10B: Calibrating Spatial Interaction Models with R

In this exercise, we will learn to calibrate Spatial Interaction Models (SIMs) using various regression methods to determine factors affecting public bus passenger flows during the morning peak in Singapore.

Author

Teng Kok Wai (Walter)

Published

September 29, 2024

Modified

October 14, 2024

1 Exercise 10B Reference

R for Geospatial Data Science and Analytics - 16 Calibrating Spatial Interaction Models with R

2 Overview

Spatial Interaction Models (SIMs) are mathematical models for estimating flows between spatial entities developed by Alan Wilson in the late 1960s and early 1970, with considerable uptake and refinement for transport modelling since then Boyce and Williams (2015).

There are four main types of traditional SIMs (Wilson 1971):

Unconstrained
Production-constrained
Attraction-constrained
Doubly-constrained

Ordinary least square (OLS), log-normal, Poisson and negative binomial (NB) regression methods have been used extensively to calibrate OD flow models by processing flow data as different types of dependent variables. In this chapter, you will gain hands-on experiences on using appropriate R packages to calibrate SIM by using there four regression methods.

Note

Calibration is the process of adjusting parameters in the model to try and get the estimates to agree with the observed data as much as possible. Adjusting the parameters is the sort of iterative process that computers are particularly good at and the goodness-of-fit statistics can be used to indicate when the optimum solution is found. Historically this process required a researcher with the requisite programming skills to write a computer algorithm to iteratively adjust each parameter, check the goodness-of-fit, and then start all over again until the goodness-of-fit statistic was maximised/minimised. (Adam Dennett, 2018)

3 Learning Outcome

Understand and apply different types of Spatial Interaction Models (SIMs).
Calibrate SIMs using Ordinary Least Squares (OLS), log-normal, Poisson, and Negative Binomial (NB) regression methods.
Import and prepare geospatial data using R packages such as sf and tidyverse.
Compute a distance matrix for spatial data using the sp package.
Visualize and compare model performance using ggplot2 and performance packages.

4 The Data

This exercise is a continuation of Hands-on Exercise 10A and the following datasets will be used in this exercise:

Data Set	Description	Format
od_data.rds	Weekday morning peak passenger flows at the planning subzone level.	RDS
mpsz.rds	URA Master Plan 2019 Planning Subzone boundary in simple feature tibble data frame format.	RDS
pop.csv	Additional attribute data file providing population information.	CSV

These datasets will be utilized to calibrate and visualize the Spatial Interaction Models.

This exercise is a continuation from Hands-on Exercise 9A.

5 Installing and Launching the R Packages

The following R packages will be used in this exercise:

Package	Purpose	Use Case in Exercise
sf	Imports, integrates, processes, and transforms vector-based geospatial data.	Handling vector geospatial data, such as the URA Master Plan 2019 Planning Subzone boundary.
tidyverse	A collection of packages for data science tasks such as data manipulation, visualization, and modeling.	Importing CSV files, wrangling data, and performing relational joins.
tmap	Creates static and interactive thematic maps using cartographic quality elements.	Visualizing regional development indicators and plotting maps showing spatial relationships and patterns.
performance	Provides tools to assess and compare the performance of regression models.	Comparing model performance metrics, such as R-squared and RMSE, for different spatial interaction models.
sp	Provides functions for spatial dependence analysis, including spatial weights and spatial autocorrelation.	Computing spatial weights and distance matrices for geospatial data.
ggplot2	Creates data visualizations using a layered grammar of graphics.	Plotting histograms, scatter plots, and visualizing model fits and residuals for calibrated Spatial Interaction Models.
reshape2	Provides functions to reshape data between wide and long formats.	Pivoting distance matrices into long format for model calibration and analysis.
ggpubr	Provides tools for creating and customizing publication-ready plots in ggplot2.	Combining multiple plots into a single visual for comparing different models.

To install and load these packages, use the following code:

pacman::p_load(tmap, sf, sp, performance, reshape2, ggpubr, tidyverse)

5.1 Computing Distance Matrix

In spatial interaction, a distance matrix displays the distance between pairs of locations. For example, the Euclidean distance between two locations like MESZ01 and RVSZ05 is 3926.0025, and between MESZ01 and SRSZ01 is 3939.1079. An entry of 0 on the diagonal indicates that the location is compared with itself.

First, import mpsz.rds into R:

mpsz <- read_rds("data/rds/mpsz.rds")
mpsz

Simple feature collection with 332 features and 6 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21 / Singapore TM
First 10 features:
                 SUBZONE_N SUBZONE_C       PLN_AREA_N PLN_AREA_C       REGION_N
1              MARINA EAST    MESZ01      MARINA EAST         ME CENTRAL REGION
2         INSTITUTION HILL    RVSZ05     RIVER VALLEY         RV CENTRAL REGION
3           ROBERTSON QUAY    SRSZ01  SINGAPORE RIVER         SR CENTRAL REGION
4  JURONG ISLAND AND BUKOM    WISZ01  WESTERN ISLANDS         WI    WEST REGION
5             FORT CANNING    MUSZ02           MUSEUM         MU CENTRAL REGION
6         MARINA EAST (MP)    MPSZ05    MARINE PARADE         MP CENTRAL REGION
7                   SUDONG    WISZ03  WESTERN ISLANDS         WI    WEST REGION
8                  SEMAKAU    WISZ02  WESTERN ISLANDS         WI    WEST REGION
9           SOUTHERN GROUP    SISZ02 SOUTHERN ISLANDS         SI CENTRAL REGION
10                 SENTOSA    SISZ01 SOUTHERN ISLANDS         SI CENTRAL REGION
   REGION_C                       geometry
1        CR MULTIPOLYGON (((33222.98 29...
2        CR MULTIPOLYGON (((28481.45 30...
3        CR MULTIPOLYGON (((28087.34 30...
4        WR MULTIPOLYGON (((14557.7 304...
5        CR MULTIPOLYGON (((29542.53 31...
6        CR MULTIPOLYGON (((35279.55 30...
7        WR MULTIPOLYGON (((15772.59 21...
8        WR MULTIPOLYGON (((19843.41 21...
9        CR MULTIPOLYGON (((30870.53 22...
10       CR MULTIPOLYGON (((26879.04 26...

Note that it is a sf tibble dataframe object class.

5.2 Converting from sf data.table to SpatialPolygonsDataFrame

There are at least two ways to compute the required distance matrix. One is based on sf and the other is based on sp. Past experience shown that computing distance matrix by using sf function took relatively longer time that sp method especially the data set is large. In view of this, sp method is used in the code chunks below.

First as.Spatial() will be used to convert mpsz from sf tibble data frame to SpatialPolygonsDataFrame of sp object as shown in the code chunk below.

mpsz_sp <- as(mpsz, "Spatial")
mpsz_sp

class       : SpatialPolygonsDataFrame
features    : 332
extent      : 2667.538, 56396.44, 15748.72, 50256.33  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables   : 6
names       : SUBZONE_N, SUBZONE_C, PLN_AREA_N, PLN_AREA_C,       REGION_N, REGION_C
min values  : ADMIRALTY,    AMSZ01, ANG MO KIO,         AM, CENTRAL REGION,       CR
max values  :    YUNNAN,    YSSZ09,     YISHUN,         YS,    WEST REGION,       WR

5.3 Computing the Distance Matrix

Now, we compute the Euclidean distance between centroids of planning subzones using spDists() from the sp package.

Q&A

Do you know why the distance is calculated between two centroids of a pair of spatial polygons?

Centroids simplify distance calculations by representing each polygon with a single point, avoiding the complexity of measuring distances between all boundary points of irregular shapes

dist <- spDists(mpsz_sp, longlat = FALSE)

head(dist, n = c(10, 10))

           [,1]       [,2]      [,3]      [,4]       [,5]      [,6]      [,7]
 [1,]     0.000  3926.0025  3939.108 20252.964  2989.9839  1431.330 19211.836
 [2,]  3926.003     0.0000   305.737 16513.865   951.8314  5254.066 16242.523
 [3,]  3939.108   305.7370     0.000 16412.062  1045.9088  5299.849 16026.146
 [4,] 20252.964 16513.8648 16412.062     0.000 17450.3044 21665.795  7229.017
 [5,]  2989.984   951.8314  1045.909 17450.304     0.0000  4303.232 17020.916
 [6,]  1431.330  5254.0664  5299.849 21665.795  4303.2323     0.000 20617.082
 [7,] 19211.836 16242.5230 16026.146  7229.017 17020.9161 20617.082     0.000
 [8,] 14960.942 12749.4101 12477.871 11284.279 13336.0421 16281.453  5606.082
 [9,]  7515.256  7934.8082  7649.776 18427.503  7801.6163  8403.896 14810.930
[10,]  6391.342  4975.0021  4669.295 15469.566  5226.8731  7707.091 13111.391
           [,8]      [,9]     [,10]
 [1,] 14960.942  7515.256  6391.342
 [2,] 12749.410  7934.808  4975.002
 [3,] 12477.871  7649.776  4669.295
 [4,] 11284.279 18427.503 15469.566
 [5,] 13336.042  7801.616  5226.873
 [6,] 16281.453  8403.896  7707.091
 [7,]  5606.082 14810.930 13111.391
 [8,]     0.000  9472.024  8575.490
 [9,]  9472.024     0.000  3780.800
[10,]  8575.490  3780.800     0.000

The output dist is a matrix without labeled rows and columns for the planning subzone codes.

5.4 Labeling the Distance Matrix

We will label the rows and columns of the distance matrix using the planning subzone codes.

sz_names <- mpsz$SUBZONE_C
colnames(dist) <- sz_names
rownames(dist) <- sz_names

5.5 Pivoting the Distance Matrix

Next, we pivot the distance matrix into a long format, where rows represent the origin and destination pairs.

distPair <- melt(dist) %>%
  rename(dist = value)
head(distPair, 10)

     Var1   Var2      dist
1  MESZ01 MESZ01     0.000
2  RVSZ05 MESZ01  3926.003
3  SRSZ01 MESZ01  3939.108
4  WISZ01 MESZ01 20252.964
5  MUSZ02 MESZ01  2989.984
6  MPSZ05 MESZ01  1431.330
7  WISZ03 MESZ01 19211.836
8  WISZ02 MESZ01 14960.942
9  SISZ02 MESZ01  7515.256
10 SISZ01 MESZ01  6391.342

Note

Observations:

The within zone distance is 0.

5.6 Updating Intra-Zonal Distances

In this section, we are going to append a constant value to replace the intra-zonal distance of 0.

We will select and find out the minimum value of the distance by using summary().

distPair %>%
  filter(dist > 0) %>%
  summary()

      Var1             Var2             dist
 MESZ01 :   331   MESZ01 :   331   Min.   :  173.8
 RVSZ05 :   331   RVSZ05 :   331   1st Qu.: 7149.5
 SRSZ01 :   331   SRSZ01 :   331   Median :11890.0
 WISZ01 :   331   WISZ01 :   331   Mean   :12229.4
 MUSZ02 :   331   MUSZ02 :   331   3rd Qu.:16401.7
 MPSZ05 :   331   MPSZ05 :   331   Max.   :49894.4
 (Other):107906   (Other):107906

Then, we replace intra-zonal distances (which are 0) with a constant value of 50m.

distPair$dist <- ifelse(distPair$dist == 0, 50, distPair$dist)
distPair %>%
  summary()

      Var1             Var2             dist
 MESZ01 :   332   MESZ01 :   332   Min.   :   50
 RVSZ05 :   332   RVSZ05 :   332   1st Qu.: 7097
 SRSZ01 :   332   SRSZ01 :   332   Median :11864
 WISZ01 :   332   WISZ01 :   332   Mean   :12193
 MUSZ02 :   332   MUSZ02 :   332   3rd Qu.:16388
 MPSZ05 :   332   MPSZ05 :   332   Max.   :49894
 (Other):108232   (Other):108232

5.7 Renaming Fields and Saving

Next, qe rename the origin and destination fields for clarity.

distPair <- distPair %>%
  rename(orig = Var1, dest = Var2)

Finally, save the updated distance pair dataframe for future use.

write_rds(distPair, "data/rds/distPair.rds")
distPair <- read_rds("data/rds/distPair.rds")

6 Preparing Flow Data

First, we import the od_data from Hands-on Exercise 9A.

od_data_fii <- read_rds("data/rds/od_data_fii.rds")

6.1 Computing Total Passenger Trips

Next, compute the total passenger trips between and within planning subzones.

flow_data <- od_data_fii %>%
  group_by(ORIGIN_SZ, DESTIN_SZ) %>%
  summarize(TRIPS = sum(MORNING_PEAK))

Display the first 10 rows of flow_data:

head(flow_data, 10)

6.2 Separating Intra-Zonal Flows

The code below adds two fields to flow_data, separating intra-zonal trips.

flow_data$FlowNoIntra <- ifelse(flow_data$ORIGIN_SZ == flow_data$DESTIN_SZ, 0, flow_data$TRIPS)
flow_data$offset <- ifelse(flow_data$ORIGIN_SZ == flow_data$DESTIN_SZ, 0.000001, 1)

6.3 Combining Passenger Volume and Distance Data

Before we can join flow_data and distPair, we need to convert data value type of ORIGIN_SZ and DESTIN_SZ fields of flow_data dataframe into factor data type.

flow_data$ORIGIN_SZ <- as.factor(flow_data$ORIGIN_SZ)
flow_data$DESTIN_SZ <- as.factor(flow_data$DESTIN_SZ)

Then, we can perform left join on flow_data with distPair to combine passenger volumes with distances.

flow_data1 <- flow_data %>%
  left_join(distPair, by = c("ORIGIN_SZ" = "orig", "DESTIN_SZ" = "dest"))

7 Preparing Origin and Destination Attributes

In this section, we will prepare the origin and destination attribute data.

7.1 Importing Population Data

pop <- read_csv("data/aspatial/pop.csv")

7.2 Geospatial Data Wrangling

Join the population data with mpsz.

pop <- pop %>%
  left_join(mpsz, by = c("PA" = "PLN_AREA_N", "SZ" = "SUBZONE_N")) %>%
  select(1:6) %>%
  rename(SZ_NAME = SZ, SZ = SUBZONE_C)

7.3 Adding Origin Attributes

Join population data with flow_data1 for origin attributes.

flow_data1 <- flow_data1 %>%
  left_join(pop, by = c(ORIGIN_SZ = "SZ")) %>%
  rename(ORIGIN_AGE7_12 = AGE7_12, ORIGIN_AGE13_24 = AGE13_24, ORIGIN_AGE25_64 = AGE25_64) %>%
  select(-c(PA, SZ_NAME))

7.4 Adding Destination Attributes

Join population data with flow_data1 for destination attributes.

flow_data1 <- flow_data1 %>%
  left_join(pop, by = c(DESTIN_SZ = "SZ")) %>%
  rename(DESTIN_AGE7_12 = AGE7_12, DESTIN_AGE13_24 = AGE13_24, DESTIN_AGE25_64 = AGE25_64) %>%
  select(-c(PA, SZ_NAME))

7.5 Saving Processed Data

The final output will be saved as SIM_data in RDS format.

write_rds(flow_data1, "data/rds/flow_data_6-9.rds")

8 Calibrating Spatial Interaction Models

In this section, you will learn how to calibrate Spatial Interaction Models by using Poisson Regression method.

8.1 Importing the Modelling Data

Firstly, we will import the saved modelling data.

SIM_data <- read_rds("data/rds/flow_data_6-9.rds")

8.2 Visualizing the Dependent Variable

To visualize the dependent variable, we will plot the distribution of TRIPS using a histogram.

ggplot(data = SIM_data, aes(x = TRIPS)) +
  geom_histogram()

Note

Observations:

Notice that the distribution is highly skewed and not resemble bell shape or also known as normal distribution.

We can also visualize the relationship between TRIPS and distance using a scatter plot.

ggplot(data = SIM_data, aes(x = dist, y = TRIPS)) +
  geom_point() +
  geom_smooth(method = lm)

Note

Observations:

the relationship between TRIPS and distance hardly resemble a linear relationship.

We can perform log-transformation on both variables to make he relationship appears more linear.

ggplot(data = SIM_data, aes(x = log(dist), y = log(TRIPS))) +
  geom_point() +
  geom_smooth(method = lm)

8.3 Handling Zero Values in Variables

Since Poisson Regression is based of log and log 0 is undefined, it is important for us to ensure that no 0 values in the explanatory variables.

We will compute the summary statistics of all variables in SIM_data data frame.

summary(SIM_data)

  ORIGIN_SZ          DESTIN_SZ             TRIPS         FlowNoIntra
 Length:20916       Length:20916       Min.   :     1   Min.   :     0.0
 Class :character   Class :character   1st Qu.:    17   1st Qu.:    15.0
 Mode  :character   Mode  :character   Median :    87   Median :    81.0
                                       Mean   :  1178   Mean   :   980.1
                                       3rd Qu.:   454   3rd Qu.:   417.0
                                       Max.   :344039   Max.   :218070.0
     offset              dist       ORIGIN_AGE7_12   ORIGIN_AGE13_24
 Min.   :0.000001   Min.   :   50   Min.   :   0.0   Min.   :    0
 1st Qu.:1.000000   1st Qu.: 3373   1st Qu.:  50.0   1st Qu.:  100
 Median :1.000000   Median : 6172   Median : 510.0   Median : 1130
 Mean   :0.986087   Mean   : 6992   Mean   : 888.6   Mean   : 1954
 3rd Qu.:1.000000   3rd Qu.: 9918   3rd Qu.:1360.0   3rd Qu.: 3010
 Max.   :1.000000   Max.   :26136   Max.   :6340.0   Max.   :16380
 ORIGIN_AGE25_64 DESTIN_AGE7_12   DESTIN_AGE13_24 DESTIN_AGE25_64
 Min.   :    0   Min.   :   0.0   Min.   :    0   Min.   :    0
 1st Qu.:  730   1st Qu.:  10.0   1st Qu.:   60   1st Qu.:  630
 Median : 5730   Median : 510.0   Median : 1100   Median : 5710
 Mean   : 9092   Mean   : 854.6   Mean   : 1896   Mean   : 8829
 3rd Qu.:14180   3rd Qu.:1350.0   3rd Qu.: 2920   3rd Qu.:13830
 Max.   :74610   Max.   :6340.0   Max.   :16380   Max.   :74610

The print report above reveals that variables ORIGIN_AGE7_12, ORIGIN_AGE13_24, ORIGIN_AGE25_64,DESTIN_AGE7_12, DESTIN_AGE13_24, DESTIN_AGE25_64 consist of 0 values.

SIM_data$DESTIN_AGE7_12 <- ifelse(SIM_data$DESTIN_AGE7_12 == 0, 0.99, SIM_data$DESTIN_AGE7_12)
SIM_data$DESTIN_AGE13_24 <- ifelse(SIM_data$DESTIN_AGE13_24 == 0, 0.99, SIM_data$DESTIN_AGE13_24)
SIM_data$DESTIN_AGE25_64 <- ifelse(SIM_data$DESTIN_AGE25_64 == 0, 0.99, SIM_data$DESTIN_AGE25_64)
SIM_data$ORIGIN_AGE7_12 <- ifelse(SIM_data$ORIGIN_AGE7_12 == 0, 0.99, SIM_data$ORIGIN_AGE7_12)
SIM_data$ORIGIN_AGE13_24 <- ifelse(SIM_data$ORIGIN_AGE13_24 == 0, 0.99, SIM_data$ORIGIN_AGE13_24)
SIM_data$ORIGIN_AGE25_64 <- ifelse(SIM_data$ORIGIN_AGE25_64 == 0, 0.99, SIM_data$ORIGIN_AGE25_64)

summary(SIM_data)

  ORIGIN_SZ          DESTIN_SZ             TRIPS         FlowNoIntra
 Length:20916       Length:20916       Min.   :     1   Min.   :     0.0
 Class :character   Class :character   1st Qu.:    17   1st Qu.:    15.0
 Mode  :character   Mode  :character   Median :    87   Median :    81.0
                                       Mean   :  1178   Mean   :   980.1
                                       3rd Qu.:   454   3rd Qu.:   417.0
                                       Max.   :344039   Max.   :218070.0
     offset              dist       ORIGIN_AGE7_12    ORIGIN_AGE13_24
 Min.   :0.000001   Min.   :   50   Min.   :   0.99   Min.   :    0.99
 1st Qu.:1.000000   1st Qu.: 3373   1st Qu.:  50.00   1st Qu.:  100.00
 Median :1.000000   Median : 6172   Median : 510.00   Median : 1130.00
 Mean   :0.986087   Mean   : 6992   Mean   : 888.77   Mean   : 1954.26
 3rd Qu.:1.000000   3rd Qu.: 9918   3rd Qu.:1360.00   3rd Qu.: 3010.00
 Max.   :1.000000   Max.   :26136   Max.   :6340.00   Max.   :16380.00
 ORIGIN_AGE25_64    DESTIN_AGE7_12    DESTIN_AGE13_24    DESTIN_AGE25_64
 Min.   :    0.99   Min.   :   0.99   Min.   :    0.99   Min.   :    0.99
 1st Qu.:  730.00   1st Qu.:  10.00   1st Qu.:   60.00   1st Qu.:  630.00
 Median : 5730.00   Median : 510.00   Median : 1100.00   Median : 5710.00
 Mean   : 9092.13   Mean   : 854.82   Mean   : 1896.17   Mean   : 8829.27
 3rd Qu.:14180.00   3rd Qu.:1350.00   3rd Qu.: 2920.00   3rd Qu.:13830.00
 Max.   :74610.00   Max.   :6340.00   Max.   :16380.00   Max.   :74610.00

All the 0 values have been replaced by 0.99.

8.4 Unconstrained Spatial Interaction Model

In this section, we will calibrate an unconstrained spatial interaction model by using glm() of Base Stats.

Note

The general formula of Unconstrained Spatial Interaction Model

\(\lambda_{ij} = \exp \left( k + \mu \ln V_i + \alpha \ln W_j - \beta \ln d_{ij} \right)\)

The explanatory variables are origin population by different age cohort, destination population by different age cohort (i.e. ORIGIN_AGE25_64) and distance between origin and destination in km (i.e. dist). To fit an unconstrained Spatial Interaction Model using Poisson regression:

uncSIM <- glm(formula = TRIPS ~ log(ORIGIN_AGE25_64) + log(DESTIN_AGE25_64) + log(dist),
              family = poisson(link = "log"),
              data = SIM_data,
              na.action = na.exclude)
uncSIM


Call:  glm(formula = TRIPS ~ log(ORIGIN_AGE25_64) + log(DESTIN_AGE25_64) +
    log(dist), family = poisson(link = "log"), data = SIM_data,
    na.action = na.exclude)

Coefficients:
         (Intercept)  log(ORIGIN_AGE25_64)  log(DESTIN_AGE25_64)
            10.58389               0.26347               0.02567
           log(dist)
            -0.73018

Degrees of Freedom: 20915 Total (i.e. Null);  20912 Residual
Null Deviance:      105500000
Residual Deviance: 62360000     AIC: 62490000

8.5 Calculating R-squared for Unconstrained SIM

In order to measure how much variation of the trips can be accounted by the model, we will define a function to calculate R-squared and apply it to the model.

CalcRSquared <- function(observed, estimated){
  r <- cor(observed, estimated)
  r^2
}

Then, we will calculate the R-squared of the unconstrained model.

CalcRSquared(uncSIM$data$TRIPS, uncSIM$fitted.values)

[1] 0.1938202

Alternatively, we can calculate McFadden’s R-squared.

r2_mcfadden(uncSIM)

# R2 for Generalized Linear Regression
       R2: 0.408
  adj. R2: 0.408

Note

McFadden’s R-squared

Interpretation: McFadden’s R-squared is used for logistic regression models and measures the improvement of the fitted model over the null model. It is not directly comparable to the traditional R-squared.
Range: McFadden’s R-squared values typically range from 0 to just under 1.
- 0: The model is no better than the null model.
- 1: The model perfectly predicts the outcome.
Good Range:
- 0.2 to 0.4: Considered good for logistic regression models.
- Above 0.4: Indicates a very strong model.

8.6 Origin (Production) Constrained SIM

In this section, we will fit an origin constrained SIM, where the trips are constrained by the origin.

Note

The general formula of Origin Constrained Spatial Interaction Model

\(\lambda_{ij} = \exp\left( k + \mu_i + \alpha \ln W_j - \beta \ln d_{ij} \right)\)

Notice that the difference between Unconstrained Spatial Interaction Model and this formula lies in the second term. In the Unconstrained Spatial Interaction Model formula, it is \(\mu \ln V_i\) as compared to \(\mu_i\) in Origin Constrained Spatial Interaction Model.

orcSIM <- glm(formula = TRIPS ~ ORIGIN_SZ + log(DESTIN_AGE25_64) + log(dist),
              family = poisson(link = "log"),
              data = SIM_data,
              na.action = na.exclude)
summary(orcSIM)


Call:
glm(formula = TRIPS ~ ORIGIN_SZ + log(DESTIN_AGE25_64) + log(dist),
    family = poisson(link = "log"), data = SIM_data, na.action = na.exclude)

Coefficients:
                       Estimate Std. Error   z value Pr(>|z|)
(Intercept)           1.251e+01  3.144e-03  3977.962  < 2e-16 ***
ORIGIN_SZAMSZ02       1.062e+00  3.687e-03   288.178  < 2e-16 ***
ORIGIN_SZAMSZ03       6.217e-01  3.774e-03   164.747  < 2e-16 ***
ORIGIN_SZAMSZ04      -2.100e-02  4.265e-03    -4.924 8.46e-07 ***
ORIGIN_SZAMSZ05      -1.778e-01  4.850e-03   -36.661  < 2e-16 ***
ORIGIN_SZAMSZ06       3.340e-01  4.387e-03    76.126  < 2e-16 ***
ORIGIN_SZAMSZ07      -1.141e+00  7.375e-03  -154.672  < 2e-16 ***
ORIGIN_SZAMSZ08      -8.224e-01  6.804e-03  -120.871  < 2e-16 ***
ORIGIN_SZAMSZ09       2.187e-01  4.538e-03    48.186  < 2e-16 ***
ORIGIN_SZAMSZ10       5.139e-01  3.975e-03   129.304  < 2e-16 ***
ORIGIN_SZAMSZ11      -1.476e+00  8.890e-03  -166.019  < 2e-16 ***
ORIGIN_SZAMSZ12      -1.512e+00  8.974e-03  -168.491  < 2e-16 ***
ORIGIN_SZBDSZ01       1.035e+00  3.647e-03   283.888  < 2e-16 ***
ORIGIN_SZBDSZ02       5.114e-01  4.195e-03   121.896  < 2e-16 ***
ORIGIN_SZBDSZ03       1.013e+00  3.733e-03   271.518  < 2e-16 ***
ORIGIN_SZBDSZ04       1.738e+00  3.265e-03   532.437  < 2e-16 ***
ORIGIN_SZBDSZ05       7.463e-01  3.713e-03   200.998  < 2e-16 ***
ORIGIN_SZBDSZ06       9.612e-01  3.743e-03   256.801  < 2e-16 ***
ORIGIN_SZBDSZ07      -9.204e-01  6.791e-03  -135.530  < 2e-16 ***
ORIGIN_SZBDSZ08      -8.772e-01  6.694e-03  -131.044  < 2e-16 ***
ORIGIN_SZBKSZ01      -2.932e-01  5.277e-03   -55.567  < 2e-16 ***
ORIGIN_SZBKSZ02       4.668e-01  4.229e-03   110.382  < 2e-16 ***
ORIGIN_SZBKSZ03       8.058e-01  3.924e-03   205.345  < 2e-16 ***
ORIGIN_SZBKSZ04      -1.653e-03  4.828e-03    -0.342    0.732
ORIGIN_SZBKSZ05       2.118e-02  4.537e-03     4.669 3.03e-06 ***
ORIGIN_SZBKSZ06       1.408e-01  4.826e-03    29.172  < 2e-16 ***
ORIGIN_SZBKSZ07       8.012e-01  3.674e-03   218.081  < 2e-16 ***
ORIGIN_SZBKSZ08       1.203e-01  4.328e-03    27.805  < 2e-16 ***
ORIGIN_SZBKSZ09       8.939e-02  4.527e-03    19.745  < 2e-16 ***
ORIGIN_SZBLSZ01      -1.359e+00  1.077e-02  -126.206  < 2e-16 ***
ORIGIN_SZBLSZ02      -2.083e+00  1.559e-02  -133.643  < 2e-16 ***
ORIGIN_SZBLSZ03      -3.206e+00  3.300e-02   -97.132  < 2e-16 ***
ORIGIN_SZBLSZ04      -1.772e+00  1.577e-02  -112.360  < 2e-16 ***
ORIGIN_SZBMSZ01       1.833e-01  3.973e-03    46.130  < 2e-16 ***
ORIGIN_SZBMSZ02      -1.192e+00  5.701e-03  -209.042  < 2e-16 ***
ORIGIN_SZBMSZ03      -3.159e-01  4.531e-03   -69.718  < 2e-16 ***
ORIGIN_SZBMSZ04       2.399e-02  4.011e-03     5.980 2.23e-09 ***
ORIGIN_SZBMSZ05      -1.274e+00  6.120e-03  -208.104  < 2e-16 ***
ORIGIN_SZBMSZ06      -1.756e+00  9.559e-03  -183.732  < 2e-16 ***
ORIGIN_SZBMSZ07      -3.696e-01  4.489e-03   -82.352  < 2e-16 ***
ORIGIN_SZBMSZ08      -4.495e-01  4.511e-03   -99.631  < 2e-16 ***
ORIGIN_SZBMSZ09      -1.141e+00  5.807e-03  -196.571  < 2e-16 ***
ORIGIN_SZBMSZ10      -1.193e+00  6.140e-03  -194.266  < 2e-16 ***
ORIGIN_SZBMSZ11      -8.546e-01  5.373e-03  -159.060  < 2e-16 ***
ORIGIN_SZBMSZ12      -1.026e+00  6.975e-03  -147.165  < 2e-16 ***
ORIGIN_SZBMSZ13      -2.195e-02  4.435e-03    -4.948 7.50e-07 ***
ORIGIN_SZBMSZ14      -5.738e-01  5.292e-03  -108.430  < 2e-16 ***
ORIGIN_SZBMSZ15      -3.203e-01  4.798e-03   -66.749  < 2e-16 ***
ORIGIN_SZBMSZ16      -1.296e+00  6.081e-03  -213.135  < 2e-16 ***
ORIGIN_SZBMSZ17      -1.756e+00  9.297e-03  -188.888  < 2e-16 ***
ORIGIN_SZBPSZ01       2.254e-01  4.507e-03    50.006  < 2e-16 ***
ORIGIN_SZBPSZ02       2.384e-01  4.944e-03    48.214  < 2e-16 ***
ORIGIN_SZBPSZ03       5.272e-01  4.549e-03   115.887  < 2e-16 ***
ORIGIN_SZBPSZ04       5.440e-01  4.077e-03   133.420  < 2e-16 ***
ORIGIN_SZBPSZ05       6.051e-01  3.781e-03   160.027  < 2e-16 ***
ORIGIN_SZBPSZ06      -9.856e-01  6.582e-03  -149.746  < 2e-16 ***
ORIGIN_SZBPSZ07      -8.273e-01  6.535e-03  -126.592  < 2e-16 ***
ORIGIN_SZBSSZ01       2.322e-03  4.357e-03     0.533    0.594
ORIGIN_SZBSSZ02       4.265e-01  3.924e-03   108.702  < 2e-16 ***
ORIGIN_SZBSSZ03       3.176e-01  3.876e-03    81.942  < 2e-16 ***
ORIGIN_SZBTSZ01       9.283e-02  4.241e-03    21.890  < 2e-16 ***
ORIGIN_SZBTSZ02      -9.336e-01  6.055e-03  -154.184  < 2e-16 ***
ORIGIN_SZBTSZ03      -8.606e-02  4.554e-03   -18.899  < 2e-16 ***
ORIGIN_SZBTSZ04      -6.871e-01  7.481e-03   -91.845  < 2e-16 ***
ORIGIN_SZBTSZ05      -1.472e+00  8.267e-03  -178.077  < 2e-16 ***
ORIGIN_SZBTSZ06      -6.835e-01  5.683e-03  -120.257  < 2e-16 ***
ORIGIN_SZBTSZ07      -1.856e+00  8.657e-03  -214.406  < 2e-16 ***
ORIGIN_SZBTSZ08      -1.019e+00  6.701e-03  -152.085  < 2e-16 ***
ORIGIN_SZCCSZ01      -1.594e+00  1.065e-02  -149.672  < 2e-16 ***
ORIGIN_SZCHSZ01      -1.120e+00  9.221e-03  -121.456  < 2e-16 ***
ORIGIN_SZCHSZ02      -6.325e-01  6.780e-03   -93.284  < 2e-16 ***
ORIGIN_SZCHSZ03       6.495e-01  4.608e-03   140.937  < 2e-16 ***
ORIGIN_SZCKSZ01       4.387e-01  4.047e-03   108.404  < 2e-16 ***
ORIGIN_SZCKSZ02       9.345e-01  4.067e-03   229.779  < 2e-16 ***
ORIGIN_SZCKSZ03       9.446e-01  3.708e-03   254.769  < 2e-16 ***
ORIGIN_SZCKSZ04       1.385e+00  3.782e-03   366.289  < 2e-16 ***
ORIGIN_SZCKSZ05       1.101e+00  4.378e-03   251.560  < 2e-16 ***
ORIGIN_SZCKSZ06       1.306e+00  4.171e-03   312.991  < 2e-16 ***
ORIGIN_SZCLSZ01      -3.752e-01  5.690e-03   -65.937  < 2e-16 ***
ORIGIN_SZCLSZ02      -1.661e+00  1.055e-02  -157.475  < 2e-16 ***
ORIGIN_SZCLSZ03      -3.461e-01  5.238e-03   -66.071  < 2e-16 ***
ORIGIN_SZCLSZ04       8.445e-01  3.627e-03   232.834  < 2e-16 ***
ORIGIN_SZCLSZ05      -1.645e+00  1.013e-02  -162.395  < 2e-16 ***
ORIGIN_SZCLSZ06       9.372e-01  3.478e-03   269.509  < 2e-16 ***
ORIGIN_SZCLSZ07      -1.060e-01  4.398e-03   -24.110  < 2e-16 ***
ORIGIN_SZCLSZ08       3.363e-01  4.833e-03    69.592  < 2e-16 ***
ORIGIN_SZCLSZ09      -1.714e+00  1.353e-02  -126.689  < 2e-16 ***
ORIGIN_SZDTSZ01      -1.720e+00  7.004e-03  -245.518  < 2e-16 ***
ORIGIN_SZDTSZ02      -1.539e+00  6.422e-03  -239.640  < 2e-16 ***
ORIGIN_SZDTSZ03      -2.810e+00  1.301e-02  -215.941  < 2e-16 ***
ORIGIN_SZDTSZ04      -3.750e+00  9.290e-02   -40.363  < 2e-16 ***
ORIGIN_SZDTSZ05      -3.057e+00  2.143e-02  -142.634  < 2e-16 ***
ORIGIN_SZDTSZ06      -2.975e+00  1.737e-02  -171.216  < 2e-16 ***
ORIGIN_SZDTSZ07      -1.877e+00  1.905e-02   -98.511  < 2e-16 ***
ORIGIN_SZDTSZ08      -2.324e+00  9.698e-03  -239.632  < 2e-16 ***
ORIGIN_SZDTSZ09      -3.034e+00  2.054e-02  -147.746  < 2e-16 ***
ORIGIN_SZDTSZ10      -2.158e+00  1.036e-02  -208.287  < 2e-16 ***
ORIGIN_SZDTSZ11      -2.329e+00  1.070e-02  -217.712  < 2e-16 ***
ORIGIN_SZDTSZ12      -3.593e+00  2.551e-02  -140.861  < 2e-16 ***
ORIGIN_SZDTSZ13      -2.397e+00  1.212e-02  -197.773  < 2e-16 ***
ORIGIN_SZGLSZ01      -1.317e+00  7.123e-03  -184.848  < 2e-16 ***
ORIGIN_SZGLSZ02       2.116e-01  4.067e-03    52.013  < 2e-16 ***
ORIGIN_SZGLSZ03       2.050e-01  4.059e-03    50.498  < 2e-16 ***
ORIGIN_SZGLSZ04       9.827e-01  3.416e-03   287.645  < 2e-16 ***
ORIGIN_SZGLSZ05       6.442e-01  3.624e-03   177.781  < 2e-16 ***
ORIGIN_SZHGSZ01       2.595e-01  3.998e-03    64.896  < 2e-16 ***
ORIGIN_SZHGSZ02       6.162e-01  3.842e-03   160.374  < 2e-16 ***
ORIGIN_SZHGSZ03       2.897e-01  4.204e-03    68.905  < 2e-16 ***
ORIGIN_SZHGSZ04       9.444e-01  3.590e-03   263.071  < 2e-16 ***
ORIGIN_SZHGSZ05       1.265e+00  3.527e-03   358.688  < 2e-16 ***
ORIGIN_SZHGSZ06       1.062e-01  4.281e-03    24.815  < 2e-16 ***
ORIGIN_SZHGSZ07       7.740e-01  3.700e-03   209.178  < 2e-16 ***
ORIGIN_SZHGSZ08       2.462e-01  4.275e-03    57.586  < 2e-16 ***
ORIGIN_SZHGSZ09      -5.094e-01  5.808e-03   -87.708  < 2e-16 ***
ORIGIN_SZHGSZ10      -2.665e+00  3.983e-02   -66.910  < 2e-16 ***
ORIGIN_SZJESZ01       3.998e-01  4.103e-03    97.427  < 2e-16 ***
ORIGIN_SZJESZ02       2.755e-01  4.118e-03    66.890  < 2e-16 ***
ORIGIN_SZJESZ03       2.744e-01  4.341e-03    63.207  < 2e-16 ***
ORIGIN_SZJESZ04      -9.169e-01  7.072e-03  -129.657  < 2e-16 ***
ORIGIN_SZJESZ05      -1.967e+00  1.188e-02  -165.550  < 2e-16 ***
ORIGIN_SZJESZ06       3.359e-01  4.037e-03    83.202  < 2e-16 ***
ORIGIN_SZJESZ07      -1.738e+00  9.337e-03  -186.144  < 2e-16 ***
ORIGIN_SZJESZ08      -5.989e-01  8.299e-03   -72.169  < 2e-16 ***
ORIGIN_SZJESZ09       4.456e-01  4.208e-03   105.910  < 2e-16 ***
ORIGIN_SZJESZ10      -1.968e+00  1.632e-02  -120.552  < 2e-16 ***
ORIGIN_SZJESZ11      -2.124e+00  1.681e-02  -126.318  < 2e-16 ***
ORIGIN_SZJWSZ01       2.286e-01  5.369e-03    42.578  < 2e-16 ***
ORIGIN_SZJWSZ02       8.915e-01  3.789e-03   235.292  < 2e-16 ***
ORIGIN_SZJWSZ03       1.293e+00  3.515e-03   367.960  < 2e-16 ***
ORIGIN_SZJWSZ04       1.351e+00  3.578e-03   377.520  < 2e-16 ***
ORIGIN_SZJWSZ05      -1.267e+00  1.009e-02  -125.555  < 2e-16 ***
ORIGIN_SZJWSZ06      -1.005e+00  8.836e-03  -113.734  < 2e-16 ***
ORIGIN_SZJWSZ07      -2.587e+00  2.233e-02  -115.857  < 2e-16 ***
ORIGIN_SZJWSZ08       1.978e+00  3.429e-03   576.877  < 2e-16 ***
ORIGIN_SZJWSZ09       1.944e+00  3.247e-03   598.514  < 2e-16 ***
ORIGIN_SZKLSZ01       1.947e-01  3.918e-03    49.689  < 2e-16 ***
ORIGIN_SZKLSZ02      -4.677e-01  4.893e-03   -95.577  < 2e-16 ***
ORIGIN_SZKLSZ03      -4.055e-01  4.886e-03   -82.983  < 2e-16 ***
ORIGIN_SZKLSZ04      -1.639e+00  6.977e-03  -234.885  < 2e-16 ***
ORIGIN_SZKLSZ05      -8.975e-01  6.538e-03  -137.284  < 2e-16 ***
ORIGIN_SZKLSZ06      -5.593e-01  4.587e-03  -121.911  < 2e-16 ***
ORIGIN_SZKLSZ07      -9.375e-01  6.042e-03  -155.160  < 2e-16 ***
ORIGIN_SZKLSZ08      -8.039e-01  5.230e-03  -153.703  < 2e-16 ***
ORIGIN_SZKLSZ09      -1.545e+00  6.681e-03  -231.271  < 2e-16 ***
ORIGIN_SZLKSZ01      -2.964e+00  2.862e-02  -103.566  < 2e-16 ***
ORIGIN_SZMDSZ01      -2.428e+00  2.166e-02  -112.061  < 2e-16 ***
ORIGIN_SZMDSZ02      -1.119e+00  9.861e-03  -113.450  < 2e-16 ***
ORIGIN_SZMDSZ03      -1.830e+00  1.316e-02  -138.979  < 2e-16 ***
ORIGIN_SZMPSZ01      -9.639e-01  6.419e-03  -150.164  < 2e-16 ***
ORIGIN_SZMPSZ02      -4.675e-01  5.349e-03   -87.390  < 2e-16 ***
ORIGIN_SZMPSZ03       1.260e-01  4.248e-03    29.670  < 2e-16 ***
ORIGIN_SZMSSZ01      -7.329e+00  2.673e-01   -27.422  < 2e-16 ***
ORIGIN_SZMUSZ01      -1.240e+00  5.753e-03  -215.628  < 2e-16 ***
ORIGIN_SZMUSZ02      -3.100e+00  1.434e-02  -216.222  < 2e-16 ***
ORIGIN_SZMUSZ03      -1.786e+00  6.783e-03  -263.255  < 2e-16 ***
ORIGIN_SZNTSZ01      -2.286e+00  2.415e-02   -94.639  < 2e-16 ***
ORIGIN_SZNTSZ02      -2.401e+00  1.230e-02  -195.268  < 2e-16 ***
ORIGIN_SZNTSZ03      -7.543e-01  5.755e-03  -131.074  < 2e-16 ***
ORIGIN_SZNTSZ05      -3.114e+00  3.721e-02   -83.706  < 2e-16 ***
ORIGIN_SZNTSZ06      -3.415e+00  4.002e-02   -85.338  < 2e-16 ***
ORIGIN_SZNVSZ01       6.885e-01  3.551e-03   193.889  < 2e-16 ***
ORIGIN_SZNVSZ02      -4.131e-01  4.796e-03   -86.147  < 2e-16 ***
ORIGIN_SZNVSZ03      -1.054e+00  5.897e-03  -178.743  < 2e-16 ***
ORIGIN_SZNVSZ04      -1.291e+00  7.055e-03  -182.959  < 2e-16 ***
ORIGIN_SZNVSZ05      -2.522e+00  1.287e-02  -195.929  < 2e-16 ***
ORIGIN_SZORSZ01      -2.858e+00  2.619e-02  -109.108  < 2e-16 ***
ORIGIN_SZORSZ02      -1.160e+00  5.665e-03  -204.808  < 2e-16 ***
ORIGIN_SZORSZ03      -1.632e+00  6.782e-03  -240.706  < 2e-16 ***
ORIGIN_SZOTSZ01      -1.691e+00  7.205e-03  -234.655  < 2e-16 ***
ORIGIN_SZOTSZ02      -1.755e+00  8.076e-03  -217.286  < 2e-16 ***
ORIGIN_SZOTSZ03      -7.979e-01  5.279e-03  -151.147  < 2e-16 ***
ORIGIN_SZOTSZ04      -7.263e-01  8.350e-03   -86.984  < 2e-16 ***
ORIGIN_SZPGSZ01      -6.693e-01  9.124e-03   -73.359  < 2e-16 ***
ORIGIN_SZPGSZ02      -3.145e-01  5.659e-03   -55.571  < 2e-16 ***
ORIGIN_SZPGSZ03       1.146e+00  3.625e-03   315.999  < 2e-16 ***
ORIGIN_SZPGSZ04       1.228e+00  3.627e-03   338.445  < 2e-16 ***
ORIGIN_SZPGSZ05       4.787e-01  4.510e-03   106.145  < 2e-16 ***
ORIGIN_SZPLSZ01      -4.797e-01  7.758e-03   -61.839  < 2e-16 ***
ORIGIN_SZPLSZ02      -1.394e+00  1.121e-02  -124.392  < 2e-16 ***
ORIGIN_SZPLSZ03      -2.963e+00  3.138e-02   -94.406  < 2e-16 ***
ORIGIN_SZPLSZ04      -3.465e+00  3.524e-02   -98.323  < 2e-16 ***
ORIGIN_SZPLSZ05      -2.266e+00  1.760e-02  -128.773  < 2e-16 ***
ORIGIN_SZPNSZ01       1.532e+00  3.821e-03   400.912  < 2e-16 ***
ORIGIN_SZPNSZ02      -5.587e-01  9.699e-03   -57.611  < 2e-16 ***
ORIGIN_SZPNSZ03      -1.975e+00  1.746e-02  -113.152  < 2e-16 ***
ORIGIN_SZPNSZ04      -2.601e+00  2.552e-02  -101.923  < 2e-16 ***
ORIGIN_SZPNSZ05      -1.741e+00  1.729e-02  -100.694  < 2e-16 ***
ORIGIN_SZPRSZ01      -6.841e-01  9.325e-03   -73.364  < 2e-16 ***
ORIGIN_SZPRSZ02       1.110e+00  3.790e-03   292.835  < 2e-16 ***
ORIGIN_SZPRSZ03       8.811e-01  3.786e-03   232.709  < 2e-16 ***
ORIGIN_SZPRSZ04      -2.439e-01  6.099e-03   -39.986  < 2e-16 ***
ORIGIN_SZPRSZ05       1.308e+00  3.625e-03   360.972  < 2e-16 ***
ORIGIN_SZPRSZ06      -4.679e-01  6.803e-03   -68.781  < 2e-16 ***
ORIGIN_SZPRSZ07      -2.530e+00  1.680e-02  -150.629  < 2e-16 ***
ORIGIN_SZPRSZ08       1.336e-01  5.012e-03    26.655  < 2e-16 ***
ORIGIN_SZQTSZ01      -3.690e-01  5.363e-03   -68.812  < 2e-16 ***
ORIGIN_SZQTSZ02      -6.760e-01  5.029e-03  -134.429  < 2e-16 ***
ORIGIN_SZQTSZ03      -2.233e-01  4.569e-03   -48.869  < 2e-16 ***
ORIGIN_SZQTSZ04      -1.136e+00  6.204e-03  -183.037  < 2e-16 ***
ORIGIN_SZQTSZ05      -1.992e-01  4.562e-03   -43.664  < 2e-16 ***
ORIGIN_SZQTSZ06      -5.709e-01  5.252e-03  -108.705  < 2e-16 ***
ORIGIN_SZQTSZ07      -1.532e+00  7.795e-03  -196.606  < 2e-16 ***
ORIGIN_SZQTSZ08      -2.386e-01  4.733e-03   -50.403  < 2e-16 ***
ORIGIN_SZQTSZ09      -5.253e-01  5.308e-03   -98.976  < 2e-16 ***
ORIGIN_SZQTSZ10      -3.837e-01  5.196e-03   -73.846  < 2e-16 ***
ORIGIN_SZQTSZ11      -1.313e+00  7.588e-03  -173.079  < 2e-16 ***
ORIGIN_SZQTSZ12      -6.748e-01  6.337e-03  -106.488  < 2e-16 ***
ORIGIN_SZQTSZ13      -8.076e-02  4.804e-03   -16.811  < 2e-16 ***
ORIGIN_SZQTSZ14      -1.335e+00  7.113e-03  -187.705  < 2e-16 ***
ORIGIN_SZQTSZ15      -7.194e-01  8.216e-03   -87.558  < 2e-16 ***
ORIGIN_SZRCSZ01      -6.379e-01  5.283e-03  -120.740  < 2e-16 ***
ORIGIN_SZRCSZ02      -2.358e+00  1.482e-02  -159.094  < 2e-16 ***
ORIGIN_SZRCSZ03      -1.470e+00  7.322e-03  -200.791  < 2e-16 ***
ORIGIN_SZRCSZ04      -2.219e+00  1.109e-02  -200.111  < 2e-16 ***
ORIGIN_SZRCSZ05      -2.770e+00  1.341e-02  -206.629  < 2e-16 ***
ORIGIN_SZRCSZ06      -4.466e-01  6.935e-03   -64.404  < 2e-16 ***
ORIGIN_SZRCSZ08      -2.558e+00  1.610e-02  -158.880  < 2e-16 ***
ORIGIN_SZRCSZ09      -1.978e+00  1.217e-02  -162.512  < 2e-16 ***
ORIGIN_SZRCSZ10      -1.811e+00  6.993e-03  -259.015  < 2e-16 ***
ORIGIN_SZRVSZ01      -2.814e+00  1.312e-02  -214.462  < 2e-16 ***
ORIGIN_SZRVSZ02      -1.124e+00  6.715e-03  -167.428  < 2e-16 ***
ORIGIN_SZRVSZ03      -1.888e+00  9.780e-03  -193.068  < 2e-16 ***
ORIGIN_SZRVSZ04      -1.973e+00  1.378e-02  -143.142  < 2e-16 ***
ORIGIN_SZRVSZ05      -2.176e+00  1.213e-02  -179.378  < 2e-16 ***
ORIGIN_SZSBSZ01       7.906e-01  4.386e-03   180.258  < 2e-16 ***
ORIGIN_SZSBSZ02      -5.268e-01  6.431e-03   -81.909  < 2e-16 ***
ORIGIN_SZSBSZ03       1.010e+00  3.843e-03   262.897  < 2e-16 ***
ORIGIN_SZSBSZ04       7.772e-01  4.361e-03   178.202  < 2e-16 ***
ORIGIN_SZSBSZ05      -7.765e-02  5.496e-03   -14.128  < 2e-16 ***
ORIGIN_SZSBSZ06      -1.650e+00  1.359e-02  -121.412  < 2e-16 ***
ORIGIN_SZSBSZ07      -7.893e-01  9.127e-03   -86.478  < 2e-16 ***
ORIGIN_SZSBSZ08      -9.503e-01  9.362e-03  -101.512  < 2e-16 ***
ORIGIN_SZSBSZ09      -4.758e-01  7.016e-03   -67.806  < 2e-16 ***
ORIGIN_SZSESZ02       1.165e+00  3.595e-03   323.972  < 2e-16 ***
ORIGIN_SZSESZ03       1.289e+00  3.463e-03   372.373  < 2e-16 ***
ORIGIN_SZSESZ04       1.019e+00  3.931e-03   259.250  < 2e-16 ***
ORIGIN_SZSESZ05      -1.106e-01  4.827e-03   -22.907  < 2e-16 ***
ORIGIN_SZSESZ06       9.972e-01  3.761e-03   265.133  < 2e-16 ***
ORIGIN_SZSESZ07      -2.195e+00  1.404e-02  -156.357  < 2e-16 ***
ORIGIN_SZSGSZ01      -8.301e-01  6.827e-03  -121.592  < 2e-16 ***
ORIGIN_SZSGSZ02      -1.135e+00  8.180e-03  -138.755  < 2e-16 ***
ORIGIN_SZSGSZ03       3.108e-01  4.304e-03    72.220  < 2e-16 ***
ORIGIN_SZSGSZ04       3.315e-01  3.954e-03    83.844  < 2e-16 ***
ORIGIN_SZSGSZ05      -1.574e+00  8.178e-03  -192.514  < 2e-16 ***
ORIGIN_SZSGSZ06       4.707e-01  3.752e-03   125.452  < 2e-16 ***
ORIGIN_SZSGSZ07      -5.168e-01  4.973e-03  -103.909  < 2e-16 ***
ORIGIN_SZSKSZ01      -1.388e-01  6.497e-03   -21.359  < 2e-16 ***
ORIGIN_SZSKSZ02       3.980e-01  4.751e-03    83.770  < 2e-16 ***
ORIGIN_SZSKSZ03      -4.619e-01  6.153e-03   -75.064  < 2e-16 ***
ORIGIN_SZSKSZ04      -2.355e+00  2.132e-02  -110.454  < 2e-16 ***
ORIGIN_SZSKSZ05      -1.229e+00  1.245e-02   -98.702  < 2e-16 ***
ORIGIN_SZSLSZ01      -3.172e+00  2.571e-02  -123.358  < 2e-16 ***
ORIGIN_SZSLSZ04      -3.140e-01  5.780e-03   -54.333  < 2e-16 ***
ORIGIN_SZSRSZ01      -1.508e+00  7.084e-03  -212.819  < 2e-16 ***
ORIGIN_SZSRSZ02      -1.783e+00  7.222e-03  -246.939  < 2e-16 ***
ORIGIN_SZSRSZ03      -2.801e+00  1.514e-02  -184.996  < 2e-16 ***
ORIGIN_SZSVSZ01      -2.798e+00  2.815e-02   -99.409  < 2e-16 ***
ORIGIN_SZTHSZ01      -2.986e+00  4.617e-02   -64.671  < 2e-16 ***
ORIGIN_SZTHSZ03      -1.556e+00  1.269e-02  -122.621  < 2e-16 ***
ORIGIN_SZTHSZ04      -3.011e+00  2.435e-02  -123.655  < 2e-16 ***
ORIGIN_SZTHSZ06      -1.631e+00  1.084e-02  -150.482  < 2e-16 ***
ORIGIN_SZTMSZ01       1.080e+00  4.087e-03   264.249  < 2e-16 ***
ORIGIN_SZTMSZ02       2.226e+00  3.176e-03   700.906  < 2e-16 ***
ORIGIN_SZTMSZ03       1.522e+00  3.389e-03   449.101  < 2e-16 ***
ORIGIN_SZTMSZ04       9.516e-01  3.881e-03   245.213  < 2e-16 ***
ORIGIN_SZTMSZ05      -2.458e-01  6.138e-03   -40.038  < 2e-16 ***
ORIGIN_SZTNSZ01      -1.047e+00  5.897e-03  -177.587  < 2e-16 ***
ORIGIN_SZTNSZ02      -1.015e+00  5.720e-03  -177.473  < 2e-16 ***
ORIGIN_SZTNSZ03      -1.518e+00  7.506e-03  -202.275  < 2e-16 ***
ORIGIN_SZTNSZ04      -6.469e-01  5.491e-03  -117.823  < 2e-16 ***
ORIGIN_SZTPSZ01      -6.208e-01  5.132e-03  -120.979  < 2e-16 ***
ORIGIN_SZTPSZ02       4.640e-01  3.614e-03   128.411  < 2e-16 ***
ORIGIN_SZTPSZ03      -4.501e-01  5.102e-03   -88.218  < 2e-16 ***
ORIGIN_SZTPSZ04      -2.797e-01  4.821e-03   -58.026  < 2e-16 ***
ORIGIN_SZTPSZ05      -2.178e-01  4.987e-03   -43.675  < 2e-16 ***
ORIGIN_SZTPSZ06       2.260e-01  5.019e-03    45.026  < 2e-16 ***
ORIGIN_SZTPSZ07      -1.387e-01  5.015e-03   -27.651  < 2e-16 ***
ORIGIN_SZTPSZ08      -6.925e-01  6.545e-03  -105.811  < 2e-16 ***
ORIGIN_SZTPSZ09      -4.794e-01  5.358e-03   -89.465  < 2e-16 ***
ORIGIN_SZTPSZ10      -3.714e-01  5.362e-03   -69.266  < 2e-16 ***
ORIGIN_SZTPSZ11       2.625e-01  4.218e-03    62.229  < 2e-16 ***
ORIGIN_SZTPSZ12      -4.955e-01  5.288e-03   -93.700  < 2e-16 ***
ORIGIN_SZTSSZ01      -3.428e+00  3.961e-02   -86.524  < 2e-16 ***
ORIGIN_SZTSSZ02       5.153e-01  5.790e-03    89.000  < 2e-16 ***
ORIGIN_SZTSSZ03       5.407e-01  5.827e-03    92.803  < 2e-16 ***
ORIGIN_SZTSSZ04       5.158e-01  6.000e-03    85.964  < 2e-16 ***
ORIGIN_SZTSSZ05      -9.510e-01  1.122e-02   -84.796  < 2e-16 ***
ORIGIN_SZTSSZ06      -1.155e+00  1.360e-02   -84.933  < 2e-16 ***
ORIGIN_SZWCSZ01       3.529e-01  5.612e-03    62.877  < 2e-16 ***
ORIGIN_SZWCSZ02      -2.680e+00  2.497e-02  -107.302  < 2e-16 ***
ORIGIN_SZWCSZ03      -4.218e+00  1.325e-01   -31.839  < 2e-16 ***
ORIGIN_SZWDSZ01       1.430e+00  3.467e-03   412.347  < 2e-16 ***
ORIGIN_SZWDSZ02       1.111e+00  3.926e-03   283.079  < 2e-16 ***
ORIGIN_SZWDSZ03       2.250e+00  3.358e-03   669.831  < 2e-16 ***
ORIGIN_SZWDSZ04       1.192e+00  4.101e-03   290.750  < 2e-16 ***
ORIGIN_SZWDSZ05       6.377e-01  4.136e-03   154.160  < 2e-16 ***
ORIGIN_SZWDSZ06       1.295e+00  3.808e-03   340.102  < 2e-16 ***
ORIGIN_SZWDSZ07       2.689e-01  5.407e-03    49.740  < 2e-16 ***
ORIGIN_SZWDSZ08      -1.872e-01  6.132e-03   -30.533  < 2e-16 ***
ORIGIN_SZWDSZ09       1.826e+00  3.498e-03   521.848  < 2e-16 ***
ORIGIN_SZYSSZ01      -5.305e-02  4.658e-03   -11.389  < 2e-16 ***
ORIGIN_SZYSSZ02       1.003e+00  4.134e-03   242.541  < 2e-16 ***
ORIGIN_SZYSSZ03       1.997e+00  3.458e-03   577.479  < 2e-16 ***
ORIGIN_SZYSSZ04       9.186e-01  3.725e-03   246.606  < 2e-16 ***
ORIGIN_SZYSSZ05       1.984e-01  4.595e-03    43.178  < 2e-16 ***
ORIGIN_SZYSSZ06      -8.206e-01  7.460e-03  -109.992  < 2e-16 ***
ORIGIN_SZYSSZ07      -5.298e-01  7.220e-03   -73.381  < 2e-16 ***
ORIGIN_SZYSSZ08      -1.277e-02  5.142e-03    -2.483    0.013 *
ORIGIN_SZYSSZ09       1.405e+00  3.560e-03   394.739  < 2e-16 ***
log(DESTIN_AGE25_64)  2.458e-02  6.646e-05   369.857  < 2e-16 ***
log(dist)            -7.137e-01  1.016e-04 -7023.324  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 105499054  on 20915  degrees of freedom
Residual deviance:  45982898  on 20605  degrees of freedom
AIC: 46115907

Number of Fisher Scoring iterations: 7

Similarly, we can calculate R-squared for the origin-constrained model:

CalcRSquared(orcSIM$data$TRIPS, orcSIM$fitted.values)

[1] 0.3847416

8.7 Destination Constrained

In this section, We will fit a destination-constrained Spatial Interaction Model (SIM).

Note

The general formula of Destination Constrained Spatial Interaction Model

\(\lambda_{ij} = \exp \left( k + \mu \ln V_i + \alpha_i - \beta \ln d_{ij} \right)\)

Notice that the difference between Unconstrained Spatial Interaction Model and this formula lies in the third term. In the Unconstrained Spatial Interaction Model formula, it is \(\alpha \ln W_j\) as compared to \(\alpha_i\) in Destination Constrained Spatial Interaction Model.

decSIM <- glm(formula = TRIPS ~
                DESTIN_SZ +
                log(ORIGIN_AGE25_64) +
                log(dist),
              family = poisson(link = "log"),
              data = SIM_data,
              na.action = na.exclude)
summary(decSIM)


Call:
glm(formula = TRIPS ~ DESTIN_SZ + log(ORIGIN_AGE25_64) + log(dist),
    family = poisson(link = "log"), data = SIM_data, na.action = na.exclude)

Coefficients:
                       Estimate Std. Error   z value Pr(>|z|)
(Intercept)          11.1886479  0.0027928  4006.284  < 2e-16 ***
DESTIN_SZAMSZ02       0.2047468  0.0035739    57.290  < 2e-16 ***
DESTIN_SZAMSZ03       0.2696890  0.0034700    77.720  < 2e-16 ***
DESTIN_SZAMSZ04      -0.9029147  0.0051608  -174.956  < 2e-16 ***
DESTIN_SZAMSZ05      -0.9208381  0.0048845  -188.522  < 2e-16 ***
DESTIN_SZAMSZ06      -0.7523009  0.0047388  -158.753  < 2e-16 ***
DESTIN_SZAMSZ07      -1.8644282  0.0085251  -218.699  < 2e-16 ***
DESTIN_SZAMSZ08      -0.9320542  0.0053993  -172.625  < 2e-16 ***
DESTIN_SZAMSZ09      -0.9606544  0.0051384  -186.955  < 2e-16 ***
DESTIN_SZAMSZ10       0.4367176  0.0035636   122.549  < 2e-16 ***
DESTIN_SZAMSZ11       0.1387581  0.0060108    23.085  < 2e-16 ***
DESTIN_SZAMSZ12      -0.0968912  0.0044113   -21.965  < 2e-16 ***
DESTIN_SZBDSZ01       0.5325773  0.0032062   166.106  < 2e-16 ***
DESTIN_SZBDSZ02      -0.1583330  0.0040652   -38.949  < 2e-16 ***
DESTIN_SZBDSZ03       0.0326536  0.0036002     9.070  < 2e-16 ***
DESTIN_SZBDSZ04       1.0265974  0.0029357   349.691  < 2e-16 ***
DESTIN_SZBDSZ05       0.4370921  0.0032622   133.986  < 2e-16 ***
DESTIN_SZBDSZ06       0.1411228  0.0036420    38.749  < 2e-16 ***
DESTIN_SZBDSZ07      -0.8768697  0.0072058  -121.689  < 2e-16 ***
DESTIN_SZBDSZ08      -1.4154555  0.0073628  -192.244  < 2e-16 ***
DESTIN_SZBKSZ01      -1.0792597  0.0053008  -203.605  < 2e-16 ***
DESTIN_SZBKSZ02      -0.1605882  0.0043097   -37.262  < 2e-16 ***
DESTIN_SZBKSZ03      -0.4113294  0.0044445   -92.548  < 2e-16 ***
DESTIN_SZBKSZ04       0.1040345  0.0039514    26.328  < 2e-16 ***
DESTIN_SZBKSZ05      -0.6361868  0.0045293  -140.461  < 2e-16 ***
DESTIN_SZBKSZ06      -0.8459238  0.0050627  -167.089  < 2e-16 ***
DESTIN_SZBKSZ07       0.2936470  0.0033772    86.949  < 2e-16 ***
DESTIN_SZBKSZ08      -1.0336173  0.0057054  -181.166  < 2e-16 ***
DESTIN_SZBKSZ09      -0.1990406  0.0040672   -48.938  < 2e-16 ***
DESTIN_SZBLSZ01      -0.3332097  0.0058008   -57.442  < 2e-16 ***
DESTIN_SZBLSZ02       0.6867036  0.0055489   123.755  < 2e-16 ***
DESTIN_SZBLSZ03       1.5117931  0.0059625   253.551  < 2e-16 ***
DESTIN_SZBLSZ04       0.0399457  0.0108522     3.681 0.000232 ***
DESTIN_SZBMSZ01      -0.1382645  0.0037137   -37.231  < 2e-16 ***
DESTIN_SZBMSZ02      -0.4926028  0.0039199  -125.667  < 2e-16 ***
DESTIN_SZBMSZ03      -0.9056345  0.0047824  -189.366  < 2e-16 ***
DESTIN_SZBMSZ04      -0.5703442  0.0040984  -139.162  < 2e-16 ***
DESTIN_SZBMSZ05      -0.5916186  0.0049394  -119.775  < 2e-16 ***
DESTIN_SZBMSZ06      -1.6992029  0.0083045  -204.612  < 2e-16 ***
DESTIN_SZBMSZ07      -0.1189131  0.0036137   -32.906  < 2e-16 ***
DESTIN_SZBMSZ08      -1.0778761  0.0047721  -225.870  < 2e-16 ***
DESTIN_SZBMSZ09      -2.0275228  0.0075864  -267.257  < 2e-16 ***
DESTIN_SZBMSZ10      -1.5564968  0.0059936  -259.692  < 2e-16 ***
DESTIN_SZBMSZ11      -1.5778427  0.0059658  -264.483  < 2e-16 ***
DESTIN_SZBMSZ12      -1.0550880  0.0061596  -171.292  < 2e-16 ***
DESTIN_SZBMSZ13      -0.2495049  0.0038737   -64.410  < 2e-16 ***
DESTIN_SZBMSZ14      -1.0137466  0.0060702  -167.005  < 2e-16 ***
DESTIN_SZBMSZ15      -1.2476855  0.0056991  -218.927  < 2e-16 ***
DESTIN_SZBMSZ16      -1.6211588  0.0061126  -265.216  < 2e-16 ***
DESTIN_SZBMSZ17      -1.5085672  0.0071513  -210.951  < 2e-16 ***
DESTIN_SZBPSZ01      -0.4821870  0.0044490  -108.381  < 2e-16 ***
DESTIN_SZBPSZ02      -1.4688886  0.0071354  -205.860  < 2e-16 ***
DESTIN_SZBPSZ03      -1.1618519  0.0067027  -173.342  < 2e-16 ***
DESTIN_SZBPSZ04      -0.6445302  0.0049336  -130.641  < 2e-16 ***
DESTIN_SZBPSZ05       0.5220193  0.0032426   160.988  < 2e-16 ***
DESTIN_SZBPSZ06      -0.7705496  0.0060773  -126.790  < 2e-16 ***
DESTIN_SZBPSZ07      -0.5975374  0.0062143   -96.156  < 2e-16 ***
DESTIN_SZBSSZ01      -0.1159850  0.0036849   -31.476  < 2e-16 ***
DESTIN_SZBSSZ02      -0.7414329  0.0043019  -172.349  < 2e-16 ***
DESTIN_SZBSSZ03       0.3124431  0.0031917    97.893  < 2e-16 ***
DESTIN_SZBTSZ01       0.1570827  0.0034455    45.591  < 2e-16 ***
DESTIN_SZBTSZ02      -0.7950768  0.0055915  -142.194  < 2e-16 ***
DESTIN_SZBTSZ03      -0.1947127  0.0040495   -48.083  < 2e-16 ***
DESTIN_SZBTSZ04      -1.6039884  0.0079447  -201.895  < 2e-16 ***
DESTIN_SZBTSZ05      -0.7366517  0.0056153  -131.188  < 2e-16 ***
DESTIN_SZBTSZ06      -0.8638970  0.0050164  -172.216  < 2e-16 ***
DESTIN_SZBTSZ07      -1.9314011  0.0079429  -243.162  < 2e-16 ***
DESTIN_SZBTSZ08      -1.2455761  0.0066997  -185.914  < 2e-16 ***
DESTIN_SZCCSZ01      -0.4157153  0.0052731   -78.838  < 2e-16 ***
DESTIN_SZCHSZ01      -1.0527185  0.0073777  -142.689  < 2e-16 ***
DESTIN_SZCHSZ02      -0.0306697  0.0045975    -6.671 2.54e-11 ***
DESTIN_SZCHSZ03       1.6952057  0.0032057   528.814  < 2e-16 ***
DESTIN_SZCKSZ01      -0.1484317  0.0040290   -36.840  < 2e-16 ***
DESTIN_SZCKSZ02      -0.4018663  0.0044007   -91.319  < 2e-16 ***
DESTIN_SZCKSZ03       0.7189825  0.0032425   221.737  < 2e-16 ***
DESTIN_SZCKSZ04      -0.6740941  0.0051090  -131.943  < 2e-16 ***
DESTIN_SZCKSZ05      -0.3100378  0.0053366   -58.096  < 2e-16 ***
DESTIN_SZCKSZ06       0.7865808  0.0037571   209.361  < 2e-16 ***
DESTIN_SZCLSZ01       0.4766029  0.0038956   122.345  < 2e-16 ***
DESTIN_SZCLSZ02      -2.2614707  0.0109131  -207.225  < 2e-16 ***
DESTIN_SZCLSZ03      -0.9662790  0.0060017  -161.000  < 2e-16 ***
DESTIN_SZCLSZ04       0.1515855  0.0035858    42.274  < 2e-16 ***
DESTIN_SZCLSZ05      -1.2509920  0.0070287  -177.983  < 2e-16 ***
DESTIN_SZCLSZ06       0.2066348  0.0033754    61.218  < 2e-16 ***
DESTIN_SZCLSZ07      -0.5707110  0.0043519  -131.139  < 2e-16 ***
DESTIN_SZCLSZ08      -0.4633052  0.0048981   -94.589  < 2e-16 ***
DESTIN_SZCLSZ09       0.3993821  0.0054058    73.880  < 2e-16 ***
DESTIN_SZDTSZ01      -0.8694081  0.0044649  -194.722  < 2e-16 ***
DESTIN_SZDTSZ02      -0.8521954  0.0043451  -196.127  < 2e-16 ***
DESTIN_SZDTSZ03      -1.0743465  0.0051507  -208.582  < 2e-16 ***
DESTIN_SZDTSZ04      -1.1505259  0.0110647  -103.981  < 2e-16 ***
DESTIN_SZDTSZ05      -1.0093612  0.0085677  -117.810  < 2e-16 ***
DESTIN_SZDTSZ06      -1.1371739  0.0058133  -195.615  < 2e-16 ***
DESTIN_SZDTSZ07      -1.9214590  0.0180651  -106.363  < 2e-16 ***
DESTIN_SZDTSZ08      -0.6971968  0.0042214  -165.156  < 2e-16 ***
DESTIN_SZDTSZ09      -1.5178484  0.0094552  -160.530  < 2e-16 ***
DESTIN_SZDTSZ10      -1.3018755  0.0075368  -172.737  < 2e-16 ***
DESTIN_SZDTSZ11      -0.8241052  0.0044947  -183.351  < 2e-16 ***
DESTIN_SZDTSZ12      -2.4402256  0.0146441  -166.636  < 2e-16 ***
DESTIN_SZDTSZ13      -1.9939705  0.0092374  -215.859  < 2e-16 ***
DESTIN_SZGLSZ01       0.0262850  0.0041498     6.334 2.39e-10 ***
DESTIN_SZGLSZ02      -0.3074574  0.0038496   -79.867  < 2e-16 ***
DESTIN_SZGLSZ03       0.3982838  0.0032437   122.787  < 2e-16 ***
DESTIN_SZGLSZ04       0.3933413  0.0031685   124.141  < 2e-16 ***
DESTIN_SZGLSZ05       0.1442632  0.0033403    43.189  < 2e-16 ***
DESTIN_SZHGSZ01       0.4273977  0.0032226   132.624  < 2e-16 ***
DESTIN_SZHGSZ02      -0.5494812  0.0044023  -124.816  < 2e-16 ***
DESTIN_SZHGSZ03      -1.0856653  0.0052758  -205.781  < 2e-16 ***
DESTIN_SZHGSZ04      -0.2538606  0.0037451   -67.784  < 2e-16 ***
DESTIN_SZHGSZ05      -0.1395323  0.0037621   -37.089  < 2e-16 ***
DESTIN_SZHGSZ06      -0.6632432  0.0044051  -150.564  < 2e-16 ***
DESTIN_SZHGSZ07       0.2923291  0.0033875    86.296  < 2e-16 ***
DESTIN_SZHGSZ08      -0.2730755  0.0039832   -68.557  < 2e-16 ***
DESTIN_SZHGSZ09       0.2735423  0.0041633    65.704  < 2e-16 ***
DESTIN_SZHGSZ10      -3.1736449  0.0284659  -111.489  < 2e-16 ***
DESTIN_SZJESZ01      -0.1053234  0.0041515   -25.370  < 2e-16 ***
DESTIN_SZJESZ02      -0.4383285  0.0042949  -102.057  < 2e-16 ***
DESTIN_SZJESZ03      -0.5711346  0.0046960  -121.621  < 2e-16 ***
DESTIN_SZJESZ04      -0.0238346  0.0048249    -4.940 7.81e-07 ***
DESTIN_SZJESZ05      -0.7798774  0.0074938  -104.070  < 2e-16 ***
DESTIN_SZJESZ06       0.3824640  0.0034475   110.939  < 2e-16 ***
DESTIN_SZJESZ07      -0.8847081  0.0061908  -142.907  < 2e-16 ***
DESTIN_SZJESZ08      -0.5789622  0.0064625   -89.589  < 2e-16 ***
DESTIN_SZJESZ09      -0.4456534  0.0046887   -95.049  < 2e-16 ***
DESTIN_SZJESZ10       0.7168959  0.0061692   116.205  < 2e-16 ***
DESTIN_SZJESZ11       1.0389409  0.0054888   189.285  < 2e-16 ***
DESTIN_SZJWSZ01      -0.3405028  0.0052968   -64.284  < 2e-16 ***
DESTIN_SZJWSZ02      -0.3557479  0.0045025   -79.011  < 2e-16 ***
DESTIN_SZJWSZ03       0.6822220  0.0033463   203.873  < 2e-16 ***
DESTIN_SZJWSZ04       1.0313249  0.0031431   328.118  < 2e-16 ***
DESTIN_SZJWSZ05      -0.1390898  0.0050165   -27.726  < 2e-16 ***
DESTIN_SZJWSZ06       0.4283248  0.0045931    93.255  < 2e-16 ***
DESTIN_SZJWSZ07      -0.8125529  0.0196066   -41.443  < 2e-16 ***
DESTIN_SZJWSZ08       0.4721844  0.0038233   123.502  < 2e-16 ***
DESTIN_SZJWSZ09       1.4834158  0.0028734   516.263  < 2e-16 ***
DESTIN_SZKLSZ01      -0.5939223  0.0040738  -145.792  < 2e-16 ***
DESTIN_SZKLSZ02      -0.7600734  0.0046596  -163.119  < 2e-16 ***
DESTIN_SZKLSZ03      -1.2543810  0.0051402  -244.031  < 2e-16 ***
DESTIN_SZKLSZ04      -1.7311984  0.0066049  -262.109  < 2e-16 ***
DESTIN_SZKLSZ05      -1.1216909  0.0066398  -168.935  < 2e-16 ***
DESTIN_SZKLSZ06      -0.9437123  0.0044840  -210.462  < 2e-16 ***
DESTIN_SZKLSZ07      -1.0592925  0.0050379  -210.265  < 2e-16 ***
DESTIN_SZKLSZ08      -0.1414796  0.0036529   -38.731  < 2e-16 ***
DESTIN_SZKLSZ09      -1.8622768  0.0066005  -282.142  < 2e-16 ***
DESTIN_SZLKSZ01      -1.6806379  0.0192395   -87.353  < 2e-16 ***
DESTIN_SZMDSZ01      -1.2927817  0.0158848   -81.385  < 2e-16 ***
DESTIN_SZMDSZ02      -1.1068976  0.0093181  -118.790  < 2e-16 ***
DESTIN_SZMDSZ03      -2.8169619  0.0208242  -135.274  < 2e-16 ***
DESTIN_SZMPSZ01      -1.2490736  0.0064894  -192.478  < 2e-16 ***
DESTIN_SZMPSZ02      -0.8411939  0.0048205  -174.505  < 2e-16 ***
DESTIN_SZMPSZ03      -0.1963932  0.0039391   -49.858  < 2e-16 ***
DESTIN_SZMSSZ01      -3.7028759  0.0680912   -54.381  < 2e-16 ***
DESTIN_SZMUSZ01      -1.1416953  0.0048741  -234.235  < 2e-16 ***
DESTIN_SZMUSZ02      -1.4043160  0.0069637  -201.663  < 2e-16 ***
DESTIN_SZMUSZ03      -1.1249185  0.0047737  -235.647  < 2e-16 ***
DESTIN_SZNTSZ01      -2.7232365  0.0219075  -124.306  < 2e-16 ***
DESTIN_SZNTSZ02      -2.0281258  0.0087672  -231.332  < 2e-16 ***
DESTIN_SZNTSZ03      -1.1948616  0.0060363  -197.945  < 2e-16 ***
DESTIN_SZNTSZ05      -1.7337767  0.0161951  -107.056  < 2e-16 ***
DESTIN_SZNTSZ06      -3.0002577  0.0278723  -107.643  < 2e-16 ***
DESTIN_SZNVSZ01      -0.2596984  0.0035898   -72.343  < 2e-16 ***
DESTIN_SZNVSZ02      -0.4670565  0.0040847  -114.343  < 2e-16 ***
DESTIN_SZNVSZ03      -0.5798852  0.0043601  -132.998  < 2e-16 ***
DESTIN_SZNVSZ04      -2.0652499  0.0084658  -243.951  < 2e-16 ***
DESTIN_SZNVSZ05      -1.8113949  0.0072668  -249.271  < 2e-16 ***
DESTIN_SZORSZ01      -2.0102988  0.0174760  -115.032  < 2e-16 ***
DESTIN_SZORSZ02       0.0988466  0.0034722    28.468  < 2e-16 ***
DESTIN_SZORSZ03      -0.8997969  0.0047365  -189.970  < 2e-16 ***
DESTIN_SZOTSZ01      -1.5647551  0.0060744  -257.599  < 2e-16 ***
DESTIN_SZOTSZ02      -0.8001690  0.0053202  -150.401  < 2e-16 ***
DESTIN_SZOTSZ03      -1.4925670  0.0057452  -259.793  < 2e-16 ***
DESTIN_SZOTSZ04      -1.5602417  0.0082363  -189.434  < 2e-16 ***
DESTIN_SZPGSZ01      -2.1137374  0.0140083  -150.892  < 2e-16 ***
DESTIN_SZPGSZ02      -0.7896523  0.0053545  -147.475  < 2e-16 ***
DESTIN_SZPGSZ03       0.5822347  0.0032692   178.098  < 2e-16 ***
DESTIN_SZPGSZ04       0.1448694  0.0036782    39.386  < 2e-16 ***
DESTIN_SZPGSZ05      -0.8881741  0.0060384  -147.087  < 2e-16 ***
DESTIN_SZPLSZ01      -0.0795041  0.0058033   -13.700  < 2e-16 ***
DESTIN_SZPLSZ02      -1.1900069  0.0102914  -115.631  < 2e-16 ***
DESTIN_SZPLSZ03      -0.1247849  0.0084378   -14.789  < 2e-16 ***
DESTIN_SZPLSZ04      -0.2719495  0.0081384   -33.416  < 2e-16 ***
DESTIN_SZPLSZ05      -0.6771576  0.0097439   -69.496  < 2e-16 ***
DESTIN_SZPNSZ01       1.1397906  0.0042414   268.733  < 2e-16 ***
DESTIN_SZPNSZ02       1.6933075  0.0055624   304.420  < 2e-16 ***
DESTIN_SZPNSZ03       0.9586915  0.0062893   152.432  < 2e-16 ***
DESTIN_SZPNSZ04       1.7374816  0.0063491   273.659  < 2e-16 ***
DESTIN_SZPNSZ05       0.8918270  0.0092596    96.314  < 2e-16 ***
DESTIN_SZPRSZ01      -0.6266318  0.0060975  -102.768  < 2e-16 ***
DESTIN_SZPRSZ02      -0.0691388  0.0040915   -16.898  < 2e-16 ***
DESTIN_SZPRSZ03       0.7957648  0.0031463   252.920  < 2e-16 ***
DESTIN_SZPRSZ04      -0.7720323  0.0069948  -110.373  < 2e-16 ***
DESTIN_SZPRSZ05      -0.0147428  0.0038961    -3.784 0.000154 ***
DESTIN_SZPRSZ06       0.4832118  0.0041522   116.374  < 2e-16 ***
DESTIN_SZPRSZ07      -1.5779831  0.0103602  -152.313  < 2e-16 ***
DESTIN_SZPRSZ08      -0.8342815  0.0056390  -147.948  < 2e-16 ***
DESTIN_SZQTSZ01      -1.5074560  0.0081347  -185.311  < 2e-16 ***
DESTIN_SZQTSZ02      -1.4974563  0.0059956  -249.759  < 2e-16 ***
DESTIN_SZQTSZ03      -0.8830042  0.0053218  -165.922  < 2e-16 ***
DESTIN_SZQTSZ04      -1.1472088  0.0055287  -207.500  < 2e-16 ***
DESTIN_SZQTSZ05      -0.9139073  0.0048233  -189.476  < 2e-16 ***
DESTIN_SZQTSZ06      -1.1834757  0.0052284  -226.353  < 2e-16 ***
DESTIN_SZQTSZ07      -1.6922635  0.0088214  -191.837  < 2e-16 ***
DESTIN_SZQTSZ08       0.0925004  0.0037722    24.521  < 2e-16 ***
DESTIN_SZQTSZ09      -0.3797599  0.0045808   -82.903  < 2e-16 ***
DESTIN_SZQTSZ10      -0.4079167  0.0042387   -96.235  < 2e-16 ***
DESTIN_SZQTSZ11       0.2574301  0.0040515    63.539  < 2e-16 ***
DESTIN_SZQTSZ12      -0.3365383  0.0051991   -64.730  < 2e-16 ***
DESTIN_SZQTSZ13       0.1248056  0.0039363    31.706  < 2e-16 ***
DESTIN_SZQTSZ14      -0.1509191  0.0044507   -33.909  < 2e-16 ***
DESTIN_SZQTSZ15       0.0690162  0.0054153    12.745  < 2e-16 ***
DESTIN_SZRCSZ01      -1.0906712  0.0051802  -210.548  < 2e-16 ***
DESTIN_SZRCSZ02      -2.2359026  0.0135798  -164.649  < 2e-16 ***
DESTIN_SZRCSZ03      -1.1174100  0.0070391  -158.744  < 2e-16 ***
DESTIN_SZRCSZ04      -2.4568960  0.0102337  -240.080  < 2e-16 ***
DESTIN_SZRCSZ05      -2.3552607  0.0095366  -246.970  < 2e-16 ***
DESTIN_SZRCSZ06      -2.1550501  0.0124313  -173.356  < 2e-16 ***
DESTIN_SZRCSZ08      -2.0841001  0.0102626  -203.077  < 2e-16 ***
DESTIN_SZRCSZ09      -1.5747291  0.0094616  -166.434  < 2e-16 ***
DESTIN_SZRCSZ10      -1.1849003  0.0051385  -230.594  < 2e-16 ***
DESTIN_SZRVSZ01      -2.1529608  0.0085900  -250.637  < 2e-16 ***
DESTIN_SZRVSZ02      -2.4492143  0.0117753  -207.996  < 2e-16 ***
DESTIN_SZRVSZ03      -2.4222717  0.0100597  -240.790  < 2e-16 ***
DESTIN_SZRVSZ04      -1.7371692  0.0114377  -151.881  < 2e-16 ***
DESTIN_SZRVSZ05      -2.0839854  0.0111068  -187.632  < 2e-16 ***
DESTIN_SZSBSZ01      -0.0294186  0.0046753    -6.292 3.13e-10 ***
DESTIN_SZSBSZ02      -0.9958381  0.0062222  -160.046  < 2e-16 ***
DESTIN_SZSBSZ03       0.7158194  0.0034449   207.793  < 2e-16 ***
DESTIN_SZSBSZ04       0.1191755  0.0043864    27.169  < 2e-16 ***
DESTIN_SZSBSZ05      -0.7522878  0.0057118  -131.708  < 2e-16 ***
DESTIN_SZSBSZ06      -2.7324967  0.0215545  -126.771  < 2e-16 ***
DESTIN_SZSBSZ07      -0.7273121  0.0152820   -47.593  < 2e-16 ***
DESTIN_SZSBSZ08       1.5312821  0.0041649   367.666  < 2e-16 ***
DESTIN_SZSBSZ09       0.8296969  0.0041401   200.405  < 2e-16 ***
DESTIN_SZSESZ02      -0.1094068  0.0038531   -28.395  < 2e-16 ***
DESTIN_SZSESZ03       0.7051751  0.0030797   228.975  < 2e-16 ***
DESTIN_SZSESZ04      -0.4959627  0.0043902  -112.970  < 2e-16 ***
DESTIN_SZSESZ05      -0.1095390  0.0037931   -28.879  < 2e-16 ***
DESTIN_SZSESZ06      -0.5019247  0.0047087  -106.594  < 2e-16 ***
DESTIN_SZSESZ07      -2.8784581  0.0198656  -144.897  < 2e-16 ***
DESTIN_SZSGSZ01      -0.4217244  0.0048327   -87.264  < 2e-16 ***
DESTIN_SZSGSZ02       0.0522081  0.0043245    12.073  < 2e-16 ***
DESTIN_SZSGSZ03      -0.3606981  0.0040326   -89.445  < 2e-16 ***
DESTIN_SZSGSZ04      -0.3212470  0.0039814   -80.687  < 2e-16 ***
DESTIN_SZSGSZ05      -1.9946206  0.0072648  -274.558  < 2e-16 ***
DESTIN_SZSGSZ06       0.4334467  0.0031411   137.991  < 2e-16 ***
DESTIN_SZSGSZ07      -0.3911286  0.0040499   -96.577  < 2e-16 ***
DESTIN_SZSISZ01      -0.9199242  0.0131178   -70.128  < 2e-16 ***
DESTIN_SZSKSZ01      -0.0948233  0.0058707   -16.152  < 2e-16 ***
DESTIN_SZSKSZ02       0.7988244  0.0041751   191.329  < 2e-16 ***
DESTIN_SZSKSZ03       0.0431353  0.0049234     8.761  < 2e-16 ***
DESTIN_SZSKSZ04      -0.6283844  0.0124899   -50.311  < 2e-16 ***
DESTIN_SZSKSZ05       0.0853350  0.0093562     9.121  < 2e-16 ***
DESTIN_SZSLSZ01      -0.3676636  0.0065766   -55.905  < 2e-16 ***
DESTIN_SZSLSZ04      -0.4556080  0.0052586   -86.641  < 2e-16 ***
DESTIN_SZSRSZ01      -1.6857168  0.0063800  -264.220  < 2e-16 ***
DESTIN_SZSRSZ02      -1.6369037  0.0074875  -218.618  < 2e-16 ***
DESTIN_SZSRSZ03      -1.5531939  0.0067905  -228.730  < 2e-16 ***
DESTIN_SZSVSZ01      -2.1265870  0.0278669   -76.312  < 2e-16 ***
DESTIN_SZTHSZ01      -2.5637337  0.0281096   -91.205  < 2e-16 ***
DESTIN_SZTHSZ03      -1.5761506  0.0171972   -91.652  < 2e-16 ***
DESTIN_SZTHSZ04      -2.2051773  0.0180679  -122.050  < 2e-16 ***
DESTIN_SZTHSZ06      -1.2196441  0.0115873  -105.257  < 2e-16 ***
DESTIN_SZTMSZ01       0.0307882  0.0042806     7.192 6.36e-13 ***
DESTIN_SZTMSZ02       1.7151835  0.0027605   621.327  < 2e-16 ***
DESTIN_SZTMSZ03       0.7346828  0.0031466   233.481  < 2e-16 ***
DESTIN_SZTMSZ04       0.8505032  0.0032061   265.273  < 2e-16 ***
DESTIN_SZTMSZ05       0.6510306  0.0040091   162.390  < 2e-16 ***
DESTIN_SZTNSZ01      -0.7109821  0.0044663  -159.188  < 2e-16 ***
DESTIN_SZTNSZ02      -1.4620229  0.0058144  -251.447  < 2e-16 ***
DESTIN_SZTNSZ03      -1.4562412  0.0070409  -206.827  < 2e-16 ***
DESTIN_SZTNSZ04      -1.0851666  0.0054825  -197.934  < 2e-16 ***
DESTIN_SZTPSZ01      -0.5817555  0.0045298  -128.429  < 2e-16 ***
DESTIN_SZTPSZ02       0.1808566  0.0031446    57.513  < 2e-16 ***
DESTIN_SZTPSZ03      -0.4988927  0.0045255  -110.240  < 2e-16 ***
DESTIN_SZTPSZ04      -1.5628293  0.0061959  -252.235  < 2e-16 ***
DESTIN_SZTPSZ05      -0.8873261  0.0047840  -185.479  < 2e-16 ***
DESTIN_SZTPSZ06      -0.2876381  0.0053863   -53.402  < 2e-16 ***
DESTIN_SZTPSZ07      -1.7906770  0.0089538  -199.991  < 2e-16 ***
DESTIN_SZTPSZ08      -1.3194188  0.0064723  -203.855  < 2e-16 ***
DESTIN_SZTPSZ09      -0.6211985  0.0049347  -125.883  < 2e-16 ***
DESTIN_SZTPSZ10      -0.6393478  0.0059480  -107.489  < 2e-16 ***
DESTIN_SZTPSZ11      -0.3804245  0.0040839   -93.152  < 2e-16 ***
DESTIN_SZTPSZ12      -0.8566449  0.0050890  -168.331  < 2e-16 ***
DESTIN_SZTSSZ01      -0.6057724  0.0191686   -31.602  < 2e-16 ***
DESTIN_SZTSSZ02       1.0228252  0.0073989   138.240  < 2e-16 ***
DESTIN_SZTSSZ03       1.7210049  0.0054964   313.117  < 2e-16 ***
DESTIN_SZTSSZ04       1.6552387  0.0055719   297.069  < 2e-16 ***
DESTIN_SZTSSZ05       1.8953793  0.0058191   325.717  < 2e-16 ***
DESTIN_SZTSSZ06       0.9420685  0.0094085   100.130  < 2e-16 ***
DESTIN_SZWCSZ01       1.6641119  0.0036267   458.847  < 2e-16 ***
DESTIN_SZWCSZ02       0.1145004  0.0081389    14.068  < 2e-16 ***
DESTIN_SZWCSZ03      -1.2456001  0.0229509   -54.272  < 2e-16 ***
DESTIN_SZWDSZ01       1.6721786  0.0029145   573.752  < 2e-16 ***
DESTIN_SZWDSZ02      -0.2568974  0.0047005   -54.653  < 2e-16 ***
DESTIN_SZWDSZ03       1.2710250  0.0030587   415.540  < 2e-16 ***
DESTIN_SZWDSZ04       0.2153072  0.0044239    48.669  < 2e-16 ***
DESTIN_SZWDSZ05       0.2343907  0.0042605    55.015  < 2e-16 ***
DESTIN_SZWDSZ06       0.5955289  0.0033837   176.001  < 2e-16 ***
DESTIN_SZWDSZ07       1.2712699  0.0042065   302.217  < 2e-16 ***
DESTIN_SZWDSZ08       0.7528739  0.0050831   148.112  < 2e-16 ***
DESTIN_SZWDSZ09       0.7921131  0.0036639   216.196  < 2e-16 ***
DESTIN_SZYSSZ01       1.2635195  0.0031437   401.923  < 2e-16 ***
DESTIN_SZYSSZ02       0.2534000  0.0041190    61.519  < 2e-16 ***
DESTIN_SZYSSZ03      -0.0477465  0.0043261   -11.037  < 2e-16 ***
DESTIN_SZYSSZ04       0.0517178  0.0040947    12.631  < 2e-16 ***
DESTIN_SZYSSZ05      -1.5878543  0.0085780  -185.108  < 2e-16 ***
DESTIN_SZYSSZ06      -1.2417883  0.0064833  -191.535  < 2e-16 ***
DESTIN_SZYSSZ07      -0.8080207  0.0079004  -102.276  < 2e-16 ***
DESTIN_SZYSSZ08       0.7024251  0.0032453   216.443  < 2e-16 ***
DESTIN_SZYSSZ09       0.3893030  0.0033390   116.593  < 2e-16 ***
log(ORIGIN_AGE25_64)  0.2272793  0.0001037  2191.851  < 2e-16 ***
log(dist)            -0.7146129  0.0001018 -7020.061  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 105499054  on 20915  degrees of freedom
Residual deviance:  44679882  on 20604  degrees of freedom
AIC: 44812892

Number of Fisher Scoring iterations: 7

Next, we can examine how the constraints hold for destinations this time.

CalcRSquared(decSIM$data$TRIPS, decSIM$fitted.values)

[1] 0.485937

8.8 Doubly Constrained

In this section, We will fit a doubly constrained SIM.

Note

The general formula of Doubly Constrained Spatial Interaction Model

\(\lambda_{ij} = \exp \left( k + \mu_i + \alpha_i - \beta \ln d_{ij} \right)\)

Notice that the difference between Unconstrained Spatial Interaction Model and this formula lies in the second and third term. Comparing Unconstrained Spatial Interaction Model formula and Doubly Constrained Spatial Interaction Model, it is \(\mu \ln V_i\) compared to \(\mu_i\) and \(\alpha \ln W_j\) compared to \(\alpha_i\) respectively.

dbcSIM <- glm(formula = TRIPS ~
                ORIGIN_SZ +
                DESTIN_SZ +
                log(dist),
              family = poisson(link = "log"),
              data = SIM_data,
              na.action = na.exclude)
summary(dbcSIM)


Call:
glm(formula = TRIPS ~ ORIGIN_SZ + DESTIN_SZ + log(dist), family = poisson(link = "log"),
    data = SIM_data, na.action = na.exclude)

Coefficients:
                  Estimate Std. Error   z value Pr(>|z|)
(Intercept)     12.8209944  0.0037017  3463.498  < 2e-16 ***
ORIGIN_SZAMSZ02  1.0211738  0.0037920   269.300  < 2e-16 ***
ORIGIN_SZAMSZ03  0.5918913  0.0038618   153.267  < 2e-16 ***
ORIGIN_SZAMSZ04  0.1495750  0.0043158    34.658  < 2e-16 ***
ORIGIN_SZAMSZ05  0.0342910  0.0049206     6.969 3.20e-12 ***
ORIGIN_SZAMSZ06  0.4990749  0.0044867   111.235  < 2e-16 ***
ORIGIN_SZAMSZ07 -0.7632253  0.0074431  -102.541  < 2e-16 ***
ORIGIN_SZAMSZ08 -0.6370000  0.0069109   -92.174  < 2e-16 ***
ORIGIN_SZAMSZ09  0.4389776  0.0046271    94.871  < 2e-16 ***
ORIGIN_SZAMSZ10  0.4528313  0.0040674   111.331  < 2e-16 ***
ORIGIN_SZAMSZ11 -1.5882086  0.0091196  -174.153  < 2e-16 ***
ORIGIN_SZAMSZ12 -1.6680341  0.0090745  -183.815  < 2e-16 ***
ORIGIN_SZBDSZ01  0.7900065  0.0037852   208.710  < 2e-16 ***
ORIGIN_SZBDSZ02  0.4100908  0.0043348    94.604  < 2e-16 ***
ORIGIN_SZBDSZ03  0.8165741  0.0038527   211.949  < 2e-16 ***
ORIGIN_SZBDSZ04  1.4216001  0.0033859   419.856  < 2e-16 ***
ORIGIN_SZBDSZ05  0.5426736  0.0038342   141.535  < 2e-16 ***
ORIGIN_SZBDSZ06  0.8337532  0.0038916   214.246  < 2e-16 ***
ORIGIN_SZBDSZ07 -0.8141464  0.0070275  -115.852  < 2e-16 ***
ORIGIN_SZBDSZ08 -0.6895439  0.0067720  -101.822  < 2e-16 ***
ORIGIN_SZBKSZ01 -0.1417253  0.0054077   -26.208  < 2e-16 ***
ORIGIN_SZBKSZ02  0.5709571  0.0044624   127.949  < 2e-16 ***
ORIGIN_SZBKSZ03  1.0136255  0.0041269   245.613  < 2e-16 ***
ORIGIN_SZBKSZ04 -0.0550877  0.0050301   -10.952  < 2e-16 ***
ORIGIN_SZBKSZ05  0.1809565  0.0046783    38.680  < 2e-16 ***
ORIGIN_SZBKSZ06  0.3522991  0.0050200    70.179  < 2e-16 ***
ORIGIN_SZBKSZ07  0.8090159  0.0038448   210.417  < 2e-16 ***
ORIGIN_SZBKSZ08  0.3293552  0.0044605    73.839  < 2e-16 ***
ORIGIN_SZBKSZ09  0.1029554  0.0047049    21.883  < 2e-16 ***
ORIGIN_SZBLSZ01 -1.7475447  0.0109717  -159.277  < 2e-16 ***
ORIGIN_SZBLSZ02 -2.8734452  0.0158781  -180.969  < 2e-16 ***
ORIGIN_SZBLSZ03 -5.0152771  0.0334310  -150.019  < 2e-16 ***
ORIGIN_SZBLSZ04 -2.2662350  0.0166319  -136.258  < 2e-16 ***
ORIGIN_SZBMSZ01  0.2535848  0.0040863    62.058  < 2e-16 ***
ORIGIN_SZBMSZ02 -1.0064943  0.0057776  -174.207  < 2e-16 ***
ORIGIN_SZBMSZ03 -0.0690787  0.0046447   -14.872  < 2e-16 ***
ORIGIN_SZBMSZ04  0.3074282  0.0041340    74.365  < 2e-16 ***
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ORIGIN_SZBMSZ06 -1.2327549  0.0097168  -126.869  < 2e-16 ***
ORIGIN_SZBMSZ07 -0.2831713  0.0046087   -61.443  < 2e-16 ***
ORIGIN_SZBMSZ08 -0.1874351  0.0046013   -40.736  < 2e-16 ***
ORIGIN_SZBMSZ09 -0.7228122  0.0058864  -122.793  < 2e-16 ***
ORIGIN_SZBMSZ10 -0.8077952  0.0062270  -129.725  < 2e-16 ***
ORIGIN_SZBMSZ11 -0.4924012  0.0054629   -90.136  < 2e-16 ***
ORIGIN_SZBMSZ12 -0.6909135  0.0072100   -95.827  < 2e-16 ***
ORIGIN_SZBMSZ13  0.0512890  0.0045905    11.173  < 2e-16 ***
ORIGIN_SZBMSZ14 -0.2568311  0.0054867   -46.810  < 2e-16 ***
ORIGIN_SZBMSZ15  0.0870108  0.0049432    17.602  < 2e-16 ***
ORIGIN_SZBMSZ16 -0.9610030  0.0061491  -156.284  < 2e-16 ***
ORIGIN_SZBMSZ17 -1.4357633  0.0093761  -153.131  < 2e-16 ***
ORIGIN_SZBPSZ01  0.5477033  0.0046878   116.836  < 2e-16 ***
ORIGIN_SZBPSZ02  0.8280377  0.0051841   159.727  < 2e-16 ***
ORIGIN_SZBPSZ03  1.1025706  0.0048263   228.452  < 2e-16 ***
ORIGIN_SZBPSZ04  0.7644581  0.0042529   179.748  < 2e-16 ***
ORIGIN_SZBPSZ05  0.5761734  0.0039555   145.663  < 2e-16 ***
ORIGIN_SZBPSZ06 -0.8038843  0.0067269  -119.503  < 2e-16 ***
ORIGIN_SZBPSZ07 -0.7081274  0.0067298  -105.223  < 2e-16 ***
ORIGIN_SZBSSZ01  0.0623814  0.0044582    13.993  < 2e-16 ***
ORIGIN_SZBSSZ02  0.5536861  0.0039874   138.858  < 2e-16 ***
ORIGIN_SZBSSZ03  0.2359262  0.0039552    59.649  < 2e-16 ***
ORIGIN_SZBTSZ01  0.0704870  0.0043712    16.125  < 2e-16 ***
ORIGIN_SZBTSZ02 -0.7549336  0.0061482  -122.789  < 2e-16 ***
ORIGIN_SZBTSZ03 -0.0350744  0.0047023    -7.459 8.72e-14 ***
ORIGIN_SZBTSZ04 -0.2858571  0.0078090   -36.606  < 2e-16 ***
ORIGIN_SZBTSZ05 -1.3311102  0.0083964  -158.534  < 2e-16 ***
ORIGIN_SZBTSZ06 -0.5039784  0.0057973   -86.934  < 2e-16 ***
ORIGIN_SZBTSZ07 -1.5400199  0.0087117  -176.777  < 2e-16 ***
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ORIGIN_SZCHSZ03 -0.4472993  0.0049038   -91.215  < 2e-16 ***
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ORIGIN_SZCKSZ02  1.2022570  0.0043552   276.053  < 2e-16 ***
ORIGIN_SZCKSZ03  0.8860008  0.0039444   224.624  < 2e-16 ***
ORIGIN_SZCKSZ04  1.8272911  0.0041197   443.546  < 2e-16 ***
ORIGIN_SZCKSZ05  1.5316688  0.0051392   298.034  < 2e-16 ***
ORIGIN_SZCKSZ06  1.1564174  0.0054508   212.157  < 2e-16 ***
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ORIGIN_SZCLSZ02 -1.3058599  0.0106228  -122.929  < 2e-16 ***
ORIGIN_SZCLSZ03 -0.1976445  0.0053882   -36.681  < 2e-16 ***
ORIGIN_SZCLSZ04  0.8670957  0.0037702   229.989  < 2e-16 ***
ORIGIN_SZCLSZ05 -1.4675258  0.0102663  -142.946  < 2e-16 ***
ORIGIN_SZCLSZ06  0.9061705  0.0036171   250.525  < 2e-16 ***
ORIGIN_SZCLSZ07 -0.0547981  0.0045274   -12.104  < 2e-16 ***
ORIGIN_SZCLSZ08  0.2882434  0.0051775    55.672  < 2e-16 ***
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ORIGIN_SZDTSZ05 -2.8339289  0.0214468  -132.138  < 2e-16 ***
ORIGIN_SZDTSZ06 -2.8287483  0.0173910  -162.656  < 2e-16 ***
ORIGIN_SZDTSZ07 -1.4138308  0.0191288   -73.911  < 2e-16 ***
ORIGIN_SZDTSZ08 -2.0630511  0.0097602  -211.373  < 2e-16 ***
ORIGIN_SZDTSZ09 -2.6480874  0.0206165  -128.445  < 2e-16 ***
ORIGIN_SZDTSZ10 -1.8351696  0.0104263  -176.013  < 2e-16 ***
ORIGIN_SZDTSZ11 -1.9555077  0.0108002  -181.061  < 2e-16 ***
ORIGIN_SZDTSZ12 -3.1193610  0.0255389  -122.141  < 2e-16 ***
ORIGIN_SZDTSZ13 -2.0406987  0.0121657  -167.742  < 2e-16 ***
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ORIGIN_SZGLSZ02  0.2358565  0.0041565    56.744  < 2e-16 ***
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ORIGIN_SZGLSZ04  0.9786034  0.0035154   278.375  < 2e-16 ***
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ORIGIN_SZHGSZ04  0.9678527  0.0036761   263.282  < 2e-16 ***
ORIGIN_SZHGSZ05  1.2917000  0.0036294   355.904  < 2e-16 ***
ORIGIN_SZHGSZ06  0.2142855  0.0043515    49.244  < 2e-16 ***
ORIGIN_SZHGSZ07  0.6701110  0.0037993   176.380  < 2e-16 ***
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ORIGIN_SZHGSZ10 -3.1262897  0.0398756   -78.401  < 2e-16 ***
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ORIGIN_SZJESZ02  0.3712710  0.0042775    86.796  < 2e-16 ***
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ORIGIN_SZJESZ05 -1.8580763  0.0120151  -154.645  < 2e-16 ***
ORIGIN_SZJESZ06  0.2384917  0.0042026    56.749  < 2e-16 ***
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ORIGIN_SZJESZ09  0.4198064  0.0044417    94.515  < 2e-16 ***
ORIGIN_SZJESZ10 -2.7745302  0.0169190  -163.989  < 2e-16 ***
ORIGIN_SZJESZ11 -3.2086375  0.0173519  -184.916  < 2e-16 ***
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ORIGIN_SZJWSZ02  0.9856003  0.0039792   247.690  < 2e-16 ***
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ORIGIN_SZJWSZ07 -2.2397234  0.0229427   -97.623  < 2e-16 ***
ORIGIN_SZJWSZ08  1.9992609  0.0037950   526.820  < 2e-16 ***
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ORIGIN_SZTMSZ04  0.3986725  0.0041038    97.147  < 2e-16 ***
ORIGIN_SZTMSZ05 -0.8775543  0.0064806  -135.412  < 2e-16 ***
ORIGIN_SZTNSZ01 -0.7901703  0.0059867  -131.989  < 2e-16 ***
ORIGIN_SZTNSZ02 -0.6947947  0.0058037  -119.716  < 2e-16 ***
ORIGIN_SZTNSZ03 -1.1794385  0.0075996  -155.198  < 2e-16 ***
ORIGIN_SZTNSZ04 -0.2906944  0.0056078   -51.837  < 2e-16 ***
ORIGIN_SZTPSZ01 -0.4220000  0.0052288   -80.706  < 2e-16 ***
ORIGIN_SZTPSZ02  0.5186540  0.0036928   140.449  < 2e-16 ***
ORIGIN_SZTPSZ03 -0.4340206  0.0052061   -83.368  < 2e-16 ***
ORIGIN_SZTPSZ04 -0.0658329  0.0049011   -13.432  < 2e-16 ***
ORIGIN_SZTPSZ05  0.0876853  0.0051031    17.183  < 2e-16 ***
ORIGIN_SZTPSZ06  0.5914622  0.0057688   102.528  < 2e-16 ***
ORIGIN_SZTPSZ07  0.0863881  0.0051162    16.885  < 2e-16 ***
ORIGIN_SZTPSZ08 -0.3835720  0.0066615   -57.580  < 2e-16 ***
ORIGIN_SZTPSZ09 -0.4862839  0.0054787   -88.759  < 2e-16 ***
ORIGIN_SZTPSZ10 -0.1484374  0.0054913   -27.031  < 2e-16 ***
ORIGIN_SZTPSZ11  0.3115335  0.0043348    71.868  < 2e-16 ***
ORIGIN_SZTPSZ12 -0.4386990  0.0053693   -81.705  < 2e-16 ***
ORIGIN_SZTSSZ01 -3.4734418  0.0409852   -84.749  < 2e-16 ***
ORIGIN_SZTSSZ02  0.1327694  0.0076669    17.317  < 2e-16 ***
ORIGIN_SZTSSZ03 -0.1394126  0.0079630   -17.508  < 2e-16 ***
ORIGIN_SZTSSZ04 -0.4834844  0.0081910   -59.026  < 2e-16 ***
ORIGIN_SZTSSZ05 -2.6972890  0.0129593  -208.135  < 2e-16 ***
ORIGIN_SZTSSZ06 -3.1376055  0.0185958  -168.727  < 2e-16 ***
ORIGIN_SZWCSZ01 -0.8838079  0.0063970  -138.159  < 2e-16 ***
ORIGIN_SZWCSZ02 -2.7133955  0.0253131  -107.194  < 2e-16 ***
ORIGIN_SZWCSZ03 -4.6049194  0.1325023   -34.753  < 2e-16 ***
ORIGIN_SZWDSZ01  0.8560664  0.0036494   234.575  < 2e-16 ***
ORIGIN_SZWDSZ02  0.9855869  0.0041492   237.535  < 2e-16 ***
ORIGIN_SZWDSZ03  1.7133802  0.0037306   459.281  < 2e-16 ***
ORIGIN_SZWDSZ04  1.1377918  0.0045405   250.585  < 2e-16 ***
ORIGIN_SZWDSZ05  0.4888027  0.0043787   111.633  < 2e-16 ***
ORIGIN_SZWDSZ06  0.9765953  0.0040953   238.467  < 2e-16 ***
ORIGIN_SZWDSZ07 -0.2293223  0.0058436   -39.243  < 2e-16 ***
ORIGIN_SZWDSZ08 -0.6573590  0.0064698  -101.604  < 2e-16 ***
ORIGIN_SZWDSZ09  1.6829017  0.0038504   437.071  < 2e-16 ***
ORIGIN_SZYSSZ01 -0.4848312  0.0048344  -100.288  < 2e-16 ***
ORIGIN_SZYSSZ02  0.9943936  0.0045034   220.811  < 2e-16 ***
ORIGIN_SZYSSZ03  2.2977837  0.0037334   615.459  < 2e-16 ***
ORIGIN_SZYSSZ04  0.8979911  0.0038661   232.276  < 2e-16 ***
ORIGIN_SZYSSZ05  0.4287521  0.0046808    91.598  < 2e-16 ***
ORIGIN_SZYSSZ06 -0.5936855  0.0075695   -78.432  < 2e-16 ***
ORIGIN_SZYSSZ07 -0.4146970  0.0078162   -53.056  < 2e-16 ***
ORIGIN_SZYSSZ08 -0.3883423  0.0053350   -72.792  < 2e-16 ***
ORIGIN_SZYSSZ09  1.1966606  0.0037486   319.226  < 2e-16 ***
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DESTIN_SZAMSZ03  0.1352549  0.0035587    38.006  < 2e-16 ***
DESTIN_SZAMSZ04 -0.9163012  0.0052190  -175.571  < 2e-16 ***
DESTIN_SZAMSZ05 -0.8273596  0.0049522  -167.070  < 2e-16 ***
DESTIN_SZAMSZ06 -0.7114591  0.0048518  -146.637  < 2e-16 ***
DESTIN_SZAMSZ07 -1.6525934  0.0085950  -192.274  < 2e-16 ***
DESTIN_SZAMSZ08 -0.8286756  0.0054861  -151.051  < 2e-16 ***
DESTIN_SZAMSZ09 -1.0100389  0.0052362  -192.896  < 2e-16 ***
DESTIN_SZAMSZ10  0.0577538  0.0036539    15.806  < 2e-16 ***
DESTIN_SZAMSZ11  0.0264569  0.0062075     4.262 2.03e-05 ***
DESTIN_SZAMSZ12  0.0699142  0.0044763    15.619  < 2e-16 ***
DESTIN_SZBDSZ01  0.4079216  0.0033431   122.020  < 2e-16 ***
DESTIN_SZBDSZ02 -0.2983737  0.0042145   -70.798  < 2e-16 ***
DESTIN_SZBDSZ03 -0.1081401  0.0037322   -28.975  < 2e-16 ***
DESTIN_SZBDSZ04  0.7139597  0.0030744   232.229  < 2e-16 ***
DESTIN_SZBDSZ05  0.4298721  0.0033834   127.052  < 2e-16 ***
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DESTIN_SZBDSZ07 -0.5446859  0.0074833   -72.787  < 2e-16 ***
DESTIN_SZBDSZ08 -1.4299619  0.0074446  -192.080  < 2e-16 ***
DESTIN_SZBKSZ01 -1.2418700  0.0054274  -228.816  < 2e-16 ***
DESTIN_SZBKSZ02 -0.3747029  0.0045775   -81.858  < 2e-16 ***
DESTIN_SZBKSZ03 -0.8106383  0.0046912  -172.798  < 2e-16 ***
DESTIN_SZBKSZ04 -0.0447919  0.0041811   -10.713  < 2e-16 ***
DESTIN_SZBKSZ05 -0.6469806  0.0046863  -138.059  < 2e-16 ***
DESTIN_SZBKSZ06 -1.0379007  0.0052498  -197.703  < 2e-16 ***
DESTIN_SZBKSZ07  0.1008434  0.0035689    28.256  < 2e-16 ***
DESTIN_SZBKSZ08 -1.1235248  0.0058606  -191.708  < 2e-16 ***
DESTIN_SZBKSZ09 -0.1769490  0.0042448   -41.686  < 2e-16 ***
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DESTIN_SZBLSZ02  0.4289945  0.0057776    74.251  < 2e-16 ***
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DESTIN_SZBLSZ04 -0.4365000  0.0114393   -38.158  < 2e-16 ***
DESTIN_SZBMSZ01 -0.0437992  0.0038281   -11.442  < 2e-16 ***
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DESTIN_SZBMSZ03 -0.5831276  0.0048985  -119.042  < 2e-16 ***
DESTIN_SZBMSZ04 -0.3088132  0.0042301   -73.004  < 2e-16 ***
DESTIN_SZBMSZ05 -0.4009981  0.0050362   -79.623  < 2e-16 ***
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DESTIN_SZBMSZ07  0.1723604  0.0037269    46.248  < 2e-16 ***
DESTIN_SZBMSZ08 -0.7228516  0.0048637  -148.622  < 2e-16 ***
DESTIN_SZBMSZ09 -1.5648751  0.0076588  -204.324  < 2e-16 ***
DESTIN_SZBMSZ10 -1.0792725  0.0060780  -177.571  < 2e-16 ***
DESTIN_SZBMSZ11 -1.2337029  0.0060444  -204.105  < 2e-16 ***
DESTIN_SZBMSZ12 -0.6038761  0.0063848   -94.580  < 2e-16 ***
DESTIN_SZBMSZ13 -0.0069636  0.0040263    -1.730 0.083716 .
DESTIN_SZBMSZ14 -0.6078154  0.0063169   -96.220  < 2e-16 ***
DESTIN_SZBMSZ15 -0.9452800  0.0058509  -161.562  < 2e-16 ***
DESTIN_SZBMSZ16 -1.2893936  0.0061729  -208.880  < 2e-16 ***
DESTIN_SZBMSZ17 -1.3174691  0.0072184  -182.516  < 2e-16 ***
DESTIN_SZBPSZ01 -0.8121560  0.0046763  -173.675  < 2e-16 ***
DESTIN_SZBPSZ02 -1.7589943  0.0074436  -236.311  < 2e-16 ***
DESTIN_SZBPSZ03 -1.6224673  0.0071034  -228.408  < 2e-16 ***
DESTIN_SZBPSZ04 -0.9099544  0.0051560  -176.486  < 2e-16 ***
DESTIN_SZBPSZ05  0.2822083  0.0034510    81.776  < 2e-16 ***
DESTIN_SZBPSZ06 -0.7067413  0.0062576  -112.941  < 2e-16 ***
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DESTIN_SZBSSZ01  0.0907030  0.0037765    24.018  < 2e-16 ***
DESTIN_SZBSSZ02 -0.7477704  0.0043714  -171.058  < 2e-16 ***
DESTIN_SZBSSZ03  0.3866005  0.0032621   118.514  < 2e-16 ***
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DESTIN_SZBTSZ03 -0.0353772  0.0041931    -8.437  < 2e-16 ***
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DESTIN_SZPRSZ05 -0.3237481  0.0041005   -78.954  < 2e-16 ***
DESTIN_SZPRSZ06  0.2039871  0.0043057    47.376  < 2e-16 ***
DESTIN_SZPRSZ07 -1.1457123  0.0104668  -109.461  < 2e-16 ***
DESTIN_SZPRSZ08 -0.8729830  0.0057992  -150.534  < 2e-16 ***
DESTIN_SZQTSZ01 -1.4677400  0.0083689  -175.380  < 2e-16 ***
DESTIN_SZQTSZ02 -1.2697556  0.0060732  -209.077  < 2e-16 ***
DESTIN_SZQTSZ03 -0.6967149  0.0054986  -126.708  < 2e-16 ***
DESTIN_SZQTSZ04 -0.7466019  0.0056216  -132.809  < 2e-16 ***
DESTIN_SZQTSZ05 -0.5993615  0.0049649  -120.719  < 2e-16 ***
DESTIN_SZQTSZ06 -0.8812866  0.0053353  -165.181  < 2e-16 ***
DESTIN_SZQTSZ07 -1.4915992  0.0089009  -167.578  < 2e-16 ***
DESTIN_SZQTSZ08  0.2165260  0.0039176    55.270  < 2e-16 ***
DESTIN_SZQTSZ09 -0.4146415  0.0046976   -88.266  < 2e-16 ***
DESTIN_SZQTSZ10 -0.2193841  0.0043617   -50.298  < 2e-16 ***
DESTIN_SZQTSZ11  0.3378311  0.0041782    80.855  < 2e-16 ***
DESTIN_SZQTSZ12 -0.2029944  0.0054158   -37.482  < 2e-16 ***
DESTIN_SZQTSZ13  0.2270464  0.0041252    55.038  < 2e-16 ***
DESTIN_SZQTSZ14  0.1102927  0.0045744    24.111  < 2e-16 ***
DESTIN_SZQTSZ15  0.1633004  0.0057721    28.291  < 2e-16 ***
DESTIN_SZRCSZ01 -0.8071584  0.0052748  -153.022  < 2e-16 ***
DESTIN_SZRCSZ02 -2.1569024  0.0136086  -158.495  < 2e-16 ***
DESTIN_SZRCSZ03 -1.0009513  0.0071508  -139.978  < 2e-16 ***
DESTIN_SZRCSZ04 -2.1570287  0.0102663  -210.107  < 2e-16 ***
DESTIN_SZRCSZ05 -2.1689708  0.0095674  -226.704  < 2e-16 ***
DESTIN_SZRCSZ06 -1.8254095  0.0124663  -146.428  < 2e-16 ***
DESTIN_SZRCSZ08 -1.5012102  0.0103539  -144.990  < 2e-16 ***
DESTIN_SZRCSZ09 -1.2443962  0.0095075  -130.885  < 2e-16 ***
DESTIN_SZRCSZ10 -0.8072282  0.0051969  -155.330  < 2e-16 ***
DESTIN_SZRVSZ01 -1.6546043  0.0086377  -191.555  < 2e-16 ***
DESTIN_SZRVSZ02 -2.2642314  0.0118550  -190.994  < 2e-16 ***
DESTIN_SZRVSZ03 -1.9241528  0.0101322  -189.905  < 2e-16 ***
DESTIN_SZRVSZ04 -1.3647507  0.0114905  -118.772  < 2e-16 ***
DESTIN_SZRVSZ05 -1.3337470  0.0113096  -117.931  < 2e-16 ***
DESTIN_SZSBSZ01 -0.3783598  0.0053692   -70.468  < 2e-16 ***
DESTIN_SZSBSZ02 -0.9714858  0.0063803  -152.263  < 2e-16 ***
DESTIN_SZSBSZ03  0.5849635  0.0037871   154.461  < 2e-16 ***
DESTIN_SZSBSZ04  0.0628994  0.0048666    12.925  < 2e-16 ***
DESTIN_SZSBSZ05 -0.7903141  0.0059734  -132.306  < 2e-16 ***
DESTIN_SZSBSZ06 -2.4749704  0.0221461  -111.757  < 2e-16 ***
DESTIN_SZSBSZ07 -2.0802889  0.0163700  -127.079  < 2e-16 ***
DESTIN_SZSBSZ08  1.0512838  0.0045018   233.528  < 2e-16 ***
DESTIN_SZSBSZ09  0.4794881  0.0044114   108.692  < 2e-16 ***
DESTIN_SZSESZ02 -0.5041790  0.0039964  -126.157  < 2e-16 ***
DESTIN_SZSESZ03  0.3999040  0.0031970   125.086  < 2e-16 ***
DESTIN_SZSESZ04 -0.8064748  0.0046027  -175.216  < 2e-16 ***
DESTIN_SZSESZ05 -0.2274573  0.0038846   -58.554  < 2e-16 ***
DESTIN_SZSESZ06 -0.8726109  0.0049007  -178.058  < 2e-16 ***
DESTIN_SZSESZ07 -3.2557191  0.0198930  -163.662  < 2e-16 ***
DESTIN_SZSGSZ01 -0.2147069  0.0049537   -43.343  < 2e-16 ***
DESTIN_SZSGSZ02 -0.2178068  0.0043929   -49.582  < 2e-16 ***
DESTIN_SZSGSZ03 -0.4519684  0.0041222  -109.641  < 2e-16 ***
DESTIN_SZSGSZ04 -0.3205081  0.0040661   -78.824  < 2e-16 ***
DESTIN_SZSGSZ05 -1.8375273  0.0072964  -251.842  < 2e-16 ***
DESTIN_SZSGSZ06  0.3861484  0.0032161   120.067  < 2e-16 ***
DESTIN_SZSGSZ07 -0.3839443  0.0041174   -93.249  < 2e-16 ***
DESTIN_SZSISZ01 -0.4807184  0.0132026   -36.411  < 2e-16 ***
DESTIN_SZSKSZ01 -0.6652944  0.0062595  -106.285  < 2e-16 ***
DESTIN_SZSKSZ02  0.2704252  0.0046528    58.121  < 2e-16 ***
DESTIN_SZSKSZ03 -0.4653217  0.0051732   -89.948  < 2e-16 ***
DESTIN_SZSKSZ04 -0.8906681  0.0133805   -66.564  < 2e-16 ***
DESTIN_SZSKSZ05 -0.4150470  0.0111530   -37.214  < 2e-16 ***
DESTIN_SZSLSZ01 -0.7565998  0.0066751  -113.347  < 2e-16 ***
DESTIN_SZSLSZ04 -0.8814465  0.0053640  -164.327  < 2e-16 ***
DESTIN_SZSRSZ01 -1.1903322  0.0064592  -184.284  < 2e-16 ***
DESTIN_SZSRSZ02 -1.3027415  0.0075504  -172.539  < 2e-16 ***
DESTIN_SZSRSZ03 -1.3475702  0.0068354  -197.144  < 2e-16 ***
DESTIN_SZSVSZ01 -1.0127677  0.0413591   -24.487  < 2e-16 ***
DESTIN_SZTHSZ01 -3.3280757  0.0281757  -118.119  < 2e-16 ***
DESTIN_SZTHSZ03 -2.1899228  0.0177909  -123.092  < 2e-16 ***
DESTIN_SZTHSZ04 -2.5247657  0.0181864  -138.827  < 2e-16 ***
DESTIN_SZTHSZ06 -1.6350566  0.0118054  -138.501  < 2e-16 ***
DESTIN_SZTMSZ01 -0.3684293  0.0045579   -80.833  < 2e-16 ***
DESTIN_SZTMSZ02  1.3042469  0.0029248   445.921  < 2e-16 ***
DESTIN_SZTMSZ03  0.4611332  0.0033186   138.952  < 2e-16 ***
DESTIN_SZTMSZ04  0.8108143  0.0034018   238.351  < 2e-16 ***
DESTIN_SZTMSZ05  0.6733671  0.0042473   158.540  < 2e-16 ***
DESTIN_SZTNSZ01 -0.4410888  0.0045518   -96.905  < 2e-16 ***
DESTIN_SZTNSZ02 -1.0328372  0.0058953  -175.197  < 2e-16 ***
DESTIN_SZTNSZ03 -1.0414947  0.0071340  -145.990  < 2e-16 ***
DESTIN_SZTNSZ04 -0.9341618  0.0055851  -167.258  < 2e-16 ***
DESTIN_SZTPSZ01 -0.3698813  0.0046189   -80.079  < 2e-16 ***
DESTIN_SZTPSZ02  0.2467295  0.0032238    76.534  < 2e-16 ***
DESTIN_SZTPSZ03 -0.2439122  0.0046552   -52.395  < 2e-16 ***
DESTIN_SZTPSZ04 -1.5337020  0.0062616  -244.936  < 2e-16 ***
DESTIN_SZTPSZ05 -0.9305243  0.0048843  -190.515  < 2e-16 ***
DESTIN_SZTPSZ06 -0.4458706  0.0063048   -70.720  < 2e-16 ***
DESTIN_SZTPSZ07 -1.8014933  0.0091025  -197.912  < 2e-16 ***
DESTIN_SZTPSZ08 -1.1930009  0.0065955  -180.881  < 2e-16 ***
DESTIN_SZTPSZ09 -0.3469701  0.0050671   -68.476  < 2e-16 ***
DESTIN_SZTPSZ10 -0.9026251  0.0060902  -148.210  < 2e-16 ***
DESTIN_SZTPSZ11 -0.3080454  0.0042040   -73.275  < 2e-16 ***
DESTIN_SZTPSZ12 -0.7156204  0.0051754  -138.273  < 2e-16 ***
DESTIN_SZTSSZ01 -0.8543639  0.0200859   -42.536  < 2e-16 ***
DESTIN_SZTSSZ02 -0.5825767  0.0092558   -62.942  < 2e-16 ***
DESTIN_SZTSSZ03  0.2500858  0.0075201    33.256  < 2e-16 ***
DESTIN_SZTSSZ04  0.4593958  0.0074856    61.370  < 2e-16 ***
DESTIN_SZTSSZ05  1.5286175  0.0074395   205.473  < 2e-16 ***
DESTIN_SZTSSZ06  1.6286875  0.0134475   121.114  < 2e-16 ***
DESTIN_SZWCSZ01  1.3710389  0.0041810   327.920  < 2e-16 ***
DESTIN_SZWCSZ02 -0.7273251  0.0084384   -86.192  < 2e-16 ***
DESTIN_SZWCSZ03 -1.9636127  0.0230329   -85.253  < 2e-16 ***
DESTIN_SZWDSZ01  0.9399623  0.0031066   302.572  < 2e-16 ***
DESTIN_SZWDSZ02 -0.7981392  0.0049869  -160.048  < 2e-16 ***
DESTIN_SZWDSZ03  0.5407450  0.0034954   154.700  < 2e-16 ***
DESTIN_SZWDSZ04 -0.4445645  0.0050274   -88.428  < 2e-16 ***
DESTIN_SZWDSZ05 -0.1434480  0.0045552   -31.491  < 2e-16 ***
DESTIN_SZWDSZ06  0.1461975  0.0036640    39.901  < 2e-16 ***
DESTIN_SZWDSZ07  0.2388041  0.0046521    51.332  < 2e-16 ***
DESTIN_SZWDSZ08 -0.0098378  0.0054437    -1.807 0.070733 .
DESTIN_SZWDSZ09 -0.2806644  0.0041193   -68.135  < 2e-16 ***
DESTIN_SZYSSZ01  0.9357382  0.0033642   278.144  < 2e-16 ***
DESTIN_SZYSSZ02 -0.3420741  0.0045422   -75.310  < 2e-16 ***
DESTIN_SZYSSZ03 -1.2656914  0.0047488  -266.528  < 2e-16 ***
DESTIN_SZYSSZ04 -0.3567482  0.0043080   -82.811  < 2e-16 ***
DESTIN_SZYSSZ05 -1.9189376  0.0087174  -220.126  < 2e-16 ***
DESTIN_SZYSSZ06 -1.3204784  0.0065783  -200.734  < 2e-16 ***
DESTIN_SZYSSZ07 -0.6894968  0.0085776   -80.383  < 2e-16 ***
DESTIN_SZYSSZ08  0.6192002  0.0033756   183.436  < 2e-16 ***
DESTIN_SZYSSZ09 -0.0120655  0.0035260    -3.422 0.000622 ***
log(dist)       -0.6933493  0.0001075 -6451.265  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 105499054  on 20915  degrees of freedom
Residual deviance:  36022972  on 20297  degrees of freedom
AIC: 36156596

Number of Fisher Scoring iterations: 7

We can examine how the constraints hold this time.

CalcRSquared(dbcSIM$data$TRIPS, dbcSIM$fitted.values)

[1] 0.5614171

Note

Observations:

There is a relatively greater improvement in the \(R^2\) value using the doubly constrained model.

8.9 Model Comparison

Another useful model performance measure for continuous dependent variable is Root Mean Squared Error. We can use compare_performance() of performance package for this purpose.

First, we will create a list of models:

model_list <- list(unconstrained=uncSIM,
                   originConstrained=orcSIM,
                   destinationConstrained=decSIM,
                   doublyConstrained=dbcSIM)

Then, compare their RMSE values:

compare_performance(model_list,
                    metrics = "RMSE")

# Comparison of Model Performance Indices

Name                   | Model |     RMSE
-----------------------------------------
unconstrained          |   glm | 5485.538
originConstrained      |   glm | 4849.984
destinationConstrained |   glm | 4409.233
doublyConstrained      |   glm | 4231.607

Note

Observations:

The doubly constrained SIM has the lowest RMSE, making it the best model.

8.10 Visualizing Fitted Values

We can extract and visualize the fitted values for each model. To do so, start by extracting the fitted values for each model and adding their fitted values to SIM_data:

df <- as.data.frame(uncSIM$fitted.values) %>%
  round(digits = 0)
SIM_data <- SIM_data %>%
  cbind(df) %>%
  rename(uncTRIPS = "uncSIM$fitted.values")

df <- as.data.frame(orcSIM$fitted.values) %>%
  round(digits = 0)

SIM_data <- SIM_data %>%
  cbind(df) %>%
  rename(orcTRIPS = "orcSIM$fitted.values")

df <- as.data.frame(decSIM$fitted.values) %>%
  round(digits = 0)

SIM_data <- SIM_data %>%
  cbind(df) %>%
  rename(decTRIPS = "decSIM$fitted.values")

df <- as.data.frame(dbcSIM$fitted.values) %>%
  round(digits = 0)

SIM_data <- SIM_data %>%
  cbind(df) %>%
  rename(dbcTRIPS = "dbcSIM$fitted.values")

Create scatter plots for each model and present in single plot:

unc_p <- ggplot(data = SIM_data,
                aes(x = uncTRIPS,
                    y = TRIPS)) +
  geom_point() +
  geom_smooth(method = lm)

orc_p <- ggplot(data = SIM_data,
                aes(x = orcTRIPS,
                    y = TRIPS)) +
  geom_point() +
  geom_smooth(method = lm)

dec_p <- ggplot(data = SIM_data,
                aes(x = decTRIPS,
                    y = TRIPS)) +
  geom_point() +
  geom_smooth(method = lm)

dbc_p <- ggplot(data = SIM_data,
                aes(x = dbcTRIPS,
                    y = TRIPS)) +
  geom_point() +
  geom_smooth(method = lm)

ggarrange(unc_p, orc_p, dec_p, dbc_p,
          ncol = 2,
          nrow = 2)

Note

Observations:

The doubly constrained model provides the best fit to the data, as seen by the closer clustering around the diagonal line.
Both the origin and destination-constrained models improve over the unconstrained model but still show some deviations.
The unconstrained model performs the worst, with a weak relationship between predicted and actual values.