Text from the interactive explanation

A Systematic Sample Size Analysis of ML-Based Path Loss Prediction for LPWAN

Robert Bitterling, Christian Nettersheim, Jörn Hees, Michael Rademacher

Bonn's radio map, learned from garbage trucks

Cities are wiring up sensors with LoRa , a radio that sends tiny messages kilometres to rooftop receivers called gateways. Before installing anything, planners need to know where the signal will reach. For 230 days, radios on Bonn's garbage trucks measured exactly that. This study asks how few of those measurements a computer needs to predict the rest.

Dots: real measured positions · Colours: live k-NN prediction from real measurements · Moving trucks and packet arcs: illustrative

1 · The problem

A signal that fades, but not evenly

A LoRa sensor sends a short packet; a gateway on a rooftop or mast picks it up. On the way, the signal weakens enormously. Engineers call this weakening path loss and measure it in decibels (dB) .

Decibels in one slider

Every 10 dB means the signal arrives ten times weaker. The links measured in Bonn lost between about 86 and 160 dB.

The simplest prediction is: the farther away, the weaker. The scatter below shows all 64,294 measured links from Bonn. Try to describe them with one curve of the textbook form loss = loss at 1 km + 10·n·log₁₀(distance).

The Bonn model is a log-distance curve fitted in an earlier study of this same measurement campaign (loss at 1 km 132.41 dB, n = 1.58).

Same distance, different fate

Tap a dot in the scatter to compare two links of nearly the same length.

At any given distance, measured path loss spreads over 30 dB or more. What lies between sensor and gateway — buildings, terrain, vegetation — differs from street to street, and a curve that only knows distance cannot see it. That leftover scatter is why the best possible single curve still misses by about 9.6 dB.

So what if, instead of a formula, we learned from the measurements themselves? The paper tests two simple ways to do that.

2 · Model one

Ask the neighbours

The k-nearest-neighbours method (k-NN) is about the simplest learner there is. To predict a link, it finds the k measured links with the most similar coordinates — where the sensor was and where the receiving gateway stands — and averages their path loss, giving closer ones more weight. It needs nothing but four numbers per link: two coordinates for each end.

With 100 links spread over eight gateways and a whole city, the 31 "nearest" links are mostly far away, often across town or from other gateways. Averaging them blurs everything towards a city-wide mean. That is why the paper also evaluated k = 7 for small training sets: at 100 links its mean error is 9.56 dB versus 10.02 dB for k = 31. With tens of thousands of links, the 31 neighbours sit close by and averaging more of them smooths out measurement noise, so k = 31 wins (6.15 vs 6.32 dB).

3 · Model two

Read the terrain

The second learner, a random forest (RF), never sees where a link is. Instead, for every link the researchers cut a height profile out of laser-scanned ( LiDAR ) elevation data of Bonn and summarised it in a handful of numbers: how high the antennas sit above sea level, how far apart they are, the angles between them, the loss expected in empty space, and how tall the receiving antenna would have to be to clear the obstacles in between.

Vertical scale strongly exaggerated. The shaded lens is the first Fresnel zone, the region around the straight line that carries most of the radio energy at 868 MHz. In the study, these features come from real LiDAR data of Bonn, processed with the SignalServer propagation tool.

A crowd of decision trees

Each tree in a random forest learns a chain of yes/no questions about these numbers ("does the line of sight clear the rooftops?", "is the link longer than a few kilometres?") from a random slice of the training data, and ends in a path-loss guess. The forest averages many such guesses, so no single tree's quirks dominate. The paper used an off-the-shelf random forest with a standard configuration and no heavy tuning.

Schematic: the questions above are examples of the kind of rule a tree can learn, not the paper's fitted rules.

● Neighbour lookup (k-NN)

Needs only the coordinates of sensor and gateway. Cheap to set up, but it only knows places that have been measured.

● Terrain reader (RF)

Needs LiDAR elevation data, a terrain profile for every link and feature extraction. More effort, but its inputs describe the physics of each path.

Which effort pays off, and at what amount of data? That is the paper's central experiment.

4 · The experiment

The race: who learns faster?

The researchers set aside 16,074 links as a fixed exam . They trained each model on random handfuls of the remaining links, from 100 up to all 48,220, and repeated every size 100 times with fresh random picks. The score is the RMSE , the typical size of a miss in dB. Lower is better: a 6 dB miss means the prediction is off by a factor of about four in power.

Predict first. With only 200 measured links, which predicts best?

And with all 48,220 links?

What the curves say

With a few hundred links, the terrain reader leads clearly. Around 3,250 links both neighbour variants catch up, and near 4,750 k = 31 overtakes k = 7. At the full 48,220 links, all three sit below 6.5 dB (RF 6.34, k = 7 6.32, k = 31 6.15) against 9.68 dB for the best formula. The spread between runs also shrinks as data grow.

Diminishing returns

For the random forest, going from 100 to 500 links gains 0.62 dB, from 500 to 15,000 another 1.08 dB, and the last 33,000 links only 0.31 dB. More data alone has a practical limit.

Each column shows how often a model missed by a given amount on the exam links (one run; 5 dB bins; brighter = more links, logarithmic scale). Positive values mean the model predicted less loss than was measured (it underestimated path loss). The formulas mostly underestimate; ITM is centred near zero but very wide; ITWOM is shifted towards underestimation. The learned models are compact, and the Bonn formula's spread resembles theirs at 500 links. Rare large misses remain for every model.

5 · The stress test

A gateway it has never seen

In the race, exam links and training links came from the same eight gateways, so the models only had to fill gaps inside an existing network. A city that adds a new gateway has no measurements from it yet. To probe this, the researchers left one gateway out at a time: train on all links of the other seven, then predict every link of the eighth.

Predict first: which gateway do you expect to be hardest to predict without its own data?

Random forest error for each held-out gateway

Bars: random forest. Diamonds: k-NN (k = 31). Labels: antenna height and mean link distance. Dotted line: random pooled error (≈6.3 dB); dashed line: the random forest's mean over all eight held-out gateways (8.9 dB).

Before → after hiding a gateway

Random pooled split vs. leave-one-gateway-out, at the largest training size (Table II).

The neighbour lookup roughly doubles its error, from about 6.3 to about 13.5 dB. It is a spatial lookup: no training link shares the hidden gateway's position, so it can only borrow links heard by other gateways. The random forest degrades far less, from 6.34 to 8.92 dB, because its terrain features describe the geometry of each path and partly carry over.

The average hides big differences. For five of eight gateways, the forest stays between 6.6 and 7.6 dB, close to its pooled score. Three sites push the mean up: gw 3 with an unusually tall 55 m antenna (14.2 dB), gw 1 whose links average 4.9 km (11.2 dB) and gw 2 with a 30 m antenna (10.4 dB). The authors treat these site traits as descriptive indicators, not confirmed causes.

6 · Back to the city

So, how many measurements does a city need?

Put yourself in the place of a city planning its own LoRa network. The paper's guidance depends on how much data you have and what you want to predict.

Numbers are the paper's measured results in Bonn (mean RMSE; leave-one-gateway-out was checked at the largest training size).

Back to the question we started with. Inside a network that has already been measured, a few hundred links let a terrain-aware model beat the general-purpose formulas, and a few thousand let even a plain coordinate lookup do the same. With the full data set both reach errors below 6.5 dB, against 9.7 dB for the best formula. Every extra link helps less than the one before.

Predicting for a gateway that has never been measured is harder. Terrain features carry over partly, and how well depends on where the gateway stands.

The authors point to two next steps: designing small, smart measurement campaigns that pick representative locations to reach a target accuracy with as few links as possible, and repeating the study in other cities, frequencies and radio technologies. These results come from one city, one technology (LoRa at 868 MHz) and models with limited tuning.

Sources & credits

Where everything comes from

Paper: Robert Bitterling, Christian Nettersheim, Jörn Hees, Michael Rademacher. "A Systematic Sample Size Analysis of ML-Based Path Loss Prediction for LPWAN." arXiv:2608.11083v1 [cs.NI], 11 August 2026. arxiv.org/abs/2608.11083. Fraunhofer FKIE and Hochschule Bonn-Rhein-Sieg. The arXiv listing notes acceptance at the IEEE Conference on Local Computer Networks (LCN).

Code, prepared data and run results: github.com/mclab-hbrs/lora-bonn-ml-pathloss (code under the MIT licence).

Measurement data: M. Rademacher, H. Linka, T. Horstmann, M. Henze, "Path Loss in Urban LoRa Networks: A Large-Scale Measurement Study," IEEE VTC2021-Fall, doi:10.1109/VTC2021-Fall52928.2021.9625531. Data provider: Hochschule Bonn-Rhein-Sieg. Licence: Data licence Germany – attribution – version 2.0 (dl-de/by-2-0).

Modifications for this page: selected columns of the prepared data set; coordinates rounded to about 1 m; path loss rounded to 0.1 dB; run results summarised as means and 95 % ranges; all charts re-drawn. Truck routes are illustrative paths through measured positions, not GPS tracks. The terrain cross-section is illustrative.

Live computations: the k-NN maps and exam scores are recomputed in your browser from the released data with the paper's settings (coordinate features, distance weighting, a fixed 16,074-link exam), so they track the paper's values closely but are not identical to any single run in it.

Terrain data in the study: LiDAR elevation data provided by the State of North Rhine-Westphalia.

Fonts: Space Grotesk and Inter, SIL Open Font License.