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{{Primary sources|date=April 2012}}
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{{notability|date=June 2012}}
 
In [[statistics]], '''scaled correlation''' is a form of a coefficient of [[correlation]] applicable to data that have a temporal component such as [[time series]]. If the signals have multiple components (slow and fast), scaled coefficient of correlation can be computed only for the fast components of the signals, ignoring the contributions of the slow components.<ref name = "Nikolicetal">Nikolić D, Muresan RC, Feng W, Singer W (2012) Scaled correlation analysis: a better way to compute a cross-correlogram. ''European Journal of Neuroscience'', pp. 1–21, doi:10.1111/j.1460-9568.2011.07987.x http://www.danko-nikolic.com/wp-content/uploads/2012/03/Scaled-correlation-analysis.pdf</ref> This has the advantages of not having to make assumptions about the sinusoidal nature of the signals.
 
For example, in the studies of brain signals researchers are often interested in the high-frequency components (beta and gamma range; 25–80&nbsp;Hz), and may not be interested in lower frequency ranges (alpha, theta, etc.). In that case scaled correlation can be computed only for frequencies higher than 25 Hz by choosing the scale of the analysis, ''s'', to correspond to the period of that frequency (e.g., ''s''&nbsp;=&nbsp;40 ms for 25&nbsp;Hz oscillation).
 
==Definition==
Scaled correlation between two signals is defined as the average correlation computed across short segments of those signals. First, it is necessary to determine the number of segments <math>K</math> that can fit into the total length of the signals <math>T</math> for a given scale <math>s</math>:
 
:<math>K = \operatorname{round}\left(\frac{T}{s}\right).</math>
 
Next, if <math>r_k</math> is [[Pearson's coefficient of correlation]] for segment <math>k</math>, the scaled correlation across the entire signals <math>\bar{r}_s</math> is computed as
 
:<math>\bar{r}_s = \frac{1}{K} \sum\limits_{k=1}^K r_k.</math>
 
==Efficiency==
In a detailed analysis, Nikolić et al.<ref name = "Nikolicetal"/> showed that the degree to which the contributions of the slow components will be attenuated depends on three factors, the choice of the scale, the amplitude ratios between the slow and the fast component, and the differences in their oscillation frequencies. The larger the differences in oscillation frequencies, the more efficiently will the contributions of the slow components be removed from the computed correlation coefficient. Similarly, the smaller the power of slow components relative to the fast components, the better will scaled correlation perform.
 
==Application to cross-correlation==
[[Image:Classical vs scaled CCH.png|thumb|right|350px| Example of a cross-correlogram between spike trains computed in a classical manner (left) and by using scaled correlation (right; <math>s</math> = 200 ms). Scaled correlation removes the slow component from the cross-correlogram.]]
Scaled correlation can be applied to [[autocorrelation|auto-]] and [[cross-correlation]] in order to investigate how correlations of high-frequency components change at different temporal delays. To compute cross-scaled-correlation for every time shift properly, it is necessary to segment the signals anew after each time shift. In other words, signals are always shifted ''before'' the segmentation is applied.
 
==Advantages over filtering methods==
Scaled correlation should be in many cases preferred over signal filtering based on spectral methods. The advantage of scaled correlation is that it does not make assumptions about the spectral properties of the signal (e.g., sinusoidal shapes of signals). Nikolić et al.<ref name = "Nikolicetal"/> have shown that the use of [[Wiener–Khinchin theorem]] to remove slow components is inferior to results obtained by scaled correlation. These advantages become obvious especially when the signals are non-periodic or when they consist of discrete events such as the time stamps at which neuronal action potentials have been detected.
 
==Matlab code==
Assume that we have two signals ''a'' and ''b'' with length(''a'') = length(''b'') = ''s''*''K''.
In Matlab, scaled correlation can be computed as follows:
 
    % segment the two data vectors
    Y = reshape(a, s, K);
    X = reshape(b, s, K);
   
    % demean
    onesMat =  ones(s,K);
    X      =  X - onesMat*diag(mean(X));
    Y      =  Y - onesMat*diag(mean(Y));
   
    % compute variances and co-variances for all segments
    varX = mean(X .^ 2);
    varY = mean(Y .^ 2);
    cov  = mean(X .* Y);
   
    % compute correlations coefficients for all segments
    corrSegments = cov ./ sqrt(varX .* varY);
   
    % remove not-a-number correlations
    corrSegments(isnan(corrSegments)) = [];
   
    % average all correlation coefficients to obtain scaled correlation
    scaledCorr = mean(corrSegments)
 
==See also==
* [[Autocorrelation]]
* [[Coherence (signal processing)]]
* [[Convolution]]
* [[Correlation]]
* [[Cross-correlation]]
* [[Phase correlation]]
* [[Spectral density]]
* [[Cross-spectrum]]
* [[Wiener–Khinchin theorem]]
 
==References==
{{reflist}}
 
==Free sources==
A free source code for computing scaled correlation and an interface for MATLAB can be downloaded here:&nbsp;http://www.raulmuresan.ro/sources/corrlib/
 
[[Category:Covariance and correlation]]

Latest revision as of 14:46, 23 October 2014

In the overall performance parts business, individuals tend to use the words parts and accessories interchangeably, while other people see a difference. So whats the distinction, for instance, in between truck parts and truck accessories if several of them serve the exact same standard function? It depends on who you ask.

Due to the fact the words parts and accessories each and every encompass a huge quantity of items that overlap and can as a result be categorized as either, people tend to the use the words interchangeably without having any regard to the distinction. In reality, when dealing with the performance parts market, the word only overlaps in specific instances, and even then opinions will differ as to which factors apply. Clicking article perhaps provides suggestions you might tell your uncle. Truck parts and truck accessories every have their personal list of included products, which meet in the middle based on their intended use, as properly as their categorization as an OEM or aftermarket item.

A truck parts advisor for example, may inform you that a truck component has a a lot more generic definition, and contains almost everything from repair, maintenance, and restoration, to interior and exterior enhancement. Such items might contain oil filters, air filters, shocks, spoilers, or headlamps, as they in effect, are portion of the truck. Visit garage door installation northridge to discover when to ponder it. So long as they are the original part of the truck, or even a replacement or repair item, it can be described as a portion whereas an aftermarket component created to boost the car right after the initial obtain tends to fall into the accessory category.

In the meantime, the exact same parts advisor might tell you that the word accessory is synonymous with the word portion when referring to a item in common, but the particular make and use of the product will determine which category it falls into.

Auto covers, sun shields, make-precise paraphernalia and such products that serve primarily to boost a car or truck (such as lift kits) would most likely be considered solely accessories. A sport utility rack could potentially fall into either.

Even oil filters, brake pads or rotors, or air filters could fall into either category. The common, OEM version will usually be considered a truck part whereas a performance-based aftermarket version thereof will far more most likely be thought of a Truck accessory (such as K&N oil filters, which take advantage of advanced filtration technologies you wont locate in its standard OEM counterpart). A lift kit, which isnt some thing thats included with the truck but rather utilized as an enhancement will usually be classified as an accessory, even though some will nevertheless call it a truck component.

The items that tend to be categorized especially as accessories are aftermarket goods created to boost (whether it be efficiency or aesthetic). For instance aftermarket air filters, high top quality seat covers, and floor mats for trucks are usually considered truck accessories if they are not the original OEM portion. The problem becomes even much more clouded the moment you get into OEM-made accessories vs. To check up more, please consider having a gander at: open in a new browser. aftermarket accessories or even aftermarket parts, but thats a fully various post.

In the end, the difference in between the two will constantly differ from person to person, even amongst truck enthusiasts and professionals within the repair or efficiency parts business..

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