Good morning, I'm trying to perform a 2D FFT as 2 1-Dimensional FFT.
The problem setup is the following:
There's a matrix of complex numbers generated by an inverse FFT on an array of real numbers, lets call it arr[-nx..+nx][-nz..+nz].
Now, since the original array was made up of real numbers, I exploit the symmetry and reduce my array to be arr[0..nx][-nz..+nz].
My problem starts here, with arr[0..nx][-nz..nz] provided.
Now I should come back in the domain of real numbers.
The question is what kind of transformation I should use in the 2 directions?
In x I use the fftw_plan_r2r_1d( .., .., .., FFTW_HC2R, ..), called Half complex to Real transformation because in that direction I've exploited the symmetry, and that's ok I think.
But in z direction I can't figure out if I should use the same transformation or, the Complex to complex (C2C) transformation?
What is the correct once and why?
In case of needing here, at page 11, the HC2R transformation is briefly described
Thank you
"To easily retrieve a result comparable to that of fftw_plan_dft_r2c_2d(), you can chain a call to fftw_plan_dft_r2c_1d() and a call to the complex-to-complex dft fftw_plan_many_dft(). The arguments howmany and istride can easily be tuned to match the pattern of the output of fftw_plan_dft_r2c_1d(). Contrary to fftw_plan_dft_r2c_1d(), the r2r_1d(...FFTW_HR2C...) separates the real and complex component of each frequency. A second FFTW_HR2C can be applied and would be comparable to fftw_plan_dft_r2c_2d() but not exactly similar.
As quoted on the page 11 of the documentation that you judiciously linked,
'Half of these column transforms, however, are of imaginary parts, and should therefore be multiplied by I and combined with the r2hc transforms of the real columns to produce the 2d DFT amplitudes; ... Thus, ... we recommend using the ordinary r2c/c2r interface.'
Since you have an array of complex numbers, you can either use c2r transforms or unfold real/imaginary parts and try to use HC2R transforms. The former option seems the most practical.Which one might solve your issue?"
-#Francis
I was wondering if anybody had suggestions as to how I could analyze an output bitstring that is being permuted by a genetic algorithm. In particular it would be nice if I could try to identify patterns of bits (I'm calling them genes here) that seem to yield a desirable cv score. The difficulty comes in trying to examine these datasets because there are a lot of them (I have probably already something like 30 million bitstrings that are 140 bits long and I'll probably hit over 100 million pretty quickly), so after I sort out the desirable data there is still ALOT of potential datasets and doing similarity comparisons by eye is out of the question. My questions are:
How should I compare for similarity between these bitstrings?
How can I identify "genes" in these bitstrings in an algorithmic (aka programmable) way?
As you want to extract common gene-patterns, what about looking at the intersection of the two strings. So if you have
set1 = 11011101110011...
set2 = 11001100000110...
# apply bitwise '=='
set1 && set2 == 11101110000010...
The result now shows what genes are the same, and could be used in further analysis.
For the similarity part you need to do an exclusive-or (XOR). The result of this bit-wise operation will give you the difference between two bit strings, and is probably the most efficient and easy way of doing it (for pair comparison). As an example:
>>> from bitarray import bitarray
>>> a = bitarray('0001100111')
>>> b = bitarray('0100110110')
>>> a ^ b
bitarray('0101010001')
Then you can either count the differences, inspect quickly where the differences lie, etc.
For the second part, it depends on the representation of course, and on the programming language (PL) chosen for the implementation. Most PL libraries will have a search function, that retrieves all or at least the first of the indexes where some pattern is found in a string (or bitstring, or bitstream...). You just have to refer to the documentation of your chosen PL to know more about the performance if you have more than one option for the task.
I'm having issues 'describing each step' when creating an NFA from a regular expression. The question is as follows:
Convert the following regular expression to a non-deterministic finite-state automaton (NFA), clearly describing the steps of the algorithm that you use:
(b|a)*b(a|b)
I've made a simple 3-state machine but it's very much from intuition.
This is a question from a past exam written by my lecturer, who also wrote the following explanation of Thompson's algorithm: http://www.cs.may.ie/staff/jpower/Courses/Previous/parsing/node5.html
Can anyone clear up how to 'describe each step clearly'? It just seems like a set of basic rules rather than an algorithm with steps to follow.
Maybe there's an algorithm I've glossed over somewhere but so far I've just created them with an educated guess.
Short version for general approach.
There's an algo out there called the Thompson-McNaughton-Yamada Construction Algorithm or sometimes just "Thompson Construction." One builds intermediate NFAs, filling in the pieces along the way, while respecting operator precedence: first parentheses, then Kleene Star (e.g., a*), then concatenation (e.g., ab), followed by alternation (e.g., a|b).
Here's an in-depth walkthrough for building (b|a)*b(a|b)'s NFA
Building the top level
Handle parentheses. Note: In actual implementation, it can make sense to handling parentheses via a recursive call on their contents. For the sake of clarity, I'll defer evaluation of anything inside of parens.
Kleene Stars: only one * there, so we build a placeholder Kleene Star machine called P (which will later contain b|a).
Intermediate result:
Concatenation: Attach P to b, and attach b to a placeholder machine called Q (which will contain (a|b). Intermediate result:
There's no alternation outside of parentheses, so we skip it.
Now we're sitting on a P*bQ machine. (Note that our placeholders P and Q are just concatenation machines.) We replace the P edge with the NFA for b|a, and replace the Q edge with the NFA for a|b via recursive application of the above steps.
Building P
Skip. No parens.
Skip. No Kleene stars.
Skip. No contatenation.
Build the alternation machine for b|a. Intermediate result:
Integrating P
Next, we go back to that P*bQ machine and we tear out the P edge. We have the source of the P edge serve as the starting state for the P machine, and the destination of the P edge serve as the destination state for the P machine. We also make that state reject (take away its property of being an accept state). The result looks like this:
Building Q
Skip. No parens.
Skip. No Kleene stars.
Skip. No contatenation.
Build the alternation machine for a|b. Incidentally, alternation is commutative, so a|b is logically equivalent to b|a. (Read: skipping this minor footnote diagram out of laziness.)
Integrating Q
We do what we did with P above, except replacing the Q edge with the intermedtae b|a machine we constructed. This is the result:
Tada! Er, I mean, QED.
Want to know more?
All the images above were generated using an online tool for automatically converting regular expressions to non-deterministic finite automata. You can find its source code for the Thompson-McNaughton-Yamada Construction algorithm online.
The algorithm is also addressed in Aho's Compilers: Principles, Techniques, and Tools, though its explanation is sparse on implementation details. You can also learn from an implementation of the Thompson Construction in C by the excellent Russ Cox, who described it some detail in a popular article about regular expression matching.
In the GitHub repository below, you can find a Java implementation of Thompson's construction where first an NFA is being created from the regex and then an input string is being matched against that NFA:
https://github.com/meghdadFar/regex
https://github.com/White-White/RegSwift
No more tedious words. Check out this repo, it translates your regular expression to an NFA and visually shows you the state transitions of an NFA.
I wonder if there's any tool/website where I can plot some run times as graphs. Because, the asymptotic notation is often not what I wanted i.e. Don't want to ignore constants.
For example, suppose I have two notations, like:
1) O = (n * log n).
2) O = (n * log n * log n)/5.
It's obvious that 1st one is asymptotically better. But what I want to see how they perform and at which point the second one starts to become better.
A graphical notation where I can enter different equations and plot them them to see how they vary would be greatly useful for this purpose. In my search I found this site where they have some plots. I am looking for something similar but I also want to input my equations to plot to analyse the performance for various 'n' values.
As soon as you stop "ignoring constants", you're no longer graphing "Big O" notation, but just performing a standard XY plot. As such, any graphing program, even online graphing calculators, would let you display this, just replace "n" for "X" and you'll get the proper graph.
Would this or this help?
If you use a 3d grapher, you can use the other dimension (say y) as a constant replacement.
This way you would be able to interpret results as:
when y is greater than 5, n*log(n)*log(n)/y is better than n*log(n) starting from n = (actual value)
Also, you can ignore the 3rd dimension. Or use it if you have a complexity depending on 2 variables.
Just input the difference between the complexities. In this case, ignoring the 3rd dimension and considering log(x) = ln(x), the equation is:
z = x*ln(x) - x*ln(x)*ln(x)/5
An you can interpret that as x*ln(x) is more efficient when z is negative.
If you want to see how they perform then you have to implement the algorithms and execute them on various graph. Modern processors with memory locality and cache misses make it really hard to come up with an equation that gives you a reasonable estimation.
I can guarantee you that oyu won't measure what you would expect.
I'm working with a couple of AI algorithms at school and I find people use the words Fuzzy Logic to explain any situation that they can solve with a couple of cases. When I go back to the books I just read about how instead of a state going from On to Off it's a diagonal line and something can be in both states but in different "levels".
I've read the wikipedia entry and a couple of tutorials and even programmed stuff that "uses fuzzy logic" (an edge detector and a 1-wheel self-controlled robot) and still I find it very confusing going from Theory to Code... for you, in the less complicated definition, what is fuzzy logic?
Fuzzy logic is logic where state membership is, essentially, a float with range 0..1 instead of an int 0 or 1. The mileage you get out of it is that things like, for example, the changes you make in a control system are somewhat naturally more fine-tuned than what you'd get with naive binary logic.
An example might be logic that throttles back system activity based on active TCP connections. Say you define "a little bit too many" TCP connections on your machine as 1000 and "a lot too many" as 2000. At any given time, your system has a "too many TCP connections" state from 0 (<= 1000) to 1 (>= 2000), which you can use as a coefficient in applying whatever throttling mechanisms you have available. This is much more forgiving and responsive to system behavior than naive binary logic that only knows how to determine "too many", and throttle completely, or "not too many", and not throttle at all.
I'd like to add to the answers (that have been modded up) that, a good way to visualize fuzzy logic is follows:
Traditionally, with binary logic you would have a graph whose membership function is true or false whereas in a fuzzy logic system, the membership function is not.
1|
| /\
| / \
| / \
0|/ \
------------
a b c d
Assume for a second that the function is "likes peanuts"
a. kinda likes peanuts
b. really likes peanuts
c. kinda likes peanuts
d. doesn't like peanuts
The function itself doesn't have to be triangular and often isn't (it's just easier with ascii art).
A fuzzy system will likely have many of these, some even overlapping (even opposites) like so:
1| A B
| /\ /\ A = Likes Peanuts
| / \/ \ B = Doesn't Like Peanuts
| / /\ \
0|/ / \ \
------------
a b c d
so now c is "kind likes peanuts, kinda doesn't like peanuts" and d is "really doesn't like peanuts"
And you can program accordingly based on that info.
Hope this helps for the visual learners out there.
The best definition of fuzzy logic is given by its inventor Lotfi Zadeh:
“Fuzzy logic means of representing problems to computers in a way akin to the way human solve them and the essence of fuzzy logic is that everything is a matter of degree.”
The meaning of solving problems with computers akin to the way human solve can easily be explained with a simple example from a basketball game; if a player wants to guard another player firstly he should consider how tall he is and how his playing skills are. Simply if the player that he wants to guard is tall and plays very slow relative to him then he will use his instinct to determine to consider if he should guard that player as there is an uncertainty for him. In this example the important point is the properties are relative to the player and there is a degree for the height and playing skill for the rival player. Fuzzy logic provides a deterministic way for this uncertain situation.
There are some steps to process the fuzzy logic (Figure-1). These steps are; firstly fuzzification where crisp inputs get converted to fuzzy inputs secondly these inputs get processed with fuzzy rules to create fuzzy output and lastly defuzzification which results with degree of result as in fuzzy logic there can be more than one result with different degrees.
Figure 1 – Fuzzy Process Steps (David M. Bourg P.192)
To exemplify the fuzzy process steps, the previous basketball game situation could be used. As mentioned in the example the rival player is tall with 1.87 meters which is quite tall relative to our player and can dribble with 3 m/s which is slow relative to our player. Addition to these data some rules are needed to consider which are called fuzzy rules such as;
if player is short but not fast then guard,
if player is fast but not short then don’t guard
If player is tall then don’t guard
If player is average tall and average fast guard
Figure 2 – how tall
Figure 3- how fast
According to the rules and the input data an output will be created by fuzzy system such as; the degree for guard is 0.7, degree for sometimes guard is 0.4 and never guard is 0.2.
Figure 4-output fuzzy sets
On the last step, defuzzication, is using for creating a crisp output which is a number which may determine the energy that we should use to guard the player during game. The centre of mass is a common method to create the output. On this phase the weights to calculate the mean point is totally depends on the implementation. On this application it is considered to give high weight to guard or not guard but low weight given to sometimes guard. (David M. Bourg, 2004)
Figure 5- fuzzy output (David M. Bourg P.204)
Output = [0.7 * (-10) + 0.4 * 1 + 0.2 * 10] / (0.7 + 0.4 + 0.2) ≈ -3.5
As a result fuzzy logic is using under uncertainty to make a decision and to find out the degree of decision. The problem of fuzzy logic is as the number of inputs increase the number of rules increase exponential.
For more information and its possible application in a game I wrote a little article check this out
To build off of chaos' answer, a formal logic is nothing but an inductively defined set that maps sentences to a valuation. At least, that's how a model theorist thinks of logic. In the case of a sentential boolean logic:
(basis clause) For all A, v(A) in {0,1}
(iterative) For the following connectives,
v(!A) = 1 - v(A)
v(A & B) = min{v(A), v(B)}
v(A | B) = max{v(A), v(B)}
(closure) All sentences in a boolean sentential logic are evaluated per above.
A fuzzy logic changes would be inductively defined:
(basis clause) For all A, v(A) between [0,1]
(iterative) For the following connectives,
v(!A) = 1 - v(A)
v(A & B) = min{v(A), v(B)}
v(A | B) = max{v(A), v(B)}
(closure) All sentences in a fuzzy sentential logic are evaluated per above.
Notice the only difference in the underlying logic is the permission to evaluate a sentence as having the "truth value" of 0.5. An important question for a fuzzy logic model is the threshold that counts for truth satisfaction. This is to ask: for a valuation v(A), for what value D it is the case the v(A) > D means that A is satisfied.
If you really want to found out more about non-classical logics like fuzzy logic, I would recommend either An Introduction to Non-Classical Logic: From If to Is or Possibilities and Paradox
Putting my coder hat back on, I would be careful with the use of fuzzy logic in real world programming, because of the tendency for a fuzzy logic to be undecidable. Maybe it's too much complexity for little gain. For instance a supervaluational logic may do just fine to help a program model vagueness. Or maybe probability would be good enough. In short, I need to be convinced that the domain model dovetails with a fuzzy logic.
Maybe an example clears up what the benefits can be:
Let's say you want to make a thermostat and you want it to be 24 degrees.
This is how you'd implement it using boolean logic:
Rule1: heat up at full power when
it's colder than 21 degrees.
Rule2:
cool down at full power when it's
warmer than 27 degrees.
Such a system will only once and a while be 24 degrees, and it will be very inefficient.
Now, using fuzzy logic, it would be like something like this:
Rule1: For each degree that it's colder than 24 degrees, turn up the heater one notch (0 at 24).
Rule2: For each degree that it's warmer than 24 degress, turn up the cooler one notch (0 at 24).
This system will always be somewhere around 24 degrees, and it only once and will only once and a while make a tiny adjustment. It will also be more energy-efficient.
Well, you could read the works of Bart Kosko, one of the 'founding fathers'. 'Fuzzy Thinking: The New Science of Fuzzy Logic' from 1994 is readable (and available quite cheaply secondhand via Amazon). Apparently, he has a newer book 'Noise' from 2006 which is also quite approachable.
Basically though (in my paraphrase - not having read the first of those books for several years now), fuzzy logic is about how to deal with the world where something is perhaps 10% cool, 50% warm, and 10% hot, where different decisions may be made on the degree to which the different states are true (and no, it wasn't entirely an accident that those percentages don't add up to 100% - though I'd accept correction if needed).
A very good explanation, with a help of Fuzzy Logic Washing Machines.
I know what you mean about it being difficult to go from concept to code. I'm writing a scoring system that looks at the values of sysinfo and /proc on Linux systems and comes up with a number between 0 and 10, 10 being the absolute worst. A simple example:
You have 3 load averages (1, 5, 15 minute) with (at least) three possible states, good, getting bad, bad. Expanding that, you could have six possible states per average, adding 'about to' to the three that I just noted. Yet, the result of all 18 possibilities can only deduct 1 from the score. Repeat that with swap consumed, actual VM allocated (committed) memory and other stuff .. and you have one big bowl of conditional spaghetti :)
Its as much a definition as it is an art, how you implement the decision making process is always more interesting than the paradigm itself .. whereas in a boolean world, its rather cut and dry.
It would be very easy for me to say if load1 < 2 deduct 1, but not very accurate at all.
If you can teach a program to do what you would do when evaluating some set of circumstances and keep the code readable, you have implemented a good example of fuzzy logic.
Fuzzy Logic is a problem-solving methodology that lends itself to implementation in systems ranging from simple, small, embedded micro-controllers to large, networked, multi-channel PC or workstation-based data acquisition and control systems. It can be implemented in hardware, software, or a combination of both. Fuzzy Logic provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. Fuzzy Logic approach to control problems mimics how a person would make decisions, only much faster.
Fuzzy logic has proved to be particularly useful in expert system and other artificial intelligence applications. It is also used in some spell checkers to suggest a list of probable words to replace a misspelled one.
To learn more, just check out: http://en.wikipedia.org/wiki/Fuzzy_logic.
The following is sort of an empirical answer.
A simple (possibly simplistic answer) is that "fuzzy logic" is any logic that returns values other than straight true / false, or 1 / 0. There are a lot of variations on this and they tend to be highly domain specific.
For example, in my previous life I did search engines that used "content similarity searching" as opposed to then common "boolean search". Our similarity system used the Cosine Coefficient of weighted-attribute vectors representing the query and the documents and produced values in the range 0..1. Users would supply "relevance feedback" which was used to shift the query vector in the direction of desirable documents. This is somewhat related to the training done in certain AI systems where the logic gets "rewarded" or "punished" for results of trial runs.
Right now Netflix is running a competition to find a better suggestion algorithm for their company. See http://www.netflixprize.com/. Effectively all of the algorithms could be characterized as "fuzzy logic"
Fuzzy logic is calculating algorithm based on human like way of thinking. It is particularly useful when there is a large number of input variables. One online fuzzy logic calculator for two variables input is given:
http://www.cirvirlab.com/simulation/fuzzy_logic_calculator.php