/
detection.py
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detection.py
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# -*- coding: utf-8 -*-
# ---
# jupyter:
# jupytext:
# formats: ipynb,py:light
# text_representation:
# extension: .py
# format_name: light
# format_version: '1.5'
# jupytext_version: 1.6.0
# kernelspec:
# display_name: Python [conda env:root] *
# language: python
# name: conda-root-py
# ---
# %run detection_utils.py
# %config InlineBackend.figure_format = 'retina'
# ## Case study on French insurer data
halfhours, daily = load_call_data()
daily["numcalls"].plot(linewidth=1)
print("Number of Calls Each Day")
sns.despine()
fname = "../Figures/calls-by-day.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
dailyWD = daily.copy()
dailyWD.loc[dailyWD.index.weekday == 5, "numcalls"] = np.NaN
dailyWD.loc[dailyWD.holiday, "numcalls"] = np.NaN
dailyWD["numcalls"].plot(linewidth=1);
print("Number of Calls each Weekday")
sns.despine()
# plt.savefig("../Figures/calls-by-weekday.pdf")
plt.figure(figsize=[5,3])
dailyWD["numcalls"].plot(linewidth=1);
sns.despine()
fname = "../Figures/calls-by-weekday-squarer.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# +
df = daily["2016"].copy()
df.loc[df.index.weekday == 5, "numcalls"] = np.NaN
df.loc[df.holiday, "numcalls"] = np.NaN
plt.figure(figsize=[5,3])
df["numcalls"].plot(linewidth=1);
print("Number of Calls each Weekday in 2016")
plt.gca().xaxis.set_minor_locator(plt.NullLocator())
sns.despine()
fname = "../Figures/calls-by-weekday-2016-squarer.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
xt = range(6), ["Mon", "Tues", "Wed", "Thur", "Fri", "Sat"]
dd = daily.copy()
dd.loc[dd.holiday, "numcalls"] = np.NaN
by_week(dd.first("M").numcalls).plot()
print("Number of Calls by Day of Week")
plt.xticks(*xt)
plt.legend(frameon=False, title="Week")
sns.despine()
# plt.savefig("../Figures/calls-by-day-of-week-2015-04.pdf");
# +
if COLOUR:
halfhours.numcalls.first("W").groupby(pd.Grouper(freq="D")).apply(plt.plot)
else:
week = halfhours.numcalls.first("W")
week = add_nans_to_days(week)
week.plot()
plt.gca().margins(x=0.1)
put_days_between_ticks(numletters=3, dateskip=1)
d1, d2 = min_max_dates(halfhours.first("W"))
print(f"First Week ({d1} to {d2})");
sns.despine()
# plt.savefig(f"../Figures/calls-during-{d1}-week-side-colorful.pdf")
# +
plt.figure(figsize=[5,3])
if COLOUR:
halfhours.numcalls.first("W").groupby(pd.Grouper(freq="D")).apply(plt.plot)
else:
week = halfhours.numcalls.first("W")
week = add_nans_to_days(week)
week.plot()
plt.gca().margins(x=0.1)
put_days_between_ticks(numletters=3, dateskip=1)
d1, d2 = min_max_dates(halfhours.first("W"))
print(f"First Week ({d1} to {d2})")
sns.despine()
fname = f"../Figures/calls-during-{d1}-week-side-squarer.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
d1, d2 = plot_by_day(halfhours.first("W"))
# plt.savefig(f"../Figures/calls-during-{d1}-week.pdf")
# +
plt.figure(figsize=[5, 3])
def print_day(series):
series.index -= pd.Timedelta(days=series.index.day[0])
series.plot()
workWeek = halfhours.numcalls.first("5D")
workWeek.groupby(workWeek.index.date).apply(print_day)
plt.xlabel("")
sns.despine()
fname = f"../Figures/calls-during-{d1}-work-week-squarer.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
hh = halfhours[halfhours.index.day_name() != "Saturday"].copy()
hh.loc[hh.holiday, "numcalls"] = np.NaN
plot_median(hh["numcalls"]);
# plt.savefig("../Figures/calls-by-halfhour-weekdays-quantiles.pdf")
hh = halfhours[halfhours.index.day_name() == "Saturday"].copy()
hh.loc[hh.holiday, "numcalls"] = np.NaN
plot_median(hh["numcalls"]);
# plt.savefig("../Figures/calls-by-halfhour-saturdays-quantiles.pdf")
# +
# hhNan = add_nans_to_days(halfhours)
# hhNan.numcalls.plot(linewidth=1);
# plt.margins(x=0.01)
# put_months_between_ticks(dateskip=3)
# sns.despine()
## fname = "../Figures/calls-by-halfhour.pdf"
## plt.savefig(fname)
# ## !pdfcrop $fname $fname
# +
hhNan = add_nans_to_days(halfhours)
hhNanSub = hhNan.loc[(hhNan.index.minute == 0) | (hhNan.numcalls > 90)]
hhNanSub = hhNanSub.loc[((hhNanSub.index.hour % 3) == 0) | (hhNanSub.numcalls > 90)]
print(hhNan.shape, hhNanSub.shape)
hhNanSub.numcalls.plot(linewidth=1);
plt.margins(x=0.01)
put_months_between_ticks(dateskip=3)
sns.despine()
fname = "../Figures/calls-by-halfhour.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
# # Fit a GLM to the daily resolution data
# Adding factors/features for the linear regression to train on.
dd = daily.copy()
# +
# Features relating to the date-time
dd["workday"] = dd.index.weekday < 5
dd["day"] = dd.index.weekday
dd["month"] = dd.index.month
dd["relativedays"] = (dd.index - dd.index[0]).days
# Flag whether this date follows holiday period
dd["prevhol"] = False
dd["prev2hol"] = False
holDates = np.unique(dd.loc[dd.holiday].index.date)
dd.loc[dd.index.isin(holDates + pd.Timedelta("1D")), "prevhol"] = True
dd.loc[dd.index.isin(holDates + pd.Timedelta("2D")), "prev2hol"] = True
# +
# Pull out the indices on which we will train and test.
dt_train = dd.loc[:"2017-09-26"].index
dt_test = dd.loc["2017-11-02":].index
# N.B. Don't want to train on dates which are known holidays.
# They will obviously have ~0 arrivals, so will treat separately.
holInd = dd.loc[dd["holiday"]].index
dt_train = dt_train.difference(holInd)
dt_test = dt_test.difference(holInd)
# Also, can remove all Sundays and other missing
# dates (like the 8 days in 07/2016).
dt_train = dt_train[~dd.loc[dt_train].numcalls.isna()]
dt_test = dt_test[~dd.loc[dt_test].numcalls.isna()]
dt_both = dt_train.union(dt_test)
# Extract the target, the number of calls which actually did
# arrive at each time period, for the train and test periods.
y_train = dd.loc[dt_train, ["numcalls"]].to_numpy()
y_test = dd.loc[dt_test, ["numcalls"]].to_numpy()
# +
df_train = dd.loc[dt_train]
df_test = dd.loc[dt_test]
df_both = dd.loc[dt_both]
df = df_train
# -
# ### For NuPIC, export some of the same data
hh = halfhours.copy()
# +
# Features relating to the date-time
hh["day"] = hh.index.weekday
hh["month"] = hh.index.month
hh["relativedays"] = (hh.index - hh.index[0]).days
# Flag whether this date follows holiday period
hh["prevhol"] = 0
holDates = np.unique(hh.loc[hh.holiday].index.date)
hh.loc[pd.DatetimeIndex(hh.index.date).isin(holDates + pd.Timedelta("1D")), "prevhol"] = 1
# +
# Pull out the indices on which we will train and test.
dt_train_hh = hh.loc[:"2017-09-26"].index
dt_test_hh = hh.loc["2017-11-02":].index
# N.B. Don't want to train on dates which are known holidays.
# They will obviously have ~0 arrivals, so will treat separately.
holInd = hh.loc[hh["holiday"]].index
dt_train_hh = dt_train_hh.difference(holInd)
dt_test_hh = dt_test_hh.difference(holInd)
# Also, can remove all Sundays and other missing
# dates (like the 8 days in 07/2016).
dt_train_hh = dt_train_hh[~hh.loc[dt_train_hh].numcalls.isna()]
dt_test_hh = dt_test_hh[~hh.loc[dt_test_hh].numcalls.isna()]
dt_both_hh = dt_train_hh.union(dt_test_hh)
# -
df_train_hh = hh.loc[dt_train_hh]
df_test_hh = hh.loc[dt_test_hh]
def fit_and_save_glm(df, form, first=False, fname="glm_models.txt"):
y, X = patsy.dmatrices(form, df, return_type='matrix')
model = sm.GLM(y, X, family=sm.families.Poisson())
fit = model.fit()
print(f"Loglikelihood {fit.llf}, BIC {fit.bic}")
mode = "w" if first else "a"
with open(fname, mode) as fh:
fh.write(f"Model: {form}\n\n")
fh.write(fit.summary().as_text())
fh.write(f"\n\n")
return fit
form = "numcalls ~ workday"
fit_and_save_glm(df, form, first=True);
form = "numcalls ~ workday + relativedays"
fit_and_save_glm(df, form);
form = "numcalls ~ workday + relativedays + C(month)"
fit_and_save_glm(df, form);
form = "numcalls ~ relativedays + C(month) + C(day)"
fit_and_save_glm(df, form);
form = "numcalls ~ relativedays + C(month) + C(day) + prevhol"
fit_and_save_glm(df, form);
form = "numcalls ~ relativedays + C(month) + C(day) + " \
+ "prevhol + prev2hol"
fit_and_save_glm(df, form);
# #### Final daily model
# +
finalForm = "numcalls ~ relativedays + C(month) + C(day) + prevhol"
finalFit = fit_and_save_glm(df, finalForm, first=True, fname="final_glm_model.txt")
with open('final_glm_model.csv', 'w') as fh:
fh.write(finalFit.summary().as_csv())
# +
X_train = patsy.dmatrices(finalForm, df_train, return_type='matrix')[1]
X_test = patsy.dmatrices(finalForm, df_test, return_type='matrix')[1]
df_train["pred"] = finalFit.predict(X_train)
df_test["pred"] = finalFit.predict(X_test)
# -
newind = pd.date_range(df_train.index.min(), df_train.index.max())
df_plot = df_train[["pred", "numcalls"]].reindex(newind)
df_plot.plot()
plt.legend(["Predicted", "Observed"], loc=9, frameon=False, ncol=2);
sns.despine()
# +
newind = pd.date_range(df_train.index.min(), df_train.index.max())
df_plot = df_train[df_train.workday][["pred", "numcalls"]].reindex(newind)
df_plot.plot(linewidth=1)
plt.legend(["Predicted", "Observed"], loc=9, ncol=2, frameon=False);
sns.despine()
fname = "../Figures/linear-regression-train.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
plt.figure(figsize=[4,4])
plt.scatter(df_train["numcalls"], df_train["pred"], alpha=0.25)
xl, xr = plt.xlim()
plt.plot([xl, xr], [xl, xr], color="k", ls="--")
sns.despine()
plt.tight_layout()
fname = "../Figures/linear-regression-train-scatter.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# +
# newind = pd.date_range(df_test.index.min(), df_test.index.max())
# df_plot = df_test[df_test.workday][["pred", "numcalls"]].reindex(newind)
# df_plot.plot(legend=False)
# sns.despine()
# -
plt.figure(figsize=[4,4])
plt.scatter(df_test["numcalls"], df_test["pred"], alpha=0.25)
xl, xr = plt.xlim()
plt.plot([xl, xr], [xl, xr], color="k", ls="--")
sns.despine()
plt.tight_layout()
fname = "../Figures/linear-regression-test-scatter.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# ## Try to estimate the fraction of daily calls which arrive during each half-hour period
hh = halfhours.copy()
# +
# Pull out the indices on which we will train and test.
dt_train_hh = hh.loc[:"2017-09-26"].index
dt_test_hh = hh.loc["2017-11-02":].index
# N.B. Don't want to train on dates which are known holidays.
# They will obviously have ~0 arrivals, so will treat separately.
holInd = hh.loc[hh["holiday"]].index
dt_train_hh = dt_train_hh.difference(holInd)
dt_test_hh = dt_test_hh.difference(holInd)
# Also, can remove all Sundays and other missing
# dates (like the 8 days in 07/2016).
dt_train_hh = dt_train_hh[~hh.loc[dt_train_hh].numcalls.isna()]
dt_test_hh = dt_test_hh[~hh.loc[dt_test_hh].numcalls.isna()]
dt_both_hh = dt_train_hh.union(dt_test_hh)
# +
fracs = hh[["numcalls"]].copy()
fracs["frac"] = fracs["numcalls"]
fracs["dayscalls"] = fracs["numcalls"]
dates = [str(d) for d in np.unique(fracs.index.date)]
for date in dates:
fracs.loc[date, "dayscalls"] = dd.loc[date, "numcalls"]
fracs.loc[date, "frac"] /= dd.loc[date, "numcalls"]
# -
# #### Separate the weekdays from saturdays
# +
fracsWD = fracs[fracs.index.weekday < 5].copy()
fracsSat = fracs[fracs.index.weekday == 5].copy()
fracsSat.loc[:,"frac"] /= 2
l1 = plot_median(fracsWD.loc[fracsWD.index.intersection(dt_train_hh), "frac"])
l2 = plot_median(fracsSat.loc[fracsSat.index.intersection(dt_train_hh), "frac"])
plt.legend([l1[0], l2[0]], ["Weekdays", "Saturdays"], ncol=2, frameon=False)
fname = "../Figures/frac-calls-halfhour.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
if COLOUR:
fracsWD = fracs[fracs.index.weekday < 5].copy()
fracsSat = fracs[fracs.index.weekday == 5].copy()
fracsWD = fracsWD.loc[fracsWD.index.intersection(dt_train_hh)]
fracsSat = fracsSat.loc[fracsSat.index.intersection(dt_train_hh)]
low1 = fracsWD.frac.groupby(fracsWD.index.time).quantile(0.05)
med1 = fracsWD.frac.groupby(fracsWD.index.time).quantile(0.5)
up1 = fracsWD.frac.groupby(fracsWD.index.time).quantile(0.95)
dtIndex = time_to_datetime_index(low1.index)
low1.index = dtIndex
med1.index = dtIndex
up1.index = dtIndex
low2 = fracsSat.frac.groupby(fracsSat.index.time).quantile(0.05)
med2 = fracsSat.frac.groupby(fracsSat.index.time).quantile(0.5)
up2 = fracsSat.frac.groupby(fracsSat.index.time).quantile(0.95)
dtIndex = time_to_datetime_index(low2.index)
low2.index = dtIndex
med2.index = dtIndex
up2.index = dtIndex
fig, ax1 = plt.subplots()
l0 = ax1.plot(med1, lw=2)
c0 = l0[0].get_color()
ax1.fill_between(low1.index, low1, up1, alpha=0.25, color=c0, lw=0)
ax1.tick_params(axis='y', labelcolor=c0)
ax2 = ax1.twinx() # instantiate a second axes that shares the same x-axis
c = '#ff7f0e'
l = ax2.plot(med2, lw=2, color=c)
ax2.fill_between(low2.index, low2, up2, alpha=0.25, color=c, lw=0)
ax2.tick_params(axis='y', labelcolor=c)
plt.gca().xaxis.set_major_locator(mdates.HourLocator())
plt.gca().xaxis.set_major_formatter(mdates.DateFormatter("%H:%M"))
ax1.set_ylim([0, 0.12])
ax2.set_ylim([0, 0.24])
ax1.set_yticks([0.02, 0.04, 0.06, 0.08, 0.1])
ax2.set_yticks([0.04, 0.08, 0.12, 0.16, 0.2])
fig.tight_layout() # otherwise the right y-label is slightly clipped
plt.legend([l0[0], l[0]], ['Weekdays', 'Saturdays'], frameon=False)
sns.despine(right=False)
fname = "../Figures/frac-calls-halfhour-dual-axes.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# #### Okay, treat all the same, but pretend that twice as many calls arrive on each Saturday to make things even
fracsAdj = fracs.copy()
fracsAdj.loc[fracsAdj.index.weekday == 5, "frac"] /= 2
fracsAdj.loc[fracsAdj.index.weekday == 5, "dayscalls"] *= 2
plot_median(fracsAdj.frac);
# +
fracsAdjTrain = fracsAdj.loc[dt_train_hh]
plot_median(fracsAdjTrain.frac);
mean = fracsAdjTrain.frac.groupby(fracsAdjTrain.index.time).mean()
mean /= np.sum(mean)
mean.index = time_to_datetime_index(mean.index)
plt.plot(mean, lw=2);
# -
fracPerHH = fracsAdjTrain.frac.groupby(fracsAdjTrain.index.time).mean()
fracPerHH /= fracPerHH.sum()
fracPerHH.sum()
# #### Is the hourly pattern different on busy days to quiet days?
quartiles = dd.numcalls.quantile([0, 0.25, 0.5, 0.75, 1]).to_numpy()
if COLOUR:
lines = []
l = plot_median(fracsAdj.frac)
lines.append(l)
for q1, q2 in zip(quartiles[:-1], quartiles[1:]):
inds = (fracsAdj.dayscalls > q1) & (fracsAdj.dayscalls < q2)
fracsAdjQ = fracsAdj.loc[inds]
l = plot_median(fracsAdjQ.frac)
lines.append(l)
y1, y2 = plt.ylim()
plt.ylim([y1, y2*1.1])
lines = [l[0] for l in lines]
plt.legend(lines, ["0%-25%", "25%-50%", "50%-75%", "75%-100%"], loc=9, ncol=4, frameon=False);
fname = "../Figures/frac-calls-by-busyness.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
if not COLOUR:
with plt.style.context({'axes.prop_cycle': monochrome[1:]}):
for q1, q2 in zip(quartiles[:-1], quartiles[1:]):
inds = (fracsAdj.dayscalls > q1) & (fracsAdj.dayscalls < q2)
fracsAdjQ = fracsAdj.loc[inds]
plot_median(fracsAdjQ.frac, quantile=False)
y1, y2 = plt.ylim()
plt.ylim([y1, y2*1.14])
plt.legend(["0%-25%", "25%-50%", "50%-75%", "75%-100%"], loc=9, ncol=4, frameon=False);
fname = "../Figures/frac-calls-by-busyness.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# ## Make our predictions based on the linear regression over the daily dataset, and subdivided into half-hours using the estimated fractions above
# +
hh = halfhours.copy()
hh["constpred"] = 0.0
hh["pred"] = 0.0
ddNotNA = dd.dropna()
X_dd = patsy.dmatrices(finalForm, ddNotNA, return_type='matrix')[1]
preds = pd.Series(finalFit.predict(X_dd), index=ddNotNA.index)
for d in np.unique(ddNotNA.index.date):
if str(d) in hh.index:
dDay = hh.loc[str(d)].dropna()
inds = dDay.index
hhDist = fracPerHH[inds.time].to_numpy()
hhDist /= np.sum(hhDist)
hh.loc[inds, "constpred"] = preds[d] / len(inds)
hh.loc[inds, "pred"] = preds[d] * hhDist
# -
df_train_hh = hh.loc[dt_train_hh]
df_test_hh = hh.loc[dt_test_hh]
df_both_hh = hh.loc[dt_train_hh.union(dt_test_hh)]
plt.figure(figsize=[5,5])
plt.scatter(df_both_hh["numcalls"], df_both_hh["pred"], alpha=0.25)
mr = min(plt.xlim()[1], plt.ylim()[1])
plt.plot([0, mr], [0, mr], color="k", ls="--")
plt.xlim((-10, mr * 1.1)); plt.ylim((-10, mr * 1.1))
plt.show()
# ## Choosing the threshold for CUSUM to raise alarms at
ρ = 1.2
df_train_hh = hh.loc[dt_train_hh]
df_test_hh = hh.loc[dt_test_hh]
df_both_hh = hh.loc[dt_train_hh.union(dt_test_hh)]
# Let's simulate some CUSUM processes $V_t$ under the pre-change distribution. We need this to calibrate the threshold $m$ at which to raise alarms $T = \inf\{ t: V_t > m \}$.
# Simulate `R` i.i.d. CUSUM processes for `tMax` timesteps each.
# +
# %%time
tMax = 10**5
R = 10**3
λ = halfhours.numcalls.dropna().mean()
rng = default_rng(1)
Ns = np.empty((tMax, R), dtype=np.int64)
Vs = np.empty((tMax, R), dtype=np.float64)
Ms = np.empty((tMax, R), dtype=np.float64)
β = (ρ-1)/(np.log(ρ))
for r in trange(R):
dN = rng.poisson(λ, size=tMax)
U = np.cumsum(dN - β*λ)
V = np.maximum(U - np.insert(np.minimum.accumulate(U), 0, 0)[:-1], 0)
Ns[:,r] = np.cumsum(dN)
Vs[:,r] = V
Ms[:,r] = np.maximum.accumulate(V)
# -
# Plot some extracts of the CUSUM simulated processes.
plt.plot(Vs[:200,:10]); sns.despine();
# In the process, we calculated $M_t = \max_{s \le t} V_s$ the running maximum of each CUSUM process. This is useful for finding when an alarm would have been raised for a certain $m$ threshold.
#
# Also, we can find the largest $m$ threshold which is exceeded by every simulated CUSUM process:
np.min(Ms[-1])
# So let's calculate some different expected false positive values when CUSUM is run with $m$ taking values in a grid.
msConst = np.arange(1, np.floor(np.min(Ms[-1]))+1)
msConst[0], msConst[-1]
# The generalised Lorden criterion is the constraint that $\mathbb{E}[N_T] \ge \pi$ where $T = \inf\{ t : V_t > m \}$. We calculate and plot the relationship between $m$ and $\mathbb{E}[N_T]$ over the grid of $m$ values above.
ExpN_TConst = gen_lorden_criterion(msConst, Ns, Ms)
πConst = daily["2018"].numcalls.sum()
πToM = interp1d(ExpN_TConst, msConst)
mConst = πToM(πConst)
πConst, mConst
# +
plt.plot(msConst, ExpN_TConst)
plt.title("$\mathbb{E}[N_T]$ for different CUSUM thresholds")
plt.xlabel("CUSUM Threshold $m$")
plt.plot([0, mConst], [πConst, πConst], 'r--', alpha=0.5)
plt.plot([mConst, mConst], [0, πConst], 'r--', alpha=0.5)
plt.xticks([mConst])
plt.yticks([πConst])
sns.despine()
plt.xlim(1, mConst*1.25)
plt.ylim(0, πConst*2)
plt.show()
# -
plt.plot(msConst, np.log(ExpN_TConst))
plt.plot(msConst[[10,-1]], np.log(ExpN_TConst[[10,-1]]))
# ## Find the $m$ applicable to the GLM adaptive intensity case
# +
# %%time
rng = default_rng(1)
R = 10**4
λ = df_test_hh["pred"].to_numpy()
tMax = len(λ)
Ns = np.empty((tMax, R), dtype=np.int64)
Vs = np.empty((tMax, R), dtype=np.float64)
Ms = np.empty((tMax, R), dtype=np.float64)
β = (ρ-1)/(np.log(ρ))
for r in trange(R):
dN = rng.poisson(λ)
U = np.cumsum(dN - β*λ)
V = np.maximum(U - np.insert(np.minimum.accumulate(U), 0, 0)[:-1], 0)
Ns[:,r] = np.cumsum(dN)
Vs[:,r] = V
Ms[:,r] = np.maximum.accumulate(V)
# -
ms = np.arange(1, np.floor(np.min(Ms[-1]))+1)
ms[0], ms[-1]
ExpN_T = gen_lorden_criterion(ms, Ns, Ms)
avgWeek = df_both.groupby(pd.Grouper(freq="W")).numcalls.sum().mean()
π = avgWeek * 7
π = 30000
πToM = interp1d(ExpN_T, ms)
m = πToM(π)
π, m
# +
plt.plot(ms, np.log(ExpN_T))
mToLogπ = interp1d(ms, np.log(ExpN_T), fill_value="extrapolate")
msBig = np.arange(ms[-1], ms[-1]*2)
LogExpN_TBig = mToLogπ(msBig)
ExpN_TBig = np.exp(LogExpN_TBig)
plt.plot(msBig, mToLogπ(msBig), ls="--");
# -
π = πConst
LogπToM = interp1d(np.log(ExpN_T), ms, fill_value="extrapolate")
m = LogπToM(np.log(π))
π, m
# +
line, = plt.plot(ms, ExpN_T)
plt.plot(msBig, ExpN_TBig, ls="--", c=line.get_color());
plt.title("$\mathbb{E}[N_T]$ for different CUSUM thresholds")
plt.xlabel("CUSUM Threshold $m$")
plt.plot([0, m], [π, π], 'r--', alpha=0.5)
plt.plot([m, m], [0, π], 'r--', alpha=0.5)
plt.xticks([m])
plt.yticks([π])
sns.despine()
plt.xlim(1, m*1.1)
plt.ylim(0, π*1.1)
plt.show()
# -
# # Running CUSUM
# ### Doing a stupid CUSUM with constant expected arrival intensity
# +
β = (ρ - 1) / np.log(ρ)
# Set constant arrival intensity
λ = df_train_hh.numcalls.mean()
csNaive = pd.DataFrame({"λ": λ, "N": df_test_hh.numcalls})
N_t = np.cumsum(csNaive["N"])
Lambda_t = np.cumsum(csNaive["λ"])
U_t = N_t - β * Lambda_t
csNaive["V"] = U_t - U_t.shift(1).cummin().fillna(0)
csNaive.loc[csNaive["V"] < 0, "V"] = 0
csNaive.index.name = ""
# +
if COLOUR:
csColors = [c['color'] for c in mpl.rcParams['axes.prop_cycle']]
csColors.pop(3)
csColors = csColors[:4]
else:
stylesList = [s for s in monochrome]
styles = {
"λ": {"c": "k", "marker": "", "ls":"-", "label": "$\lambda_t$"},
"N": {"c": "k", "marker": "", "ls":"-", "alpha": 0.4, "label": "$\Delta N_t$"},
"m": {"c": "k", "marker": "", "ls":"--", "label": "$m$"},
"V": {"c": "k", "marker": "", "ls":":", "lw": 1, "label": "$V_t$"},
"NuPIC": {"c": "k", "marker": "", "ls":"-.", "label": "NuPIC"}
}
stylesNL = {
col: {key: styles[col].get(key) for key in ("c", "marker", "ls", "alpha")}
for col in styles.keys()
}
stylesNL
# +
csNaive["N"].plot(**styles["N"])
csNaive["V"].iloc[::25].plot(**styles["V"])
plt.axhline(mConst, **styles["m"])
plt.margins(x=0.01)
put_months_between_ticks()
plt.legend(frameon=False)
sns.despine()
fname = "../Figures/naive-cusum.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
np.mean(csNaive.loc["2018", "V"] > mConst)
# +
csStart = add_nans_to_days(csNaive["2018-01-08":].first("6D").copy())
csStart.loc[pd.Timestamp("2018-01-07 12:00")] = [0, np.NaN, 0]
csStart.loc[pd.Timestamp("2018-01-13 18:00")] = [0, np.NaN, 0]
csStart = csStart.sort_index()
newTime = stretch_time(csStart.index, scale=1.85)
for col in csStart.columns:
plt.plot(newTime, csStart[col], **styles[col])
plt.axhline(mConst, **styles["m"])
plt.margins(x=0.01)
put_days_between_ticks(numletters=3, dateskip=1)
plt.legend(ncol=2, frameon=False)
sns.despine()
# Add annotations to the plot describing when the alarms are raised
alarmed = csStart[csStart["V"] > mConst]
# Annotation for the beginning of the busy period
xTextDate = pd.Timestamp("2018-01-07 23:00")
xText = mdates.date2num(xTextDate)
yText = 170
xDate = alarmed.index[0] - pd.Timedelta("0.35H")
x = mdates.date2num(stretch_time([xDate]))
plt.annotate("Busy period\ndetected", xytext=(xText, yText), xy=(x, mConst*1.5),
arrowprops=dict(arrowstyle="->", mutation_scale=15, connectionstyle="arc3,rad=.1"), ha='center')
# Annotation for the end of the busy period
xTextDate = pd.Timestamp("2018-01-13 10:00")
xText = mdates.date2num(xTextDate)
yText = 170
xDate = alarmed.index[-1] + pd.Timedelta("0.25H")
x = mdates.date2num(stretch_time([xDate]))
plt.annotate("Busy period\nending", xytext=(xText, yText), xy=(x, mConst*1.5),
arrowprops=dict(arrowstyle="->", mutation_scale=15, connectionstyle="arc3,rad=-.1"), ha='center')
fname = "../Figures/naive-cusum-2018-start.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
# ## Using the adaptive intensity based on the GLM predictions
# +
cs = df_test_hh[["pred", "numcalls"]].copy()
cs.columns = ["λ", "N"]
cs.index.name = ""
N_t = np.cumsum(cs["N"])
Lambda_t = np.cumsum(cs["λ"])
U_t = N_t - β * Lambda_t
cs["V"] = U_t - U_t.shift(1).cummin().fillna(0)
cs.loc[cs["V"] < 0, "V"] = 0
print(np.mean(cs["N"] > cs["λ"]))
print(np.mean(cs.loc["2018", "V"] > m))
# +
cs["N"].plot(**styles["N"])
cs["V"].iloc[::20].plot(**styles["V"])
plt.axhline(m, **styles["m"])
plt.margins(x=0.01)
put_months_between_ticks()
plt.legend(loc=9, ncol=4, frameon=False)
sns.despine()
# plt.savefig("../Figures/cusum-big-picture.pdf")
# +
startDate = dt.date(2018, 10, 12)
csTrun = cs[str(startDate):].first("16D").copy()
csTrun = add_nans_to_days(csTrun)
newTime = stretch_time(csTrun.index, scale=1.85)
bax = broken_axis(csTrun)
for col in csTrun.columns:
bax.plot(newTime, csTrun[col], **styles[col])
bax.axhline(m, zorder=2, **styles["m"])
prevV = csTrun.loc["2018-10-12 11:30", "V"]
normalPred = hh.loc["2018-10-12 11:30":"2018-10-12", "pred"]
missingTime = stretch_time(normalPred.index, scale=1.85)
bax.plot(missingTime, normalPred, **stylesNL["λ"])
bax.plot(missingTime[[0,-1]], [0,0], **stylesNL["N"])
bax.plot(missingTime[[0,-1]], [prevV, prevV], **stylesNL["V"])
# bax.legend(loc=2)
put_days_between_ticks(numletters=3, skiplast=False, dateskip=1, ax=bax.axs[0])
[put_days_between_ticks(numletters=3, dateskip=1, ax=ax) for ax in bax.axs[1:]]
bax.legend(loc=2, ncol=2, frameon=False)
# Add annotations to the plot describing when the alarms are raised
alarmed = csTrun[csTrun["V"] > m]
# Annotation for the beginning of the busy period
xTextDate = pd.Timestamp("2018-10-14 06")
xText = mdates.date2num(xTextDate)
yText = 160
xDate = alarmed.loc["2018-10-13":].index[0]
x = mdates.date2num(stretch_time([xDate]))
bax.annotate("Busy period\ndetected", xytext=(xText, yText), xy=(x, m*1.1),
arrowprops=dict(arrowstyle="->", mutation_scale=15, connectionstyle="arc3,rad=.3"), ha='center')
# Annotation for the end of the busy period
xTextDate = pd.Timestamp("2018-10-25 12")
xText = mdates.date2num(xTextDate)
yText = 160
xDate = alarmed.index[-1] - pd.Timedelta("0.9H")
x = mdates.date2num(stretch_time([xDate]))
bax.annotate("Busy period\nending", xytext=(xText, yText), xy=(x, m),
arrowprops=dict(arrowstyle="->", mutation_scale=15, connectionstyle="arc3,rad=.2"), ha='center')
fname = "../Figures/cusum-case-study-2018-10-12.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# +
start = "2018-06-15"
print(start)
csTrun = cs.loc[start].copy()
csTrun = add_nans_to_days(csTrun)
for col in csTrun.columns:
csTrun[col].plot(**styles[col])
# ax = csTrun.plot()
# ax.get_lines()[2].set_zorder(3)
plt.axhline(m, **styles["m"])
plt.legend(loc=9, ncol=4, frameon=False)
sns.despine()
fname = "../Figures/cusum-case-study-2018-06-15.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# -
# ## Running CUSUM on the modified data regime (Stéphane's idea)
# +
csMod = df_test_hh[["pred", "numcalls"]].copy()
tuesdaysToEdit = np.unique(csMod.index[csMod.index.weekday == 1].date)[::3]
for date in tuesdaysToEdit:
mornCalls = csMod.loc[str(date):str(date)+" 11:59", "numcalls"].sum()
csMod.loc[str(date):str(date)+" 11:59", "numcalls"] = 0
aftCalls = csMod.loc[str(date)+" 12:00":str(date), "numcalls"].sum()
csMod.loc[str(date)+" 12:00":str(date), "numcalls"] = np.round(
csMod.loc[str(date)+" 12:00":str(date), "numcalls"] * (mornCalls + aftCalls) / aftCalls)
# +
csMod.columns = ["λ", "N"]
csMod.index.name = ""
N_t = np.cumsum(csMod["N"])
Lambda_t = np.cumsum(csMod["λ"])
U_t = N_t - β * Lambda_t
csMod["V"] = U_t - U_t.shift(1).cummin().fillna(0)
csMod.loc[csMod["V"] < 0, "V"] = 0
print(np.mean(csMod["N"] > csMod["λ"]))
print(np.mean(csMod.loc["2018", "V"] > m))
# -
# Add a few calls to the modified Tuesday mornings so
# the 0's can be seen above the x axis in the plots
for date in tuesdaysToEdit:
csMod.loc[str(date):str(date)+" 11:59", "N"] = 3
dts = []
maxs = []
for tues in tuesdaysToEdit:
dts.append(mdates.date2num(pd.Timestamp(str(tues) + " 14:00")))
maxs.append(csMod.loc[str(tues), "N"].max())
# +
csMod["N"].plot(**styles["N"])
csMod["V"].iloc[::25].plot(**styles["V"])
plt.axhline(m, **styles["m"])
plt.margins(x=0.01)
put_months_between_ticks()
plt.legend(loc=9, ncol=4, frameon=False)
plt.plot(dts, maxs, "x")
sns.despine()
fname = "../Figures/modified-tuesdays-big-picture.pdf"
plt.savefig(fname)
# !pdfcrop $fname $fname
# +
csTrun = csMod.loc[str(tuesdaysToEdit[0])].copy()
csTrun = add_nans_to_days(csTrun)
firstIndex = csTrun.index