Method for separating 'randomness' and 'non-randomness' Let's assume I have a random two signals: 
Sin(t)  

R(t)

Sin(t) is of course the trignometric function, but R(t) is a random process.
So let's now assume I create a third function in which I sum these two functions.
Z(t) = Sin(t) + R(t)

Because R(t) was a random process, Z(t) is now considered a random process. However, I see that there is a non-random part within this function. That of course is Sin(t).
So my question: Are there method for separating the random and non-random parts of a random process? Do there exist general methods for this separation, or does it always come down to making assumptions about the non-random signal?
I'm somewhat familiar with Kalman filtering in which we attempt to find the 'true' signal by taking into consideration preconceived notions of the signal along with the amount of randomness we expect.
Is it required that we make assumptions in order to have any hopes of separating random and non-random components?
 A: Your best bet is using a discrete wavelet transform without knowing more about your stochastic process. If it is stationary (i.e. the freqencies of the signal do not change with respect to time), normal Fourier methods (i.e. STFT) will suffice. Do you have access to the Wavelet toolbox in MATLAB? I'll post more information in the morning when I have a chance
Here is some documentation to get you started: 
http://web.mit.edu/1.130/WebDocs/wavelet_ug.pdf
I do not know what your background is so I'll start from the basics of the theory so I apologize if this gets long here. Anyways, the wavelet transform(s) (there are many) are a method of decomposing a signal into various scales or resolution. First, for the theoretical part, assume your signal is truncated so that it goes to zero outside of some finite interval such that our signal $f(t) \in L^{2}(\mathbb{R})$. Then we define the multiresolution analysis on $L^{2}(\mathbb{R})$ which is a nesting of subspaces of $L^{2}(\mathbb{R})$ which we denote $\mathcal{V}_{j}$ where $j \in \mathbb{Z}$such that:
$f(t) \in \mathcal{V}_{j} \implies f(2t) \in \mathcal{V}_{j+1}$
$\{0\}=\mathcal{V}_{-\infty} \cup \cdots \cup \mathcal{V}_{0} \cup \mathcal{V}_{1} \cup \cdots \cup \mathcal{V}_{j} \cup \cdots \cup \mathcal{V}_{\infty}=L^{2}(\mathbb{R})$ (i.e. each subspace contains the elements of all of the subspaces of lower resolution)
Now, define a scaling function $\phi(t)$ whose integer translates $\phi_{k}(t)=\phi(t-k)$ span the subspace $\mathcal{V}_{0}$ (i.e. $\mathcal{V}_{0} = \text{span}_{k}\{\phi_{k}(t)\})$
So then for $\mathcal{V}_{j}$, we have $\mathcal{V}_{j}=\text{span}_{k}\{\phi(2^{j}t-k)\}$. Now, this nesting of spans allows us to relate $\phi(t)$ to the the scaled and translated functions $\phi_{j,k}(t)=\phi(2^{j}t-k)$ via the following (functional) equation:
$$\phi(t)=\sum_{i=-\infty}^{\infty} h_{i} \sqrt{2} \phi(2t-i)$$
ansd we call the coefficents $h_{i}$ the "scaling function coefficients'', while the $\sqrt{2}$ term just serves to normalize the scaling function. Don't worry about solving this equation as it these are often very difficult to solve. We only need this equation to relate the scaling function to the wavelet function anyways.
So now we need to define a different set of functions $\psi_{j,k}(t)$ which are orthogonal to the scaling functions in $V_{j}$, which we will call $\mathcal{W}_{j}$ and has the property that for some $j$, say $\mathcal{V_{0}}$, then we have
$$
\mathcal{V}_{1}=\mathcal{V}_{0} \oplus \mathcal{W}_{0}
$$
but note that we can start at any value of $j$ (usually the lowest resolution of interest for a given signal in practice) and the same relationship will hold. Now since the wavelet functions $\psi(t)$ are in the space spanned by the scaling function of one higher resolution we have that $\mathcal{W}_{0} \subset \mathcal{V}_{1}$ and we can represent the wavelet function much like we did the scaling function where we have
$$
\psi(t)=\sum_{i=-\infty}^{\infty}h^{*}_{i} \sqrt{2} \phi(2t-i)
$$
where $h^{*}_{i}$ is given by
$$
h^{*}_{i}=(-1^{i})h_{1-i}
$$
and recall that $h_{i}$ is the scaling function coefficent.
Now, this allows us to define the prototypical "mother wavelet" functions which are of the form
$$
\psi_{j,k}(t)=2^{j/2}\psi(2^{j}t-k)
$$
and since $\phi_{k}(t)$ and $\psi_{j,k}(t)$ span all of $L^{2}(\mathbb{R})$
we can then write any $f(t) \in L^{2}(\mathbb{R})$ as 
$$
g(t)=\sum^{\infty}_{k=-\infty} \langle f(t),\phi_{k}(t) \rangle \phi_{k}(t)+ \sum^{\infty}_{k=-\infty} \sum^{\infty}_{j=-\infty} \langle f(t),\psi_{j,k}(t)\rangle \psi_{j,k}(t)
$$
Also note that we can generalize this because of the fact that 
$$L^{2}=\mathcal{V}_{j_{0}} \oplus \mathcal{W}_{j_{0}} \oplus \mathcal{W}_{j_{0}+1} \oplus \cdots$$
if we choose any $j=j_{0}$ then our decomposition becomes 
$$
 g(t)=\sum^{\infty}_{k=-\infty} \langle f(t),\phi_{j_{0},k}(t) \rangle \phi_{j_{0},k}(t)+ \sum^{\infty}_{k=-\infty} \sum^{\infty}_{j=j_{0}} \langle f(t),\psi_{j_{0},k}(t)\rangle \psi_{j_{0},k}(t)$$
This approach theoretically allows us to decompose any real world signal into "approximation" (low resolution) and "detail" (high resolution) components. Like one of the posters said before, using a low pass filter is a good idea, although wavelets (the scaling function coefficient acts as a low pass filter) offer the advantage of decomposing according to scale rather than frequency since Brownian motion is scale invariant and you should see that the deterministic component of the signal will only contribute at coarser scales. Now if your stochastic process has a richer structure, then you may have to resort to other methods.
You can also use the DWT to denoise signals and the MATLAB Wavelet toolbox is very helpful in this. In fact, I think your example is included as a demo. Hope this helps. 
A: At some point you have to decide what is "signal" and what is "noise".  If I want to know the magnitude of the tides given second-accurate water height data, then I want the "Sin(t)" in your description.  If I want to know the magnitude of the individual waves, then I want to know the "R(t)" in your description.  What is signal and what is noise depends on what you want to measure.
In spite of this, there are methods of automatic signal segregation.  A couple of examples:


*

*If your signal is the convolution of an "interesting signal" and a "blur function", you may be interested in blind deconvolution.

*If your signal is a linear combination of random processes that have different noise structure, you might be interested in independent component analysis.

*If you believe a (locally) low-dimensional process is being presented as a high-dimensional signal (for instance, the Lorenz attractor is very nearly a union of two flat attractors) you may be interested in (local) singular value decomposition.


There's actually rather a lot of literature on blind signal separation.
