# Moving extended P² quantile estimator

In the previous posts, I discussed the P² quantile estimator (a sequential estimator which takes $$O(1)$$ memory and estimates a single predefined quantile), the moving P² quantile estimator (a moving modification of P² which estimates quantiles within the moving window), and the extended P² quantile estimator (a sequential estimator which takes $$O(m)$$ memory and estimates $$m$$ predefined quantiles).

Now it’s time to build the moving modification of the extended P² quantile estimator which estimates $$m$$ predefined quantiles using $$O(m)$$ memory within the moving window.

### The approach

The idea is the same that was used for the the moving P² quantile estimator. We should reuse the described “movification” approach using the extended P² quantile estimator instead of the original P² quantile estimator and apply this logic to each requested quantile.

In this approach, in addition to the “target” (the moving window that contains the last $$L$$ elements of our stream), we maintain two consequent moving windows of the same size: the “previous” window and the “current” window (see the above figure). The union of the “previous” and “current” windows always contain the “target” window. Quantile estimations in the “previous” window are previously calculated using the extended P² quantile estimator. Quantile estimations in the “current” window should be updated on each new stream element also using the extended P² quantile estimator. Quantile estimations in the “target” window could be obtained as a weighted sum of the “previous” and the “current” window. All the additional details are presented in the post about the moving P² quantile estimator.

### Reference implementation

public class ExtendedP2QuantileEstimator
{

public int Count { get; private set; }

public ExtendedP2QuantileEstimator(params double[] probabilities)
{
this.Probabilities = probabilities;
m = probabilities.Length;
markerCount = 2 * m + 3;
n = new int[markerCount];
ns = new double[markerCount];
Q = new double[markerCount];
}

private void UpdateNs(int maxIndex)
{
// Principal markers
ns[0] = 0;
for (int i = 0; i < m; i++)
ns[i * 2 + 2] = maxIndex * Probabilities[i];
ns[markerCount - 1] = maxIndex;

// Middle markers
ns[1] = maxIndex * Probabilities[0] / 2;
for (int i = 1; i < m; i++)
ns[2 * i + 1] = maxIndex * (Probabilities[i - 1] + Probabilities[i]) / 2;
ns[markerCount - 2] = maxIndex * (1 + Probabilities[m - 1]) / 2;
}

{
if (Count < markerCount)
{
Q[Count++] = value;
if (Count == markerCount)
{
Array.Sort(Q);

UpdateNs(markerCount - 1);
for (int i = 0; i < markerCount; i++)
n[i] = (int)Math.Round(ns[i]);

Array.Copy(Q, ns, markerCount);
for (int i = 0; i < markerCount; i++)
Q[i] = ns[n[i]];
UpdateNs(markerCount - 1);
}

return;
}

int k = -1;
if (value < Q[0])
{
Q[0] = value;
k = 0;
}
else
{
for (int i = 1; i < markerCount; i++)
if (value < Q[i])
{
k = i - 1;
break;
}

if (k == -1)
{
Q[markerCount - 1] = value;
k = markerCount - 2;
}
}

for (int i = k + 1; i < markerCount; i++)
n[i]++;
UpdateNs(Count);

int leftI = 1, rightI = markerCount - 2;
while (leftI <= rightI)
{
int i;
if (Math.Abs(ns[leftI] / Count - 0.5) <= Math.Abs(ns[rightI] / Count - 0.5))
i = leftI++;
else
i = rightI--;
}

Count++;
}

{
double d = ns[i] - n[i];
if (d >= 1 && n[i + 1] - n[i] > 1 || d <= -1 && n[i - 1] - n[i] < -1)
{
int dInt = Math.Sign(d);
double qs = Parabolic(i, dInt);
if (Q[i - 1] < qs && qs < Q[i + 1])
Q[i] = qs;
else
Q[i] = Linear(i, dInt);
n[i] += dInt;
}
}

private double Parabolic(int i, double d)
{
return Q[i] + d / (n[i + 1] - n[i - 1]) * (
(n[i] - n[i - 1] + d) * (Q[i + 1] - Q[i]) / (n[i + 1] - n[i]) +
(n[i + 1] - n[i] - d) * (Q[i] - Q[i - 1]) / (n[i] - n[i - 1])
);
}

private double Linear(int i, int d)
{
return Q[i] + d * (Q[i + d] - Q[i]) / (n[i + d] - n[i]);
}

public double GetQuantile(double p)
{
if (Count == 0)
throw new InvalidOperationException("Sequence contains no elements");
if (Count <= markerCount)
{
Array.Sort(Q, 0, Count);
int index = (int)Math.Round((Count - 1) * p);
return Q[index];
}

for (int i = 0; i < m; i++)
if (Probabilities[i] == p)
return Q[2 * i + 2];

throw new InvalidOperationException($"Target quantile ({p}) wasn't requested in the constructor"); } public void Clear() { Count = 0; } } public class MovingExtendedP2QuantileEstimator { private readonly ExtendedP2QuantileEstimator estimator; private readonly int windowSize; private int n; private readonly double[] previousWindowEstimations; public MovingExtendedP2QuantileEstimator(double[] probabilities, int windowSize) { this.windowSize = windowSize; estimator = new ExtendedP2QuantileEstimator(probabilities); previousWindowEstimations = new double[probabilities.Length]; } public void Add(double value) { n++; if (n % windowSize == 0) { for (int i = 0; i < estimator.Probabilities.Length; i++) previousWindowEstimations[i] = estimator.Q[2 * i + 2]; estimator.Clear(); } estimator.Add(value); } public double GetQuantile(double p) { if (n == 0) throw new InvalidOperationException("Sequence contains no elements"); if (n < windowSize) return estimator.GetQuantile(p); for (int i = 0; i < estimator.Probabilities.Length; i++) if (estimator.Probabilities[i] == p) { double estimation1 = previousWindowEstimations[i]; double estimation2 = estimator.Q[2 * i + 2]; double w2 = (n % windowSize + 1) * 1.0 / windowSize; double w1 = 1.0 - w2; return w1 * estimation1 + w2 * estimation2; } throw new InvalidOperationException($"Target quantile ({p}) wasn't requested in the constructor");
}
}

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