I am using libSVM.
Say my feature values are in the following format:
instance1 : f11, f12, f13, f14
instance2 : f21, f22, f23, f24
instance3 : f31, f32, f33, f34
instance4 : f41, f42, f43, f44
..............................
instanceN : fN1, fN2, fN3, fN4
I think there are two scaling can be applied.
scale each instance vector such that each vector has zero mean and unit variance.
( (f11, f12, f13, f14) - mean((f11, f12, f13, f14) ). /std((f11, f12, f13, f14) )
scale each colum of the above matrix to a range. for example [-1, 1]
According to my experiments with RBF kernel (libSVM) I found that the second scaling (2) improves the results by about 10%. I did not understand the reason why (2) gives me a improved results.
Could anybody explain me what is the reason for applying scaling and why the second option gives me improved results?
The standard thing to do is to make each dimension (or attribute, or column (in your example)) have zero mean and unit variance.
This brings each dimension of the SVM into the same magnitude. From http://www.csie.ntu.edu.tw/~cjlin/papers/guide/guide.pdf:
The main advantage of scaling is to avoid attributes in greater numeric
ranges dominating those in smaller numeric ranges. Another advantage is to avoid
numerical diculties during the calculation. Because kernel values usually depend on
the inner products of feature vectors, e.g. the linear kernel and the polynomial ker-
nel, large attribute values might cause numerical problems. We recommend linearly
scaling each attribute to the range [-1,+1] or [0,1].
I believe that it comes down to your original data a lot.
If your original data has SOME extreme values for some columns, then in my opinion you lose some definition when scaling linearly, for example in the range [-1,1].
Let's say that you have a column where 90% of values are between 100-500 and in the remaining 10% the values are as low as -2000 and as high as +2500.
If you scale this data linearly, then you'll have:
-2000 -> -1 ## <- The min in your scaled data
+2500 -> +1 ## <- The max in your scaled data
100 -> -0.06666666666666665
234 -> -0.007111111111111068
500 -> 0.11111111111111116
You could argue that the discernibility between what was originally 100 and 500 is smaller in the scaled data in comparison to what it was in the original data.
At the end, I believe it very much comes down to the specifics of your data and I believe the 10% improved performance is very coincidental, you will certainly not see a difference of this magnitude in every dataset you try both scaling methods on.
At the same time, in the paper in the link listed in the other answer, you can clearly see that the authors recommend data to be scaled linearly.
I hope someone finds this useful!
The accepted answer speaks of "Standard Scaling", which is not efficient for high-dimensional data stored in sparse matrices (text data is a use-case); in such cases, you may resort to "Max Scaling" and its variants, which works with sparse matrices.
I'm not a DSP expert, but I understand that there are two ways that I can apply a discrete time-domain filter to a discrete time-domain waveform. The first is to convolve them in the time domain, and the second is to take the FFT of both, multiply both complex spectrums, and take IFFT of the result. One key difference in these methods is the second approach is subject to circular convolution.
As an example, if the filter and waveforms are both N points long, the first approach (i.e. convolution) produces a result that is N+N-1 points long, where the first half of this response is the filter filling up and the 2nd half is the filter emptying. To get a steady-state response, the filter needs to have fewer points than the waveform to be filtered.
Continuing this example with the second approach, and assuming the discrete time-domain waveform data is all real (not complex), the FFT of the filter and the waveform both produce FFTs of N points long. Multiplying both spectrums IFFT'ing the result produces a time-domain result also N points long. Here the response where the filter fills up and empties overlap each other in the time domain, and there's no steady state response. This is the effect of circular convolution. To avoid this, typically the filter size would be smaller than the waveform size and both would be zero-padded to allow space for the frequency convolution to expand in time after IFFT of the product of the two spectrums.
My question is, I often see work in the literature from well-established experts/companies where they have a discrete (real) time-domain waveform (N points), they FFT it, multiply it by some filter (also N points), and IFFT the result for subsequent processing. My naive thinking is this result should contain no steady-state response and thus should contain artifacts from the filter filling/emptying that would lead to errors in interpreting the resulting data, but I must be missing something. Under what circumstances can this be a valid approach?
Any insight would be greatly appreciated
The basic problem is not about zero padding vs the assumed periodicity, but that Fourier analysis decomposes the signal into sine waves which, at the most basic level, are assumed to be infinite in extent. Both approaches are correct in that the IFFT using the full FFT will return the exact input waveform, and both approaches are incorrect in that using less than the full spectrum can lead to effects at the edges (that usually extend a few wavelengths). The only difference is in the details of what you assume fills in the rest of infinity, not in whether you are making an assumption.
Back to your first paragraph: Usually, in DSP, the biggest problem I run into with FFTs is that they are non-causal, and for this reason I often prefer to stay in the time domain, using, for example, FIR and IIR filters.
Update:
In the question statement, the OP correctly points out some of the problems that can arise when using FFTs to filter signals, for example, edge effects, that can be particularly problematic when doing a convolution that is comparable in the length (in the time domain) to the sampled waveform. It's important to note though that not all filtering is done using FFTs, and in the paper cited by the OP, they are not using FFT filters, and the problems that would arise with an FFT filter implementation do not arise using their approach.
Consider, for example, a filter that implements a simple average over 128 sample points, using two different implementations.
FFT: In the FFT/convolution approach one would have a sample of, say, 256, points and convolve this with a wfm that is constant for the first half and goes to zero in the second half. The question here is (even after this system has run a few cycles), what determines the value of the first point of the result? The FFT assumes that the wfm is circular (i.e. infinitely periodic) so either: the first point of the result is determined by the last 127 (i.e. future) samples of the wfm (skipping over the middle of the wfm), or by 127 zeros if you zero-pad. Neither is correct.
FIR: Another approach is to implement the average with an FIR filter. For example, here one could use the average of the values in a 128 register FIFO queue. That is, as each sample point comes in, 1) put it in the queue, 2) dequeue the oldest item, 3) average all of the 128 items remaining in the queue; and this is your result for this sample point. This approach runs continuously, handling one point at a time, and returning the filtered result after each sample, and has none of the problems that occur from the FFT as it's applied to finite sample chunks. Each result is just the average of the current sample and the 127 samples that came before it.
The paper that OP cites takes an approach much more similar to the FIR filter than to the FFT filter (note though that the filter in the paper is more complicated, and the whole paper is basically an analysis of this filter.) See, for example, this free book which describes how to analyze and apply different filters, and note also that the Laplace approach to analysis of the FIR and IIR filters is quite similar what what's found in the cited paper.
Here's an example of convolution without zero padding for the DFT (circular convolution) vs linear convolution. This is the convolution of a length M=32 sequence with a length L=128 sequence (using Numpy/Matplotlib):
f = rand(32); g = rand(128)
h1 = convolve(f, g)
h2 = real(ifft(fft(f, 128)*fft(g)))
plot(h1); plot(h2,'r')
grid()
The first M-1 points are different, and it's short by M-1 points since it wasn't zero padded. These differences are a problem if you're doing block convolution, but techniques such as overlap and save or overlap and add are used to overcome this problem. Otherwise if you're just computing a one-off filtering operation, the valid result will start at index M-1 and end at index L-1, with a length of L-M+1.
As to the paper cited, I looked at their MATLAB code in appendix A. I think they made a mistake in applying the Hfinal transfer function to the negative frequencies without first conjugating it. Otherwise, you can see in their graphs that the clock jitter is a periodic signal, so using circular convolution is fine for a steady-state analysis.
Edit: Regarding conjugating the transfer function, the PLLs have a real-valued impulse response, and every real-valued signal has a conjugate symmetric spectrum. In the code you can see that they're just using Hfinal[N-i] to get the negative frequencies without taking the conjugate. I've plotted their transfer function from -50 MHz to 50 MHz:
N = 150000 # number of samples. Need >50k to get a good spectrum.
res = 100e6/N # resolution of single freq point
f = res * arange(-N/2, N/2) # set the frequency sweep [-50MHz,50MHz), N points
s = 2j*pi*f # set the xfer function to complex radians
f1 = 22e6 # define 3dB corner frequency for H1
zeta1 = 0.54 # define peaking for H1
f2 = 7e6 # define 3dB corner frequency for H2
zeta2 = 0.54 # define peaking for H2
f3 = 1.0e6 # define 3dB corner frequency for H3
# w1 = natural frequency
w1 = 2*pi*f1/((1 + 2*zeta1**2 + ((1 + 2*zeta1**2)**2 + 1)**0.5)**0.5)
# H1 transfer function
H1 = ((2*zeta1*w1*s + w1**2)/(s**2 + 2*zeta1*w1*s + w1**2))
# w2 = natural frequency
w2 = 2*pi*f2/((1 + 2*zeta2**2 + ((1 + 2*zeta2**2)**2 + 1)**0.5)**0.5)
# H2 transfer function
H2 = ((2*zeta2*w2*s + w2**2)/(s**2 + 2*zeta2*w2*s + w2**2))
w3 = 2*pi*f3 # w3 = 3dB point for a single pole high pass function.
H3 = s/(s+w3) # the H3 xfer function is a high pass
Ht = 2*(H1-H2)*H3 # Final transfer based on the difference functions
subplot(311); plot(f, abs(Ht)); ylabel("abs")
subplot(312); plot(f, real(Ht)); ylabel("real")
subplot(313); plot(f, imag(Ht)); ylabel("imag")
As you can see, the real component has even symmetry and the imaginary component has odd symmetry. In their code they only calculated the positive frequencies for a loglog plot (reasonable enough). However, for calculating the inverse transform they used the values for the positive frequencies for the negative frequencies by indexing Hfinal[N-i] but forgot to conjugate it.
I can shed some light to the reason why "windowing" is applied before FFT is applied.
As already pointed out the FFT assumes that we have a infinite signal. When we take a sample over a finite time T this is mathematically the equivalent of multiplying the signal with a rectangular function.
Multiplying in the time domain becomes convolution in the frequency domain. The frequency response of a rectangle is the sync function i.e. sin(x)/x. The x in the numerator is the kicker, because it dies down O(1/N).
If you have frequency components which are exactly multiples of 1/T this does not matter as the sync function is zero in all points except that frequency where it is 1.
However if you have a sine which fall between 2 points you will see the sync function sampled on the frequency point. It lloks like a magnified version of the sync function and the 'ghost' signals caused by the convolution die down with 1/N or 6dB/octave. If you have a signal 60db above the noise floor, you will not see the noise for 1000 frequencies left and right from your main signal, it will be swamped by the "skirts" of the sync function.
If you use a different time window you get a different frequency response, a cosine for example dies down with 1/x^2, there are specialized windows for different measurements. The Hanning window is often used as a general purpose window.
The point is that the rectangular window used when not applying any "windowing function" creates far worse artefacts than a well chosen window. i.e by "distorting" the time samples we get a much better picture in the frequency domain which closer resembles "reality", or rather the "reality" we expect and want to see.
Although there will be artifacts from assuming that a rectangular window of data is periodic at the FFT aperture width, which is one interpretation of what circular convolution does without sufficient zero padding, the differences may or may not be large enough to swamp the data analysis in question.
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I'm aware of the gradient descent and the back-propagation algorithm. What I don't get is: when is using a bias important and how do you use it?
For example, when mapping the AND function, when I use two inputs and one output, it does not give the correct weights. However, when I use three inputs (one of which is a bias), it gives the correct weights.
I think that biases are almost always helpful. In effect, a bias value allows you to shift the activation function to the left or right, which may be critical for successful learning.
It might help to look at a simple example. Consider this 1-input, 1-output network that has no bias:
The output of the network is computed by multiplying the input (x) by the weight (w0) and passing the result through some kind of activation function (e.g. a sigmoid function.)
Here is the function that this network computes, for various values of w0:
Changing the weight w0 essentially changes the "steepness" of the sigmoid. That's useful, but what if you wanted the network to output 0 when x is 2? Just changing the steepness of the sigmoid won't really work -- you want to be able to shift the entire curve to the right.
That's exactly what the bias allows you to do. If we add a bias to that network, like so:
...then the output of the network becomes sig(w0*x + w1*1.0). Here is what the output of the network looks like for various values of w1:
Having a weight of -5 for w1 shifts the curve to the right, which allows us to have a network that outputs 0 when x is 2.
A simpler way to understand what the bias is: it is somehow similar to the constant b of a linear function
y = ax + b
It allows you to move the line up and down to fit the prediction with the data better.
Without b, the line always goes through the origin (0, 0) and you may get a poorer fit.
Here are some further illustrations showing the result of a simple 2-layer feed forward neural network with and without bias units on a two-variable regression problem. Weights are initialized randomly and standard ReLU activation is used. As the answers before me concluded, without the bias the ReLU-network is not able to deviate from zero at (0,0).
Two different kinds of parameters can
be adjusted during the training of an
ANN, the weights and the value in the
activation functions. This is
impractical and it would be easier if
only one of the parameters should be
adjusted. To cope with this problem a
bias neuron is invented. The bias
neuron lies in one layer, is connected
to all the neurons in the next layer,
but none in the previous layer and it
always emits 1. Since the bias neuron
emits 1 the weights, connected to the
bias neuron, are added directly to the
combined sum of the other weights
(equation 2.1), just like the t value
in the activation functions.1
The reason it's impractical is because you're simultaneously adjusting the weight and the value, so any change to the weight can neutralize the change to the value that was useful for a previous data instance... adding a bias neuron without a changing value allows you to control the behavior of the layer.
Furthermore the bias allows you to use a single neural net to represent similar cases. Consider the AND boolean function represented by the following neural network:
(source: aihorizon.com)
w0 corresponds to b.
w1 corresponds to x1.
w2 corresponds to x2.
A single perceptron can be used to
represent many boolean functions.
For example, if we assume boolean values
of 1 (true) and -1 (false), then one
way to use a two-input perceptron to
implement the AND function is to set
the weights w0 = -3, and w1 = w2 = .5.
This perceptron can be made to
represent the OR function instead by
altering the threshold to w0 = -.3. In
fact, AND and OR can be viewed as
special cases of m-of-n functions:
that is, functions where at least m of
the n inputs to the perceptron must be
true. The OR function corresponds to
m = 1 and the AND function to m = n.
Any m-of-n function is easily
represented using a perceptron by
setting all input weights to the same
value (e.g., 0.5) and then setting the
threshold w0 accordingly.
Perceptrons can represent all of the
primitive boolean functions AND, OR,
NAND ( 1 AND), and NOR ( 1 OR). Machine Learning- Tom Mitchell)
The threshold is the bias and w0 is the weight associated with the bias/threshold neuron.
The bias is not an NN term. It's a generic algebra term to consider.
Y = M*X + C (straight line equation)
Now if C(Bias) = 0 then, the line will always pass through the origin, i.e. (0,0), and depends on only one parameter, i.e. M, which is the slope so we have less things to play with.
C, which is the bias takes any number and has the activity to shift the graph, and hence able to represent more complex situations.
In a logistic regression, the expected value of the target is transformed by a link function to restrict its value to the unit interval. In this way, model predictions can be viewed as primary outcome probabilities as shown:
Sigmoid function on Wikipedia
This is the final activation layer in the NN map that turns on and off the neuron. Here also bias has a role to play and it shifts the curve flexibly to help us map the model.
A layer in a neural network without a bias is nothing more than the multiplication of an input vector with a matrix. (The output vector might be passed through a sigmoid function for normalisation and for use in multi-layered ANN afterwards, but that’s not important.)
This means that you’re using a linear function and thus an input of all zeros will always be mapped to an output of all zeros. This might be a reasonable solution for some systems but in general it is too restrictive.
Using a bias, you’re effectively adding another dimension to your input space, which always takes the value one, so you’re avoiding an input vector of all zeros. You don’t lose any generality by this because your trained weight matrix needs not be surjective, so it still can map to all values previously possible.
2D ANN:
For a ANN mapping two dimensions to one dimension, as in reproducing the AND or the OR (or XOR) functions, you can think of a neuronal network as doing the following:
On the 2D plane mark all positions of input vectors. So, for boolean values, you’d want to mark (-1,-1), (1,1), (-1,1), (1,-1). What your ANN now does is drawing a straight line on the 2d plane, separating the positive output from the negative output values.
Without bias, this straight line has to go through zero, whereas with bias, you’re free to put it anywhere.
So, you’ll see that without bias you’re facing a problem with the AND function, since you can’t put both (1,-1) and (-1,1) to the negative side. (They are not allowed to be on the line.) The problem is equal for the OR function. With a bias, however, it’s easy to draw the line.
Note that the XOR function in that situation can’t be solved even with bias.
When you use ANNs, you rarely know about the internals of the systems you want to learn. Some things cannot be learned without a bias. E.g., have a look at the following data: (0, 1), (1, 1), (2, 1), basically a function that maps any x to 1.
If you have a one layered network (or a linear mapping), you cannot find a solution. However, if you have a bias it's trivial!
In an ideal setting, a bias could also map all points to the mean of the target points and let the hidden neurons model the differences from that point.
Modification of neuron WEIGHTS alone only serves to manipulate the shape/curvature of your transfer function, and not its equilibrium/zero crossing point.
The introduction of bias neurons allows you to shift the transfer function curve horizontally (left/right) along the input axis while leaving the shape/curvature unaltered.
This will allow the network to produce arbitrary outputs different from the defaults and hence you can customize/shift the input-to-output mapping to suit your particular needs.
See here for graphical explanation:
http://www.heatonresearch.com/wiki/Bias
In a couple of experiments in my masters thesis (e.g. page 59), I found that the bias might be important for the first layer(s), but especially at the fully connected layers at the end it seems not to play a big role.
This might be highly dependent on the network architecture / dataset.
If you're working with images, you might actually prefer to not use a bias at all. In theory, that way your network will be more independent of data magnitude, as in whether the picture is dark, or bright and vivid. And the net is going to learn to do it's job through studying relativity inside your data. Lots of modern neural networks utilize this.
For other data having biases might be critical. It depends on what type of data you're dealing with. If your information is magnitude-invariant --- if inputting [1,0,0.1] should lead to the same result as if inputting [100,0,10], you might be better off without a bias.
Bias determines how much angle your weight will rotate.
In a two-dimensional chart, weight and bias can help us to find the decision boundary of outputs.
Say we need to build a AND function, the input(p)-output(t) pair should be
{p=[0,0], t=0},{p=[1,0], t=0},{p=[0,1], t=0},{p=[1,1], t=1}
Now we need to find a decision boundary, and the ideal boundary should be:
See? W is perpendicular to our boundary. Thus, we say W decided the direction of boundary.
However, it is hard to find correct W at first time. Mostly, we choose original W value randomly. Thus, the first boundary may be this:
Now the boundary is parallel to the y axis.
We want to rotate the boundary. How?
By changing the W.
So, we use the learning rule function: W'=W+P:
W'=W+P is equivalent to W' = W + bP, while b=1.
Therefore, by changing the value of b(bias), you can decide the angle between W' and W. That is "the learning rule of ANN".
You could also read Neural Network Design by Martin T. Hagan / Howard B. Demuth / Mark H. Beale, chapter 4 "Perceptron Learning Rule"
In simpler terms, biases allow for more and more variations of weights to be learnt/stored... (side-note: sometimes given some threshold). Anyway, more variations mean that biases add richer representation of the input space to the model's learnt/stored weights. (Where better weights can enhance the neural net’s guessing power)
For example, in learning models, the hypothesis/guess is desirably bounded by y=0 or y=1 given some input, in maybe some classification task... i.e some y=0 for some x=(1,1) and some y=1 for some x=(0,1). (The condition on the hypothesis/outcome is the threshold I talked about above. Note that my examples setup inputs X to be each x=a double or 2 valued-vector, instead of Nate's single valued x inputs of some collection X).
If we ignore the bias, many inputs may end up being represented by a lot of the same weights (i.e. the learnt weights mostly occur close to the origin (0,0).
The model would then be limited to poorer quantities of good weights, instead of the many many more good weights it could better learn with bias. (Where poorly learnt weights lead to poorer guesses or a decrease in the neural net’s guessing power)
So, it is optimal that the model learns both close to the origin, but also, in as many places as possible inside the threshold/decision boundary. With the bias we can enable degrees of freedom close to the origin, but not limited to origin's immediate region.
In neural networks:
Each neuron has a bias
You can view bias as a threshold (generally opposite values of threshold)
Weighted sum from input layers + bias decides activation of a neuron
Bias increases the flexibility of the model.
In absence of bias, the neuron may not be activated by considering only the weighted sum from the input layer. If the neuron is not activated, the information from this neuron is not passed through rest of the neural network.
The value of bias is learnable.
Effectively, bias = — threshold. You can think of bias as how easy it is to get the neuron to output a 1 — with a really big bias, it’s very easy for the neuron to output a 1, but if the bias is very negative, then it’s difficult.
In summary: bias helps in controlling the value at which the activation function will trigger.
Follow this video for more details.
Few more useful links:
geeksforgeeks
towardsdatascience
Expanding on zfy's explanation:
The equation for one input, one neuron, one output should look:
y = a * x + b * 1 and out = f(y)
where x is the value from the input node and 1 is the value of the bias node;
y can be directly your output or be passed into a function, often a sigmoid function. Also note that the bias could be any constant, but to make everything simpler we always pick 1 (and probably that's so common that zfy did it without showing & explaining it).
Your network is trying to learn coefficients a and b to adapt to your data.
So you can see why adding the element b * 1 allows it to fit better to more data: now you can change both slope and intercept.
If you have more than one input your equation will look like:
y = a0 * x0 + a1 * x1 + ... + aN * 1
Note that the equation still describes a one neuron, one output network; if you have more neurons you just add one dimension to the coefficient matrix, to multiplex the inputs to all nodes and sum back each node contribution.
That you can write in vectorized format as
A = [a0, a1, .., aN] , X = [x0, x1, ..., 1]
Y = A . XT
i.e. putting coefficients in one array and (inputs + bias) in another you have your desired solution as the dot product of the two vectors (you need to transpose X for the shape to be correct, I wrote XT a 'X transposed')
So in the end you can also see your bias as is just one more input to represent the part of the output that is actually independent of your input.
To think in a simple way, if you have y=w1*x where y is your output and w1 is the weight, imagine a condition where x=0 then y=w1*x equals to 0.
If you want to update your weight you have to compute how much change by delw=target-y where target is your target output. In this case 'delw' will not change since y is computed as 0. So, suppose if you can add some extra value it will help y = w1x + w01, where bias=1 and weight can be adjusted to get a correct bias. Consider the example below.
In terms of line slope, intercept is a specific form of linear equations.
y = mx + b
Check the image
image
Here b is (0,2)
If you want to increase it to (0,3) how will you do it by changing the value of b the bias.
For all the ML books I studied, the W is always defined as the connectivity index between two neurons, which means the higher connectivity between two neurons.
The stronger the signals will be transmitted from the firing neuron to the target neuron or Y = w * X as a result to maintain the biological character of neurons, we need to keep the 1 >=W >= -1, but in the real regression, the W will end up with |W| >=1 which contradicts how the neurons are working.
As a result, I propose W = cos(theta), while 1 >= |cos(theta)|, and Y= a * X = W * X + b while a = b + W = b + cos(theta), b is an integer.
Bias acts as our anchor. It's a way for us to have some kind of baseline where we don't go below that. In terms of a graph, think of like y=mx+b it's like a y-intercept of this function.
output = input times the weight value and added a bias value and then apply an activation function.
The term bias is used to adjust the final output matrix as the y-intercept does. For instance, in the classic equation, y = mx + c, if c = 0, then the line will always pass through 0. Adding the bias term provides more flexibility and better generalisation to our neural network model.
The bias helps to get a better equation.
Imagine the input and output like a function y = ax + b and you need to put the right line between the input(x) and output(y) to minimise the global error between each point and the line, if you keep the equation like this y = ax, you will have one parameter for adaptation only, even if you find the best a minimising the global error it will be kind of far from the wanted value.
You can say the bias makes the equation more flexible to adapt to the best values