huff.lambda: Fitting the distance parameter lambda in the Huff model
In MCI: Multiplicative Competitive Interaction (MCI) Model
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
Description
This function estimates a distance decay parameter from observed total store/location data (e.g. complete annual turnovers) using bisection or "trial and error"
Usage
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huff.lambda(huffdataset, origins, locations, attrac, dist, gamma = 1, atype = "pow",
gamma2 = NULL, lambda_startv = -1, lambda_endv = -3, dtype = "pow",
localmarket_dataset, origin_id, localmarket,
location_dataset, location_id, location_total,
method = "bisection", iterations = 10, output = "matrix",
plotVal = FALSE, show_proc = FALSE, check_df = TRUE)
Arguments
huffdataset
an interaction matrix which is a data.frame containing the origins, locations and the explanatory variables (attraction, transport costs)
origins
the column in the interaction matrix huffdataset containing the origins (e.g. ZIP codes)
locations
the column in the interaction matrix huffdataset containing the locations (e.g. store codes)
attrac
the column in the interaction matrix huffdataset containing the attraction variable (e.g. sales area)
dist
the column in the interaction matrix huffdataset containing the transport costs (e.g. travelling time or street distance)
gamma
a single numeric value of γ for the exponential weighting of the attraction variable (default: 1)
atype
Type of attraction weighting function: atype = "pow" (power function), atype = "exp" (exponential function) or atype = "logistic" (default: atype = "pow")
gamma2
if atype = "logistic" a second γ parameter is needed
lambda_startv
Start value for λ search
lambda_endv
End value for λ search
dtype
Type of distance weighting function: dtype = "pow" (power function), dtype = "exp" (exponential function) or dtype = "logistic" (default: dtype = "pow")
localmarket_dataset
A data.frame containing the origins saved in a column which has the same name as in huffdataset and another column containing the local market potential
origin_id
the column in the dataset localmarket_dataset containing the origins (e.g. statistical districts, ZIP codes)
localmarket
the column in the dataset localmarket_dataset containing the local market potential (e.g. purchasing power, number of customers)
location_dataset
A data.frame containing the suppliers/locations and their observed total values
location_id
the column in the dataset location_dataset containing the locations (e.g. store codes), j, according to the codes in huffdataset
location_total
the column in the dataset location_dataset containing the observed total values of suppliers/locations, T_{j,obs} (e.g. annual sales, total number of customers)
method
If method = "bisection" (default value), the optimal λ is found by bisection. If method = "compare", several values are tried and the best one is found ("trial and error")
iterations
a single numeric value for the desired number of iterations
output
If output = "iterations", the results for every single iteration is shown. If output = "total", total sales and market shares (or total market area) of the suppliers are shown.
plotVal
If plotVal = TRUE, the function shows a simple plot of λ values against residuals
show_proc
logical argument that indicates if the function prints messages about the state of process during the work (e.g. “Processing variable xyz ...”). Default: show_proc = FALSE (messages off)
check_df
logical argument that indicates if the given dataset is checked for correct input, only for internal use, should not be deselected (default: TRUE)
Details
In many cases, only total empirical values of the suppliers/locations (e.g. annual turnover) can be used for market area estimation. This function fits the Huff model by estimating the λ parameter iteratively using an optimization algorithm based on the idea of Klein (1988). The fitting process in the huff.lambda includes of given number of iterations, while the fit gets better with every iteration, measured using the sum of squared residuals of observed vs. expected total values. The iterative optimization can be done via bisection (see Kaw et al. 2011, ch. 03.03) or "trial and error" (see Fuelop et al. 2011).
Value
The function output can be controlled by the function argument output. If output = "iterations", the results for every single iteration is shown (data.frame). If output = "total", total sales and market shares (or total market area) of the suppliers are shown (data.frame). The default output is a list with γ and λ.
Note
Note that the iterations can be time-consuming and depend on the number of suppliers/locations. Use show_proc = TRUE for monitoring the iteration process.
Author(s)
Thomas Wieland
References
Fuelop, G./Kopetsch, T./Schoepe, P. (2011): “Catchment areas of medical practices and the role played by geographical distance in the patient's choice of doctor”. In: The Annals of Regional Science, 46, 3, p. 691-706.
Kaw, A. K./Kalu, E. E./Nguyen, D. (2011): “Numerical Methods with Applications”. http://nm.mathforcollege.com/topics/textbook_index.html
Klein, R. (1988): “Der Lebensmittel-Einzelhandel im Raum Verden. Raeumliches Einkaufsverhalten unter sich wandelnden Bedingungen”. Flensburger Arbeitspapiere zur Landeskunde und Raumordnung, 6. Flensburg.
See Also
huff.attrac, huff.shares, huff.decay, huff.fit
Examples
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data(DIY1)
data(DIY2)
data(DIY3)
# Loading the three DIY store datasets
DIY_alldata <- merge (DIY1, DIY2, by.x = "j_destination", by.y = "j_destination")
# Add store data to distance matrix
huff_DIY <- huff.shares (DIY_alldata, "i_origin", "j_destination", "A_j_salesarea_sqm",
"t_ij_min", gamma = 1, lambda = -2)
# Calculating Huff local market shares
# Gamma = 1, Lambda = -2
huff_DIY <- merge (huff_DIY, DIY3, by.x = "i_origin", by.y = "district")
# Add data for origins
huff_DIY_total <- shares.total (huff_DIY, "i_origin", "j_destination", "p_ij",
"population")
# Calculating total market areas (=sums of customers)
colnames(DIY3) <- c("district", "pop")
# Change column name to "pop" (must be other name)
huff.lambda (huff_DIY, "i_origin", "j_destination", "A_j_salesarea_sqm",
"t_ij_min", gamma = 1, atype = "pow", gamma2 = NULL,
lambda_startv = -1, lambda_endv = -2.5, dtype= "pow",
DIY3, "district", "pop", huff_DIY_total, "suppliers_single", "sum_E_j",
method = "bisection", iterations = 10)
# Iterative search for the best lambda value using bisection
# Output: gamma and lambda
huff.lambda (huff_DIY, "i_origin", "j_destination", "A_j_salesarea_sqm",
"t_ij_min", gamma = 1, atype = "pow", gamma2 = NULL,
lambda_startv = -1, lambda_endv = -2.5, dtype= "pow",
DIY3, "district", "pop", huff_DIY_total, "suppliers_single", "sum_E_j",
method = "bisection", iterations = 10, output = "iterations", show_proc = TRUE)
# Same procedure, output: single iterations
huff.lambda (huff_DIY, "i_origin", "j_destination", "A_j_salesarea_sqm",
"t_ij_min", gamma = 1, atype = "pow", gamma2 = NULL,
lambda_startv = -1, lambda_endv = -2.5, dtype= "pow",
DIY3, "district", "pop", huff_DIY_total, "suppliers_single", "sum_E_j",
method = "compare", iterations = 10, output = "iterations", show_proc = TRUE, plotVal = TRUE)
# Using compare method, output: single iterations and plot
Example output
$gamma
[1] 1
$lambda
[1] -2.000488
Iteration 1 of 10 ...
Iteration 2 of 10 ...
Iteration 3 of 10 ...
Iteration 4 of 10 ...
Iteration 5 of 10 ...
Iteration 6 of 10 ...
Iteration 7 of 10 ...
Iteration 8 of 10 ...
Iteration 9 of 10 ...
Iteration 10 of 10 ...
Iteration Lambda
1 1 -1.750000
2 2 -2.125000
3 3 -1.937500
4 4 -2.031250
5 5 -1.984375
6 6 -2.007812
7 7 -1.996094
8 8 -2.001953
9 9 -1.999023
10 10 -2.000488
Iteration 1 of 151 ...
Iteration 2 of 151 ...
Iteration 3 of 151 ...
Iteration 4 of 151 ...
Iteration 5 of 151 ...
Iteration 6 of 151 ...
Iteration 7 of 151 ...
Iteration 8 of 151 ...
Iteration 9 of 151 ...
Iteration 10 of 151 ...
Iteration 11 of 151 ...
Iteration 12 of 151 ...
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Iteration 145 of 151 ...
Iteration 146 of 151 ...
Iteration 147 of 151 ...
Iteration 148 of 151 ...
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Iteration 150 of 151 ...
Iteration 151 of 151 ...
Iteration Lambda
1 1 -1.00
2 2 -1.01
3 3 -1.02
4 4 -1.03
5 5 -1.04
6 6 -1.05
7 7 -1.06
8 8 -1.07
9 9 -1.08
10 10 -1.09
11 11 -1.10
12 12 -1.11
13 13 -1.12
14 14 -1.13
15 15 -1.14
16 16 -1.15
17 17 -1.16
18 18 -1.17
19 19 -1.18
20 20 -1.19
21 21 -1.20
22 22 -1.21
23 23 -1.22
24 24 -1.23
25 25 -1.24
26 26 -1.25
27 27 -1.26
28 28 -1.27
29 29 -1.28
30 30 -1.29
31 31 -1.30
32 32 -1.31
33 33 -1.32
34 34 -1.33
35 35 -1.34
36 36 -1.35
37 37 -1.36
38 38 -1.37
39 39 -1.38
40 40 -1.39
41 41 -1.40
42 42 -1.41
43 43 -1.42
44 44 -1.43
45 45 -1.44
46 46 -1.45
47 47 -1.46
48 48 -1.47
49 49 -1.48
50 50 -1.49
51 51 -1.50
52 52 -1.51
53 53 -1.52
54 54 -1.53
55 55 -1.54
56 56 -1.55
57 57 -1.56
58 58 -1.57
59 59 -1.58
60 60 -1.59
61 61 -1.60
62 62 -1.61
63 63 -1.62
64 64 -1.63
65 65 -1.64
66 66 -1.65
67 67 -1.66
68 68 -1.67
69 69 -1.68
70 70 -1.69
71 71 -1.70
72 72 -1.71
73 73 -1.72
74 74 -1.73
75 75 -1.74
76 76 -1.75
77 77 -1.76
78 78 -1.77
79 79 -1.78
80 80 -1.79
81 81 -1.80
82 82 -1.81
83 83 -1.82
84 84 -1.83
85 85 -1.84
86 86 -1.85
87 87 -1.86
88 88 -1.87
89 89 -1.88
90 90 -1.89
91 91 -1.90
92 92 -1.91
93 93 -1.92
94 94 -1.93
95 95 -1.94
96 96 -1.95
97 97 -1.96
98 98 -1.97
99 99 -1.98
100 100 -1.99
101 101 -2.00
102 102 -2.01
103 103 -2.02
104 104 -2.03
105 105 -2.04
106 106 -2.05
107 107 -2.06
108 108 -2.07
109 109 -2.08
110 110 -2.09
111 111 -2.10
112 112 -2.11
113 113 -2.12
114 114 -2.13
115 115 -2.14
116 116 -2.15
117 117 -2.16
118 118 -2.17
119 119 -2.18
120 120 -2.19
121 121 -2.20
122 122 -2.21
123 123 -2.22
124 124 -2.23
125 125 -2.24
126 126 -2.25
127 127 -2.26
128 128 -2.27
129 129 -2.28
130 130 -2.29
131 131 -2.30
132 132 -2.31
133 133 -2.32
134 134 -2.33
135 135 -2.34
136 136 -2.35
137 137 -2.36
138 138 -2.37
139 139 -2.38
140 140 -2.39
141 141 -2.40
142 142 -2.41
143 143 -2.42
144 144 -2.43
145 145 -2.44
146 146 -2.45
147 147 -2.46
148 148 -2.47
149 149 -2.48
150 150 -2.49
151 151 -2.50
MCI documentation built on May 2, 2019, 6:02 a.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
Description
This function estimates a distance decay parameter from observed total store/location data (e.g. complete annual turnovers) using bisection or "trial and error"
Usage
1 2 3 4 5 6 | huff.lambda(huffdataset, origins, locations, attrac, dist, gamma = 1, atype = "pow",
gamma2 = NULL, lambda_startv = -1, lambda_endv = -3, dtype = "pow",
localmarket_dataset, origin_id, localmarket,
location_dataset, location_id, location_total,
method = "bisection", iterations = 10, output = "matrix",
plotVal = FALSE, show_proc = FALSE, check_df = TRUE)
|
Arguments
huffdataset |
an interaction matrix which is a |
origins |
the column in the interaction matrix |
locations |
the column in the interaction matrix |
attrac |
the column in the interaction matrix |
dist |
the column in the interaction matrix |
gamma |
a single numeric value of γ for the exponential weighting of the attraction variable (default: 1) |
atype |
Type of attraction weighting function: |
gamma2 |
if |
lambda_startv |
Start value for λ search |
lambda_endv |
End value for λ search |
dtype |
Type of distance weighting function: |
localmarket_dataset |
A |
origin_id |
the column in the dataset |
localmarket |
the column in the dataset |
location_dataset |
A |
location_id |
the column in the dataset |
location_total |
the column in the dataset |
method |
If |
iterations |
a single numeric value for the desired number of iterations |
output |
If |
plotVal |
If |
show_proc |
logical argument that indicates if the function prints messages about the state of process during the work (e.g. “Processing variable xyz ...”). Default: |
check_df |
logical argument that indicates if the given dataset is checked for correct input, only for internal use, should not be deselected (default: |
Details
In many cases, only total empirical values of the suppliers/locations (e.g. annual turnover) can be used for market area estimation. This function fits the Huff model by estimating the λ parameter iteratively using an optimization algorithm based on the idea of Klein (1988). The fitting process in the huff.lambda includes of given number of iterations, while the fit gets better with every iteration, measured using the sum of squared residuals of observed vs. expected total values. The iterative optimization can be done via bisection (see Kaw et al. 2011, ch. 03.03) or "trial and error" (see Fuelop et al. 2011).
Value
The function output can be controlled by the function argument output. If output = "iterations", the results for every single iteration is shown (data.frame). If output = "total", total sales and market shares (or total market area) of the suppliers are shown (data.frame). The default output is a list with γ and λ.
Note
Note that the iterations can be time-consuming and depend on the number of suppliers/locations. Use show_proc = TRUE for monitoring the iteration process.
Author(s)
Thomas Wieland
References
Fuelop, G./Kopetsch, T./Schoepe, P. (2011): “Catchment areas of medical practices and the role played by geographical distance in the patient's choice of doctor”. In: The Annals of Regional Science, 46, 3, p. 691-706.
Kaw, A. K./Kalu, E. E./Nguyen, D. (2011): “Numerical Methods with Applications”. http://nm.mathforcollege.com/topics/textbook_index.html
Klein, R. (1988): “Der Lebensmittel-Einzelhandel im Raum Verden. Raeumliches Einkaufsverhalten unter sich wandelnden Bedingungen”. Flensburger Arbeitspapiere zur Landeskunde und Raumordnung, 6. Flensburg.
See Also
huff.attrac, huff.shares, huff.decay, huff.fit
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | data(DIY1)
data(DIY2)
data(DIY3)
# Loading the three DIY store datasets
DIY_alldata <- merge (DIY1, DIY2, by.x = "j_destination", by.y = "j_destination")
# Add store data to distance matrix
huff_DIY <- huff.shares (DIY_alldata, "i_origin", "j_destination", "A_j_salesarea_sqm",
"t_ij_min", gamma = 1, lambda = -2)
# Calculating Huff local market shares
# Gamma = 1, Lambda = -2
huff_DIY <- merge (huff_DIY, DIY3, by.x = "i_origin", by.y = "district")
# Add data for origins
huff_DIY_total <- shares.total (huff_DIY, "i_origin", "j_destination", "p_ij",
"population")
# Calculating total market areas (=sums of customers)
colnames(DIY3) <- c("district", "pop")
# Change column name to "pop" (must be other name)
huff.lambda (huff_DIY, "i_origin", "j_destination", "A_j_salesarea_sqm",
"t_ij_min", gamma = 1, atype = "pow", gamma2 = NULL,
lambda_startv = -1, lambda_endv = -2.5, dtype= "pow",
DIY3, "district", "pop", huff_DIY_total, "suppliers_single", "sum_E_j",
method = "bisection", iterations = 10)
# Iterative search for the best lambda value using bisection
# Output: gamma and lambda
huff.lambda (huff_DIY, "i_origin", "j_destination", "A_j_salesarea_sqm",
"t_ij_min", gamma = 1, atype = "pow", gamma2 = NULL,
lambda_startv = -1, lambda_endv = -2.5, dtype= "pow",
DIY3, "district", "pop", huff_DIY_total, "suppliers_single", "sum_E_j",
method = "bisection", iterations = 10, output = "iterations", show_proc = TRUE)
# Same procedure, output: single iterations
huff.lambda (huff_DIY, "i_origin", "j_destination", "A_j_salesarea_sqm",
"t_ij_min", gamma = 1, atype = "pow", gamma2 = NULL,
lambda_startv = -1, lambda_endv = -2.5, dtype= "pow",
DIY3, "district", "pop", huff_DIY_total, "suppliers_single", "sum_E_j",
method = "compare", iterations = 10, output = "iterations", show_proc = TRUE, plotVal = TRUE)
# Using compare method, output: single iterations and plot
|
Example output
$gamma
[1] 1
$lambda
[1] -2.000488
Iteration 1 of 10 ...
Iteration 2 of 10 ...
Iteration 3 of 10 ...
Iteration 4 of 10 ...
Iteration 5 of 10 ...
Iteration 6 of 10 ...
Iteration 7 of 10 ...
Iteration 8 of 10 ...
Iteration 9 of 10 ...
Iteration 10 of 10 ...
Iteration Lambda
1 1 -1.750000
2 2 -2.125000
3 3 -1.937500
4 4 -2.031250
5 5 -1.984375
6 6 -2.007812
7 7 -1.996094
8 8 -2.001953
9 9 -1.999023
10 10 -2.000488
Iteration 1 of 151 ...
Iteration 2 of 151 ...
Iteration 3 of 151 ...
Iteration 4 of 151 ...
Iteration 5 of 151 ...
Iteration 6 of 151 ...
Iteration 7 of 151 ...
Iteration 8 of 151 ...
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Iteration Lambda
1 1 -1.00
2 2 -1.01
3 3 -1.02
4 4 -1.03
5 5 -1.04
6 6 -1.05
7 7 -1.06
8 8 -1.07
9 9 -1.08
10 10 -1.09
11 11 -1.10
12 12 -1.11
13 13 -1.12
14 14 -1.13
15 15 -1.14
16 16 -1.15
17 17 -1.16
18 18 -1.17
19 19 -1.18
20 20 -1.19
21 21 -1.20
22 22 -1.21
23 23 -1.22
24 24 -1.23
25 25 -1.24
26 26 -1.25
27 27 -1.26
28 28 -1.27
29 29 -1.28
30 30 -1.29
31 31 -1.30
32 32 -1.31
33 33 -1.32
34 34 -1.33
35 35 -1.34
36 36 -1.35
37 37 -1.36
38 38 -1.37
39 39 -1.38
40 40 -1.39
41 41 -1.40
42 42 -1.41
43 43 -1.42
44 44 -1.43
45 45 -1.44
46 46 -1.45
47 47 -1.46
48 48 -1.47
49 49 -1.48
50 50 -1.49
51 51 -1.50
52 52 -1.51
53 53 -1.52
54 54 -1.53
55 55 -1.54
56 56 -1.55
57 57 -1.56
58 58 -1.57
59 59 -1.58
60 60 -1.59
61 61 -1.60
62 62 -1.61
63 63 -1.62
64 64 -1.63
65 65 -1.64
66 66 -1.65
67 67 -1.66
68 68 -1.67
69 69 -1.68
70 70 -1.69
71 71 -1.70
72 72 -1.71
73 73 -1.72
74 74 -1.73
75 75 -1.74
76 76 -1.75
77 77 -1.76
78 78 -1.77
79 79 -1.78
80 80 -1.79
81 81 -1.80
82 82 -1.81
83 83 -1.82
84 84 -1.83
85 85 -1.84
86 86 -1.85
87 87 -1.86
88 88 -1.87
89 89 -1.88
90 90 -1.89
91 91 -1.90
92 92 -1.91
93 93 -1.92
94 94 -1.93
95 95 -1.94
96 96 -1.95
97 97 -1.96
98 98 -1.97
99 99 -1.98
100 100 -1.99
101 101 -2.00
102 102 -2.01
103 103 -2.02
104 104 -2.03
105 105 -2.04
106 106 -2.05
107 107 -2.06
108 108 -2.07
109 109 -2.08
110 110 -2.09
111 111 -2.10
112 112 -2.11
113 113 -2.12
114 114 -2.13
115 115 -2.14
116 116 -2.15
117 117 -2.16
118 118 -2.17
119 119 -2.18
120 120 -2.19
121 121 -2.20
122 122 -2.21
123 123 -2.22
124 124 -2.23
125 125 -2.24
126 126 -2.25
127 127 -2.26
128 128 -2.27
129 129 -2.28
130 130 -2.29
131 131 -2.30
132 132 -2.31
133 133 -2.32
134 134 -2.33
135 135 -2.34
136 136 -2.35
137 137 -2.36
138 138 -2.37
139 139 -2.38
140 140 -2.39
141 141 -2.40
142 142 -2.41
143 143 -2.42
144 144 -2.43
145 145 -2.44
146 146 -2.45
147 147 -2.46
148 148 -2.47
149 149 -2.48
150 150 -2.49
151 151 -2.50
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