I have an optimization problem: I have a 2-D square like 0<x<X and 0<y<Y where X and Y are the boundaries of the square, and I have a 2-D array in this region Arr[x,y], which has positive and negative values.
Now, the task is to find the optimal region inside this square so that when I integrate the given array within this region, I get the maximum possible number.
Which direction should I look? Which techniques are available for this type of question?
Related
Suppose there's a Poisson equation with circle region and the boundary condition is zero. I'd like to solve it numerically using fininte difference method. However, as far as I konw, finite difference is widely use in renctangular region. So, if I still use finite difference in circle region, what should I do?
Maybe perform polar change of variables and express Poisson's equation in these polar coordinates. Then the circular region becomes rectangular. Now, try finite differences.
Is there any available DM script that can compare two images and know the difference?
I mean the script can compare two or more images, and it can determine the similarity of two images, for example the 95% area of one image is same as another image, then the similarity of these two images is 95%.
The script can compare brightness and contrast distribution of images.
Thanks,
This question is a bit ill-defined, as "similarity" between images depends a lot on what you want.
If by "95% of the area is the same" you mean that 95% of the pixels are of identical value in images A & B, you can simply create a mask and sum() it to count the number of pixels, i.e.:
sum( abs(A-B)==0 ? 1 : 0 )
However, this will utterly fail if the images A & B are shifted with respect to each other even by a single pixel. It will also fail, if A & B are of same contrast but different absolute value.
I guess the intended question was to find similarity of two images in a fuzzy way.
For these, one way is to do crosscorrelation. DM has this function. Like this,
image xcorr= CrossCorrelate(ref,img)
From xcorr, the peak position gives x- and y- shift between the two, the peak intensity gives "similarity" of the two.
If you know there is no shift between the two, you can just do the sum and multiplication,
number similarity1=sum(img1*img2)
Another way to do similarity is calculate Euclidian distance of the two:
number similarity2=sqrt(sum((img1-img2)**2)).
"similarity2" calculates the "pure" similarity. "similarity1" is the pure similarity plus the mean intensity of img1 and img2. The difference is essentially this,
(a-b)**2=a**2+b**2-2*a*b.
The left term is "similarity2", the last term on the right is the "crosscorrelation" or "similarity1".
I think "similarity1" is called cross-correlation, "similarity2" is called correlation coefficient.
In example comparing two diffraction patterns, if you want to compute the degree of similarity, use "similarity2". If you want to compute the degree of similarity plus a certain character of the diffraction pattern, use "similarity1".
I have some data that tells me the amount of hours water is available for particular towns.
You can see it here
I want to use train a Multilayer Perceptron based on that data, to take a set of coordinates and indicate the approximate number of hours for which that coordinate will have water.
Does this make sense?
If so, am I correct in saying, there has to be two input layers? One for lat and one for long. And the output layer should be the number of hours.
Would love some guidance.
I would solve that differently:
Just create an ArrayList of WaterInfo:
WaterInfo contains lat,lon, waterHours.
Then for a given coordinate search the closest WaterInfo in the list.
Since you have not many elements, just do a brute force search, to find the closest.
You further can optimize, to find the three closest WaterInfo points, and calculate the weithted average of WaterHours. As weight you use the air distance from current position to Waterinfo position.
To answer your question:
"Does this makes sense"?
From the goal to get a working solution: NO!
Ask yourself, why do you want to use MLP for this task.
Further i doubt that using two layers for lat / long makes sense.
A coordinate (lat/lon) is one point on the world, so that should be one layer in the model. You can convert the lat/lon coord to a cell identifier: Span a grid over Brazil; with cell width 10 or 50km; now convert a lat/long coordinate to a cellId: Like E4 on a chess board, you will calculate one integer value representing the cell. (There are other solutions to get an unique number, too, choose one you like)
Now you have a modell geoCellID -> waterHours, which better represents the real world situation.
Basically, I have a set of up to 100 co-ordinates, along with the desired tangents to the curve at the first and last point.
I have looked into various methods of curve-fitting, by which I mean an algorithm with takes the inputted data points and tangents, and outputs the equation of the cure, such as the gaussian method and interpolation, but I really struggled understanding them.
I am not asking for code (If you choose to give it, thats acceptable though :) ), I am simply looking for help into this algorithm. It will eventually be converted to Objective-C for an iPhone app, if that changes anything..
EDIT:
I know the order of all of the points. They are not too close together, so passing through all points is necessary - aka interpolation (unless anyone can suggest something else). And as far as I know, an algebraic curve is what I'm looking for. This is all being done on a 2D plane by the way
I'd recommend to consider cubic splines. There is some explanation and code to calculate them in plain C in Numerical Recipes book (chapter 3.3)
Most interpolation methods originally work with functions: given a set of x and y values, they compute a function which computes a y value for every x value, meeting the specified constraints. As a function can only ever compute a single y value for every x value, such an curve cannot loop back on itself.
To turn this into a real 2D setup, you want two functions which compute x resp. y values based on some parameter that is conventionally called t. So the first step is computing t values for your input data. You can usually get a good approximation by summing over euclidean distances: think about a polyline connecting all your points with straight segments. Then the parameter would be the distance along this line for every input pair.
So now you have two interpolation problem: one to compute x from t and the other y from t. You can formulate this as a spline interpolation, e.g. using cubic splines. That gives you a large system of linear equations which you can solve iteratively up to the desired precision.
The result of a spline interpolation will be a piecewise description of a suitable curve. If you wanted a single equation, then a lagrange interpolation would fit that bill, but the result might have odd twists and turns for many sets of input data.
I have a set of vectors in multidimensional space (may be several thousands of dimensions). In this space, I can calculate distance between 2 vectors (as a cosine of the angle between them, if it matters). What I want is to visualize these vectors keeping the distance. That is, if vector a is closer to vector b than to vector c in multidimensional space, it also must be closer to it on 2-dimensional plot. Is there any kind of diagram that can clearly depict it?
I don't think so. Imagine any twodimensional picture of a tetrahedron. There is no way of depicting the four vertices in two dimensions with equal distances from each other. So you will have a hard time trying to depict more than three n-dimensional vectors in 2 dimensions conserving their mutual distances.
(But right now I can't think of a rigorous proof.)
Update:
Ok, second idea, maybe it's dumb: If you try and find clusters of closer associated objects/texts, then calculate the center or mean vector of each cluster. Then you can reduce the problem space. At first find a 2D composition of the clusters that preserves their relative distances. Then insert the primary vectors, only accounting for their relative distances within a cluster and their distance to the center of to two or three closest clusters.
This approach will be ok for a large number of vectors. But it will not be accurate in that there always will be somewhat similar vectors ending up at distant places.