This is another atmospheric modeling question thread: http://www.sciforums.com/showthread.php?t=97064 In any case this is a question regarding an iterative model and initial value uncertainties. The nature of the model is moderately irrelevant. Say initial conditions have a certainty of +-.01 - however, the model holds float values. After the first iteration all values are held with an uncertainty of +-.00000001. (This is an uncertainty from its assumed value, not actual value) The models 12th iteration gets new initial values of +-.01 certainty. Typically for said model these values are significantly divergent from the new initial values. Here's the proposal: What if the original initial values, (i1), were reset to fit the new initial values, (i2). This was done continuously for a large number of 'initial values' such that by the 100th new values, the set of i1 to i2 were fine tuned to the point where divergence was almost non existent. Wouldn't this greatly increase the accuracy of future modeling to the point that its assumed values were more accurate than the +-.01 uncertainty of the input values. - Is this bad mathematical intuition? - Does such a process exist? If so, what is it called? FOR MOD Can you change "Turning" to "Tuning"
Okay I'll propose an idea I had for this - does anyone understand the goal, or was my previous explanation poor? A set of differentials like... \(a_0y^{(n)} +a_1y^{(n-1)} ...a_{(n-1)}y' + a_{(n)}y = f(y,t)\) n being the iterative step function calculated like such. Code: y(t)(n=5) dx y(t)(n=4) dx dx y(t)''' dx dx dx y(t)'' dx dx dx dx y(t)' dx dx dx dx dx y(t) x1 x2 x3 x4 x5 x6 So when \(n \to Limit\) a function can be found in the form \(y = f(x,t)\) You would actually have a best fit to f(x,t) as a function of something like this... \(f(x,t) = Acos(\omega_n t)+Bcos(\omega_{n-1}t)...\) Which would be the various regression values of known climatological cycles. The RMS values of each would give an upper and lower limit. For example: \(\alpha=.1\) would be considered the significance value in the regression analysis of a given point. A So your specific max and min values might be a result of 2 standard deviations... or \(\bar x + 2\sigma = MAX\) \(\bar x - 2\sigma = MIN\) Thus the functional equivalents to be used in the linear solutions. \(f^{MAX}(x,t) = Acos(\omega^{MAX}_n t)+Bcos(\omega^{MAX}_{n-1}t)...\) \(f^{MIN}(x,t) = Acos(\omega^{MIN}_n t)+Bcos(\omega^{MIN}_{n-1}t)...\) Then you would find linear differential solution values: \(a_0y^{(n)} +a_1y^{(n-1)} ...a_{(n-1)}y' + a_{(n)}y = f^{MAX}(y,t) \equiv y^{MAX}(x,t)\) \(a_0y^{(n)} +a_1y^{(n-1)} ...a_{(n-1)}y' + a_{(n)}y = f(y,t)=\equiv y(x,t)\) \(a_0y^{(n)} +a_1y^{(n-1)} ...a_{(n-1)}y' + a_{(n)}y = f^{MIN}\equiv (y,t)y^{MIN}(x,t)\) Since the information to this point is a 0-Dimensional value, you then might then find the actualized values through a process described in a previous thread using Laplacian matrix. The information of the laplacian algorithm is being copied from the OP's link \(\nabla_{i_n j_n} = sqrt(\frac{4i_nj_n^2 + (i_{(n-1)}j_{(n)})^2 +(i_{(n+1)}j_{(n)})^2 + (i_{(n)}j_{(n+1)})^2 + (i_{(n)}j_{(n-1)})^2}{4}) \to \nabla_{i_{(n+1)}j_n}\) Map Code: i1 i2 i3 i4 i5 i6 j1 ................... j2 ................... j3 ................... j4 ................... j5 ................... j6 ............. i6j6 Algorithm Pattern Code: i1 i2 i3 i4 i5 i6 j1 1 2 3 4 5 6 j2 20 21 22 23 24 7 j3 19 32 33 34 25 8 j4 18 31 36 35 26 9 j5 17 30 29 28 27 10 j6 16 15 14 13 12 11 The agorithm would continue to iterate until the final values would be the result of this algorithm such that for every point... \([y^{MIN}(x,t)_{(i,j)} < y^(x,t)_{(i,j)}<y^{MAX}(x,t)_{(i,j)}]\forall y(x,t)_{(i,j)\)
On the face of it, this sounds very much like negative feedback used to improve the output accuracy of nonlinear signal amplification. But you have another post...
I'm not seeing how this reduces (regresses) over time... Your differentials above, yes, but not this. Does it not? That follows so far... Missing a few steps, or am I rusty? This seems very much along the lines of a matrix approach to biological decay, except you're using series functions based on polynomials instead of nat logs... Helps us, here. You appear to be searching for error reduction among recursive data involving climatological cycles. Am I correct? I think what you're looking for it a recursive modification of previous values based upon known, better values. Although never found in nature, this is sometimes necessary in mathematical modeling due to errant (even if only ever so slightly) data in chaotic systems. Recursing actual measurments into the predictive algorithm performas a smoothing function which makes subsequent predictions more accurate without starting with a massive remeasurement of the entire initial conditions. It's perfect for ongoing weather prediction, and I see (now) your "best fit to f(x,t)" fuction includes provisions for non-diminishing cycles. Need more input... :bugeye:
whoa,what are these ancient/alien/egyptian symbols :$ What's this all about? Even the title is confusing.
