I have build my own neural net and I have a weird problem with it.
The net is quite a simple feed-forward 1-N-1 net with back propagation learning. Sigmoid is used as activation function.
My training set is generated with random values between [-PI, PI] and their [0,1]-scaled sine values (This is because the "Sigmoid-net" produces only values between [0,1] and unscaled sine -function produces values between [-1,1]).
With that training-set, and the net set to 1-10-1 with learning rate of 0.5, everything works great and the net learns sin-function as it should. BUT.. if I do everything exately the same way for COSINE -function, the net won't learn it. Not with any setup of hidden layer size or learning rate.
Any ideas? Am I missing something?
EDIT: My problem seems to be similar than can be seen with this applet. It won't seem to learn sine-function unless something "easier" is taught for the weights first (like 1400 cycles of quadratic function). All the other settings in the applet can be left as they initially are. So in the case of sine or cosine it seems that the weights need some boosting to atleast partially right direction before a solution can be found. Why is this?
I'm struggling to see how this could work.
You have, as far as I can see, 1 input, N nodes in 1 layer, then 1 output. So there is no difference between any of the nodes in the hidden layer of the net. Suppose you have an input x, and a set of weights wi. Then the output node y will have the value:
y = Σi w_i x
= x . Σi w_i
So this is always linear.
In order for the nodes to be able to learn differently, they must be wired differently and/or have access to different inputs. So you could supply inputs of the value, the square root of the value (giving some effect of scale), etc and wire different hidden layer nodes to different inputs, and I suspect you'll need at least one more hidden layer anyway.
The neural net is not magic. It produces a set of specific weights for a weighted sum. Since you can derive a set weights to approximate a sine or cosine function, that must inform your idea of what inputs the neural net will need in order to have some chance of succeeding.
An explicit example: the Taylor series of the exponential function is:
exp(x) = 1 + x/1! + x^2/2! + x^3/3! + x^4/4! ...
So if you supplied 6 input notes with 1, x1, x2 etc, then a neural net that just received each input to one corresponding node, and multiplied it by its weight then fed all those outputs to the output node would be capable of the 6-term taylor expansion of the exponential:
in hid out
1 ---- h0 -\
x -- h1 --\
x^2 -- h2 ---\
x^3 -- h3 ----- y
x^4 -- h4 ---/
x^5 -- h5 --/
Not much of a neural net, but you get the point.
Further down the wikipedia page on Taylor series, there are expansions for sin and cos, which are given in terms of odd powers of x and even powers of x respectively (think about it, sin is odd, cos is even, and yes it is that straightforward), so if you supply all the powers of x I would guess that the sin and cos versions will look pretty similar with alternating zero weights. (sin: 0, 1, 0, -1/6..., cos: 1, 0, -1/2...)
I think you can always compute sine and then compute cosine externally. I think your concern here is why the neural net is not learning the cosine function when it can learn the sine function. Assuming that this artifact if not because of your code; I would suggest the following:
It definitely looks like an error in the learning algorithm. Could be because of your starting point. Try starting with weights that gives the correct result for the first input and then march forward.
Check if there is heavy bias in your learning - more +ve than -ve
Since cosine can be computed by sine 90 minus angle, you could find the weights and then recompute the weights in 1 step for cosine.
Related
I am learning about Two Dimensional Neuron Network so I am facing many obstacles but I believe it is worth it and I am really enjoying this learning process.
Here's my plan: To make a 2-D NN work on recognizing images of digits. Images are 5 by 3 grids and I prepared 10 images from zero to nine. For Example this would be number 7:
Number 7 has indexes 0,1,2,5,8,11,14 as 1s (or 3,4,6,7,9,10,12,13 as 0s doesn't matter) and so on. Therefore, my input layer will be a 5 by 3 neuron layer and I will be feeding it zeros OR ones only (not in between and the indexes depends on which image I am feeding the layer).
My output layer however will be one dimensional layer of 10 neurons. Depends on which digit was recognized, a certain neuron will fire a value of one and the rest should be zeros (shouldn't fire).
I am done with implementing everything, I have a problem in computing though and I would really appreciate any help. I am getting an extremely high error rate and an extremely low (negative) output values on all output neurons and values (error and output) do not change even on the 10,000th pass.
I would love to go further and post my Backpropagation methods since I believe the problem is in it. However to break down my work I would love to hear some comments first, I want to know if my design is approachable.
Does my plan make sense?
All the posts are speaking about ranges ( 0->1, -1 ->+1, 0.01 -> 0.5 etc ), will it work for either { 0 | .OR. | 1 } on the output layer and not a range? if yes, how can I control that?
I am using TanHyperbolic as my transfer function. Does it make a difference between this and sigmoid, other functions.. etc?
Any ideas/comments/guidance are appreciated and thanks in advance
Well, by the description given above, I think that the design and approach taken it's correct! With respect to the choice of the activation function, remember that those functions help to get the neurons which have the largest activation number, also, their algebraic properties, such as an easy derivative, help with the definition of Backpropagation. Taking this into account, you should not worry about your choice of activation function.
The ranges that you mention above, correspond to a process of scaling of the input, it is better to have your input images in range 0 to 1. This helps to scale the error surface and help with the speed and convergence of the optimization process. Because your input set is composed of images, and each image is composed of pixels, the minimum value and and the maximum value that a pixel can attain is 0 and 255, respectively. To scale your input in this example, it is essential to divide each value by 255.
Now, with respect to the training problems, Have you tried checking if your gradient calculation routine is correct? i.e., by using the cost function, and evaluating the cost function, J? If not, try generating a toy vector theta that contains all the weight matrices involved in your neural network, and evaluate the gradient at each point, by using the definition of gradient, sorry for the Matlab example, but it should be easy to port to C++:
perturb = zeros(size(theta));
e = 1e-4;
for p = 1:numel(theta)
% Set perturbation vector
perturb(p) = e;
loss1 = J(theta - perturb);
loss2 = J(theta + perturb);
% Compute Numerical Gradient
numgrad(p) = (loss2 - loss1) / (2*e);
perturb(p) = 0;
end
After evaluating the function, compare the numerical gradient, with the gradient calculated by using backpropagation. If the difference between each calculation is less than 3e-9, then your implementation shall be correct.
I recommend to checkout the UFLDL tutorials offered by the Stanford Artificial Intelligence Laboratory, there you can find a lot of information related to neural networks and its paradigms, it's worth to take look at it!
http://ufldl.stanford.edu/wiki/index.php/Main_Page
http://ufldl.stanford.edu/tutorial/
we're trying to generate a 3d world using a 2d perlin noise (with a recorsive/fractal technique). We have generated mountains and valleys quite fine but now we are having problems with desert and dunes because we only worked on persistence and octaves and we aren't able to make the classic shape of the dune. Has anybody already experienced that? Any solution, possibly still using perlin noise, or also other algorithms which allow you to do this?
You could give the Musgrave ridged multifractal a try. It gives nice ridged structures and you can use your existing noise algorithms for it.
The C reference implementation for it is here
Dunes are lobsided: .='\ cross section... you may want to use an initial shape of that kind
They are regular, like waves in the sea. not completely noise
they are elongated towards the wind
I didnt use the first condition, but i have made great dunes by multiplying 2 1d perin noises together, or even 2 sin/parabol functions, where they are both lined to one axis. i.e. Z, and they have a small low frequency Sin or noise wobbling them along X axis, so they aren't alined.
try this:
dunes = sin ( X + 1dperlin(Z) *.2 ) * sin ( X + 1dperlin(Z+432) *.2 );
otherwise to test it:
dunes = sin ( X + sin(Z) *.2 ) (plus or times or devided by) sin ( X + sin(Z+432) *.2 );
0.2 makes dunes 10 times longer than wide, and it's like when two straight water waves meet at almost the same angle, plus an uncertainty variable using noise for the angle.
Maybe turbulence is yet enough for what you need... Try to play with turbulence using the absolute value of your octaves return values instead of the normal values. You can also evaluate separately and combine your noise and your turbulence to mix both effects in some areas.
I'm trying to implement a RBM and I'm testing it on MNIST dataset. However, it does not seems to converge.
I've 28x28 visible units and 100 hidden units. I'm using mini-batches of size 50. For each epoch, I traverse the whole dataset. I've a learning rate of 0.01 and a momentum of 0.5. The weights are randomly generated based on a Gaussian distribution of mean 0.0 and stdev of 0.01. The visible and hidden biases are initialized to 0. I'm using a logistic sigmoid function as activation.
After each epoch, I compute the average reconstruction error of all mini-batches, here are the errors I get:
epoch 0: Reconstruction error average: 0.0481795
epoch 1: Reconstruction error average: 0.0350295
epoch 2: Reconstruction error average: 0.0324191
epoch 3: Reconstruction error average: 0.0309714
epoch 4: Reconstruction error average: 0.0300068
I plotted the histograms of the weights to check (left to right: hiddens, weights, visibles. top: weights, bottom: updates):
Histogram of the weights after epoch 3
Histogram of the weights after epoch 3 http://baptiste-wicht.com/static/finals/histogram_epoch_3.png
Histogram of the weights after epoch 4
Histogram of the weights after epoch 4 http://baptiste-wicht.com/static/finals/histogram_epoch_4.png
but, except for the hidden biases that seem a bit weird, the remaining seems OK.
I also tried to plot the hidden weights:
Weights after epoch 3
Weights after epoch 3 http://baptiste-wicht.com/static/finals/hiddens_weights_epoch_3.png
Weights after epoch 4
Weights after epoch 4 http://baptiste-wicht.com/static/finals/hiddens_weights_epoch_4.png
(they are plotted in two colors using that function:
static_cast<size_t>(value > 0 ? (static_cast<size_t>(value * 255.0) << 8) : (static_cast<size_t>(-value * 255.)0) << 16) << " ";
)
And here, they do not make sense at all...
If I go further, the reconstruction error falls a bit more, but do no go further than 0.025. Even if I change the momentum after sometime, it goes higher and then goes down a bit but not interestingly. Moreover, the weights do no make more sense after more epochs. In most example implementations I've seen, the weights were making some sense after iterating through the complete data set two or three times.
I've also tried to reconstruct an image from the visible units, but the results seems almost random.
What could I do to check what goes wrong in my implementation ? Should the weights be within some range ? Does something seems really strange in the data ?
Complete code: https://github.com/wichtounet/dbn/blob/master/include/rbm.hpp
You are using a very small learning rate. In most NNs trained by SGD you start out with a higher learning rate and decay it over time. Search for learning rate or adaptive learning rate to find more information on that.
Second, when implementing a new algorithm I would recommend finding the paper that introduced it and reproducing their results. A good paper should include most of the settings used - or the method used to determine the settings.
If a paper is unavailable, or it was tested on a data set you don't have access to - go find a working implementation and compare the outputs when using the same settings. If the implementations are not feature compatible, turn off as many features as you can that are not shared.
How can I measure this area in C++?
(update: I posted the solution and code as an answer rather than edit the question again)
The ideal line (dashed red) is the plot from starting point with the average rise added with each angle of measurement; this I obtain via average. I measured the test data in black. How can I quantify the area of the dip in blue? X-axis is unitized, so slopes and math are simplified.
I could determine a cutoff for the size of areas like this and then flag this part for retesting or failure. Rarely, there is another dip that appears closer to the right, but setting a cutoff value for standard deviation usually fails those parts.
Update
Diego's answer helped me visualize this. Now that I can see what I'm trying to do, I'll work on the algorithm to implement the "homemade dip detector". :)
Why?
I created a test bench to test throttle position sensors I'm selling. I'm trying to programatically quantify how straight the plot is by analyzing the data collected. This one particular model is vexing me.
Sample plot of a part I prefer not to sell:
The X axis are evenly spaced angles of throttle opening. The stepper motor turns the input shaft, stopping every 0.75° to measure the output on a 10 bit ADC, which gets translated to the Y axis. The plot is the translation of data[idx] to idx,value mapped to (x,y) bitmap coordinates. Then I draw lines between the points within the bitmap using Bresenham's algorithm.
My other TPS products produce amazingly linear output.
The lower (left) portion of the plot is crucial to normal usage of any motor vehicle; it's when you're driving around town, entering parking lots, etc. This particular part has a tendency to develop a dip around 15° opening and I wish to use the program to quantify this "dip" in the curve and rely less upon the tester's intuition. In the above example, the plot dips but doesn't return to what an ideal line might be.
Even though this is an embedded application, printing the report takes 10 seconds, thus I do not consider stepping through an array of 120 points of data multiple times a waste of cycles. Also, since I'm using a uC32 PIC32 microcontroller, there's plenty of memory, so I have the luxury of being able to ponder this problem within the controller.
What I'm trying already
Array of rise between test points: I dismiss the X-axis entirely, considering it unitized, and then make an array of change from one reading to the next. This array is what contributes to the report's "Min rise between points: 0 Max: 14". I call this array deltas.
I've tried using standard deviation on deltas, however, during testing I have found that a low Std Dev is not a reliable measure for this part. If the dip quickly returns to the original line implied by early data points, the Std Dev can be deceptively low (observed to be as low as 2.3) but the part is still something I wouldn't want to use. I tried setting a cutoff at 2.6, but it failed too many parts with great plots. The other, more linear part linked to above can reliably count on Std Dev for quality.
Kurtosis seems not to apply for this situation at all. I learned of Kurtosis today and found a Statistics Library which includes Kurtosis and Skewness. During continued testing, I found that of these two measures, there was not a trend of positive, negative, or amplitude which would correspond to either passing or failing. That same gentleman has shared a linear regression library, but I believe Lin Reg is unrelated to my situation, as I am comfortable with the assumption of the AVG of deltas being my ideal line. Linear Regression and R^2 are more for finding a line from less ideal data or much larger sets.
Comparing each delta to AVG and Std Dev I set up a monitor to check each delta against final average of the deltas's data. Here, too, I couldn't find a reliable metric. Too many good parts would not pass a test restricting any delta to within 2x Std Dev away from the Average. Ultimately, the only variation from AVG I could settle on is to be within AVG+Std Dev difference from the AVG itself. Anything more restrictive would fail otherwise good parts. And the elusive dip around 15° opening can sneak through this test.
Homemade dip detector When feeding deltas to the serial monitor of the computer, I observed consecutive negative deltas during the dip, so I programmed in a dip detector, but it feels very crude to me. If there are 5 or more negative deltas in a row, I sum them. I have seen that if I take that sum the dip's differences from AVG then divide by the number of negative deltas, a value over 2.9 or 3 could mean a fail. I have observed dips lasting from 6 to 15 deltas. Readily observable dips would have their differences from AVG sum up to -35.
Trending accumulated variation from the AVG The above made me think watching the summation of deltas as it wanders away from AVG could be the answer. Meaning, I step through the array and sum the differences of each delta from AVG. I thought I was on to something until a good part blew this theory. I was seeing a trend of the fewer times the running sum varied from AVG by less than 2x AVG, the more straight the line appeared. Many ideal parts would only show 8 or less delta points where the sumOfDiffs would stray from the AVG very far.
float sumOfDiffs=0.0;
for( int idx=0; idx<stop; idx++ ){
float spread = deltas[idx] - line->AdcAvgRise;
sumOfDiffs = sumOfDiffs + spread;
...
testVal = 2*line->AdcAvgRise;
if( sumOfDiffs > testVal || sumOfDiffs < -testVal ){
flag = 'S';
}
...
}
And then a part with a fantastic linear plot came through with 58 data points where sumOfDiffs was more than twice the AVG! I find this amazing, as at the end of the ~120 data points, sumOfDiffs value is -0.000057.
During testing, the final sumOfDiffs result would often register as 0.000000 and only on exceptionally bad parts would it be greater than .000100. I found this quite surprising, actually: how a "bad part" can have accumulated great accuracy.
Sample output from monitoring sumOfDiffs This below output shows a dip happening. The test watches as the running sumOfDiffs is more than 2x the AVG away from the AVG for the whole test. This dip lasts from deltas idx of 23 through 49; starts at 17.25° and lasts for 19.5°.
Avg rise: 6.75 Std dev: 2.577
idx: delta diff from avg sumOfDiffs Flag
23: 5 -1.75 -14.05 S
24: 6 -0.75 -14.80 S
25: 7 0.25 -14.55 S
26: 5 -1.75 -16.30 S
27: 3 -3.75 -20.06 S
28: 3 -3.75 -23.81 S
29: 7 0.25 -23.56 S
30: 4 -2.75 -26.31 S
31: 2 -4.75 -31.06 S
32: 8 1.25 -29.82 S
33: 6 -0.75 -30.57 S
34: 9 2.25 -28.32 S
35: 8 1.25 -27.07 S
36: 5 -1.75 -28.82 S
37: 15 8.25 -20.58 S
38: 7 0.25 -20.33 S
39: 5 -1.75 -22.08 S
40: 9 2.25 -19.83 S
41: 10 3.25 -16.58 S
42: 9 2.25 -14.34 S
43: 3 -3.75 -18.09 S
44: 6 -0.75 -18.84 S
45: 11 4.25 -14.59 S
47: 3 -3.75 -16.10 S
48: 8 1.25 -14.85 S
49: 8 1.25 -13.60 S
Final Sum of diffs: 0.000030
RunningStats analysis:
NumDataValues= 125
Mean= 6.752
StandardDeviation= 2.577
Skewness= 0.251
Kurtosis= -0.277
Sobering note about quality: what started me on this journey was learning how major automotive OEM suppliers consider a 4 point test to be the standard measure for these parts. My first test bench used an Arduino with 8k of RAM, didn't have a TFT display nor a printer, and a mechanical resolution of only 3°! Back then I simply tested deltas being within arbitrary total bounds and choosing a limit of how big any single delta could be. My 120+ point test feels high class compared to that 30 point test from before, but that test had no idea about these dips.
Premises
the mean of a set of data has the mathematical property that the sum of the deviations from the mean is 0.
this explains why both bad and good datasets alwais give almost 0.
basically the result when differs from zero is essentially an accumulations of rounding errors in the diffs and that's why unfortunately cannot hold useful informations
the thing that most clearly define what you're looking for is your image: you're looking for an AREA and this is why you're not finding the solution in this ways:
looking to a metric in the single points is too local to extract that information
looking to global accumulations or parameters (global standard deviation) is too global and you lose the data among too much information and source of variations
kurtosis (you've already told I know but is for completeness) is out of its field of applications since this is not a probability distribution
in the end the more suitable approach of your already tryied ones is the "Homemade dip detector" because thinks in a way that is local but not too much.
Last but not least:
Any Algorithm you're going to choose has its tacit points on which it stands.
So maybe one is looking for a super clever algorithm that with no parametrization and tuning automatically adapts to the problem and self define thereshods and other.
On the other side there is an algorithm that will stand on the knowledge by the writer of the tipical data behavior (good and bad) and that is itself stupid in the way that if there is another different and unespected behavior the results are unpredictable
Ok, the right way is one of this two or is in-between them depending on the application. So if it works also the "Homemade dip detectors" can be a solution. There is not reason to define it crude but it could be that is not sufficient based on applicaton needs and that's an other thing.
How to find the area
Once you have the data the first thing is to clearly define the "theoretical straight line". I give some options:
use RANSAC algorithm (formally the best option IMHO)
this give you the best fit to the aligned points disregarding the not aligned ones
it is quite difficult and maybe oversized for this work (IMHO)
consider the line defined by the first and last point
you told that the dip is almost always in the same position that is not near boundaries so first and last points can be thought as affordable
very easy to implement
this is an example of using the knowledge about expected behaviors as I told before so you need to think if and how much confidence you give to this assumption
consider a linear fit to the first 10 points and last 10 points
is only a more affordable version of previous since using more points you can be less worried that maybe just the first point or the last were affected by any measure problem and so all fails because of this
also quite easy to implement
if I were you I will use this or something inspired to this
calculate the Y value given by the straight line for each X
calculate the area between the two curves (or the areas under the function Y_dev = Y_data - Y_straight that is mathematically the same) with this procedure:
PositiveMax = 0; NegativeMax = 0;
start from first point (value can be positive or negative) and put in a temporary area accumulator tmp_Area
for each next point
if the sign is the same then accumulate the value
if it is different
stop accumulating
check if the accumulated value is the greater than PositiveMax or below NegativeMax and if it is than store as new PositiveMax or NegativeMax
in any case reset the accumulator with tmp_Area = Y_dev; to the current value starting this way a new accumulation
in the end you will have the values of the maximum overvalued contiguous area and maximum undervalued contiguous area that I think are the scores you're looking for.
if you want you can only manage the NegativeMax based on observed and expected data behaviors
you may find useful to put a thereshold so that if a value Y_dev is lower than the thereshold you do not accumulate it.
this in order to not obtain large accumulations from many points close to the straight line that can be similar to the accumulations of few points far from the line
the need of this and and the proper thereshold needs to be evaluated on some sample data
you need to find an appropriate thereshold for this contiguous area and you can have it only from observation of sample data.
again: it can be you observing and deciding the thereshold or you can build a repository of good and bad samples and write a program that automatically learn which thereshold to use. But his is not the algorithm, this is how to find its operative parameters and there is nothing wrong to do by human brain.. ..it only depends if we're looking for a method to separate bad and good things or if we're looking for and autoadaptive algorithm that does this.. ..you decide the target.
It turns out the result of my gut feeling and Diego's method is an average of the integral. I still don't like that name, so I have described the algorithm and have asked on Math.SE what to call this, which got migrated to "Cross Validated", Stats.SE .
I Updated graphs after a massive edit of my Math.SE question. It turns out I'm taking the average of a closed integral of the derivative of the data. :P First, we gather the data:
Next is the "derivative": step through the original data array to form the deltas array which is the rise of ADC values from one 0.75° step to the next. "Rise" or "slope" is what the derivative is: dy/dx.
With the "slope" or average leveled out, I can find multiple negative deltas in a row, sum them, then divide by the count at the end of the dip. The sum is an integral of the area between average and the deltas and when the dip goes back positive, I can divide the sum by the count of the dips.
During testing, I came up with a cutoff value for this average of the integral at 2.6. That was a great measure of my "gut instinct" looking at the plot thinking a part was good or bad.
In case someone else finds themselves trying to quantify this, here's the code I implemented. Note that it is only looking for negative dips. Also, dipCountLimit is defined elsewhere as 5. In addition to the dip detector/accumulator (ie Numerical Integrator) I also have a spike detector that arbitrarily flags the test as bad if any data points stray from the average by the amount of average + standard deviation. AVG+STD DEV as a spike limit was chosen arbitrarily based on the observed plots of the parts it would fail.
int dipdx=0;
// inDipFlag also counts the length of this dip
int inDipFlag=0;
float dips[140] = { 0.0 };
for( int idx=0; idx<stop; idx++ ){
const float diffFromAvg = deltas[idx] - line->AdcAvgRise;
// state machine to monitor dips
const int _stop = stop-1;
if( diffFromAvg < 0 && idx < _stop ) {
// check NEXT data point for negative diff & set dipFlag to put state in dip
const float nextDiff = deltas[idx+1] - line->AdcAvgRise;
if( nextDiff < 0 && inDipFlag == 0 )
inDipFlag = 1;
// already IN a dip, and next diff is negative
if( nextDiff < 0 && inDipFlag > 0 ) {
inDipFlag++;
}
// accumulate this dip
dips[dipdx]+= diffFromAvg;
// next data point ends this dip and we advance dipdx to next dip
if( inDipFlag > 0 && nextDiff > 0 ) {
if( inDipFlag < dipCountLimit ){
// reset the accumulator, do not advance dipdx to next entry
dips[dipdx]=0.0;
} else {
// change this entry's value from dip sum to its ratio
dips[dipdx] = -dips[dipdx]/inDipFlag;
// advance dipdx to next entry
dipdx++;
}
// Next diff isn't negative, so the dip is done
inDipFlag = 0;
}
}
}
what is the best approach to generate random samples from bivariate normal and student T distributions? In both cases sigma is one, mean 0 - so the only parameter I am really interested in is correlation (and degrees of freedom for student t). I need to have the solution in C++, so I can't unfortunately use already implemented functions from MatLab or Mathematica.
You can use the GNU GSL libraries. See here for Bivariate normal:
http://www.gnu.org/software/gsl/manual/html_node/The-Bivariate-Gaussian-Distribution.html
and Student's t-distribution here:
http://www.gnu.org/software/gsl/manual/html_node/The-t_002ddistribution.html
They are straight forward to use.
For a bivariate normal with covariance unity and zero mean, just draw two univariate normals.
If you want to draw a bivariate normal with means (m1, m2), standard deviations (s1, s2) and correlation rho, then draw two unit univariate normals X and Y and set
u = m1 + s1 * X
v = m2 + s2 * (rho X + sqrt(1 - rho^2) Y)
Then u and v are distributed as you wish.
For the Student T, you have to draw a normal variate N and a chi^2 variate V. Then, N / sqrt(V) has T distribution.
To draw the chi^2, you should use a package. Or have a look at Numerical Recipes chapter 7 for how to draw from a Gamma distribution (xhi^2 is a special case of Gamma).
You should take a look at the Boost libraries random distributions - see http://www.boost.org/doc/libs/1_41_0/libs/random/random-distributions.html. I've found them very easy to use, once you wrap your head around their basic concepts. Unfortunately, I don't know enough about statistics to tell you whether they will exactly meet your needs.