Bexar County Texas and Parcel Size vs Counts

Parcel Sizes vs Parcel Counts

Recently a question arose -- in the search for suitable real estate properties vs parcel size -- coupled with ample choice.  
The question was simple:  How many land parcels of a certain size are available for real estate development -- and re-development?

Bexar county TX provides an interesting study and test environment -- over 600,000 well defined tax parcels from the Bexar County Tax Appraisal (BCAD) GIS system.

In this post, I shall examine how parcel sub-division across an urban area trends in a pattern typically seen in language & word complexity -- and mimics something called Zipf's_Law -- and maybe something called Rank_Size_Distribution for city population vs country size ranking.

Rank-Order Distribution for Cities 

(from Wikipedia)

First, the Numbers

The decreasing number of large tax parcels is very noticeable via this "raw" data table extract from the BCAD GIS parcel quilt:

===========================
======= Table 1.0 =========
-------------------  ------
Acre-Range            Kount
-------------------  ------
10000.00 and Up           1
 5000.00 - 10000.00       2
 2000.00 -  5000.00       6
 1000.00 -  2000.00      17
  500.00 -  1000.00      36
  200.00 -   500.00     228
  100.00 -   200.00     545
   50.00 -   100.00    1131
   20.00 -    50.00    3142
   10.00 -    20.00    5284
    5.00 -    10.00    7760
    2.00 -     5.00   16591
    1.00 -     2.00   21665
    0.50 -     1.00   33188
    0.20 -     0.50  172027
    0.10 -     0.20  305560
    0.05 -     0.10   24289
    0.01 -     0.05    6995
     Less than 0.01    3094

The above table matches what life experience tells us:  Small parcels are common with large repeat counts; Large parcels are rare.  And because Bexar county is mostly an urban area -- there are a large number of "individual" sized lots -- mostly for residential and small biz operations.

The lone parcel above 10,000 acres in 2015 BCAD tax parcel  quilt was 31062 acres for Camp Bullis military reservation.  Many of the largest parcels are "tied up" by utility, quarry, water & sewer treatment operations, public green spaces, quasi-government entities and military bases & industrial parks.

For Bexar county, the most common tax parcel (lot) size was between 0.10 and 0.20 acres -- 4456 to 8712 square feet -- corresponding to square lots of about 66x66 feet to 93x93 feet.  An examination of the GIS parcel quilt confirms this large count corresponds almost exclusively to single home lots and residential areas.

Trend Surprise?

A log-log plot of Table 1 (above) produced some interesting kinks and hints:

Fig 1: Bexar Acres vs Count

(click for larger)

Dots in Fig 1 plot are the observed acres vs counts from the 2015 BCAD GIS.  

MATH MODEL? Assuming the observed data might "follow" a "power law" relation of the form:

y = a x ^b

The blue and red lines are two possible mathematical "fits" to the observed data.  WHY?  For what purpose?  To extract some numerical "intelligence" and "insight" from the data.  Sometimes "math models" provide hints of related data patterns from seemingly unrelated subject areas.

BLUE LINE was a least squares "fit" to the data from 0.5 to 31062 acres -- and manifests a "fit" to the following "power law" relation:

parcel_count = 45305 acres ^-0.9774

The coefficient 45305 is interesting because it is "close" to the Camp Bullis large acreage -- which dominates the rank-order distribution of tax parcels.  The exponent "-0.997" is close to "-1.0" -- which means that tax parcel acreage vs counts is an inverse relation -- i.e. like this equation:

y ~= 1/x

RED LINE also assumes a power law relation -- and uses the Camp Bullis acreage of 31062 acres as the "y-intercept" and "origin" for the decreasing counts -- as acres increase. Also, assuming that observed data has a strict inverse relationship:

count = 31062 acres ^-1.0

or using algebra:

count = 31062 ÷  acres

SIGNIFICANCE?  The odds of finding a large parcel in Bexar county goes down -- rapidly -- in a pattern that mimics Zipf's law for complexity of words in language.  This is also similar to the odds of large cities in countries of various sizes.