I have one table with two columns
ID Probability
A 1%
B 2%
C 3%
D 4%
I have another table, with some IDs and corresponding weights:
ID Weight
A 50%
D 25%
A 15%
B 5%
B 5%
What I'm looking for is a way, in a single formula, to find the corresponding probabilities for each of the IDs in the second table using the data from the first, multiply each by their respective weights from the second table, then sum the results.
I recognise a simple way to solve it would be to add a proxy column to the second table and list corresponding probabilities using a vlookup and multiplying by the weight, then summing the results, but I feel like there must be a more elegant solution.
I've tried entering the second table IDs as an array in both Vlookup and Index/Match formulas, but while both accept a range as a lookup value, both only execute for the first value of the range instead of cycling through the whole array.
I guess ideally the formula would
set an 1 x 5 array for the IDs,
populate a new 1 x 5 array based on the probabilities from the first table
multiply the new array by the existing 1x5 array for weights
Sum whatever is the result
[edit] So for the above example, the final result would be (50% x 1%)+(25% x 4%) + (15% x 1%) + (5% x 2%) + (5% x 2%) = 1.85%
The real tables are much, much bigger than the examples I've given so a simple Sum() function for individual vlookups is out.
Love to hear of any clever solutions?
Using the same ranges as given by Trương Ngọc Đăng Khoa:
=SUMPRODUCT(SUMIF(A1:A4,D1:D5,B1:B4),E1:E5)
Regards
You can use this formula :
{=SUM(LOOKUP(D1:D5;A1:A4;B1:B4)*E1:E5)}
With table in this :
A B C D E
1 A 1% A 50%
2 B 2% D 25%
3 C 3% A 15%
4 D 4% B 5%
5 B 5%
Great response, thanks guys!
XOR LX, your answer seemed to work in all cases, which is what I was looking for (and seems like it was much simpler than I'd originally thought). I think I misunderstood the way the SUMIF function works.
In case anyone is interested, I also found my own (stupidly complex) solution:
=SUM(IF(A1:A4=TRANSPOSE(D1:D5),1,0)*TRANSPOSE(E1:E5)*B1:B4)
Which basically works by transforming the thing into a 4 x 5 matrix instead. I think I still prefer the XOR LX solution for it's simplicity.
Appreciate the help, everyone!
Related
I'm looking for a non-VBA solution to this problem.
Say I have a graph (in the computer science sense) in a spreadsheet as follows:
A B C D
1 Vertex Neighbors Degree Avg Nghbr Deg
2 A B,C 2 2.5
3 B A,C 2 2.5
4 C A,B,D 3 1.666666667
5 D C 1 3
I've entered columns C and D by hand but I want them to be calculated automatically. I've found reasonable solutions for column C that essentially count the commas and add 1. But for column D, I can't find a solution. I've found countless articles that explain how to lookup one value multiple times in one column, and countless articles that explain how to look up multiple values once in multiple columns, but I can't figure out how to look up multiple values in ONE column, get back an array of values, and then take the average of that array. I'm sure this can be done in VBA but I'd prefer a native Excel solution if one exists.
Obviously I'd like to extend this so that I can do other analyses of a vertex's neighbors. Presumably once I know the method to analyze a "looked-up array" I will be able to use it in other functions as well.
Any help is greatly appreciated.
To get column C:
=LEN(B2)-LEN(SUBSTITUTE(B2,",",""))+1
To get column D use SUMPRODUCT with SEARCH:
=SUMPRODUCT((ISNUMBER(SEARCH("," & $A$2:$A$5 & ",","," & B2 & ",")))*$C$2:$C$5)/C2
Given a 2D array of Boolean values I want to find all patterns that consist of at least 2 columns and at least 2 rows. The problem is somewhat close to finding cliques in a graph.
In the example below green cells represent "true" bits, greys are "false". Pattern 1 contains cols 1,3,4 and 5 and rows 1 and 2. Pattern 2 contains only columns 2 and 4, and rows 2,3,4.
Business idea behind this is finding similarity patterns among various groups of social network users. In real world number of rows can go up to 3E7, and the number of columns up to 300.
Can't really figure out a solution other than brute force matching.
Please advice the proper name of the problem, so I could read more, or advice an elegant solution.
This is (equivalent to) asking for all bicliques (complete bipartite subgraphs) larger than a certain size in a bipartite graph. Here the rows are the vertices of one part A of the graph, and the columns are the vertices of the other part B, and there is an edge between u \in A and v \in B whenever the cell at row u, column v is green.
Although you say that you want to find all patterns, you probably only want to find only maximal ones -- that is, patterns that cannot be extended to become larger patterns by adding more rows or columns. (Otherwise, for any pattern with c >= 2 columns and r >= 3 rows, you will also get back the more than 2^(c-2)*2^(r-3) non-maximal patterns that can be formed by deleting some of the rows or columns.)
But even listing just the maximal patterns can take time exponential in the number of rows and columns, assuming that P != NP. That's because the problem of finding a maximum (i.e. largest-possible) pattern, in terms of the total number of green cells, has been proven to be NP-complete: if it were possible to list all maximal patterns in polynomial time, then we could simply do so, and pick the largest, thereby solving this NP-complete problem in polynomial time.
I have an m x n matrix of real numbers. I want to choose a single value from each column such that the sum of my selected values is as close as possible to a pre-specified total.
I am not an experienced programmer (although I have an experienced friend who will help). I would like to achieve this using Matlab, Mathematica or c++ (MySQL if necessary).
The code only needs to run a few times, once every few days - it does not necessarily need to be optimised. I will have 16 columns and about 12 rows.
Normally I would suggest dynamic programming, but there are a few features of this situation suggesting an alternative approach. First, the performance demands are light; this program will be run only a couple times, and it doesn't sound as though a running time on the order of hours would be a problem. Second, the matrix is fairly small. Third, the matrix contains real numbers, so it would be necessary to round and then do a somewhat sophisticated search to ensure that the optimal possibility was not missed.
Instead, I'm going to suggest the following semi-brute-force approach. 12**16 ~ 1.8e17, the total number of possible choices, is too many, but 12**9 ~ 5.2e9 is doable with brute force, and 12**7 ~ 3.6e7 fits comfortably in memory. Compute all possible choices for the first seven columns. Sort these possibilities by total. For each possible choice for the last nine columns, use an efficient search algorithm to find the best mate among the first seven. (If you have a lot of memory, you could try eight and eight.)
I would attempt a first implementation in C++, using std::sort and std::lower_bound from the <algorithm> standard header. Measure it; if it's too slow, then try an in-memory B+-tree (does Boost have one?).
I spent some more time thinking about how to implement what I wrote above in the simplest way possible. Here's an approach that will work well for a 12 by 16 matrix on a 64-bit machine with roughly 4 GB of memory.
The number of choices for the first eight columns is 12**8. Each choice is represented by a 4-byte integer between 0 and 12**8 - 1. To decode a choice index i, the row for the first column is given by i % 12. Update i /= 12;. The row for the second column now is given by i % 12, et cetera.
A vector holding all choices requires roughly 12**8 * 4 bytes, or about 1.6 GB. Two such vectors require 3.2 GB. Prepare one for the first eight columns and one for the last eight. Sort them by sum of the entries that they indicate. Use saddleback search to find the best combination. (Initialize an iterator into the first vector and a reverse iterator into the second. While neither iterator is at its end, compare the current combination against the current best and update the current best if necessary. If the current combination sums to than the target, increment the first iterator. If the sum is greater than the target, increment the second iterator.)
I would estimate that this requires less than 50 lines of C++.
Without knowing the range of values that might fill the arrays, how about something generic like this:
divide the target by the number of remaining columns.
Pick the number from that column closest to that value.
Repeat from 1. Until each column picked.
I have a simple machine learning question:
I have n (~110) elements, and a matrix of all the pairwise distances. I would like to choose the 10 elements that are most far apart. That is, I want to
Maximize:
Choose 10 different elements.
Return min distance over (all pairings within the 10).
My distance metric is symmetric and respects the triangle inequality.
What kind of algorithm can I use? My first instinct is to do the following:
Cluster the n elements into 20
clusters.
Replace each cluster with just the
element of that cluster that is
furthest from the mean element of
the original n.
Use brute force to solve the
problem on the remaining 20
candidates. Luckily, 20 choose 10 is
only 184,756.
Edit: thanks to etarion's insightful comment, changed "Return sum of (distances)" to "Return min distance" in the optimization problem statement.
Here's how you might approach this combinatorial optimization problem by taking the convex relaxation.
Let D be an upper triangular matrix with your distances on the upper triangle. I.e. where i < j, D_i,j is the distance between elements i and j. (Presumably, you'll have zeros on the diagonal, as well.)
Then your objective is to maximize x'*D*x, where x is binary valued with 10 elements set to 1 and the rest to 0. (Setting the ith entry in x to 1 is analogous to selecting the ith element as one of your 10 elements.)
The "standard" convex optimization thing to do with a combinatorial problem like this is to relax the constraints such that x need not be discrete valued. Doing so gives us the following problem:
maximize y'*D*y
subject to: 0 <= y_i <= 1 for all i, 1'*y = 10
This is (morally) a quadratic program. (If we replace D with D + D', it'll become a bona fide quadratic program and the y you get out should be no different.) You can use an off-the-shelf QP solver, or just plug it in to the convex optimization solver of your choice (e.g. cvx).
The y you get out need not be (and probably won't be) a binary vector, but you can convert the scalar values to discrete ones in a bunch of ways. (The simplest is probably to let x be 1 in the 10 entries where y_i is highest, but you might need to do something a little more complicated.) In any case, y'*D*y with the y you get out does give you an upper bound for the optimal value of x'*D*x, so if the x you construct from y has x'*D*x very close to y'*D*y, you can be pretty happy with your approximation.
Let me know if any of this is unclear, notation or otherwise.
Nice question.
I'm not sure if it can be solved exactly in an efficient manner, and your clustering based solution seems reasonable. Another direction to look at would be local search method such as simulated annealing and hill climbing.
Here's an obvious baseline I would compare any other solution against:
Repeat 100 times:
Greedily select the datapoint that whose removal decreases the objective function the least and remove it.
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