cirquit
PhD student with a focus on machine learning, distributed systems and functional programming.
Math Basics
Introduction
1
Adrian, Jonathan, Alexander
Part I
Introduction
2
Example: Grades
Grading system:
Values:
Gauss Distribution
3
Gauss Distribution
N(x∣μ,σ)=2πσ21⋅e−2σ2(x−μ)2
\mathcal{N}(x\; |\; \mu, \sigma) = \frac{1}{\sqrt{2\pi\sigma^2}} \cdot e^{-\frac{{(x-\mu)}^2}{2\sigma^2}}
Gauss Distribution
4
Probability Density Function
μ=2.55
\mu = 2.55
σ2=1.06
\sigma^2 = 1.06
Gauss Distribution
5
Probability Density Function
μ=3.0
\mu = 3.0
σ2=0.8
\sigma^2 = 0.8
Gauss Distribution
6
Model Sampling
1.0−5.0?
1.0 - 5.0?
Gauss Distribution
7
Model Sampling
⋅x+
\cdot x +
x∈(−1,1)
x \in (-1,1)
⋅0.37+=2.94
\cdot 0.37 +\;\;\;\;\;\;\; = 2.94
2.55
2.55
1.06
1.06
μ
\mu
σ
\sigma
Gauss Distribution
8
Model Sampling
Gauss Distribution
9
Model Rules
verygoodgrade→ok′ishgrade
very\; good\; grade \rightarrow ok'ish\;grade
verybadgrade→goodgrade
very\; bad\; grade \rightarrow good\; grade
ok′ishgrade→ok′ishgrade
ok'ish\; grade\rightarrow ok'ish\;grade
+
+
+
+
≈2.5
\approx 2.5
Gauss Distribution
10
Without the last value
verygoodgrade→ok′ishgrade
very\; good\; grade \rightarrow ok'ish\;grade
verybadgrade→goodgrade
very\; bad\; grade \rightarrow good\; grade
ok′ishgrade→ok′ishgrade
ok'ish\; grade\rightarrow ok'ish\;grade
+
+
+
+
Gauss Distribution
11
Another "sensor"
verygoodgrade→ok′ishgrade
very\; good\; grade \rightarrow ok'ish\;grade
verybadgrade→goodgrade
very\; bad\; grade \rightarrow good\; grade
ok′ishgrade→ok′ishgrade
ok'ish\; grade\rightarrow ok'ish\;grade
+
+
+
+
≈3.5
\approx 3.5
Gauss Distribution
12
All "sensors"
verygoodgrade→ok′ishgrade
very\; good\; grade \rightarrow ok'ish\;grade
verybadgrade→goodgrade
very\; bad\; grade \rightarrow good\; grade
ok′ishgrade→ok′ishgrade
ok'ish\; grade\rightarrow ok'ish\;grade
+
+
+
+
≈3.22
\approx 3.22
Gauss Distribution
13
Formality
verygoodgrade→ok′ishgrade
very\; good\; grade \rightarrow ok'ish\;grade
verybadgrade→goodgrade
very\; bad\; grade \rightarrow good\; grade
ok′ishgrade→ok′ishgrade
ok'ish\; grade\rightarrow ok'ish\;grade
+
+
≈3.22
\approx 3.22
+
+
xknowndata∼N(μk,σk)
x_{known\;data} \sim N(\mu_k, \sigma_k)
xt=rule(xt−1)
x_{t} = rule(x_{t-1})
xface∼N(μf,σf)
x_{face} \sim N(\mu_f, \sigma_f)
xknowndata⋅trustk
x_{known\;data} \cdot trust_k
xface⋅trustf
x_{face} \cdot trust_f
rule(
rule(
)
)
+
+
xt−1⋅trustt
x_{t-1} \cdot trust_t
+
+
Basic Physics Formulae
x,y=position
x,y = \text{position}
yt+1=(21axt2+vxt)⋅sin(ϕ)+(21ayt2+vyt)⋅cos(ϕ)+yt
y_{t+1} = (\frac{1}{2} a_x t^2 + v_x t) \cdot sin(\phi) + (\frac{1}{2}a_y t^2 + v_y t) \cdot cos(\phi) + y_t
xt+1=(21axt2+vxt)⋅cos(ϕ)+(21ayt2+vyt)⋅sin(ϕ)+xt
x_{t+1} = (\frac{1}{2} a_x t^2 + v_x t) \cdot cos(\phi) + (\frac{1}{2}a_y t^2 + v_y t) \cdot sin(\phi) + x_t
ax,ay=acceleration
a_x,a_y = \text{acceleration}
vx,vy=forward / lateral velocity
v_x,v_y = \text{forward / lateral velocity}
ϕ=yaw angle
\phi = \text{yaw angle}
Math Basics Introduction 1 Adrian, Jonathan, Alexander Part I
Math Basics Part I
By cirquit
Math Basics Part I
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