Effective reasonable indexing for numeric vector search? - search

Question

I have a long numeric table where 7 columns are a key and 4 columns is a value to find.
Actually I have rendered an object with different distances and perspective angles and have calculated Hu moments for it's contour. But this is not important to the question, just a sample to imagine.
So, when I have 7 values, I need to scan a table, find closest values in that 7 columns and extract corresponding 4 values.
So, the task aspects to consider is follows:
1) numbers have errors
2) the scale in function domain is not the same as the scale in function value; i.e. the "distance" from point in 7-dimensional space should depend on that 4 values, how it affect
3) search should be fast
So the question is follows: isn't some algorithm out there to solve this task efficiently, i.e. perform some indexing on that 7 columns, but do this no like conventional databases do, but taking into account point above.

Answer 1

If I understand the problem correctly, you might consider using scipy.cluster.vq (vector quantization):
Suppose your 7 numeric columns look like this (let's call the array code_book):
import scipy.cluster.vq as vq
import scipy.spatial as spatial
import numpy as np
np.random.seed(2013)
np.set_printoptions(precision=2)
code_book = np.random.random((3,7))
print(code_book)
# [[ 0.68 0.96 0.27 0.6 0.63 0.24 0.7 ]
# [ 0.84 0.6 0.59 0.87 0.7 0.08 0.33]
# [ 0.08 0.17 0.67 0.43 0.52 0.79 0.11]]
Suppose the associated 4 columns of values looks like this:
values = np.arange(12).reshape(3,4)
print(values)
# [[ 0 1 2 3]
# [ 4 5 6 7]
# [ 8 9 10 11]]
And finally, suppose we have some "observations" of 7-column values like this:
observations = np.random.random((5,7))
print(observations)
# [[ 0.49 0.39 0.41 0.49 0.9 0.89 0.1 ]
# [ 0.27 0.96 0.16 0.17 0.72 0.43 0.64]
# [ 0.93 0.54 0.99 0.62 0.63 0.81 0.36]
# [ 0.17 0.45 0.84 0.02 0.95 0.51 0.26]
# [ 0.51 0.8 0.2 0.9 0.41 0.34 0.36]]
To find the 7-valued row in code_book which is closest to each observation, you could use vq.vq:
index, dist = vq.vq(observations, code_book)
print(index)
# [2 0 1 2 0]
The index values refer to rows in code_book. However, if the rows in values are ordered the same way as code_book, we can "lookup" the associated value with values[index]:
print(values[index])
# [[ 8 9 10 11]
# [ 0 1 2 3]
# [ 4 5 6 7]
# [ 8 9 10 11]
# [ 0 1 2 3]]
The above assumes you have all your observations arranged in an array. Thus, to find all the indices you need only one call to vq.vq.
However, if you obtain the observations one at a time and need to find the closest row in code_book before going on to the next observation, then it would be inefficient to call vq.vq each time. Instead, generate a KDTree once, and then find the nearest neighbor(s) in the tree:
tree = spatial.KDTree(code_book)
for observation in observations:
distances, indices = tree.query(observation)
print(indices)
# 2
# 0
# 1
# 2
# 0
Note that the number of points in your code_book (N) must be large compared to the dimension of the data (e.g. N >> 2**7) for the KDTree to be fast compared to simple exhaustive search.
Using vq.vq or KDTree.query may or may not be faster than exhaustive search, depending on the size of your data (code_book and observations). To find out which is faster, be sure to benchmark these versus an exhaustive search using timeit.

Answer 2

i don't know if i understood well your question,but i will try giving an answer.
for each row K in the table compute the distance of your key from the key in that row:
( (X1-K1)^2 + (X2-K2)^2 + (X3-K3)^2 + (X4-K4)^2 + (X5-K5)^2 + (X6-K6)^2 + (X7-K7)^2 )^0.5
where {X1,X2,X3,X4,X5,X6,X7} is the key and {K1,K2,K3,K4,K5,K6,K7}is the key at row K
you could make one factor of the key more or less relevant of the others multiplying it while computing distance,for example you could replace (X1-K1)^2 in the formula above with 5*(X1-K1)^2 to make that more influent.
and store in a variable the distance ,in a second variable the row number
do the same with the following rows and if the new distance is lower then the one you stored then replace the distance and the row number.
when you have checked all the rows in your table the second variable you have used will show you the nearest row to the key
here is some pseudo-code:
int Row= 0
float Key[7] #suppose it is already filled with some values
float ClosestDistance= +infinity
int ClosestRow= 0
while Row<NumberOfRows{
NewDistance= Distance(Key,Table[Row][0:7])#suppose Distance is a function that outputs the distance and Table is the table you want to control Table[Row= NumberOfRows][Column= 7+4]
if NewDistance<ClosestDistance{
ClosestDistance= NewDistance
ClosestRow= Row}
increase row by 1}
ValueFound= Table[ClosestRow][7:11]#this should be the value you were looking for
i know it isn't fast but it is the best i could do,hope it helped.
P.S. i haven't considered measurement errors,i know.