Assume for a moment I have two input signals f1 and f2. I could add these signals to produce a third signal f3 = f1 + f2. I would then compute the spectrogram of f3 as log(|stft(f3)|^2).
Unfortunately I don't have the original signals f1 and f2. I have, however, their spectrograms A = log(|stft(f1)|^2) and B = log(|stft(f2)|^2). What I'm looking for is a way to approximate log(|stft(f3)|^2) as closely as possible using A and B. If we do some math we can derive:
log(|stft(f1 + f2)|^2) = log(|stft(f1) + stft(f2)|^2)
express stft(f1) = x1 + i * y1 & stft(f2) = x2 + i * y2 to write
... = log(|x1 + i * y1 + x2 + i * y2|^2)
... = log((x1 + x2)^2 + (y1 + y2)^2)
... = log(x1^2 + x2^2 + y1^2 + y2^2 + 2 * (x1 * x2 + y1 * y2))
... = log(|stft(f1)|^2 + |stft(f2)|^2 + 2 * (x1 * x2 + y1 * y2))
So at this point I could use the approximation:
log(|stft(f3)|^2) ~ log(exp(A) + exp(B))
but I would ignore the last part 2 * (x1 * x2 + y1 * y2). So my question is: Is there a better approximation for this?
Any ideas? Thanks.
I'm not 100% understanding your notation but I'll give it a shot. Addition in the time domain corresponds to addition in the frequency domain. Adding two time domain signals x1 and x2 produces a 3rd time domain signal x3. x1, x2 and x3 all have a frequency domain spectrum, F(x1), F(x2) and F(x3). F(x3) is also equal to F(x1) + F(x2) where the addition is performed by adding the real parts of F(x1) to the real parts of F(x2) and adding the imaginary parts of F(x1) to the imaginary parts of F(x2). So if x1[0] is 1+0j and x2[0] is 0.5+0.5j then the sum is 1.5+0.5j. Judging from your notation you are trying to add the magnitudes, which with this example would be |1+0j| + |0.5+0.5j| = sqrt(1*1) + sqrt(0.5*0.5+0.5*0.5) = sqrt(2) + sqrt(0.5). Obviously not the same thing. I think you want something like this:
log((|stft(a) + stft(b)|)^2) = log(|stft(a)|^2) + log(|stft(b)|^2)
Take the exp() of the 2 log magnitudes, add them, then take the log of the sum.
Stepping back from the math for a minute, we can see that at a fundamental level, this just isn't possible.
Consider a 1st signal f1 that is a pure tone at frequency F and amplitude A.
Consider a 2nd signal f2 that is a pure tone at frequency F and amplitude A, but perfectly out of phase with f1.
In this case, the spectrograms of f1 & f2 are identical.
Now consider two possible combined signals.
f1 added to itself is a pure tone at frequency F and amplitude 2A.
f1 added to f2 is complete silence.
From the spectrograms of f1 and f2 alone (which are identical), you've no way to know which of these very different situations you're in. And this doesn't just hold for pure tones. Any signal and its reflection about the axis suffer the same problem. Generalizing even further, there's just no way to know how much your underlying signals cancel and how much they reinforce each other. That said, there are limits. If, for a particular frequency, your underlying signals had amplitudes A1 and A2, the biggest possible amplitude is A1+A2 and the smallest possible is abs(A1-A2).
Related
I have the following two mixed models
Full model (with the three-way interaction)
m_Kunkle_pacc3_n <- lmer(pacc3_old ~ PRS_Kunkle*AgeAtVisit +
PRS_Kunkle*I(AgeAtVisit^2) +
APOE_score*AgeAtVisit + APOE_score*I(AgeAtVisit^2) + PRS_Kunkle*APOE_score + famhist +
+ gender + EdYears_Coded_Max20 + VisNo + X1 + X2 + X3 + X4 + X5 +
(1 |family/DBID),
data = WRAP_all, REML = F)
Nested model (exclude the three-way interaction (two variables are excluded: three-way interaction with linear age and quadratic age))
m_Kunkle_pacc3 <- lmer(pacc3_old ~ PRS_Kunkle*AgeAtVisit*APOE_score +
PRS_Kunkle*I(AgeAtVisit^2)*APOE_score +
+ gender + EdYears_Coded_Max20 + VisNo + famhist + X1 + X2 + X3 + X4 + X5 +
(1 |family/DBID),
data = WRAP_all, REML = F)
I used the likelihood ratio test to test the difference between full model and nested model, am I correct in testing the significance of this three-way interaction?
pacc3_LRT_Kunkle <- anova(m_Kunkle_pacc3, m_Kunkle_pacc3_n, test = "chisq")
Many thanks
If you are interested in testing the significance of the three-way interaction, I think in general you should do that within the context of a single model. You first select a model based on theoretical considerations and sometimes certain indices, and then look at the parameters of the model you pick. For example, the BIC is related to a model's negative log-likelihood penalized by its complexity (it also depends on your sample size), and you can use the BIC to select a model among competing choices. Once you pick a model that has a certain interaction term within it, you evaluate its coefficient. I should warn you that interpreting three-way interactions can be very challenging, so you should consider that in the context of your problem as well.
TLDR; comparing a model that has a term with one that does not have it (whether you look at their R^2, compare likelihoods or penalized likelihoods, etc.) will tell you something about the whole model, not the parameter itself.
I am attempting a maximisation problem subject to various constraints.
i.e. max y = x1 + x2 + x3 + .... + xn
where each xi is a vector of values over time: x1 = (x11, x12, x13,...)
Some of the constraints state that specific values of xit cannot be positive in the same time period.
i.e. if(x1t > 0), x2t = 0; if(x2t > 0), x1t = 0
For context, the constraint is equivalent to "maximise the revenue of a shop, but you cant sell product A and B on the same day"
How do I go about formulating an LP model in Excel (using solver) to solve this.
This is called a complementarity constraint. One way of modeling this is:
x(1,t) * x(2,t) = 0
x(i,t) ≥ 0
However, this is nonlinear (and in a somewhat nasty way). A linear approach, using an extra binary variable δ can look like:
x(1,t) ≤ UP(1,t) * δ(t)
x(2,t) ≤ UP(2,t) * (1-δ(t))
x(i,t) ∈ [0,UP(i,t)] 'UP is an upper bound on x'
δ(t) ∈ {0,1} 'δ is a binary variable'
I would like to investigate the effects of two independent variables on a dependent variable. Suppose we have X1, X2 independent variables, and Y dependent variable.
I use two different approaches. In the first approach, to eliminate the effect of X1 on Y, I generate the conditional distribution of Y|X1 and perform regression using the second variable X2. When I check the correlations between X2 and Y|X1, I obtain relatively high correlations (R2>0.50). However, when I perform multiple regression over a wide range of data (X1 and X2), the effect of X2 on Y is decreased and becomes insignificant. How do these approaches give conflicting results? What is the most appropriate approach to determine the effect of X2 on Y for a given X1 value? Thanks.
It could be good to see the code or the above in mathematical notation.
For instance: did you include the constant terms?
What do you see when:
Y = B0 + B1X1 + B2X2
That will be the easiest to check, and B2 will give you probably what you want.
That model is still simple, you could explore something like:
Y = B0 + B1X1 + B2X2 + B3X1X2
or
Y = B0 + B1X1 + B2X2 + B3X1X2 + B4X1^2 + B5X2^2
And see if there are changes in the coefficients and if there are new significant coefficients.
You could go further and explore Structural Equation Models
For a 3D straight line expressed in the standard form
a1*x + b1*y + c1*z + d1 = 0
a2*x + b2*y + c2*z + d2 = 0
and a given point x0,y0,z0
what is the distance from the point to the straight line?
Distance from point P0 to parametric line L(t) = Base + t * Dir is
Dist = Length(CrossProduct(Dir, P0 - Base)) / Length(Dir)
To find direction vector:
Dir = CrossProduct((a1,b1,c1), (a2,b2,c2))
To get some arbitrary base point, solve equation system with 2 equations and three unknowns (find arbitrary solution):
a1*x + b1*y + c1*z + d1 = 0
a2*x + b2*y + c2*z + d2 = 0
Check minors consisting of a and b, a and c, b and c coefficients. When minor is non-zero, corresponding variable might be taken as free one. For example, if a1 * b2 - b1 * a2 <> 0, choose variable z as free - make it zero or another value and solve system for two unknowns x and y.
(I omitted extra cases of parallel or coinciding planes)
I have a single of x-y coordinate system
This diagram should represent what you've told me.
The key point, is to express [x2],[y2] in CS1. (I can't use latex here so let's assume that [A] means the vector A, |A| is the length of the vector A)
[v2] = v2x * [x2] + v2y * [y2]
Since we have well defined [v1] and [d2], we can calculate [x']
[x`] = [d2] - [v1]
From [x'] we can calculate x2
[x2] = (|x2|/|x'|)[x`] = (|x1|/|x'|)[x'] since |x1| = |x2|
From x2 we can calculate y2, although I don't remember how. It's a simple 90° rotation.
Should be this:
y2x = - x2y
y2y = x2x
Once we have expressed x2,y2 in CS1, we can compute v2
v2 = v2x * [x2] + v2y * [y2] = v2x * (x2x*[x1]+x2y*[y1]) + v2y * (y2x*[x1]+y2y*[y1])
= (v2xx2x + v2yy2x)[x1] + (v2xx2y + v2yy2y) [y1] // Hope I didn't make any mistake here :)
And finally
[X] = [v1] + [v2]
I think the best option is to create a vector class and do all the math using vector algebra. You just need to define 3 operation: Addition, ScalarMultiplication, 90Rotation.