn't have sugar) or could be converted to binary(by possibly adding more states) then you have a lightning speed algorithm for your purpose: Bitmap Indices
Bitmap indices can do such comparisons in memory and there literally is nothing in comparison on speed with these (ANDing bits is what computers can really do the fastest).
http://en.wikipedia.org/wiki/Bitmap_index
Here's the link to the original work on this simple but amazing data structure: http://www.sciencedirect.com/science/article/pii/0306457385901086
Almost all SQL databases supoort Bitmap Indexing and there are several possible optimizations for it as well(by compression etc.):
MS SQL: http://technet.microsoft.com/en-us/library/bb522541(v=sql.105).aspx
Oracle: http://www.orafaq.com/wiki/Bitmap_index
Edit:
Apparently the original research work on bitmap indices is no longer available for free public access.
Links to recent literature on this subject:
Bitmap Index Design Choices and Their Performance
Implications
Bitmap Index Design and Evaluation
Compressing Bitmap Indexes for Faster Search Operations
This problem is known in the literature as subset query. It is equivalent to the "partial match" problem (e.g.: find all words in a dictionary matching A??PL? where ? is a "don't care" character).
One of the earliest results in this area is from this paper by Ron Rivest from 19761. This2 is a more recent paper from 2002. Hopefully, this will be enough of a starting point to do a more in-depth literature search.
Rivest, Ronald L. "Partial-match retrieval algorithms." SIAM Journal on Computing 5.1 (1976): 19-50.
Charikar, Moses, Piotr Indyk, and Rina Panigrahy. "New algorithms for subset query, partial match, orthogonal range searching, and related problems." Automata, Languages and Programming. Springer Berlin Heidelberg, 2002. 451-462.
This seems like a custom made problem for a graph database. You make a node for each set or subset, and a node for each element of a set, and then you link the nodes with a relationship Contains. E.g.:
Now you put all the elements A,B,C,D,E in an index/hash table, so you can find a node in constant time in the graph. Typical performance for a query [A,B,C] will be the order of the smallest node, multiplied by the size of a typical set. E.g. to find {A,B,C] I find the order of A is one, so I look at all the sets A is in, S1, and then I check that it has all of BC, since the order of S1 is 4, I have to do a total of 4 comparisons.
A prebuilt graph database like Neo4j comes with a query language, and will give good performance. I would imagine, provided that the typical orders of your database is not large, that its performance is far superior to the algorithms based on set representations.
Hashing is usually an efficient technique for storage and retrieval of multidimensional data. Problem is here that the number of attributes is variable and potentially very large, right? I googled it a bit and found Feature Hashing on Wikipedia. The idea is basically the following:
Construct a hash of fixed length from each data entry (aka feature vector)
The length of the hash must be much smaller than the number of available features. The length is important for the performance.
On the wikipedia page there is an implementation in pseudocode (create hash for each feature contained in entry, then increase feature-vector-hash at this index position (modulo length) by one) and links to other implementations.
Also here on SO is a question about feature hashing and amongst others a reference to a scientific paper about Feature Hashing for Large Scale Multitask Learning.
I cannot give a complete solution but you didn't want one. I'm quite convinced this is a good approach. You'll have to play around with the length of the hash as well as with different hashing functions (bloom filter being another keyword) to optimize the speed for your special case. Also there might still be even more efficient approaches if for example retrieval speed is more important than storage (balanced trees maybe?).
Why would one use kmedoids algoirthm rather then kmeans? Is it only the fact that
the number of metrics that can be used in kmeans is very limited or is there something more?
Is there an example of data, for which it makes much more sense to choose the best representatives
of cluster from the data rather then from R^n?
The problem with k-means is that it is not interpretable. By interpretability i mean the model should also be able to output the reason that why it has resulted a certain output.
lets take an example.
Suppose there is food review dataset which has two posibility that there is a +ve review or a -ve review so we can say we will have k= 2 where k is the number of clusters. Now if you go with k-means where in the algorithm the third step is updation step where you update your k-centroids based on the mean distance of the points that lie in a particular cluster. The example that we have chosen is text problem, so you would also apply some kind of text-featured vector schemes like BagOfWords(BOW), word2vec. now for every review you would get the corresponding vector. Now the generated centroid c_i that you will get after running the k-means would be the mean of the vectors present in that cluster. Now with that centroid you cannot interpret much or rather i should say nothing.
But for same problem you apply k-medoids wherein you choose your k-centroids/medoids from your dataset itself. lets say you choose x_5 point from your dataset as first medoid. From this your interpretability will increase beacuse now you have the review itself which is termed as medoid/centroid. So in k-medoids you choose the centroids from your dataset itself.
This is the foremost motivation of introducing k-mediods
Coming to the metrics part you can apply all the metrics that you apply for k-means
Hope this helps.
Why would we use k-medoids instead of k-means in case of (squared) Euclidean distance?
1. Technical justification
In case of relatively small data sets (as k-medoids complexity is greater) - to obtain a clustering more robust to noise and outliers.
Example 2D data showing that:
The graph on the left shows clusters obtained with K-medoids (sklearn_extra.cluster.KMedoids method in Python with default options) and the one on the right with K-means for K=2. Blue crosses are cluster centers.
The Python code used to generate green points:
import numpy as np
import matplotlib.pyplot as plt
rng = np.random.default_rng(seed=32)
a = rng.random((6,2))*2.35 - 3*np.ones((6,2))
b = rng.random((50,2))*0.25 - 2*np.ones((50,2))
c = rng.random((100,2))*0.5 - 1.5*np.ones((100,2))
d = rng.random((7,2))*0.55
points = np.concatenate((a, b, c, d))
plt.plot(points[:,0],points[:,1],"g.", markersize=8, alpha=0.3) # green points
2. Business case justification
Here are some example business cases showing why we would prefer k-medoids. They mostly come down to the interpretability of the results and the fact that in k-medoids the resulting cluster centers are members of the original dataset.
2.1 We have a recommender engine based only on user-item preference data and want to recommend to the user those items (e.g. movies) that other similar people enjoyed. So we assign the user to his/her closest cluster and recommend top movies that the cluster representant (actual person) watched. If the cluster representant wasn't an actual person we wouldn't possess the history of actually watched movies to recommend. Each time we'd have to search additionally e.g. for the closest person from the cluster. Example data: classic MovieLens 1M Dataset
2.2 We have a database of patients and want to pick a small representative group of size K to test a new drug with them. After clustering the patients with K-medoids, cluster representants are invited to the drug trial.
Difference between is that in k-means centroids(cluster centrum) are calculated as average of vectors containing in the cluster, and in k-medoids the medoid (cluster centrum) is record from dataset closest to centroid, so if you need to represent cluster centrum by record of your data you use k-medoids, otherwise i should use k-means (but concept of these algorithms are same)
The K-Means algorithm uses a Distance Function such as Euclidean Distance or Manhattan Distance, which are computed over vector-based instances. The K-Medoid algorithm instead uses a more general (and less constrained) distance function: aka pair-wise distance function.
This distinction works well in contexts like Complex Data Types or relational rows, where the instances have a high number of dimensions.
High dimensionality problem
In standard clustering libraries and the k-means algorithms, the distance computation phase can spend a lot of time scanning the entire vector of attributes that belongs to an instance; for instance, in the context of documents clustering, using the standard TF-IDF representation. During the computation of the cosine similarity, the distance function scans all the possible words that appear in the whole collection of documents. Which in many cases can be composed by millions of entries. This is why, in this domain, some authors [1] suggests to restrict the words considered to a subset of N most frequent word of that language.
Using K-Kedoids there is no need to represent and store the documents as vectors of word frequencies.
As an alternative representation for the documents is possible to use the set of words appearing at least twice in the document; and as a distance measure, there can be used Jaccard Distance.
In this case, vector representation is long as the number of words in your dictionary.
Heterogeneousity and Complex Data Types.
There are many domains where is considerably better to abstract the implementation of an instance:
Graph's nodes clustering;
Car driving behaviour, represented as GPS routes;
Complex data type allows the design of ad-hoc distance measures which can fit better with the proper data domain.
[1] Christopher D. Manning, Prabhakar Raghavan, and Hinrich Schütze. 2008. Introduction to Information Retrieval. Cambridge University Press, New York, NY, USA.
Source: https://github.com/eracle/Gap
I am working in a chemistry/biology project. We are building a web-application for fast matching of the user's experimental data with predicted data in a reference database. The reference database will contain up to a million entries. The data for one entry is a list (vector) of tuples containing a float value between 0.0 and 20.0 and an integer value between 1 and 18. For instance (7.2394 , 2) , (7.4011, 1) , (9.9367, 3) , ... etc.
The user will enter a similar list of tuples and the web-app must then return the - let's say - top 50 best matching database entries.
One thing is crucial: the search algorithm must allow for discrepancies between the query data and the reference data because both can contain small errors in the float values (NOT in the integer values). (The query data can contain errors because it is derived from a real-life experiment and the reference data because it is the result of a prediction.)
Edit - Moved text to answer -
How can we get an efficient ranking of 1 query on 1 million records?
You should add a physicist to the project :-) This is a very common problem to compare functions e.g. look here:
http://en.wikipedia.org/wiki/Autocorrelation
http://en.wikipedia.org/wiki/Correlation_function
In the first link you can read: "The SEQUEST algorithm for analyzing mass spectra makes use of autocorrelation in conjunction with cross-correlation to score the similarity of an observed spectrum to an idealized spectrum representing a peptide."
An efficient linear scan of 1 million records of that type should take a fraction of a second on a modern machine; a compiled loop should be able to do it at about memory bandwidth, which would transfer that in a two or three milliseconds.
But, if you really need to optimise this, you could construct a hash table of the integer values, which would divide the job by the number of integer bins. And, if the data is stored sorted by the floats, that improves the locality of matching by those; you know you can stop once you're out of tolerance. Storing the offsets of each of a number of bins would give you a position to start.
I guess I don't see the need for a fancy algorithm yet... describe the problem a bit more, perhaps (you can assume a fairly high level of chemistry and physics knowledge if you like; I'm a physicist by training)?
Ok, given the extra info, I still see no need for anything better than a direct linear search, if there's only 1 million reference vectors and the algorithm is that simple. I just tried it, and even a pure Python implementation of linear scan took only around three seconds. It took several times longer to make up some random data to test with. This does somewhat depend on the rather lunatic level of optimisation in Python's sorting library, but that's the advantage of high level languages.
from cmath import *
import random
r = [(random.uniform(0,20), random.randint(1,18)) for i in range(1000000)]
# this is a decorate-sort-undecorate pattern
# look for matches to (7,9)
# obviously, you can use whatever distance expression you want
zz=[(abs((7-x)+(9-y)),x,y) for x,y in r]
zz.sort()
# return the 50 best matches
[(x,y) for a,x,y in zz[:50]]
Can't you sort the tuples and perform binary search on the sorted array ?
I assume your database is done once for all, and the positions of the entries is not important. You can sort this array so that the tuples are in a given order. When a tuple is entered by the user, you just look in the middle of the sorted array. If the query value is larger of the center value, you repeat the work on the upper half, otherwise on the lower one.
Worst case is log(n)
If you can "map" your reference data to x-y coordinates on a plane there is a nifty technique which allows you to select all points under a given distance/tolerance (using Hilbert curves).
Here is a detailed example.
One approach we are trying ourselves which allows for the discrepancies between query and reference is by binning the float values. We are testing and want to offer the user the choice of different bin sizes. Bin sizes will be 0.1 , 0.2 , 0.3 or 0.4. So binning leaves us with between 50 and 200 bins, each with a corresponding integer value between 0 and 18, where 0 means there was no value within that bin. The reference data can be pre-binned and stored in the database. We can then take the binned query data and compare it with the reference data. One approach could be for all bins, subtract the query integer value from the reference integer value. By summing up all differences we get the similarity score, with the the most similar reference entries resulting in the lowest scores.
Another (simpler) search option we want to offer is where the user only enters the float values. The integer values in both query as reference list can then be set to 1. We then use Hamming distance to compute the difference between the query and the reference binned values. I have previously asked about an efficient algorithm for that search.
This binning is only one way of achieving our goal. I am open to other suggestions. Perhaps we can use Principal Component Analysis (PCA), as described here