mathlib-fp1.9.1
On this page

TimeSeriesLib Reference

TimeSeriesLib.TimeSeries — time series analysis for Free Pascal.

TimeSeriesLib.StateSpace adds stable 1.8 scalar and multivariate linear-Gaussian Kalman baselines. Their configuration, bounded mutable state, failure-atomic block processing, and forecast ownership are documented in the applied numerics guide.

---

State-space selection in 1.8

Use TScalarKalmanFilter when state and observation are both scalar. Its configuration holds four finite coefficients; each update and forecast step uses constant retained state.

Use TMultivariateKalmanFilter when either state or observation has multiple components. TMultivariateKalmanConfiguration.Create snapshots:

  • square state transition and process covariance matrices;
  • an observation-by-state observation matrix; and
  • a square observation measurement covariance.

Update returns the posterior state, state covariance, innovation, innovation covariance, and log likelihood contribution. Process accepts observations as matrix rows and returns every state/covariance and innovation diagnostic. Forecast returns observation means and covariances without mutating the filter. StateMean and StateCovariance return independent copies. TDenseDoubleMatrixArray is the owned covariance-series container used by the multivariate result records.

Both filters validate a complete block before committing the working state. Invalid shapes, non-finite values, asymmetric covariance input, failed positive-definite innovation solves, and overflow raise EStateSpaceError without a partial state advance. Covariance updates use symmetry restoration and checked dense factors. A filter record is mutable and must not be updated concurrently.

These are time-invariant, linear-Gaussian filtering and forecasting baselines. Missing observations, controls, smoothing, nonlinear filters, and parameter estimation are deliberately outside 1.8.

---

Quick Start


uses TimeSeriesLib.TimeSeries;



// Smooth a noisy series with a 5-point moving average

smooth := TTimeSeriesKit.SimpleMovingAverage(prices, 5);



// Fit and forecast an ARIMA(1,1,0) model

model  := TTimeSeriesKit.ARIMAFit(sales, 1, 1, 0);

fcast  := TTimeSeriesKit.ARIMAForecast(model, sales, 12);  // 12 steps ahead



// Detect anomalies more than 3 standard deviations from the mean

idx := TTimeSeriesKit.ZScoreAnomalies(readings, 3.0);



// Find the dominant seasonal period

period := TTimeSeriesKit.PeriodogramPeak(Y);

All methods are class static — no Create/Free needed.

---

Result Records


TDecomposition = record

  Trend:    TDoubleArray;   { smoothed trend }

  Seasonal: TDoubleArray;   { repeating seasonal component }

  Residual: TDoubleArray;   { remainder }

  DecompType: TDecompType;

end;



TARIMAModel = record

  ARCoeffs: TDoubleArray;   { phi_1 .. phi_p }

  MACoeffs: TDoubleArray;   { theta_1 .. theta_q }

  D:        Integer;         { integration order }

  Mu:       Double;          { mean of differenced series }

  Sigma2:   Double;          { innovation variance }

  P, Q:     Integer;

end;



TLinearTrend = record

  Intercept: Double;

  Slope:     Double;

  RSquared:  Double;

end;



TADFResult = record

  Statistic:    Double;

  Crit1Pct:     Double;   { MacKinnon 1% critical value }

  Crit5Pct:     Double;   { MacKinnon 5% critical value }

  Crit10Pct:    Double;   { MacKinnon 10% critical value }

  IsStationary: Boolean;  { True if Statistic < Crit5Pct }

end;



TCUSUMResult = record

  CUSUMValues: TDoubleArray;

  ChangePoint: Integer;   { -1 if not detected }

  Detected:    Boolean;

end;

---

Smoothing & Filtering

SimpleMovingAverage


smooth := TTimeSeriesKit.SimpleMovingAverage(Y, Window);

Centred moving average. Each output point is the mean of Window surrounding values. Edge points use whatever data is available (no padding). Output is the same length as input.

When to use: removing short-term noise while preserving long-term trend.

WeightedMovingAverage


smooth := TTimeSeriesKit.WeightedMovingAverage(Y, Window);

Like SMA but assigns linearly increasing weights to more recent values (weight ∝ position). Reacts faster to recent changes than SMA.

ExponentialSmoothing


smooth := TTimeSeriesKit.ExponentialSmoothing(Y, Alpha);

smooth := TTimeSeriesKit.ExponentialSmoothing(Y, Alpha, InitValue);

S_t = Alpha × Y_t + (1 − Alpha) × S_{t−1}

  • Alpha near 0: heavy smoothing, slow response
  • Alpha near 1: tracks data closely (Alpha=1 is identity)
  • InitValue: seed value for S_0 (defaults to Y[0])

DoubleExponentialSmoothing (Holt's Method)


smooth := TTimeSeriesKit.DoubleExponentialSmoothing(Y, Alpha, Beta);

Tracks level and trend separately. Good for data with a linear trend but no seasonality.

  • Alpha: level smoothing (0..1)
  • Beta: trend smoothing (0..1)

TripleExponentialSmoothing (Holt-Winters)


smooth := TTimeSeriesKit.TripleExponentialSmoothing(Y, Alpha, Beta, Gamma, Period);

smooth := TTimeSeriesKit.TripleExponentialSmoothing(Y, Alpha, Beta, Gamma, Period, dtMultiplicative);

Handles data with both trend and seasonality.

  • Alpha: level smoothing
  • Beta: trend smoothing
  • Gamma: seasonal smoothing
  • Period: seasonal cycle length (e.g. 12 for monthly, 7 for daily-with-weekly-cycle)
  • DType: dtAdditive (default) or dtMultiplicative

HoltWintersForecast


fcast := TTimeSeriesKit.HoltWintersForecast(Y, Alpha, Beta, Gamma, Period, H);

Projects H steps ahead using the Holt-Winters state at the end of Y. Returns array of length H. Pass Period >= 2, at least two full seasons, H >= 1, and smoothing parameters in (0,1]. TripleExponentialSmoothing validates these controls; HoltWintersForecast does not currently validate all of them.

---

Decomposition

Decompose


D := TTimeSeriesKit.Decompose(Y, Period);

D := TTimeSeriesKit.Decompose(Y, Period, dtMultiplicative);

Classical decomposition: 1. Trend = centred moving average of width Period 2. Seasonal indices = average detrended values per cycle position 3. Residual = Y − Trend − Seasonal (additive) or Y / (Trend × Seasonal) (multiplicative)

Edge values in Trend are NaN (cannot centre the window at the boundaries). Residual calculation substitutes the observed value for an undefined edge trend. Pass Period >= 2 and at least two complete periods; period validity is otherwise a caller requirement.

---

Differencing & Stationarity

Difference


d1 := TTimeSeriesKit.Difference(Y, 1);   // first differences:  Y[t] - Y[t-1]

d2 := TTimeSeriesKit.Difference(Y, 2);   // second differences: Diff(Diff(Y))

Output length = N − d. Used to convert a non-stationary series to stationary before fitting ARIMA.

Undifference


orig := TTimeSeriesKit.Undifference(DiffY, InitVals, D);

Inverts differencing. InitVals must contain at least the first D values of the original series (needed to reconstruct the integration constants). Extra values are ignored.

AugmentedDickeyFuller


adf := TTimeSeriesKit.AugmentedDickeyFuller(Y);

adf := TTimeSeriesKit.AugmentedDickeyFuller(Y, Lags);

Tests whether the series has a unit root (non-stationary).

  • H₀: series has a unit root → non-stationary
  • H₁: series is stationary
  • Reject H₀ when adf.Statistic < adf.Crit5Pct
  • adf.IsStationary = True means you can reject H₀ at the 5% level

Rule of thumb: if IsStationary = False, apply one round of differencing and test again.

---

Autocorrelation

ACF


acf := TTimeSeriesKit.ACF(Y, MaxLag);

// acf[0] = 1.0, acf[k] = autocorrelation at lag k

Use the ACF plot to identify the MA order q: ACF cuts off sharply after lag q.

PACF


pacf := TTimeSeriesKit.PACF(Y, MaxLag);

// pacf[0] = 1.0, pacf[k] = partial autocorrelation at lag k

Computed via Yule-Walker equations. Use to identify the AR order p: PACF cuts off after lag p.

LjungBox


Q := TTimeSeriesKit.LjungBox(Y, MaxLag);

Tests whether the first MaxLag autocorrelations are jointly zero. Compare to chi-squared critical value with MaxLag degrees of freedom. Rule of thumb: Q > 3.84 (χ² df=1, 5%) → series is not white noise.

---

ARIMA Models

Model Order Selection Cheat Sheet

Pattern Likely model
ACF decays, PACF cuts off at lag p AR(p)
ACF cuts off at lag q, PACF decays MA(q)
Both decay ARMA(p,q)
ADF non-stationary Difference once → try again

ARFit


model := TTimeSeriesKit.ARFit(Y, P);

// model.ARCoeffs[0..P-1] = phi coefficients

Fits AR(p) via Yule-Walker equations. Fast and exact for pure AR series.

ARForecast


fcast := TTimeSeriesKit.ARForecast(model, History, H);

Recursive H-step forecast from the last P values of History.

MAFit


model := TTimeSeriesKit.MAFit(Y, Q);

Fits MA(q) via the innovations algorithm. Returns theta coefficients and innovation variance.

ARIMAFit


model := TTimeSeriesKit.ARIMAFit(Y, P, D, Q);

Full ARIMA(p,d,q) fit: 1. Differences Y by order D 2. Fits AR(p) to differenced series 3. Fits MA(q) to AR residuals

ARIMAForecast


fcast := TTimeSeriesKit.ARIMAForecast(model, OriginalY, H);

Forecasts H steps ahead and undifferences the result back to original scale. Historical AR residuals are reconstructed to seed the MA terms; unknown future innovations are set to zero, as in the standard conditional-mean forecast. Integrated models retain the last required difference states and return exactly H forecasts on the original scale for any supported D.

---

Change-point & Anomaly Detection

CUSUMDetect


R := TTimeSeriesKit.CUSUMDetect(Y);

R := TTimeSeriesKit.CUSUMDetect(Y, Threshold, Target);

Accumulates deviations from Target (defaults to sample mean). ChangePoint = first index where |CUSUM| exceeds Threshold × StdDev. Set Threshold = 0 to return the CUSUM series without detection.

ZScoreAnomalies


idx := TTimeSeriesKit.ZScoreAnomalies(Y);

idx := TTimeSeriesKit.ZScoreAnomalies(Y, Threshold);  // default 3.0

Returns indices of points more than Threshold standard deviations from the series mean.

RollingZScore


idx := TTimeSeriesKit.RollingZScore(Y, Window);

idx := TTimeSeriesKit.RollingZScore(Y, Window, Threshold);

Like ZScoreAnomalies but computed within a rolling window. Catches local anomalies that a global z-score would miss (e.g. a spike in an otherwise flat segment of a trending series).

---

Trend & Seasonality Utilities

LinearTrend


T := TTimeSeriesKit.LinearTrend(Y);

// T.Intercept, T.Slope, T.RSquared

OLS fit of y = a + b×t (t = 0, 1, ..., N−1).

DetrendLinear


residuals := TTimeSeriesKit.DetrendLinear(Y);

Subtracts the fitted linear trend. OLS residuals always sum to zero.

SeasonalStrength


strength := TTimeSeriesKit.SeasonalStrength(Decomp);

= max(0, 1 − Var(Residual) / Var(Seasonal + Residual))

  • Near 1.0 → strong, dominant seasonality
  • Near 0.0 → little to no seasonality

PeriodogramPeak


period := TTimeSeriesKit.PeriodogramPeak(Y);

period := TTimeSeriesKit.PeriodogramPeak(Y, MinPeriod, MaxPeriod);

FFT-based dominant period detection. Returns the period (in samples) of the largest spectral peak, excluding the DC component. Non-power-of-two inputs are zero-padded to NPow, and period conversion uses that FFT length, so the same frequency-bin mapping applies to every input length.

Periodogram


psd := TTimeSeriesKit.Periodogram(Y);

// psd[k] = power at frequency k/NPow, where NPow is the next power of two

One-sided power spectrum. The input is zero-padded to NPow, the next power of two. Returned bin k therefore corresponds to k / NPow, not k / N; the result contains NPow div 2 + 1 bins, including DC and Nyquist.

---

Error Handling

ETimeSeriesError is raised by the relevant methods for:

  • empty input in SMA, WMA, and exponential smoothing
  • insufficient observations for Holt smoothing, decomposition, ADF, ACF, AR, CUSUM, or linear trend
  • invalid smoothing parameters in exponential, double, and triple smoothing
  • invalid positive window/order/horizon controls where explicitly checked
  • ACF with MaxLag >= N
  • Difference with D < 0 or an order too large for the series
  • Undifference with fewer than D initial values
  • non-finite observations or coefficients in model fitting and forecasting
  • incompatible AR/MA/ARIMA orders, history lengths, or non-positive horizons
  • invalid period bounds or series too short to contain a searchable FFT bin

---

Dependencies

  • MathBase.SharedTypes — TDoubleArray, TIntegerArray

No other external libraries required.