# Make a wavelet packet regression object from a dependent and independent time series variable. The idea here is to try and build facilities to enable a transfer function model along the lines of that described by Nason and Sapatinas 2002 in **Statistics and Computing**. The idea is to turn the `timeseries` variable into a set of nondecimated wavelet packets which are already pre-selected to have some semblance of relationship to the `response` time series. The function does not actually perform any regression, in contrast to the related `makewpstDO` but returns a data frame which the user can use to build their own models. ```r makewpstRO(timeseries, response, filter.number = 10, family = "DaubExPhase", trans = logabs, percentage = 10) ``` ## Arguments - `timeseries`: The dependent variable time series. This series is decomposed using the `wpst` function into nondecimated wavelet packets, need to be a power of two length. - `response`: The independent or response time series. - `filter.number`: The type of wavelet used within `family`, see `filter.select`. - `family`: The family of wavelet, see `filter.select` - `trans`: A transform to apply to the nondecimated wavelet packet coefficients before any selection - `percentage`: The top `percentage` of nondecimated wavelet packets that correlated best with the `response` series will be preselected. ## Details The idea behind this methodology is that a `response` time series might not be directly related to the dependent `timeseries` time series, but it might be related to the nondecimated wavelet packets of the `timeseries`, these packets can pick out various features of the `timeseries` including certain delays, oscillations and others. The best packets (the number if controlled by `percentage`), those that correlate best with `response` are selected and returned. The `response` and the best nondecimated wavelet packets are returned in a data frame object and then any convenient form of statistical modeling can be used to build a model of the `response` in terms of the packet variables. Once a model has been built it can be interpreted in the usual way, but with respect to nondecimated wavelet packets. Note that nondecimated wavelet packets are essential, as they are all of the same length as the original response series. If a decimated wavelet packet algorithm had been used then it is not clear what to do with the "gaps"! If new `timeseries` data comes along the `wpstREGR` function can be used to extract the identical packets as the ones produced by this function (as the result of this function stores the identities of these packets). Then the statistical modelling that build the model from the output of this function, can be used to predict future values of the `response` time series from future values of the `timeseries` series. ## Returns An object of class `wpstRO` containing the following items - **df**: A data frame containing the `response` time series and a number of columns/variables/packets that correlated with response series. These are all entitled "Xn" where n is some integer - **ixvec**: A packet index vector. After taking the nondecimated wavelet packet transform, all the packets are stored in a matrix. This vector indicates those that were preselected - **level**: The original level from which the preselected vectors came from - **pktix**: Another index vector, this time referring to the original wavelet packet object, not the matrix in which they subsequently got stored - **nlevelsWT**: The number of resolution levels in the original wavelet packet object - **cv**: The correlation vector. These are the values of the correlations of the packets with the response, then sorted in terms of decreasing absolute correlation - **filter**: The wavelet filter details - **trans**: The transformation function actually used ## References Nason, G.P. and Sapatinas, T. (2002) Wavelet packet transfer function modeling of nonstationary time series. **Statistics and Computing**, 12 , 45-56. ## Author(s) G P Nason ## See Also `makewpstDO`, `wpst`, `wpstREGR` ## Examples ```r data(BabyECG) baseseries <- BabyECG[1:256] # # Make up a FICTITIOUS response series! # response <- BabyECG[6:261]*3+52 # # Do the modeling # BabeModel <- makewpstRO(timeseries=baseseries, response=response) #Level: 0 .......... #1 .......... #2 .......... #3 .......... #4 ................ #5 #6 #7 # #Contains SWP coefficients #Original time series length: 256 #Number of bases: 25 #Some basis selection performed # Level Pkt Index Orig Index Score #[1,] 5 0 497 0.6729833 #[2,] 4 0 481 0.6120771 #[3,] 6 0 505 0.4550616 #[4,] 3 0 449 0.4309924 #[5,] 7 0 509 0.3779385 #[6,] 1 53 310 0.3275428 #[7,] 2 32 417 -0.3274858 #[8,] 2 59 444 -0.2912863 #[9,] 3 16 465 -0.2649679 #[10,] 1 110 367 0.2605178 #etc. etc. # # # Let's look at the data frame component # names(BabeModel$df) # [1] "response" "X1" "X2" "X3" "X4" "X5" # [7] "X6" "X7" "X8" "X9" "X10" "X11" #[13] "X12" "X13" "X14" "X15" "X16" "X17" #[19] "X18" "X19" "X20" "X21" "X22" "X23" #[25] "X24" "X25" # # Generate a formula including all of the X's (note we could use the . # argument, but we later want to be more flexible # xnam <- paste("X", 1:25, sep="") fmla1 <- as.formula(paste("response ~ ", paste(xnam, collapse= "+"))) # # Now let's fit a linear model, the response on all the Xs # Babe.lm1 <- lm(fmla1, data=BabeModel$df) # # Do an ANOVA to see what's what # anova(Babe.lm1) #Analysis of Variance Table # #Response: response # Df Sum Sq Mean Sq F value Pr(>F) #X1 1 214356 214356 265.7656 < 2.2e-16 *** #X2 1 21188 21188 26.2701 6.289e-07 *** #X3 1 30534 30534 37.8565 3.347e-09 *** #X4 1 312 312 0.3871 0.5344439 #X5 1 9275 9275 11.4999 0.0008191 *** #X6 1 35 35 0.0439 0.8343135 #X7 1 195 195 0.2417 0.6234435 #X8 1 94 94 0.1171 0.7324600 #X9 1 331 331 0.4103 0.5224746 #X10 1 0 0 0.0006 0.9810560 #X11 1 722 722 0.8952 0.3450597 #X12 1 0 0 0.0004 0.9850243 #X13 1 77 77 0.0959 0.7570769 #X14 1 2770 2770 3.4342 0.0651404 . #X15 1 6 6 0.0072 0.9326155 #X16 1 389 389 0.4821 0.4881649 #X17 1 44 44 0.0544 0.8157015 #X18 1 44 44 0.0547 0.8152640 #X19 1 4639 4639 5.7518 0.0172702 * #X20 1 490 490 0.6077 0.4364469 #X21 1 389 389 0.4823 0.4880660 #X22 1 85 85 0.1048 0.7463860 #X23 1 1710 1710 2.1198 0.1467664 #X24 1 12 12 0.0148 0.9033427 #X25 1 82 82 0.1019 0.7498804 #Residuals 230 185509 807 #--- #Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 # # Looks like X1, X2, X3, X5, X14 and X19 are "significant". Also throw in # X4 as it was a highly ranked preselected variable, and refit # fmla2 <- response ~ X1 + X2 + X3 + X4 + X5 + X14 + X19 Babe.lm2 <- lm(fmla2, data=BabeModel$df) # # Let's see the ANOVA table for this # anova(Babe.lm2) #Analysis of Variance Table # #Response: response # Df Sum Sq Mean Sq F value Pr(>F) #X1 1 214356 214356 279.8073 < 2.2e-16 *** #X2 1 21188 21188 27.6581 3.128e-07 *** #X3 1 30534 30534 39.8567 1.252e-09 *** #X4 1 312 312 0.4076 0.5238034 #X5 1 9275 9275 12.1075 0.0005931 *** #X14 1 3095 3095 4.0405 0.0455030 * #X19 1 4540 4540 5.9259 0.0156263 * #Residuals 248 189989 766 #--- #Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 # # So, let's drop X4, refit, and then do ANOVA # Babe.lm3 <- update(Babe.lm2, . ~ . -X4) anova(Babe.lm3) # # After viewing this, drop X14 # Babe.lm4 <- update(Babe.lm3, . ~ . -X14) anova(Babe.lm4) # # Let's plot the original series, and the "fitted" one # ## Not run: ts.plot(BabeModel$df[["response"]]) ## Not run: lines(fitted(Babe.lm4), col=2) # # Let's plot the wavelet packet basis functions associated with the model # ## Not run: oldpar <- par(mfrow=c(2,2)) ## Not run: z <- rep(0, 256) ## Not run: zwp <- wp(z, filter.number=BabeModel$filter$filter.number, family=BabeModel$filter$family) ## End(Not run) ## Not run: draw(zwp, level=BabeModel$level[1], index=BabeModel$pktix[1], main="", sub="") ## Not run: draw(zwp, level=BabeModel$level[2], index=BabeModel$pktix[2], main="", sub="") ## Not run: draw(zwp, level=BabeModel$level[3], index=BabeModel$pktix[3], main="", sub="") ## Not run: draw(zwp, level=BabeModel$level[5], index=BabeModel$pktix[5], main="", sub="") ## Not run: par(oldpar) # # Now let's do some prediction of future values of the response, given # future values of the baseseries # newseries <- BabyECG[257:512] # # Get the new data frame # newdfinfo <- wpstREGR(newTS = newseries, wpstRO=BabeModel) # # Now use the best model (Babe.lm4) with the new data frame (newdfinfo) # to predict new values of response # newresponse <- predict(object=Babe.lm4, newdata=newdfinfo) # # What is the "true" response, well we made up a response earlier, so let's # construct the true response for this future data (in your case you'll # have a separate genuine response variable) # trucfictresponse <- BabyECG[262:517]*3+52 # # Let's see them plotted on the same plot # ## Not run: ts.plot(trucfictresponse) ## Not run: lines(newresponse, col=2) # # On my plot they look tolerably close! # ```

makewpstRO.rd function