I am trying using multivariate regression to play basketball. Specificlly, I need to, based on X, Y, and distance from the target, predict the pitch, yaw, and cannon strength. I was thinking of using multivariate regression with multipule variables for each of the output parameter. Is there a better way to do this?
Also, should I use solve directly for the best fit, or use gradient descent?
ElKamina's answer is correct but one thing to note about this is that it is identical to doing k independent ordinary least squares regressions. That is, the same as doing a separate linear regression from X to pitch, from X to yaw, and from X to strength. This means, you are not taking advantage of correlations between the output variables. This may be fine for your application, but one alternative that does take advantage of correlations in the output is reduced rank regression(a matlab implementation here), or somewhat related, you can explicitly uncorrelate y by projecting it onto its principle components (see PCA, also called PCA whitening in this case since you aren't reducing the dimensionality).
I highly recommend chapter 6 of Izenman's textbook "Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning" for a fairly high level overview of these techniques. If you're at a University it may be available online through your library.
If those alternatives don't perform well, there are many sophisticated non-linear regression methods that have multiple output versions (although most software packages don't have the multivariate modifications) such as support vector regression, Gaussian process regression, decision tree regression, or even neural networks.
Multivariate regression is equivalent to doing the inverse of the covariance of the input variable set. Since there are many solutions to inverting the matrix (if the dimensionality is not very high. Thousand should be okay), you should go directly for the best fit instead of gradient descent.
n be the number of samples, m be the number of input variables and k be the number of output variables.
X be the input data (n,m)
Y be the target data (n,k)
A be the coefficients you want to estimate (m,k)
XA = Y
X'XA=X'Y
A = inverse(X'X)X'Y
X' is the transpose of X.
As you can see, once you find the inverse of X'X you can calculate the coefficients for any number of output variables with just a couple of matrix multiplications.
Use any simple math tools to solve this (MATLAB/R/Python..).
Related
I am reading the book "Introduction to linear algebra" by Gilbert Strang. The section is called "Orthonormal Bases and Gram-Schmidt". The author several times emphasised the fact that with orthonormal basis it's very easy and fast to calculate Least Squares solution, since Qᵀ*Q = I, where Q is a design matrix with orthonormal basis. So your equation becomes x̂ = Qᵀb.
And I got the impression that it's a good idea to every time calculate QR decomposition before applying Least Squares. But later I figured out time complexity for QR decomposition and it turned out to be that calculating QR decomposition and after that applying Least Squares is more expensive than regular x̂ = inv(AᵀA)Aᵀb.
Is that right that there is no point in using QR decomposition to speed up Least Squares? Or maybe I got something wrong?
So the only purpose of QR decomposition regarding Least Squares is numerical stability?
There are many ways to do least squares; typically these vary in applicability, accuracy and speed.
Perhaps the Rolls-Royce method is to use SVD. This can be used to solve under-determined (fewer obs than states) and singular systems (where A'*A is not invertible) and is very accurate. It is also the slowest.
QR can only be used to solve non-singular systems (that is we must have A'*A invertible, ie A must be of full rank), and though perhaps not as accurate as SVD is also a good deal faster.
The normal equations ie
compute P = A'*A
solve P*x = A'*b
is the fastest (perhaps by a large margin if P can be computed efficiently, for example if A is sparse) but is also the least accurate. This too can only be used to solve non singular systems.
Inaccuracy should not be taken lightly nor dismissed as some academic fanciness. If you happen to know that the problems ypu will be solving are nicely behaved, then it might well be fine to use an inaccurate method. But otherwise the inaccurate routine might well fail (ie say there is no solution when there is, or worse come up with a totally bogus answer).
I'm a but confused that you seem to be suggesting forming and solving the normal equations after performing the QR decomposition. The usual way to use QR in least squares is, if A is nObs x nStates:
decompose A as A = Q*(R )
(0 )
transform b into b~ = Q'*b
(here R is upper triangular)
solve R * x = b# for x,
(here b# is the first nStates entries of b~)
The following is the plot of a curve f(r), where r is the radial coordinate, and plotted for different values of a parameter as shown:
However, I don't know the functional form of the curve and I am interested to find the same. Are there any numerical methods which can be used to find the functional form of f(r) in terms of the radial coordinate and the parameter?
I had found a solution of the problem based on the suggestion by ja72 to use the Eureqa software which churns through the data to create accurate predictive models using evolutionary search algorithm.
In the question, the different curves corresponds to different values of . So, initially I obtained the best fit equation for different values of and found that the following model equation is suitable for my purpose:
Then, I repeated the process for a large number of values of and calculated the values of the four functions for different values of and then individually fitted these four functions. The following are the results that I obtained:
N.B.: Eureqa gave several other better fitting formulas than those mentioned in the answer. But the formulas that I mentioned are sufficiently accurate for my purpose and have minimum complexity.
A blind curve fit without an underlying model is a dangerous thing.
You need to have an understanding of the physical model behind the data to create a successful fit. The reason is that if r is distance and the best fit curve uses r^0.4072 for example, that dimension raised to a decimal power bears no meaning and it hides any underlying assumptions.Like some other dimension l not included in the model, whereas only the dimensionless quantity (r/l) would make sense to raise to the decimal power.
From a function analysis standpoint
These curves are not the result of any standard math function. Well I am not that familiar with bessel functions, gamma functions and legendre polynomials. But none of the standard functions you find in a scientific calculator jumps out here.
If r is assumed to be dimensionless, then you try to match the asymptotic behavior when r -> 0 and when r -> ∞. The would be the baseline curve. To me it does not look hyperbolic, but rather close to 1/LN(1+r).
So change the variables make g=1/LN(1+r) and plot f(r) against g(r) and see what that looks like. Then try another round of curve fitting in the new curves ... and so on.
Nobody can answer this question
Nobody else could effectively answer this question but you, because a) you have the data, and b) you need to make assumptions about what region is important or not, and what is acceptable deviation.
I'm working on knowledge graph, more precisely in natural language processing field. To evaluate the components of my algorithm, it is necessary to be able to classify the good and the poor candidates. For this purpose, we manually classified pairs in a dataset.
My system returns the relevant pairs according to the implementation logic. now I'm able to calculate :
Precision = X
Recall = Y
For establishing a complete curve I need the rest of points (X,Y), what should I do?:
build another dataset for test ?
split my dataset ?
or any other solution ?
Neither of your proposed two methods. In short, Precision-recall or ROC curve is designed for classifiers with probabilistic output. That is, instead of simply producing a 0 or 1 (in case of binary classification), you need classifiers that can provide a probability in [0,1] range. This is the function to do it in sklearn, note how the 2nd parameter is called probas_pred.
To turn this probabilities into concrete class prediction, you can then set a threshold, say at .5. Setting such a threshold is problematic however, since you can trade-off precision/recall by varying the threshold, and an arbitrary choice can give false impression of a classifier's performance. To circumvent this, threshold-independent measures like area under ROC or Precision-Recall curve is used. They create thresholds at different intervals, say 0.1,0.2,0.3...0.9, turn probabilities into binary classes and then compute precision-recall for each such threshold.
I am given the following problem.
I have a Set of functions which are linear combinations of the following functions (f1,f2,f3....fn) and a noisy dataset of pairs (x,y). I want to find a function from my set which approximates the dataset the best.
They key to finding the solution is to find coefficients a1,a2...an so that the resulting function f=a1*f1...an*fn approximates y well given the input x. If the data wasnt noisy, I could just choose 5 points and solve the resulting system of equations but I dont think this would work well with noisy data.
How would one find the coefficients ?
(I am asking for an algorithm and not for a program, for example matlab, that does the job for me)
In presence of noise you need to find some approximation solution, that minimizes discrepancies with ideal solution.
Such best fit problems are usually solved by optimization algorithms.
Widely used one is Levenberg–Marquardt algorithm.
I know many uniform random number generators(RNGs) based on some algorithms, physical systems and so on. Eventually, all these lead to uniformly distributed random numbers. It's interesting and important to know whether there is Gaussian RNGs, i.e. the algorithm or something else creates Gaussian random numbers. Much precisely I want to say that I don't want to use transformations such as Box–Muller or Marsaglia polar method to get Gaussian from Uniform RNGs. I am interested if there is some paper, algorithm or even idea to create Gaussian random numbers without any of use Uniform RNGs. It's just to say we pretend that we don't know there exist Uniform random number generators.
As already noted in answers/comments, by virtue of CLT some sum of any iid random number could be made into some reasonable looking gaussian. If incoming stream is uniform, this is basically Bates distribution. Ami Tavory answer is pretty much amounts to using Bates in disguise. You could look at closely related Irwin-Hall distribution, and at n=12 or higher they look a lot like gaussian.
There is one method which is used in practice and does not rely on transformation of the U(0,1) - Wallace method (Wallace, C. S. 1996. "Fast Pseudorandom Generators for Normal and Exponential Variates." ACM Transactions on Mathematical Software.), or gaussian pool method. I would advice to read description here and see if it fits your purpose
As others have noted, it's a bit unclear what is your motivation for this, and therefore I'm not sure if the following answers your question.
Nevertheless, it is possible to generate (an approximation of) this without the specific formulas transforming uniform RNGs that you mention.
As with any RNG, we have to have some source of randomness (or pseudo-randomness). I'm assuming, therefore, that there is some limitless sequence of binary bits which are independently equally likely to be 0 or 1 (note that it's possible to counter that this is a uniform discrete binary RNG, so I'm unsure if this answers your question).
Choose some large fixed n. For each invocation of the RNG, generate n such bits, sum them as x, and return
(2 x - 1) / √n
By the de Moivre–Laplace theorem this is normal with mean 0 and variance 1.