
<p>Splines, piecewise polynomials segmented by discrete points, are known to be a good approximation for many real-world scenarios. Data scientists often use spline interpolation to produce smooth graphs and estimate missing values by “filling in” the space between discrete points of data.</p>



<p>Consider the following graph, which shows the temperature every four hours in Yosemite Park in early July.</p>



<figure class="wp-block-image fancybox"><img decoding="async" src="https://cdn.sisense.com/wp-content/uploads/Yosemite-temperature.png" alt="Yosemite temperature" class="wp-image-75369"/></figure>



<p>The graph appears jagged as we have data at 4 hour intervals. Spline interpolation can be used to fill in the areas between points while producing a much smoother graph:</p>



<figure class="wp-block-image fancybox"><img decoding="async" src="https://cdn.sisense.com/wp-content/uploads/Yosemite-temperature-2.png" alt="Yosemite temperature with spline interpolation" class="wp-image-75374"/></figure>



<p>Here we explore how to interpolate the data with spline interpolation using just SQL. We make use of Common-Table Expressions (CTEs) and time windowing functions.</p>



<h2 class="wp-block-heading"><strong>The method</strong></h2>



<p>Spline interpolation produces a piecewise cubic polynomial function for each pair of points. These distinct polynomial functions join smoothly at each data point. This is distinct from a rolling average, where the line rarely intersects a data point. In practice, a rolling average is good to smooth high variance data, while spline interpolation is used to fill in sparse data. <a href="https://en.wikipedia.org/wiki/Spline_interpolation" target="_blank" rel="noreferrer noopener" aria-label=" (opens in a new tab)">This wikipedia article</a> on spline interpolation is a good introduction to the topic, and we’ll be referring to the various equations in that article here.</p>



<p>Our primary task is to solve for the variables in three equations — equation 15 constrains the polynomial functions to be smooth at each data point, and equations 16 and 17 constrain the curvature at the end points to be straight lines.</p>



<p>As the article shows, these three equations can be represented in matrix form. Conveniently, the coefficient matrix is a <a href="https://en.wikipedia.org/wiki/Tridiagonal_matrix" target="_blank" rel="noreferrer noopener" aria-label=" (opens in a new tab)">tridiagonal matrix</a> which can be solved using the Thomas Algorithm, as explained <a href="https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm" target="_blank" rel="noreferrer noopener" aria-label=" (opens in a new tab)">here</a>.</p>



<p>The interpolation algorithm normally uses the entire dataset to determine the polynomial functions. In our implementation, however, we consider a subset of four nearby data points when determining the coefficients of each polynomial (also called windowing).</p>



<p>The resulting matrices for our study (n=4) are useful for explanatory purposes and produce the smooth graphs we are looking for, but you may find the need to increase the window size when using real-world data.</p>



<h2 class="wp-block-heading"><strong>The source data</strong></h2>



<p>To begin, we first download <a href="https://www.ncdc.noaa.gov/crn/qcdatasets.html" target="_blank" rel="noreferrer noopener" aria-label=" (opens in a new tab)">temperature data</a> from the <a href="https://www.noaa.gov/" target="_blank" rel="noreferrer noopener" aria-label=" (opens in a new tab)">National Oceanic and Atmospheric Administration</a> (NOAA) website, cut it down to the first two days of July 2017, and insert into a table as follows:</p>



<pre class="wp-block-code"><code>insert into temperatures (hour, temperature) values
(1,19.5),
(2,19.5),
(3,19.7),
…
(176,14.6);</code></pre>



<p>In order to simulate a rough sampling, we extract the values every four hours:</p>



<pre class="wp-block-code"><code>with xy as (
  select 
    cast(hour as double precision) as x
    , temperature as y 
from temperatures 
where
  cast(hour as integer) % 4 = 0
order by
  hour
limit 
  12
   )</code></pre>



<p>This is what we plot to get the graph in fig. 1 above.</p>



<pre class="wp-block-code"><code>with recursive
  , x_series(input_x) as (
    select
      min(x)
    from
      xy
    union all
    select
      input_x + 0.1
    from
      x_series
    where
      input_x + 0.1 &lt; (select max(x) from x)
  )
  , x_coordinate as (
  select
    input_x
    , max(x) over(order by input_x) as previous_x
  from
    x_series
  left join
    xy on abs(x_series.input_x - xy.x) &lt; 0.001
  )</code></pre>



<h2 class="wp-block-heading"><strong>Preparing the Input Variables — The 4 Points in Each Window</strong></h2>



<p>As mentioned, we will use a window of four data points around each data point. This is easy to do by creating columns for the (x, y) coordinate for each of the four data points by leveraging the lag and lead functions.</p>



<pre class="wp-block-code"><code> fullxy as (
   select
     x
     , y
     , lag(x, 1) over (order by x) as x0
     , lag(y, 1) over (order by x) as y0
     , x as x1
     , y as y1
     , lead(x, 1) over (order by x) as x2
     , lead(y, 1) over (order by x) as y2
     , lead(x, 2) over (order by x) as x3
     , lead(y, 2) over (order by x) as y3
   from 
      xy 
   order by 
      x
 )</code></pre>



<p>This produces a table like the following:</p>



<figure class="wp-block-image fancybox"><img decoding="async" src="https://cdn.sisense.com/wp-content/uploads/table-of-numbers.png" alt="Table of numbers" class="wp-image-75379"/></figure>



<p>As we can see, for each point (x, y), we also duplicate the (x, y) coordinates of the point before it (x0, y0) and the two points following it (x2, y2) and (x3, y3), for a total of four data points per row.</p>



<h2 class="wp-block-heading"><strong>Producing the Tridiagonal Coefficients</strong></h2>



<pre class="wp-block-code"><code>, tridiagonal as (
  select
    *
    , 3*(y1-y0) /power(x1-x0, 2) as d0
    , 3*((y1-y0)/(power(x1-x0, 2)) + (y2-y1)/(power(x2-x1, 2))) as d1
    , 3*((y2-y1)/(power(x2-x1, 2)) + (y3-y2)/(power(x3-x2, 2))) as d2
    , 3*(y3-y2) /power(x3-x2, 2) as d3
    , 1 / (x1-x0) as a1
    , 1 / (x2-x1) as a2
    , 1 / (x3-x2) as a3
    , 2 / (x1-x0) as b0
    , 2 * (1/(x1-x0) + 1/(x2-x1)) as b1
    , 2 * (1/(x2-x1) + 1/(x3-x2)) as b2
    , 2 / (x3-x2) as b3
    , 1 / (x1-x0) as c0
    , 1 / (x2-x1) as c1
    , 1 / (x3-x2) as c2
   from fullxy 
order by x)</code></pre>



<p>Now, for each row, we have the necessary a, b, c, and d vectors for the tridiagonal matrix.</p>



<h2 class="wp-block-heading"><strong>Finding the coefficients for the tridiagonal matrix using the Thomas Algorithm</strong></h2>



<p>The Thomas Algorithm can now be used to solve the tridiagonal matrix we created. The algorithm uses a series of “sweeps” across the matrix to find a solution. First there is a forward sweep, where the c prime and d prime values are calculated, and then a backward substitution, where the solution slope vector is solved. We unroll the calculation into distinct sweeps and solve in a series of CTEs behaving like nested subqueries.</p>



<pre class="wp-block-code"><code>forward_sweep_1 as (
  select
    *
    , c0 / b0 as c_prime_0
    , c1 / (b1 - a1 * c0 / b0) as c_prime_1
    , c2 / (b2 - a2 * (c1 / (b1 - a1 * c0 / b0))) as c_prime_2
    , d0 / b0 as d_prime_0
    , (d1 - a1 * d0 / b0) / (b1 - a1 * c0 / b0) as d_prime_1
  from
    tridiagonal
)
, forward_sweep_2 as (
 select
    *
    , (d2 - a2 * d_prime_1) / (b2 - a2 * c_prime_1) as d_prime_2
    , (d3 - a3 * ((d2 - a2 * d_prime_1) / (b2 - a2 * c_prime_1)))
        / (b3 - a3 * c_prime_2) as d_prime_3
  from
    forward_sweep_1
)
, backwards_substitution_1 as (
  select
    *
    , d_prime_3 as k3
    , d_prime_2 - c_prime_2 *(d_prime_3) as k2
    , d_prime_1 - c_prime_1 * (d_prime_2 - c_prime_2 *(d_prime_3)) as k1
  from
    forward_sweep_2
)
, backwards_substitution_2 as (
  select
    *
    , d_prime_0 - c_prime_0 * k1 as k0
  from
    backwards_substitution_1
)</code></pre>



<p><strong>Determining the polynomial coefficients</strong></p>



<p>Now, we’re ready to determine the polynomial coefficients a and b (equations 10 &amp; 11).</p>



<pre class="wp-block-code"><code>, coefficients as (
  select 
    x, y, x0, y0, x1, y1, x2, y2, k0, k1, k2, k3
    k1  * (x2-x1) - (y2-y1)  as a,
    -k2 * (x2-x1) + (y2-y1) as b
  from 
    backwards_substitution_2
)</code></pre>



<h2 class="wp-block-heading"><strong>Final step: calculating the temperature at each interpolated point</strong></h2>



<p>Here we calculate t (equation 2) and output y values (equation 1).</p>



<pre class="wp-block-code"><code>, interpolated_table as (
  select 
    *
    , (input_x - x1) / (x2 - x1) as t
  from
     x_coordinate,
     coefficients
   where
     abs(coefficients.x1 - x_coordinate.previous_x) &lt; 0.001
)
, final_table as (
  select 
    *
    , (1-t) * y1 + t*y2 + t*(1-t)*(a*(1-t)+b*t) as output_y
  from 
    interpolated_table 
  order by input_x
)</code></pre>



<p>The following is the final <em>select</em> clause. output_y contains the interpolated values. We also include the original (x, y) coordinate as well as scatter_y so we can display the original data points on a scatter plot.</p>



<pre class="wp-block-code"><code>select
  round(cast(input_x as numeric), 2) as input_x
  , x
  , y
  , round(cast(output_y as numeric), 2) as output_y
  , case
    when abs(input_x - x) &lt; 0.01 then y
      else null
    end as scatter_y
from 
  final_table 
order by 
  x</code></pre>



<h2 class="wp-block-heading"><strong>Final considerations</strong></h2>



<p>And there you have it! Using only SQL, we have used spline interpolation to generate smooth graphs. For the sake of brevity, we excluded data points on the left and right most of the data set but they can be added back into your analysis. Using splines for your graphs makes time series easier to read and pushes the limits of SQL.&nbsp;</p>



<p>Happy scripting!</p>


Using Spline Interpolation in SQL to Analyze Sparse Data

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