Let \( (x_1, y_1), …, (x_n, y_n)\) be the data points indexed by position \( i \in {1,…,n}\) . The smoothed value at index \( i\) is the Nadaraya–Watson estimator:

with the Gaussian kernel

The bandwidth \( \sigma\) is set by the slider as \( \sigma = s \cdot n \cdot 0.25\) , where \( s \in [0,1]\) is the slider value and \( n\) is the number of data points. At \( s = 0\) , \( \sigma = 0\) and the raw data is returned unchanged. At \( s = 1\) , \( \sigma = n/4\) , meaning the kernel extends over roughly a quarter of the series.

For computational efficiency only the \( \pm 4\sigma\) window around each point is summed (weights beyond \( 4\sigma\) are \( < 0.034\\ \%\) of the peak).

The data is a scattered point cloud \( {(x_i, y_i, z_i)}\) (or \( {(x_i, y_i, z_i, w_i)}\) for 4D). The XY plane is first normalised to the unit square \( [0,1]^2\) :

The kernel weight between two points is then based on Euclidean distance in this normalised space:

The smoothed elevation (and colour channel for 4D) are:

The bandwidth is \( \sigma = s \cdot 0.5\) , so at \( s=1\) the kernel half-width equals half the unit-square diagonal.

• Boundary correction: near the edges of the domain fewer neighbours exist; the denominator adjusts so that the estimate is not biased toward zero.
• Gap handling: filtered-out (masked) points are simply skipped in both sums, so holes in the data do not distort the smooth.

With complete, uniformly-spaced data far from any boundary, \( \sum_j K_\sigma(i-j) \approx \sigma\sqrt{2\pi}\) (constant), and the estimator reduces to ordinary Gaussian convolution — an ideal low-pass filter whose frequency response is \( \hat{K}(\omega) = e^{-\sigma^2\omega^2/2}\) .

The Statistics button displays elementary statistics for all the numerical columns. Here is the math.

For a column with values \( x_1, \dots, x_n\) , sorted as \( x_{(1)} \le \dots \le x_{(n)}\) :

Mean

Sample variance and standard deviation

Median — middle value of the sorted sequence (average of the two middle values when \( n\) is even).

Quartiles Q1, Q2 = median, Q3 — values at positions \( \lfloor n/4 \rfloor\) , \( \lfloor n/2 \rfloor\) , \( \lfloor 3n/4 \rfloor\) in the sorted sequence (average of neighbours when \( n\) is divisible by 4).

Min, Max, Range

Interquartile range

For every pair of numerical columns \( j, k\) :

The result is the \( p \times p\) symmetric matrix \( \mathbf{S}\) . The diagonal entries \( S_{jj} = s_j^2\) are the sample variances (consistent with Statistics). The off-diagonal entries measure linear co-variation. No standardisation is applied: units are the product of the units of the two columns.

Here is what is done.

Step 1 — Standardise each column to mean 0 and unit sample standard deviation:

This makes PCA unit-free (correlation-based PCA). \( \mathbf{Z}\) is the \( n \times p\) standardised data matrix.

Step 2 — Correlation matrix (= sample covariance of \( \mathbf{Z}\) ):

Step 3 — Eigendecomposition via the Jacobi algorithm (exact, iterative, for symmetric matrices):

Eigenvectors \( \mathbf{v}_{k}\) (columns of \( \mathbf{V}\) ) are the principal directions (loadings). Each \( \lambda_k\) measures the variance explained by PC\( k\) . Because \( \mathbf{C}\) is a correlation matrix, \( \sum_{k} \lambda_k = p\) .

Step 4 — Variance explained:

Shown as a bar chart (scree plot) with a cumulative line.

Step 5 — Scores (coordinates of each observation in the PC space):

Loadings heatmap: displays \( \mathbf{V}\) (\( p\times p\) ), colour-coded (blue = positive, red = negative). Each entry \( v_{jk}\) is the contribution of variable \( j\) to PC\( k\) .

Biplot: scatter of \( (t_{i1}, t_{i2})\) for all observations, with arrows from the origin to \( (\alpha\,v_{j1},\,\alpha\,v_{j2})\) for each variable \( j\) , where \( \alpha = 0.8\times\max_{i}\|\mathbf{t}i\|_{2}\) scales the arrows to fit the data cloud.