Sens Estimator - rileywheadon/ffa-framework GitHub Wiki

Sen's Trend Estimator is used to estimate the slope of a regression line. Unlike Least Squares, Sen's trend estimator uses a non-parametric approach which makes it robust to outliers.

After computing the regression line using Sen's trend estimator, we use the Runs Test to determine whether the residuals from the regression are random. If the Runs test identifies a non-random pattern in the residuals, it is a strong indication that the non-stationarity in the data is non-linear.

• Null hypothesis: The residuals are distributed randomly.
• Alternative hypothesis: The residuals are not distributed randomly.

To compute Sen's trend estimator we use the following procedure:

1. Iterate over all pairs of data points $(x_{i}, y_{i})$ and $(x_{j}, y_{j})$.
2. If $x_{i} \neq x_{j}$, compute the slope $(y_{j} - y_{i})/(x_{j} - x_{i})$ and add it to a list $S$.
3. Sen's trend estimator $\hat{m}$ is the median of $S$.

After computing $\hat{m}$, we can estimate the $y$-intercept $b$ by the median of $y_{i} - \hat{m}x_{i}$ for all $i$.

Prior to applying the Runs test, the data is categorized based on whether it is above ($+$) or below ($-$) the median.

Then, we compute the number of contiguous blocks of $+$ or $-$ (or runs) in the data.

Example: Suppose that after categorization, the sequence of data is as follows:

$$ +++--+++-+- $$

This sequence has six runs with length $(3, 2, 3, 1,1, 1)$.

Let $R$ be the number of runs in $N$ data points (with category counts $N_{+}$ and $N_{-}$).

Then, $R$ is asymptotically normal with the following parameters:

$$ \mathbb{E}[R] = \frac{2N_{+}N_{-}}{N} + 1, \quad \text{Var}(R) = \frac{2N_{+}N_{-}(2N_{+}N_{-} - N)}{N^2(N - 1)} $$

For more information, see the Wikipedia entry or the R Documentation.