Negative autocorrelation values

Question

Autocorrelation is informally defined (Wikipedia article) as "the similarity between observations as a function of the time lag between them".

I create the following time series in MATLAB:

>> x = 1:1000;
>> y = zeros(1,length(x));
>> y(300:700) = x(1:401);
>> plot(x,y,'LineWidth',2.0)

And compute the autocorrelation using MATLAB autocorr function:

>> autocorr(y);

Now, as far as I understood correctly, the value of 1 at lag 0 means that when there is no lag, the signal is perfectly correlated with itself. When the lag increases, the correlation decreases.

However, I am lost at the negative autocorrelation values. As far as I understand it, I will get negative correlation value when one function increases and other decreases. However, since this signal only increases (save for the sudden drop at the end), I would not expect any negative correlation. If taking into account the drop at the end is still only a small fragment of the signal.

My question is: how is it possible for this autocorrelation to be negative? What do I understand wrong?

Answer 1

For simplicity, imagine that the function only consists of 11 measurements made at $x=0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000$

For each of these $x$ values, you have a $y$ value. Let the lagged $y$ value be $y_l$. For lag = 0, $y_l=y$:

    x      y    y_l
    0      0      0
  100      0      0
  200      0      0
  300      0      0
  400    100    100
  500    200    200
  600    300    300
  700    400    400
  800      0      0
  900      0      0
 1000      0      0

There is perfect correlation between $y$ and $y_l$, so the value is 1.

As the lag increases, the $y$ and $y_l$ values become quite different:

For lag = 300:

    x      y    y_l
    0      0      0
  100      0      0
  200      0      0
  300      0      0
  400    100      0
  500    200      0
  600    300      0
  700    400    100
  800      0    200
  900      0    300
 1000      0    400

The correlation between $y$ and $y_l$ becomes negative because $y$ is high when $y_l$ is low and vice versa.