Differentially Private Mobile Crowd Sensing Considering Sensing Errors

. 2020 May 14;20(10):2785. doi: 10.3390/s20102785

Algorithm 2 Estimation Algorithm.

Input:Y, $Y_{σ}$ , $ϵ$ , $m i n v_{o r g}$ , $m a x v_{o r g}$ , $m i n v_{r e p}$ , $m a x v_{r e p}$ , $m i n σ_{r e p}$ , $m a x σ_{r e p}$ , $b_{n}$

Output: $‡$

1:
$σ_{a v e} \leftarrow A v e r a g e (Y_{σ})$ /* Consider $σ_{a v e}$ is the standard deviation of each participant*/
2:
if standard deviation is considered as private information then
3:
$b_{v}$ and $b_{σ}$ are calculated by Equations (12) and (13), and set $θ = {σ_{a v e}, b_{v}, b_{σ}}$ .
4:
else
5:
$b_{v}$ is calculated by Equation (10), and set $θ = {σ_{a v e}, b_{v}}$ .
6:
end if
7:
$w \leftarrow (m a x v_{r e p} - m i n v_{r e p}) / b_{n}$ /*w represents the width of each bin*/
8:
$x \leftarrow$ an arbitrary real number
9:
$l e f t \leftarrow \int_{- \infty}^{x - w / 2} F (y; x, θ) d y$
10:
for $j = 1, \dots, b_{n}$ do
11:
$Q (j) \leftarrow \int_{x - w / 2 + (j - 1) * w}^{x - w / 2 + j * w} F (y; x, θ) d y$ /* $P (1, j) = Q (j)$ for $j = 2, \dots, b_{n} - 1$ . $P (1, 1) = l e f t + Q (1)$ . $P (1, b_{n}) = Q (b_{n}) + r i g h t$ .*/
12:
end for
13:
$r i g h t \leftarrow 1 - l e f t - \sum_{j = 1}^{b_{n}} Q (j)$
14:
for $i = 1, \dots, b_{n}$ do
15:
$P (i, 1) \leftarrow l e f t + Q (1) - \sum_{j = 1}^{i - 1} Q (j)$ /*Note that $\sum_{j = 1}^{0} Q (j) = 0$ */
16:
for $j = 2, \dots, b_{n} - 1$ do
17:
$P (i, j) \leftarrow Q (| i - j | + 1)$
18:
end for
19:
$P (i, b_{n}) \leftarrow \sum_{j = b_{n} - i + 1}^{b_{n}} Q (j) + r i g h t$
20:
end for
21:
Set $†_{i}$ for each i based on Y.
22:
for Repeat sufficient times do
23:
for $i = 1, \dots, b_{n}$ do
24:
$d_{i} \leftarrow 0$
25:
for $j = 1, \dots, b_{n}$ do
26:
$d_{i} \leftarrow d_{i} + P (k, j) * ‡_{k}$ /*Calcuation of the denominator of Equation (5)*/
27:
end for
28:
end for
29:
for $i = 1, \dots, b_{n}$ do
30:
for $j = 1, \dots, b_{n}$ do
31:
$‡_{i}^{'} \leftarrow y_{j} * P (i, j) / d_{j}$
32:
end for
33:
$‡_{i} \leftarrow ‡_{i} * ‡_{i}^{'}$
34:
end for
35:
for $i = 1, \dots, b_{n}$ do
36:
if $i \leq ⌈ \frac{m i n v_{o r g} - m i n v_{r e p}}{w} ⌉ - 1 OR ⌈ \frac{m a x v_{o r g} - m i n v_{r e p}}{w} ⌉ - 1 \leq i$ then
37:
$‡_{i} \leftarrow 0$
38:
end if
39:
end for
40:
end for