# Combining habitat factors

To combine multiple habitat factors into an overall habitat suitability score for each pixel, you must:

• assign weights to each habitat factor that reflect their relative importance
• choose an algorithm that combines multiple factors into a single pixel suitability score

The weighted arithmetic mean is the most commonly used algorithm to combine weights, but the weighted geometric mean better reflects a situation in which one habitat factor limits suitability in a way that cannot be compensated by other factors.

## Assigning weights

To combine multiple habitat factors into one aggregate habitat suitability model, you must first assign weights to each factor that reflect their relative importance. In our linkage designs, we found it intuitive to assign each factor a percentage weight, such that the sum of the weights is 100%. For example, land cover could be assigned a weight of 60%, topographic position a weight of 20%, and distance-to-roads a weight of 20%, making land cover three times more important than the other factors. If a habitat factor is not important for a species, it is assigned a weight of 0%.

Weighting is one of the weakest parts of our models, lacking any underlying theory or hard data. One theoretical issue, for example is this: When the scores are combined across factors, does the overall pixel score still have the same biological interpretation we established when scoring suitability for each factor? Quite honestly, we don't know, but we suspect that the biological meanings have been altered, at least a little bit. The lack of hard data is obvious: We have never built or seen a corridor model that used weights based on empirical data—100% of them are based on expert opinion.

## Selecting an algorithm to combine factors

While there are many potential ways to combine the relative influence of multiple factors, we focus on two: weighted arithmetic mean and weighted geometric mean. Under many circumstances, these algorithms will produce a similar habitat suitability model.

The practical difference between the two algorithms is this: weighted arithmetic mean allows a deficiency in one factor to be compensated by other factors, while weighted geometric mean better reflects a situation in which one habitat factor limits suitability in a way that cannot be compensated by other factors.

What are some examples of this difference?

• In a habitat model for Giant spotted whiptail, land cover received a weight of 70%, while elevation received a weight of 30%. However, the species never occurs above 5000 ft. In a weighted arithmetic mean model, a pixel occurring in favorable riparian woodland vegetation at 6000 ft would be calculated as suitable habitat. In a weighted geometric mean model the pixel would be absolutely unsuitable, because it is above 5000 ft.
• In a habitat model for pronghorn, land cover received a weight of 50%, topography received a weight of 40%, and distance-to roads received a weight of 10%. Topography is important for pronghorn, because they require gentle slopes for predator detection. In a weighted arithmetic mean model, a pixel occurring in flat, high density residential land cover would be calculated as medium suitability (unsuitable land cover and suitable topographic position average out). In a weighted geometric mean model, the pixel would be absolutely unsuitable, because the species cannot occupy high density residential land, no matter how flat it is!

### Weighted arithmetic mean

Most linkage designers have used a weighted arithmetic mean algorithm to combine multiple habitat factors. The weighted arithmetic mean is calculated by multiplying the class score times the percentage weight assigned to its factor, then adding across factors. It is equivalent to the weighted overlay function in ArcGIS, and maintains the same range of suitability scores—if you used a 0-100 scale to score classes within each factor, your weighted sum will also be scaled 0-100.

### Weighted geometric mean

While the weighted geometric mean algorithm is not used as often to build habitat models, we find the approach intuitively appealing.

Weighted geometric mean better models a situation in which a deficit in one factor cannot be compensated by high scores for other factors. For instance, if urban areas are poor habitat under all circumstances, you'd want to combine factors in a way that a pixel of urban habitat doesn't get a high score because it has ideal elevation, topography, and distance to road.

This reflects the limiting factor concept, one of the earliest ideas in ecology. As originally expressed, the idea was that a species population, or an ecosystem flow like primary productivity, is limited by whichever essential factor was most scarce relative to needs. Early ecologists called it Leibig's law of the minimum. It's the same concept you learned when you balanced the equation of a chemical reaction. The amount of chemical product produced depends on the reagent in limited supply. Because we think that habitat suitability probably is limited by the worst factor, we now use this routinely.

The bottom line is that the weighted geometric mean gives more influence to suitability scores near 0 and absolute influence to suitability scores of 0.