Mapping Space, Forecasting Time: Kriging and SARIMA for Indoor Farming

Two questions behind every microclimate dashboard

Where is the warmest part of the farm now? What might the temperature be thirty minutes from now? These questions need different evidence. A spatial map estimates conditions between sensor positions at a specified time. A forecast estimates future conditions from information available at its issue time.

Tobafarm’s 10-ft container and 27 internal sensor nodes provide the setting for this discussion. Kriging and seasonal forecasting have featured in our research discussions. This article explains their roles and an evaluation workflow; it does not report a new benchmark or declare a winning model.

Kriging: learning how locations relate

Ordinary Kriging estimates a value as a weighted sum of observations. Weights sum to one and depend on a model of spatial dependence, not distance alone. Its unknown-mean assumption is local to the modelled domain; a strong systematic gradient may need a trend model or residual approach. [1]

The semivariogram describes how differences between observations change with separation. The nugget represents unresolved variation or measurement effects, the sill describes the model’s variance level, and the range characterises correlation distance; some models approach their sill asymptotically. Direction matters when airflow creates anisotropy. [2]

For a container, coordinates must correspond to real sensor positions. A left–right temperature contrast and a top–bottom contrast need not share the same correlation scale. With only 27 locations, fitting a separate elaborate directional variogram at every timestamp can be unstable. Test a small, defensible candidate set instead of treating every smooth surface as physical evidence.

Kriging uncertainty is conditional on the chosen spatial model and assumptions. It does not automatically include calibration bias, variogram selection uncertainty, or an unobserved equipment failure. Report prediction errors at withheld sensors alongside any uncertainty map. [1,2]

IDW and RBF: essential spatial comparisons

IDW assigns more influence to nearby readings through inverse-distance weights. The power and neighbourhood affect the result; at an observed location, exact interpolation returns the observed reading. It is an accessible baseline, but proximity alone may miss directional structure. [3]

RBF builds a surface from radial kernels, often with polynomial terms. Kernel choice, smoothing, and sometimes a shape parameter affect behaviour. Exact fitting is not evidence of good prediction between sensors; a smoothed fit may be more appropriate when observations are noisy. [4]

Compare these methods on the same timestamps, sensor subsets, coordinate system, and measured targets. A Kriging advantage is meaningful only if it persists under a fair comparison. Also report failure cases, compute time, boundary behaviour, and performance when fewer sensors remain available.

ARIMA and SARIMA: modelling temporal dependence

ARIMA(p,d,q) combines autoregression, differencing, and moving-average error terms. SARIMA(p,d,q)(P,D,Q)[m] adds seasonal terms with period m measured in observations. For a daily cycle, m depends on the sampling interval; it is not automatically 24. [5]

A lighting schedule can motivate a daily candidate period, but it does not prove stable seasonality. Crop growth, changes to AC operation, and sensor replacement can alter the process. Two days of data provide only two daily cycles: hundreds of measurements do not substitute for repeated cycles across operating conditions.

Historical discussion of a particular SARIMA order is a starting candidate, not a reusable optimum. Select orders using training data, check residuals, and judge the result on future windows. If lighting, outdoor conditions, or actuator inputs are used as predictors, distinguish regression with ARIMA errors or SARIMAX from an univariate SARIMA model. Future predictor values must be known or forecast at issue time.

Validation is where the scientific claim is earned

For mapping, remove a sensor before fitting the variogram, tuning parameters, imputing inputs, or training a residual correction. Score the prediction against that sensor’s original measured value. Leave-one-sensor-out testing asks about unseen positions; holding out a zone asks a harder question about spatial gaps.

For forecasting, train on the past and evaluate later observations using rolling origins. Recompute any preprocessing with information available at each origin. Compare with persistence and a seasonal-naive forecast at exactly the same horizons. [6]

Do not impute the full dataset using every sensor and then withhold a sensor: its values may already have influenced the training inputs. Do not interpolate across future timestamps before forecasting either. Keep an observed-versus-imputed flag and calculate headline errors against genuine measurements.

Report MAE and RMSE by sensor, zone, operating period, and forecast horizon. For probabilistic forecasts, inspect interval coverage and width. Separate tuning from final evaluation; aggregate uncertainty over meaningful time blocks rather than treating adjacent samples as independent experiments.

Combining Kriging and SARIMA: a testable extension

One possible workflow forecasts each observed location, then maps those forecasts to a 3D grid. Another models a spatial trend and forecasts its residual structure. These are proposed architectures here, not demonstrated Tobafarm results.

The order of operations matters. Mapping first can introduce spatially dependent errors into the temporal model; forecasting first can introduce correlated forecast errors into the map. Kriging variance alone is not the full uncertainty of the combined forecast. Evaluate the entire pipeline against withheld locations and future times.

A useful research question is whether a spatial forecast detects future hotspots better than a room-average forecast at an acceptable computation cost. This is more specific than simply combining five algorithms. A forecast can inform supervisory decisions, but it is not itself a controller or a safety guarantee.