S&P500 Stock Price Index fSPcom with log-linear (ll)and HP growth trends Trends of fSPCOMIn Index 5.5 log S(t) HPs 4.5 log-linear trend 1945 1955 1965 1975 198 1995
S&P500 Stock Price Index FSPCOM with log-linear (LL) and HP growth trends 2.5 3 3.5 4 4.5 5 lo g S(t) 5.5 1945 1955 1965 1975 1985 1995 log S(t) HPs t Trends of FSPCOMln Index log-linear trend
Detrended cycles with varying lengths Cycles of fspcomin Index 0.6 HP IA LLDC -0.2 1^ 1945 1955 1965 1975 1985 1995
Detrended cycles with varying lengths -0.6 -0.4 -0.2 0 0.2 0.4 X(t) 0.6 1945 1955 1965 1975 1985 1995 t Cycles of FSPCOMln Index HPc FD LLDc
Different images of auto-correlations Autocorrelations of FSPCOMin Cycles HPc 0.6 FD LLD 0.2 0.2 100
Different images of auto-correlations -1 -0.6 -0.2 0.2 0.6 A C(I) 1 0 20 40 60 80 100 I Autocorrelations of FSPCOMln Cycles HPc FD LLD -
Uncertainty Principle in Signal Processing An) A△≥ 4兀
Uncertainty Principle in Signal Processing
Gabor (gaussian)Wavelet with Minimum Uncertainty 0210 h(t)=( eXI 2o>)exp(io t) 432101234
Gabor (Gaussian) Wavelet with Minimum Uncertainty ) exp( ) 2 ) exp( 1 ( ) ( 2 2 4 1 2 i t t h t