autocorrelation
Say that I have a signal that is strongly or weakly autocorrelated with itself. So what? What does this mean? How can this improve businesses? Does this mean that we can construct predictive models with time variables?
I don't understand why autocorrelation is seen as a big deal or how it helps us shift focus. Why do people care about autocorrelation so much?
I think the author is probably talking about the residuals of the model. I argue this because of his statement about adding more fourier coefficients; if, as I believe, he is fitting a fourier model, then adding more coefficients will reduce the autocorrelation of the residuals at the expense of a higher CV.
If you have trouble visualizing this, think of the following example: suppose you have the following 100 points data set, which comes from a two-coefficient fourier model with addeded white gaussian noise:
The following graph shows two fits: one done with 2 fourier coefficients, and one done with 200 fourier coefficients:
As you can see, the 200 fourier coefficients fits the DATAPOINTS better, while the 2 coefficient fit (the 'real' model) fits the MODEL better. This implies that the autocorrelation of the residuals of the model with 200 coefficients will almost surely be closer to zero at all lags than the residuals of the 2 coefficient model, because the model with 200 coefficients fits exactly almost all datapoints (i.e., the residuals will be almost all zeros). However, what would you think will happen if you leave, say, 10 datapoints out of the sample and fit the same models? The 2-coefficient model will predict better the datapoints you leaved out of the sample! Thus, it will produce a lower CV error as opossed to the 200-coefficient model; this is called overfitting. The reason behind this 'magic' is because what CV actually tries to measure is prediction error, i.e., how well your model predicts datapoints not in your dataset.
Looking at the estimator for the autocovariance function at lag $ h $ might be useful (note that the autocorrelation function is simply a scaled-down version of the autocovariance function).
$$ \hat{\gamma}(h) = \frac{1}{n} \sum_{t=1}^{n-\mid h \mid} (x_{t+h} - \bar{x})(x_t - \bar{x}) $$
The idea is that, for each lag $ h $, we go through the series and check whether the data point $ h $ time steps away covaries positively or negatively (i.e. when $ t $ goes above the mean of the series, does $ t+h $ also go above or below?).
Your series is a monotonically increasing series, and has mean $ 183 $. Let's see what happens when $ h = 130 $.
First, note that we can only compute the autocovariance function up to time point 234, since when $ t = 234 $, $ t+h=365 $.
Furthermore, note that from $ t= 1 $ up until $ t = 53 $, we have that $ t + h $ is also below the mean (since 53 + 130 = 183 which is the mean of the series).
And then, from $ t=54 $ to $ t=182 $, the estimated correlation will be negative since they covary negatively.
Finally, from $ t = 183 $ to $ t = 234 $, the estimated correlation will be positive once again, since $ t $ and $ t+h $ will both be above the mean.
Do you see how this would result in the correlation averaging out due to the approximately equal contributions to the autocovariance function from the positively covarying points and the negatively covarying points?
You might notice that there are more points that are negatively covarying than points that are positively covarying. However, intuitively, the positively covarying points are of greater magnitude (since they're further away from the mean) whereas the negatively covarying points contribute smaller magnitude to the autocovariance function since they crop up closer to the mean. Thus, this results in an autocovariance function of approximately zero.
Best Answer
Autocorrelation is a simple, reliable technique to find cyclic patterns in data.
If you have a one-hour-intervaled time series over let's say one week, you can create about 35 new time series (7 days in one week x 5 weeks) by lagging the original series by n days (n is from 1 to 35) by one day.
Next calculate R-squared for the original series and each lagged series; then plot those data points (n, or units of lag on the x-axis, and R-squared on the y-axis)
if instead of R-squared, you calculate correlation (so negative y values are possible) then plot those values, you get a correlogram and by looking at its shape it can tell you about the inter-day periodicity of that series.
so for very high R-squared (near 1.0) the original series and lagged series are back in phase; if this occurs at multiples of seven, for instance, then your data has weekly periodicity....
there's a lot of reasons why this valuable information, not the least of which is that removing cycles (and the trend and regime shift if any) is a crucial predicate before forecasting (predicting future values in a time series). So for instance, perhaps the most frequently used family of forecasting models is ARIMA; the first two letters in that acronym stand for auto-regressive.