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I know that I can find a polynomial regression's coefficients doing (X'X)^-1 * X'y (where X' is the transpose, see Wikipedia for details).
This is a way of finding the coefficients; now, there is (as far as I know) at least one other way, which is by minimizing a cost function using gradient descent. The former method seems to be the easiest to implement ( I did it in C++, I have the latter in Matlab ).
What I wanted to know is the advantage of one of these methods over the other.
Upon a particular dataset, with very few points, I found that I couldn't find a satisfactory solution using (X'X)^-1 * X'y, but gradient descent worked fine and I could get an estimation function that made sense.
So what's wrong with the matrix resolution over gradient descent ? And how would one test a regression results, having all the details hidden from the user ?
Both methods are equivalent. Iterative method is much more computationally efficient thanks to lower storage and the avoidance of matrix inverse calculation. The method outweighs the closed form (matrix equation) methods especially when X is huge and sparse.
Make sure the row number of X is larger than the column number of X to avoid the underdetermined problem. Also check out the condition number of X'X to see if the problem is ill-posedness. If that is the case, you may add a small regularization factor in the closed form ((X'X + lambda * I)^(-1) * X'y) where lambda is a small value and I is the identity matrix.
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I want to program a software that calculates the best combination of materials to use base on parameters such as its tensile strength, elastic modulus, stiffness, and results from doing certain tests from those materials. Those each factor are going to be weighted differently in a WDM. Is there an algorithm that would allow me to find the best combination without actually going through all the combinations and doing each individual calculations? I will be working with a lot of data, so efficiency is important
I tried researching algorithms like kruskal's and other things, but I'm not very fammiliar with them
First step is to write down an equation to calculate a number that you want to optimize.
If you can do that and the equation has no squares or other exponential terms then this is the classical linear programming problem https://en.wikipedia.org/wiki/Linear_programming
Your equation needs to look something like this:
max O = n1 * p1 + n2 * p2 - n3 * p3 ...
If so, then your best bet is to choose a linear programming package ( ask google ) with a good introductory tutorial and plug your problem into that. After a day or so on a steep learning curve, your problem will become almost trivial.
If you cannot do that, then you will need to use some sort of hill climbing algorithm - probably best to hire an expert to help with that.
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I would like to generate a random variable having an exponential density function:
f(x) = e^x / (e - 1), 0 <= x <= 1
I know I can use a uniform random number generator with using the inversion method for a simple function like (e^-x). But, I am not sure how to use them on the function given above.
Any suggestions?
Per Wolfram Alpha, the integral of that density function from 0 to a is (e^a-1)/(e-1), which inverts to y=log((e-1)*x+1). So the inverse transform method should work fine.
In the more general case where the integral doesn't pan out or the inversion doesn't pan out, stochastic sampling methods are the most widely applicable methods for sampling a random variable given its probability density. The easiest to understand and implement is Rejection Sampling. After that, you're looking at Metropolis-Hastings, which is immensely powerful but not necessarily the simplest to get your head around.
The first step is to integrate f(x) from 0 to x to determine the cumulative distribution function, call this function U. When you (pseudo-)randomly pick a number, put it into this function U and find x that satisfies this.
Your function appears to be simple enough that direct inversion will work. If you have a more complicated function, you would have to use a Newton-Raphson method to solve x for the given U.
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I am trying to understand basic chess algorithms. I have not read the literature in depth yet but after some cogitating here is my attempt:
1) Assign weight values to the pieces(i.e. a bishop is more valuable than a pawn)
2) Define heuristic function that attaches a value to a particular move
3) Build minimax tree to store all possible moves. Prune the tree via alpha/beta pruning.
4) Traverse the tree to find the best move for each player
Is this the core "big picture" idea of chess algorithms? Can someone point me to resources that go more in depth regarding chess algorithms?
Following is an overview of chess engine development.
1. Create a board representation.
In an object-oriented language, this will be an object that will represent a chess board in memory. The options at this stage are:
Bitboards
0x88
8x8
Bitboards is the recommended way for many reasons.
2. Create an evaluation function.
This simply takes a board and side-to-evaluate as agruments and returns a score. The method signature will look something like:
int Evaluate(Board boardPosition, int sideToEvaluateFor);
This is where you use the weights assigned to each piece. This is also where you would use any heuristics if you so desire. A simple evaluation function would add weights of sideToEvaluateFor's pieces and subtract weights of the opposite side's pieces. Such an evaluation function is of course too naive for a real chess engine.
3. Create a search function.
This will be, like you said, something on the lines of a MiniMax search with Alpha-Beta pruning. Some of the popular search algorithms are:
NegaMax
NegaScout
MTD(f)
Basic idea is to try all different variations to a certain maximum depth and choose the move recommended by the variation which results in highest score. The score for each variation is the score returned by Evaluation method for the board position at the maximum depth.
For an example of chess engine in C# have a look at https://github.com/bytefire/shutranj which I put together recently. A better open source engine to look at is StockFish (https://github.com/mcostalba/Stockfish) which is written in C++.
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I'm not sure if this is a stupid question but I couldn't really find anything on Google. Given a few data points for a function f(x) would it be possible to bruteforce what the function f(x) itself might be?
This will rely on some prior knowledge of f(x).
If you know that the function is constant, one point is enough; a line, then two points, etc. for polynomial functions.
But if you have no restrictions, this isn't possible. Assuming function here means something like a real-valued function on the real numbers, there are (uncountably) infinitely many functions which will take the specified values on any finite set of data points.
This is mostly math question. It depends on number of data points that are available. You are basically fitting data to a function. You need two data points for straight line, etc. The commercial solution is TableCurve 2D, http://en.wikipedia.org/wiki/TableCurve_2D. I would search for nonlinear fit on Google.
Fitting algorithms are also described in Numerical Recipes (http://en.wikipedia.org/wiki/Numerical_Recipes). The simplest algorithm would look for deviations between assumed function and data points. If you assume certain error on your data points, you can calculate chi-square and goodness of your fit.
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I am trying to find a way to solve an Optimization problem as follows:
I have 22 different objects that can be selected more than once. I have a evaluation function f that takes the multiplicities and calculates the total value.
f is a product over fractions of linear (affine) terms and as such, differentiable and even smooth in the allowed region.
I want to optimize f with respect to the 22 variables, with the additional conditions that certain sums may not exceed certain values (for example, if a,...,v are my variables, a + e + i + m + q + s <= 9). By this, all of the variables are bounded.
If f were strictly monotonuous, this could be solved optimally by a (minimalistically modified) knapsack solution. However, the function isnt convex. That means it is even impossible to assume that if taking an object A is better than B on an empty knapsack, that this choice holds even when adding a third object C (as C could modify B's benefit to be better than A). This means that a greedy algorithm cannot be used;
Are there similar algorithms that solve such a problem in a optimal (or at least, nearly optimal) way?
EDIT: As requested, an example of what the problem is (I chose 5 variables a,b,c,d,e for simplicity)
for example,
f(a,b,c,d,e) = e*(a*0.45+b*1.2-1)/(c+d)
(Every variable only appears once, if this helps at all)
Also, for example, a+b+c=4, d+e=3
The problem is to optimize that with respect to a,b,c,d,e as integers. There is a bunch of optimization algorithms that hold for convex functions, but very few for non-convex...